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Finance and Economics Discussion Series: 2006-15 Screen Reader version*

Do Macro Variables, Asset Markets, or Surveys Forecast Inflation Better?

Andrew Ang

Columbia University and NBER

Geert Bekaert

Columbia University CEPR, and NBER

Min Wei

Federal Reserve Board of Governors

This Version: 13 February, 2006

Keywords: ARIMA, Phillips curve, forecasting, term structure models, Livingston

Abstract:

Surveys do! We examine the forecasting power of four alternative methods of forecasting U.S. inflation out-of-sample: time-series ARIMA models; regressions using real activity measures motivated from the Phillips curve; term structure models that include linear, non-linear, and arbitrage-free specifications; and survey-based measures. We also investigate several methods of combining forecasts. Our results show that surveys outperform the other forecasting methods and that the term structure specifications perform relatively poorly. We find little evidence that combining forecasts produces superior forecasts to survey information alone. When combining forecasts, the data consistently places the highest weights on survey information.

JEL Classification: E31, E37, E43, E44

1 Introduction

Obtaining reliable and accurate forecasts of future inflation is crucial for policymakers conducting monetary and fiscal policy; for investors hedging the risk of nominal assets; for firms making investment decisions and setting prices; and for labor and management negotiating wage contracts. Consequently, it is no surprise that a considerable academic literature evaluates different inflation forecasts and forecasting methods. In particular, economists use four main methods to forecast inflation. The first method is atheoretical, using time series models of the ARIMA variety. The second method builds on the economic model of the Phillips curve, leading to forecasting regressions that use real activity measures. Third, we can forecast inflation using information embedded in asset prices, in particular the term structure of interest rates. Finally, survey-based measures use information from agents (consumers or professionals) directly to forecast inflation.

In this article, we comprehensively compare and contrast the ability of these four methods to forecast inflation out of sample. Our approach makes four main contributions to the literature. First, our analysis is the first to comprehensively compare the four methods: time-series forecasts, forecasts based on the Phillips curve, forecasts from the yield curve, and all three available surveys (the Livingston, Michigan, and SPF surveys). The previous literature has concentrated on only one or two of these different forecasting methodologies. For example, Stockton and Glassman (1987) show that pure time-series models out-perform more sophisticated macro models, but do not consider term structure models or surveys. Fama and Gibbons (1984) compare term structure forecasts with the Livingston survey, but they do not consider forecasts from macro factors. Whereas Grant and Thomas (1999), Thomas (1999) and Mehra (2002) show that surveys out-perform simple time-series benchmarks for forecasting inflation, none of these studies compares the performance of survey measures with forecasts from Phillips curve or term structure models.

The lack of a study comparing these four methods of inflation forecasting implies that there is no well-accepted set of findings regarding the superiority of a particular forecasting method. The most comprehensive study to date, Stock and Watson (1999), finds that Phillips curve-based forecasts produce the most accurate out-of-sample forecasts of U.S. inflation compared with other macro series and asset prices, using data up to 1996. However, Stock and Watson only briefly compare the Phillips-curve forecasts to the Michigan survey and to simple regressions using term structure information. Stock and Watson do not consider no-arbitrage term structure models, non-linear forecasting models, or combined forecasts from all four forecasting methods. Recent work also casts doubts on the robustness of the Stock-Watson findings. In particular, Atkeson and Ohanian (2001), Fisher, Liu and Zhou (2002), Sims (2002), and Cecchetti, Chu and Steindel (2000), among others, show that the accuracy of Phillips curve-based forecasts depends crucially on the sample period. Clark and McCracken (2006) address the issue of how instability in the output gap coefficients of the Phillips curve affects forecasting power. To assess the stability of the inflation forecasts across different samples, we consider out-of-sample forecasts over both the post-1985 and post-1995 periods.

Our second contribution is to evaluate inflation forecasts implied by arbitrage-free asset pricing models. Previous studies employing term structure data mostly use only the term spread in simple OLS regressions and usually do not use all available term structure data (see, for example, Mishkin, 1990, 1991; Jorion and Mishkin, 1991; Stock and Watson, 2003). Frankel and Lown (1994) use a simple weighted average of different term spreads, but they do not impose no-arbitrage restrictions. In contrast to these approaches, we develop forecasting models that use all available data and impose no-arbitrage restrictions. Our no-arbitrage term structure models incorporate inflation as a state variable because inflation is an integral component of nominal yields. The no-arbitrage framework allows us to extract forecasts of inflation from data on inflation and asset prices taking into account potential time-varying risk premia.

No-arbitrage constraints are reasonable in a world where hedge funds and investment banks routinely eliminate arbitrage opportunities in fixed income securities. Imposing theoretical no-arbitrage restrictions may also lead to more efficient estimation. Just as Ang, Piazzesi and Wei (2004) show that no-arbitrage models produce superior forecasts of GDP growth, no-arbitrage restrictions may also produce more accurate forecasts of inflation. In addition, this is the first article to investigate non-linear, no-arbitrage models of inflation. We investigate both an empirical regime-switching model incorporating term structure information and a no-arbitrage, non-linear term structure model following Ang, Bekaert and Wei (2006) with inflation as a state variable.

Our third contribution is that we thoroughly investigate combined forecasts. Stock and Watson (2002a, 2003), among others, show that the use of aggregate indices of many macro series measuring real activity produces better forecasts of inflation than individual macro series. To investigate this further, we also include the (Phillips curve-based) index of real activity constructed by Bernanke, Boivin and Eliasz (2005) from 65 macroeconomic series. In addition, several authors (see, e.g., Stock and Watson, 1999; Brave and Fisher, 2004; Wright, 2004) advocate combining several alternative models to forecast inflation. We investigate five different methods of combining forecasts: simple means or medians, OLS based combinations, and Bayesian estimators with equal or unit weight priors.

Finally, our main focus is forecasting inflation rates. Because of the long-standing debate in macroeconomics on the stationarity of inflation rates, we also explicitly contrast the predictive power of some non-stationary models to stationary models and consider whether forecasting inflation changes alters the relative forecasting ability of different models.

Our major empirical results can be summarized as follows. The first major result is that survey forecasts outperform the other three methods in forecasting inflation. That the median Livingston and SPF survey forecasts do well is perhaps not surprising, because presumably many of the best analysts use time-series and Phillips Curve models. However, even participants in the Michigan survey who are consumers, not professionals, produce accurate out-of-sample forecasts, which are only slightly worse than those of the professionals in the Livingston and SPF surveys. We also find that the best survey forecasts are the survey median forecasts themselves; adjustments to take into account both linear and non-linear bias yield worse out-of-sample forecasting performance.

Second, term structure information does not generally lead to better forecasts and often leads to inferior forecasts than models using only aggregate activity measures. Whereas this confirms the results in Stock and Watson (1999), our investigation of term structure models is much more comprehensive. The relatively poor forecasting performance of term structure models extends to simple regression specifications, iterated long-horizon VAR forecasts, no-arbitrage affine models, and non-linear no-arbitrage models. These results suggest that while inflation is very important for explaining the dynamics of the term structure (see, e.g., Ang, Bekaert and Wei, 2006), yield curve information is less important for forecasting future inflation.

Our third major finding is that combining forecasts does not generally lead to better out-of-sample forecasting performance than single forecasting models. In particular, simple averaging, like using the mean or median of a number of forecasts, does not necessarily improve the forecast performance, whereas linear combinations of forecasts with weights computed based on past performance and prior information generate the biggest gains. Even the Phillips curve models using the Bernanke, Boivin and Eliasz (2005) forward-looking aggregate measure of real activity mostly does not perform well relative to simpler Phillips curve models and never outperforms the survey forecasts. The strong success of the surveys in forecasting inflation out-of-sample extends to surveys dominating other models in forecast combinination methods. The data consistently place the highest weights on the survey forecasts and little weight on other forecasting methods.

The remainder of this paper is organized as follows. Section 2 describes the data set. In Section 3, we describe the time-series models, predictive macro regressions, term structure models, and forecasts from survey data, and detail the forecasting methodology. Section 4 contains the empirical out-of-sample results. We examine the robustness of our results to a non-stationary inflation specification in Section 5. Finally, Section 6 concludes.

2 Data

2.1 Inflation

We consider four different measures of inflation. The first three are consumer price index (CPI) measures, including CPI-U for All Urban Consumers, All Items (PUNEW), CPI for All Urban Consumers, All Items Less Shelter (PUXHS) and CPI for All Urban Consumers, All Items Less Food and Energy (PUXX), which is also called core CPI. The latter two measures strip out highly volatile components in order to better reflect underlying price trends (see the discussion in Quah and Vahey, 1995). The fourth measure is the Personal Consumption Expenditure deflator (PCE). While all three surveys forecast a CPI-based inflation measure, PCE inflation features prominently in policy work at the Federal Reserve. All measures are seasonally adjusted and obtained from the Bureau of Labor Statistics website. The sample period is 1952:Q2 to 2002:Q4 for PUNEW and PUXHS, 1958:Q2 to 2002:Q4 for PUXX, and 1960:Q2 to 2002:Q4 for PCE.

We define the quarterly inflation rate, $\pi_{t}$, from $t-1$ to $t$ as:

$$\pi_{t}=\ln \left( \frac{P_{t}}{P_{t-1}}\right),$$ (A-1)

where $P_{t}$ is the level of one of the four inflation indices at time $t$. We use the terms " inflation" and "inflation rate" interchangeably as defined in equation (1). We take one quarter to be our base unit for estimation purposes, but forecast annual inflation, $\pi_{t+4,4}$, from $t$ to $t+4$:

$$\pi_{t+4,4} = \pi_{t+1} + \pi_{t+2} + \pi_{t+3} + \pi_{t+4},$$ (A-2)

where $\pi_t$ is the quarterly inflation rate in equation (1).

Empirical work on inflation has failed to come to a consensus regarding its stationarity properties. For example, Bryan and Cecchetti (1993) assume a stationary inflation process, while Nelson and Schwert (1977) and Stock and Watson (1999) assume that the inflation process has a unit root. Most of our analysis assumes that inflation is stationary for two reasons. First, it is difficult to generate non-stationary inflation in standard economic models, whether they are monetary in nature, or of the New Keynesian variety (see Fuhrer and Moore, 1995; Holden and Driscoll, 2003). Second, the working paper version of Bai and Ng (2004) recently rejects the null of non-stationarity for inflation. That being said, Cogley and Sargent (2005) and Stock and Watson (2005) find evidence of changes in inflation persistence over time, with a random walk or integrated MA-process providing an accurate description of inflation dynamics during certain times. Furthermore, the use of a parsimonious non-stationary model may be attractive for forecasting. In particular, Atkeson and Ohanian (2001) have made the random walk a natural benchmark to beat in forecasting exercises. Therefore, we consider whether our results are robust to assuming non-stationary inflation in Section 5.

Table 1 reports summary statistics for all four measures of inflation for the full sample in Panel A, and the post-1985 sample and the post-1995 sample in Panels B and C, respectively. Our statistics pertain to annual inflation, $\pi_{t+4,t}$, but we sample the data quarterly. Therefore, we report the fourth autocorrelation for quarterly inflation, which corresponds to the first autocorrelation for annual inflation. Table 1 shows that all four inflation measures are lower and more stable during the last two decades, in common with many other macroeconomic series, including output (see Kim and Nelson, 1999; McConnell and Perez-Quiros, 2000; Stock and Watson, 2002b). Core CPI (PUXX) has the lowest volatility of all the inflation measures. PUXX volatility ranges from 2.56% per annum over the full sample to only 0.24% per annum post-1996. The higher variability of the other measures in the latter part of the sample must be due to food and energy price changes. In the later sample periods, PCE inflation is, on average, lower than CPI inflation, which may be partly due to its use of a chain weighting in contrast to the other CPI measures which use a fixed basket (see Clark, 1999).

Inflation is somewhat persistent (0.79% for PUNEW over the full sample), but its persistence decreases over time, as can be seen from the lower autocorrelation coefficients for the PUNEW and the PUXHS measures after 1986, and for all measures after 1995. The correlations of the four measures of inflation with each other are all over 75% over the full sample. The comovement can be clearly seen in the top panel of Figure 1. Inflation is lower prior to 1969 and after 1983, but reaches a high of around 14% during the oil crisis of 1973-1983. PUXX tracks both PUNEW and PUXHS closely, except during the 1973-1975 period, where it is about 2% lower than the other two measures, and after 1985, where it appears to be more stable than the other two measures. During the periods when inflation is decelerating, such as in 1955-1956, 1987-1988, 1998-2000 and most recently 2002-2003, PUNEW declines more gradually than PUXHS, suggesting that housing prices are less volatile than the prices of other consumption goods during these periods.

2.2 Real Activity Measures

We consider six individual series for real activity along with one composite real activity factor. We compute GDP growth (GDPG) using the seasonally adjusted data on real GDP in billions of chained 2000 dollars. The unemployment rate (UNEMP) is also seasonally adjusted and computed for the civilian labor force aged 16 years and over. Both real GDP and the unemployment rate are from the Federal Reserve Economic Data (FRED) database. We compute the output gap either as the detrended log real GDP by removing a quadratic trend as in Gali and Gertler (1999), which we term GAP1, or by using the Hodrick-Prescott (1997) filter (with the standard smoothness parameter of 1,600), which we term GAP2. At time $t$, both measures are constructed using only current and past GDP values, so the filters are run recursively. We also use the labor income share (LSHR), defined as the ratio of nominal compensation to total nominal output in the U.S. nonfarm business sector. We use two forward-looking indicators: the Stock-Watson (1989) Experimental Leading Index (XLI) and their Alternative Nonfinancial Experimental Leading Index-2 (XLI-2).

Because Stock and Watson (2002a), among others, show that aggregating the information from many factors has good forecasting power, we also use a single factor aggregating the information from 65 individual series constructed by Bernanke, Boivin and Eliasz (2005). This single real activity series, which we term FAC, aggregates real output and income, employment and hours, consumption, housing starts and sales, real inventories, and average hourly earnings. The sample period for all the real activity measures is 1952:Q2 to 2001:Q4, except the Bernanke-Boivin-Eliasz real activity factor, which spans 1959:Q1 to 2001:Q3. We use the composite real activity factor at the end of each quarter for forecasting inflation over the next year.¹

The real activity measures have the disadvantage that they may use information that is not actually available at the time of the forecast, either through data revisions, or because of full sample estimation in the case of the Bernanke-Boivin-Eliasz measure. This biases the forecasts from Phillips curve models to be better than what could be actually forecasted using a real-time data set. The use of real time economic activity measures produces much worse forecasts of future inflation compared to the use of revised economic series in Orphanides and van Norden (2001) but only slightly worse forecasts for both inflation and real activity in Bernanke and Boivin (2003). Nevertheless, our forecast errors using real activity measures are likely biased downwards.

2.3 Term Structure Data

The term structure variables are zero-coupon yields for the maturities of 1, 4, 8, 12, 16, and 20 quarters from CRSP spanning 1952:Q2 to 2001:Q4. The one-quarter rate is from the CRSP Fama risk-free rate file, while all other bond yields are from the CRSP Fama-Bliss discount bond file. All yields are continuously compounded and expressed at a quarterly frequency. We define the short rate (RATE) to be the one-quarter yield and define the term spread (SPD) to be the difference between the 20-quarter yield and the short rate. Some of our term structure models also use four-quarter and 12-quarter yields for estimation.

2.4 Surveys

We examine three inflation expectation surveys: the Livingston survey, the Survey of Professional Forecasters (SPF), and the Michigan survey.² The Livingston survey is conducted twice a year, in June and in December, and polls economists from industry, government, and academia. The Livingston survey records participants' forecasts of non-seasonally-adjusted CPI levels six and twelve months in the future and is usually conducted in the middle of the month. Unlike the Livingston survey, participants in the SPF and the Michigan survey forecast inflation rates. Participants in the SPF are drawn primarily from business, and forecast changes in the quarterly average of seasonally-adjusted CPI-U levels. The SPF is conducted in the middle of every quarter and the sample period for the SPF median forecasts is from 1981:Q3 to 2002:Q4. In contrast to the Livingston survey and SPF, the Michigan survey is conducted monthly and asks households, rather than professionals, to estimate expected price changes over the next twelve months. We use the median Michigan survey forecast of inflation over the next year at the end of each quarter from 1978:Q1 to 2002:Q4.

There are some reporting lags between the time the surveys are taken and the public dissemination of their results. For the Livingston and the SPF surveys, there is a lag of about one week between the due date of the survey and their publication. However, these reporting lags are largely inconsequential for our purposes. What matters is the information set used by the forecasters in predicting future inflation. Clearly, survey forecasts must use less up to date information than either macro-economic or term structure forecasts. For example, the Livingston survey forecasters presumably use information up to at most the beginning of June and December, and mostly do not even have the May and November official CPI numbers available when making a forecast. The SPF forecasts can only use information up to at most the middle of the quarter and while we take the final month of the quarter for the Michigan survey, consumers do not have up-to-date economic data available at the end of the quarter. But, for the economist forecasting annual inflation with the surveys, all survey data is publicly available at the end of each quarter for the SPF and Michigan surveys, and at the end of each semi-annual period for the Livingston survey. Together with the slight data advantages present in revised, fitted macro data, we are in fact biasing the results against survey forecasts.

The Livingston survey is the only survey available for our full sample. In the top panel of Figure 1, which graphs the full sample of inflation data, we also include the unadjusted median Livingston forecasts. We plot the survey forecast lagged one year, so that in December 1990, we plot inflation from December 1989 to December 1990 together with the survey forecasts of December 1989. The Livingston forecasts broadly track the movements of inflation, but there are several large movements that the Livingston survey fails to track, for example the pickup in inflation in 1956-1959, 1967-1971, 1972-1975, and 1978-1981. In the bottom panel of Figure 1, we graph all three survey forecasts of future one-year inflation together with the annual PUNEW inflation, where the survey forecasts are lagged one year for direct comparison. After 1981, all survey forecasts move reasonably closely together and track inflation movements relatively well. Nevertheless, there are still some notable failures, like the slowdowns in inflation in the early 1980s and in 1996.

3 Forecasting Models and Methodology

In this section, we describe the forecasting models and describe our statistical tests. In all our out-of-sample forecasting exercises, we forecast future annual inflation. Hence, for all our models, we compute annual inflation forecasts of:

$$\mathrm{E}_{t}(\pi_{t+4,4})= \mathrm{E}_{t}\left( \sum_{i=1}^{4}\pi_{t+i}\right) ,$$ (A-3)

where $\pi_{t+4,4}$ is annual inflation from $t$ to $t+4$ defined in equation (2).

In Sections 3.1 to 3.4, we describe our 39 forecasting models. Table 2 contains a full nomenclature. Section 3.1 focuses on time-series models of inflation, which serve as our benchmark forecasts; Section 3.2 summarizes our OLS regression models using real activity macro variables; Section 3.3 describes the term structure models incorporating inflation data; and finally, Section 3.4 describes our survey forecasts. In Section 3.5, we define the out-of-sample periods and list the criteria that we use to assess the performance of out-of-sample forecasts. Finally, Section 3.6 describes our methodology to combine model forecasts.

For all models except OLS regressions, we compute implied long-horizon forecasts from single-period (quarterly) models. While Schorfheide (2005) shows that in theory, iterated forecasts need not be superior to direct forecasts from horizon-specific models, Marcellino, Stock and Watson (2006) document the empirical superiority of iterated forecasts in predicting U.S. macroeconomic series. For the OLS models, we compute the forecasts directly from the long-horizon regression estimates.

3.1 Time-Series Models

ARIMA Models

If inflation is stationary, the Wold theorem suggests that a parsimonious ARMA($p,q$) model may perform well in forecasting. We consider two ARMA($p,q$) models: an ARMA(1,1) model and a pure autoregressive model with $p$ lags, AR($p$). The optimal lag length for the AR model is recursively selected using the Schwartz criterion (BIC) on the in-sample data. The motivation for the ARMA(1,1) model derives from a long tradition in rational expectations macroeconomics (see Hamilton, 1985) and finance (see Fama, 1975) that models inflation as the sum of expected inflation and noise. If expected inflation follows an AR(1) process, then the reduced-form model for inflation is given by an ARMA(1,1) model. The ARMA(1,1) model also nicely fits the slowly decaying autocorrelogram of inflation.

The specifications of the ARMA(1,1) model,

$$\pi_{t+1} = \mu+\phi\pi_{t}+\psi\varepsilon_{t}+\varepsilon_{t+1},$$ (A-4)

and the AR($p$) model,

$$\pi_{t+1} =\mu+\phi_{1}\pi_{t}+\phi_{2}\pi_{t-1}+\ldots+\phi_{p} \pi _{t-p+1}+\varepsilon_{t+1},$$ (A-5)

are entirely standard. The ARMA(1,1) model is estimated by maximum likelihood, conditional on a zero initial residual. We compute the implied inflation level forecast over the next year expressed at a quarterly frequency. For the ARMA(1,1) model, the forecast is:

$$\mathrm{E} _{t}(\pi_{t+4,4}) =\frac{1}{1-\phi}\left[ 4-\frac{\phi\left( 1-\phi^{4}\right) }{\left( 1-\phi\right) }\right] \mu+ \frac{\phi\left( 1-\phi^{4}\right) }{\left( 1-\phi\right) }\pi_{t} + \frac{\left( 1-\phi^{4}\right) \psi}{\left( 1-\phi\right) }\varepsilon_{t}.$$

To facilitate the forecasts of annual inflation, we write the AR($p$) model in first-order companion form:

$$X_{t+1}=A+\Phi X_{t}+U_{t+1},$$

where

$$X_{t}=\left[ \begin{array}[c]{c}% \pi_{t}\ \pi_{t-1}\ \vdots\ \pi_{t-p+1}% \end{array}\right] ,\text{ }A=\left[ \begin{array}[c]{c}% \mu\ 0\ \vdots\ 0 \end{array}\right] ,\text{ }\Phi=\left[ \begin{array}[c]{cccc}% \phi_{1} & \phi_{2} & ... & \phi_{p}\ 1 & 0 & ... & 0\ \vdots & \vdots & \ddots & \vdots\ 0 & 0 & ... & 0 \end{array}\right] \text{ and }U_{t}=\left[ \begin{array}[c]{c}% \varepsilon_{t}\ 0\ \vdots\ 0 \end{array}\right] .$$

Then, the forecast for the AR($p$) model is given by:

$$\mathrm{E}_{t}(\pi_{t+4,4}) = e_{1}^{\prime}\left( I-\Phi\right) ^{-1}\left( 4 I-\Phi\left( I-\Phi\right) ^{-1}\left( I-\Phi^{4}\right) \right) A + e_{1}^{\prime}\Phi\left( I-\Phi\right) ^{-1}\left( I-\Phi ^{4}\right) X_{t},$$

where $e_{1}$ is a $p \times1$ selection vector containing a one in the first row and zeros elsewhere.

Our third ARIMA benchmark is a random walk (RW) forecast where $\pi_{t+1}=\pi_{t}+\varepsilon_{t+1}$, and $\mathrm{E} _{t}(\pi_{t+4,4}) = 4\pi_{t}$. Inspired by Atkeson and Ohanian (2001), we also forecast inflation using a random walk model on annual inflation, where the forecast is given by $\ensuremath{{\rm E}}_{t}(\pi_{t+4,4})=\pi_{t,4}$. We denote this forecast as AORW.

Regime-Switching Models

Evans and Wachtel (1993), Evans and Lewis (1995), and Ang and Bekaert (2004), among others, document regime-switching behavior in inflation. A regime-switching model may potentially account for non-linearities and structural changes, such as a sudden shift in inflation expectations after a supply shock, or a change in inflation persistence.

We estimate the following univariate regime-switching model for inflation, which we term RGM:

$$\pi_{t+1}=\mu\left( s_{t+1}\right) +\phi\left( s_{t+1}\right) \pi_{t}+\sigma\left( s_{t+1}\right) \varepsilon_{t+1}%$$ (A-6)

The regime variable $s_{t}=1,2$ follows a Markov chain with constant transition probabilities $P=Pr(s_{t+1}=1\vert s_{t}=1)$ and $Q=Pr(s_{t+1}% =2\vert s_{t}=2)$. The model can be estimated using the Bayesian filter algorithms of Hamilton (1989) and Gray (1996). We compute the implied annual horizon forecasts of inflation from equation (6), assuming that the current regime is the regime that maximizes the probability $Pr(s_{t}\vert I_{t})$. This is a byproduct of the estimation algorithm.

3.2 Regression Forecasts Based on the Phillips Curve

In standard Phillips curve models of inflation, expected inflation is linked to some measure of the output gap. There are both forward- and backward-looking Phillips curve models, but ultimately even forward-looking models link expected inflation to the current information set. According to the Phillips curve, measures of real activity should be an important part of this information set. We avoid the debate regarding the actual measure of the output gap (see, for instance, Gali and Gertler, 1999) by taking an empirical approach and using a large number of real activity measures. We choose not to estimate structural models because the BIC criterion is likely to choose the empirical model best suitable for forecasting. Previous work often finds that models with the clearest theoretical justification often have poor predictive content (see the literature summary by Stock and Watson, 2003).

The empirical specification we estimate is:

$$\pi_{t+4,4}=\alpha+\beta(L)^{\prime}X_{t}+\varepsilon _{t+4,4}%$$ (A-7)

where $X_{t}$ combines $\pi_{t}$ and one or two real activity measures. The lag length in the lag polynomial $\beta(L)$ is selected by BIC on the in-sample data and is set to be equal across all the regressors in $X_{t}$. The chosen specification tends to have two or three lags in our forecasting exercises. We list the complete set of real activity regressors in Table 2 as PC1 to PC10.

In our next section, we extend the information set to include term structure information. Regression models where term structure information is included in $X_{t}$ along with inflation and real activity are potentially consistent with a forward-looking Phillips curve that includes inflation and real activity measures in the information set. Such models can approximate the reduced form of a more sophisticated, forward-looking rational expectations Phillips curve model of inflation (see, for instance, Bekaert, Cho and Moreno, 2005).

3.3 Models Using Term Structure Data

We consider a variety of term structure forecasts, including augmenting the simple Phillips Curve OLS regressions with short rate and term spread variables; long-horizon VAR forecasts; a regime-switching specification; affine term structure models; and term structure models incorporating regime switches. We outline each of these specifications in turn.

Linear Non-Structural Models

We begin by augmenting the OLS Phillips Curve models in equation (7) with the short rate, RATE, and the term spread, SPD, as regressors in $X_{t}$. Specifications TS1-TS8 add RATE to the Phillips Curve Curve specifications PC1-PC8. TS9 and $\emph{TS10}$ only use inflation and term structure variables as predictors. TS9 uses inflation and the lagged term spread, producing a forecasting model similar to the specification in Mishkin (1990, 1991). TS10 adds the short rate to this specification. Finally, TS11 adds GDP growth to the TS10 specification.

We also consider forecasts with a VAR(1) in $X_{t}$, where $X_{t}$ contains RATE, SPD, GDPG, and $\pi_{t}$:

$$X_{t+1}=\mu+\Phi X_{t}+\varepsilon_{t+1}.%$$ (A-8)

Although the VAR is specified at a quarterly frequency, we compute the annual horizon forecast of inflation implied by the VAR. We denote this forecasting specification as VAR. As Ang, Piazzesi and Wei (2004) and Cochrane and Piazzesi (2005) note, a VAR specification can be economically motivated from the fact that a reduced-form VAR is equivalent to a Gaussian term structure model where the term structure factors are observable yields and certain assumptions on risk premia apply. Under these restrictions, a VAR coincides with a no-arbitrage term structure model only for those yields included in the VAR. However, the VAR does not impose over-identifying restrictions generated by the term structure model for yields not included as factors in the VAR.

An Empirical Non-Linear Regime-Switching Model

A large empirical literature has documented the presence of regime switches in interest rates (see, among others, Hamilton, 1988; Gray, 1996; Bekaert, Hodrick and Marshall, 2001). In particular, Ang and Bekaert (2002) show that regime-switching models forecast interest rates better than linear models. As interest rates reflect information in expected inflation, capturing the regime-switching behavior in interest rates may help in forecasting potentially regime-switching dynamics of inflation.

We estimate a regime-switching VAR, denoted as RGMVAR:

$$X_{t+1}=\mu(s_{t+1})+\Phi X_{t}+\Sigma(s_{t+1})\varepsilon _{t+1},$$ (A-9)

where $X_{t}$ contains RATE, SPD and $\pi_{t}$. Similar to the univariate regime-switching model in equation (6), $s_{t}=1$ or 2 and follows a Markov chain with constant transition probabilities. We compute out-of-sample forecasts from equation (9) assuming that the current regime is the regime with the highest probability $Pr(s_{t}\vert I_{t})$.

No-Arbitrage Term Structure Models

We estimate two no-arbitrage term structure models. Because such models have implications for the complete yield curve, it is straightforward to incorporate additional information from the yield curve into the estimation. Such additional information is absent in the empirical VAR specified in equation (8). Concretely, both no-arbitrage models have two latent variables and quarterly inflation as state variables, denoted by $X_{t}$. We estimate the models by maximum likelihood, and following Chen and Scott (1993), assume that the one- and 20-quarter yields are measured without error, and the other four- and 12-quarter yields are measured with error. The estimated models build on Ang, Bekaert and Wei (2006), who formulate a real pricing kernel as:

$$\widehat{M}_{t+1}=\exp\left( -r_{t} - \frac{1}{2}\lambda_{t}^{\prime}\lambda_{t} -\lambda_{t}\varepsilon_{t+1}\right).$$ (A-10)

Here, $\lambda_t$ is a $3 \times 1$ real price of risk vector. The real short rate is an affine function of the state variables. The nominal pricing kernel is defined in the standard way as $M_{t+1} =\widehat{M}_{t+1} \exp(-\pi_{t+1})$. Bonds are priced using the recursion:

$$\exp(-ny_{t}^{n})= \ensuremath{{\rm E}}_{t}[M_{t+1}\exp(-(n-1)y_{t+1}^{n-1})],$$

where $y_{t}^{n}$ is the n-quarter zero-coupon bond yield.

The first no-arbitrage model (MDL1) is an affine model in the class of Duffie and Kan (1996) with affine, time-varying risk premia (see Dai and Singleton, 2002; Duffee, 2002) modelled as:

$$\lambda_{t}=\lambda_{0}+\lambda_{1}X_{t}.$$ (A-11)

where $\lambda_{0}$ is a $3 \times 1$ vector and $\lambda_{1}$ a $3\times3$ diagonal matrix. The state variables follow a linear VAR:

$$X_{t}=\mu+\Phi X_{t-1}+\Sigma\varepsilon_{t+1}.%$$ (A-12)

The second model (MDL2) incorporates regime switches and is developed by Ang, Bekaert and Wei (2006). Ang, Bekaert and Wei show that this model fits the moments of yields and inflation very well and almost exactly matches the autocorrelogram of inflation. MDL2 replaces equation (12) with the regime-switching VAR:

$$X_{t}=\mu(s_{t+1})+\Phi X_{t-1}+\Sigma(s_{t+1})\varepsilon_{t+1},$$ (A-13)

and also incorporates regime switches in the prices of risk, replacing equation (11) with

$$\lambda_{t}=\lambda_{0}(s_{t+1})+\lambda_{1}X_{t}.$$ (A-14)

There are four regime variables $s_t = 1, \dots, 4$ in the Ang, Bekaert and Wei (2006) model representing all possible combinations of two regimes of inflation and two regimes of a real latent factor.

In estimating MDL1 and MDL2, we impose the same parameter restrictions necessary for identification as Ang, Bekaert and Wei (2006) do. For both MDL1 and MDL2, we compute out-of-sample forecasts of annual inflation, but the models are estimated using quarterly data.

3.4 Survey Forecasts

We produce estimates of $\mathrm{E}_{t}(\pi_{t+4,4})$ from the Livingston, SPF, and the Michigan surveys. We denote the actual forecasts from the SPF, Livingston and Michigan surveys as SPF1, LIV1, and MCH1, respectively.

Producing Forecasts from Survey Data

Participants in the Livingston survey are asked to forecast a CPI level (not an inflation rate). Given the timing of the survey, Carlson (1977) carefully studies the forecasts of individual participants in the Livingston survey and finds that the participants generally forecast inflation over the next 14 months. We follow Thomas (1999) and Mehra (2002) and adjust the raw Livingston forecasts by a factor of 12/14 to obtain an annual inflation forecast.

Participants in both the SPF and the Michigan surveys do not forecast log year-on-year CPI levels according to the definition of inflation in equation (1). Instead, the surveys record simple expected inflation changes, $\mathrm{E} _{t}(P_{t+4}/P_{t} - 1)$. This differs from $\mathrm{E} _{t}(\log P_{t+4}/P_{t} )$ by a Jensen's inequality term. In addition, the SPF participants are asked to forecast changes in the quarterly average of seasonally-adjusted PUNEW (CPI-U), as opposed to end-of-quarter changes in CPI levels. In both the SPF and the Michigan survey, we cannot directly recover forecasts of expected log changes in CPI levels. Instead, we directly use the SPF and Michigan survey forecasts to represent forecasts of future annual inflation as defined in equation (3). We expect that the effects of these measurement problems are small.³ In any case, the Jensen's term biases our survey forecasts upwards, imparting a conservative upward bias to our Root Mean Squared Error (RMSE) statistics.

Adjusting Surveys for Bias

Several authors, including Thomas (1999), Mehra (2002), and Souleles (2004), document that survey forecasts are biased. We take into account the survey bias by estimating $\alpha_{1}$ and $\beta_{1}$ in the regressions:

$$\pi_{t+4,4}=\alpha_{1}+\beta_{1}f_{t}^{S} +\varepsilon_{t+4,4},$$ (A-15)

where $f_{t}^{S}$ is the forecast from the candidate survey $S$. For an unbiased forecasting model, $\alpha_{1}=0$ and $\beta_{1}=1$. We denote survey forecasts that are adjusted using regression (15) as SPF2, LIV2, and MCH2 for the SPF, Livingston, and Michigan surveys, respectively. The bias adjustment occurs recursively, that is, we update the regression with new data points each quarter and re-estimate the coefficients.

Table 3 provides empirical evidence regarding these biases using the full sample. For each inflation measure, the first three rows report the results from regression (15). The SPF survey forecasts produce $\beta_{1}$s that are smaller than one for all inflation measures, which are, with the exception of PUXX, significant at the 95% level. However, the point estimates of $\alpha_{1}$ are also positive, although mostly not significant, which implies that at low levels of inflation, the surveys under-predict future inflation and at high levels of inflation the surveys over-predict future inflation. The turning point is $0.852/(1-0.694) = 2.8\%$, so that the SPF survey mostly over-predicts inflation. The Livingston and Michigan surveys produce largely unbiased forecasts because the slope coefficients are insignificantly different from one and the constants are insignificantly different from zero. Nevertheless, because the intercepts are positive (negative) for the Livingston (Michigan) survey, and the slope coefficients largely smaller (larger) than one, the Livingston (Michigan) survey tends to produce mostly forecasts that are too low (high).

Thomas (1999) and Mehra (2002) suggest that the bias in the survey forecasts may vary across accelerating versus decelerating inflation environments, or across the business cycle. To take account of this possible asymmetry in the bias, we augment equation (15) with a dummy variable, $D_t$, which equals one if inflation at time $t$ exceeds its past two-year moving average,

$$\pi_{t}-\frac{1}{8}\sum_{j=0}^{7}\pi_{t-j}>0,$$

otherwise $D_{t}$ is set equal to zero. The regression becomes:

$$\pi_{t+4,4}=\alpha_{1}+\alpha_{2}D_{t}+\beta_{1} f_{t}% ^{S} +\beta_{2}D_{t}f_{t}^{S} +\varepsilon_{t+4,4}.$$ (A-16)

We denote the survey forecasts that are non-linearly bias-adjusted using equation (16) as SPF3, LIV3, and MCH3 for the SPF, Livingston, and Michigan surveys, respectively.⁴

The bottom three rows of each panel in Table 3 report results from regression (16). Non-linear biases are reflected in significant $\alpha_2$ or $\beta_2$ coefficients. For the SPF survey, there is no statistical evidence of non-linear biases. For all inflation measures, the SPF's negative $\alpha_2$ and positive $\beta_2$ coefficients indicates that accelerating inflation implies a smaller intercept and a higher slope coefficient, bringing the SPF forecasts closer to unbiasedness. For the Michigan survey, the biases are larger in magnitude (except for the PUXX measure) but there is only one significant coefficient: accelerating inflation yields a significantly higher slope coefficient for the PUXHS measure. Economically, the Michigan survey is very close to unbiasedness in decelerating inflation environments, but over- (under-) predicts future inflation at low (high) inflation levels in accelerating inflation environments.

The Livingston survey has the strongest evidence of non-linear bias, for which we also have the longest data sample. The coefficients have the same sign as for the other surveys, but now the $\beta_2$ slope coefficients significantly increase in accelerating inflation environments for all inflation measures except PUXX. As in the case of the SPF survey, the Livingston survey is closer to being unbiased in accelerating inflation environments. Without accounting for non-linearity, the Livingston survey produces largely unbiased forecasts in Table 3. However, the results of regression (16) for the Livingston survey show it produces mostly biased forecasts in decelerating inflation environments, under-predicting future inflation when inflation is relatively low, and over-predicting future inflation when inflation is relatively high.

3.5 Assessing Forecasting Models

Out-of-Sample Periods

We select two starting dates for our out-of-sample forecasts, 1985:Q4 and 1995:Q4. Our main analysis focuses on recursive out-of-sample forecasts, which use all the data available at time $t$ to forecast annual future inflation from $t$ to $t+4$. Hence, the windows used for estimation lengthen through time. We also consider out-of-sample forecasts with a fixed rolling window. All of our annual forecasts are computed at a quarterly frequency, with the exception of forecasts from the Livingston survey, where forecasts are only available for the second and fourth quarter each year.⁵ The out-of-sample periods end in 2002:Q4, except for forecasts with the composite real activity factor, which end in 2001:Q3.

Measuring Forecast Accuracy

We assess forecast accuracy with the Root Mean Squared Error (RMSE) of the forecasts produced by each model and also report the ratio of RMSEs relative to a time-series ARMA(1,1) benchmark that uses only information in the past series of inflation. We show below that the ARMA(1,1) model nearly always produces the lowest RMSE among all of the ARIMA time-series models that we examine.

To compare the out-of-sample forecasting performance of the various models, we perform a forecast comparison regression, following Stock and Watson (1999):

$$\pi_{t+4,4}=\lambda f_{t}^{ARMA}+(1-\lambda)f_{t}^{x}+\varepsilon_{t+4,4},$$ (A-17)

where $f_{t}^{ARMA}$ is the forecast of $\pi_{t+4,4}$ from the ARMA(1,1) time-series model, $f_{t}^{x}$ is the forecast from the candidate model $x$, and $\varepsilon_{t+4,4}$ is the forecast error associated with the combined forecast. If $\lambda=0$, then forecasts from the ARMA(1,1) model add nothing to the forecasts from candidate model $x$, and we thus conclude that model $x$ out-performs the ARMA(1,1) benchmark. If $\lambda=1$, then forecasts from model $x$ add nothing to forecasts from the ARMA(1,1) time-series benchmark.

Stock and Watson (1999) note that inference about $\lambda$ is complicated by the fact that the forecasts errors, $\varepsilon_{t+4,4}$, follow a MA(3) process because the overlapping annual observations are sampled at a quarterly frequency. We compute standard errors that account for the overlap by using Hansen and Hodrick (1980) standard errors. To also take into account the estimated parameter uncertainty in one or both sets of the forecasts, $f_{t}^{ARMA}$ and $f_{t}^{x}$, we also compute West (1996) standard errors. The Appendix provides a detailed description of the computations involved.

3.6 Combining Models

A long statistics literature documents that forecast combinations typically provide better forecasts than individual forecasting models.⁶ For inflation forecasts, Stock and Watson (1999) and Wright (2004), among others, show that combined forecasts using real activity and financial indicators are usually more accurate than individual forecasts. To examine if combining the information in different forecasts leads to gains in out-of-sample forecasting accuracy, we examine five different methods of combining forecasts. All these methods involve placing different weights on $n$ individual forecasting models. The five model combination methods can be summarized as follows:

 

Combination Methods
     1. Mean
     2. Median
     3. OLS
     4. Equal-Weight Prior
     5. Unit-Weight Prior

 

All our model combinations are ex-ante. That is, we compute the weights on the models using the history of out-of-sample forecasts up to time $t$. Hence, the ex-ante method assesses actual out-of-sample forecasting power of combination methods. For example, the weights used to construct the ex-ante combined forecast at 2000:Q4 is based on a regression of realized annual inflation over 1985:Q4 to 2000:Q4 on the constructed out-of-sample forecasts over the same period.

In the first two model combination methods, we simply look at the overall mean and median, respectively, over $n$ different forecasting models. Equal weighting of many forecasts has been used as early as Bates and Granger (1969) and, in practice, simple equal-weighting forecasting schemes are hard to beat. In particular, Stock and Watson (2003) show that this method produces superior out-of-sample forecasts of inflation.

In the last three combination methods, we compute different individual model weights that vary over time. These weights are estimated as slope coefficients in a regression of realized inflation on model forecasts:

$$\pi_{t+4,4}=\sum_{i=1}^{n}\omega_{t}^{i}f_{t}^{i}+\varepsilon_{t,t+4}, t=1,\ldots,T,$$ (A-18)

where $f_{t}^{i}$ is the $i$-th model forecast at time $t$. The $n\times1$ weight vector $\omega_{t} = \{\omega_{t}^{i} \}$ is estimated either by OLS, as in our third model combination specification, or using the mixed regressor method proposed by Theil and Goldberger (1961) and Theil (1963), as in Combination Methods 4 and 5.

To describe the last two combination methods, we set up some notation. Suppose we have $T$ forecast observations with $n$ individual models. Let $F$ be the $T\times n$ matrix of forecasts and $\pi$ the $T\times1$ vector of actual future inflation levels that are being forecast. Consequently, the $s$-th row of $F$ is given by $F_{s}=\left\{ f_{s}^{1},...f_{s}^{n}\right\}$. The mixed regression estimator can be viewed as a Bayesian estimator with the prior $\omega\sim N\left( \mu,\sigma_{\omega}^{2}I\right)$, where $\sigma_{\omega}^{2}$ is a scalar and $I$ the $n\times n$ identity matrix. The estimator can be derived as:

$$\widehat{\omega}=(F^{\prime}F+\gamma I)^{-1}\left( F^{\prime}\pi+\gamma \mu\right) ,$$ (A-19)

where the parameter $\gamma$ controls the amount of shrinkage towards the prior. In particular, when $\gamma=0$, the estimator simplifies to standard OLS, and when $\gamma\rightarrow\infty$, the estimator approaches the weighted average of the forecasts, with the weights given by the prior weights. It is instructive to re-write the estimator as a weighted average of the OLS estimator and the prior:

$$\widehat{\omega}=\theta_{OLS}$$ $$\omega_{OLS}+\theta_{prior}$$ $$\mu$$

with $\theta_{OLS}=\left( F^{\prime}F+\gamma I\right) ^{-1}\left( F^{\prime}F\right)$ and $\theta_{prior}=\left( F^{\prime}F+\gamma I\right) ^{-1}\left( \gamma I\right)$, so that the weights add up to the identity matrix.

We use empirical Bayes methods and estimate the shrinkage parameter as:

$$\widehat{\gamma}=\widehat{\sigma}^{2}/\widehat{\sigma }_{\omega}^{2},$$ (A-20)

where

$$\widehat{\sigma}^{2}=\frac{1}{T}\pi^{\prime}\left[ I-F\left( F^{\prime }F\right) ^{-1}F^{\prime}\right] \pi$$

and

$$\widehat{\sigma}_{\omega}^{2}=\frac{\pi^{\prime}\pi-T\widehat{\sigma}^{2}% }{trace\left( F^{\prime}F\right) }.$$

To interpret the shrinkage parameter, observe that $\widehat{\sigma}^{2}$ is simply the residual variance of the regression; the numerator of $\widehat{\sigma}_{\omega}^{2}$ is the fitted variance of the regression and the denominator is the average variance of the independent variables (the forecasts) in the regression. Consequently, the shrinkage parameter, $\gamma$, in equation (20) increases when the variance of the independent variables becomes larger, and decreases as the $R^{2}$ of the regression increases. In other words, if forecasts are (not) very variable and the regression $R^{2}$ is small (large), we trust the prior (the regression).

We examine the effect of two priors. In Model Combination 4, we use an equal-weight prior where each element of $\mu$, $\mu_{i}=1/n,i=1, \ldots, n$, which leads to the Ridge regressor used by Stock and Watson (1999). In the second prior (Model Combination 5), we assign unit weight to one type of forecast, for example, $\mu=\left\{ 0\ldots1\ldots0\right\} ^{\prime}$. One natural choice for a unit weight prior would be to choose the best performing univariate forecast model.

When we compute the model weights, we impose the constraint that the weight on each model is positive and the weights sum to one. This ensures that the weights represent the best combination of models that produce good forecasts in their own right, rather than place negative weights on models that give consistently wrong forecasts. This is also very similar to shrinkage methods of forecasting (see Stock and Watson, 2005). For example, Bayesian Model Averaging uses posterior probabilities as weights, which are, by construction, positive and sum to one.⁷

The positivity constraint is imposed by minimizing the usual loss function, $L$, associated with OLS for combination method 3:

$$L=\left( \pi-F\omega\right) ^{\prime}\left( \pi-F\omega\right) ,$$

and a loss function for the mixed regressor estimations (combination methods 4 and 5):

$$L=\frac{\left( \pi-F\omega\right) ^{\prime}\left( \pi-F\omega\right) }{\widehat{\sigma}^{2}}+\frac{\left( \omega-\mu\right) ^{\prime}\left( \omega-\mu\right) }{\widehat{\sigma}_{\omega}^{2}},$$

subject to the positivity constraints. These are standard constrained quadratic programming problems.

4 Empirical Results

Section 4.1 lays out our main empirical results for the forecasts of time-series models, OLS Phillips curve regressions, term structure models, and survey forecasts. We summarize these results in Section 4.2. Section 4.3 investigates how consistently the best models perform through time and Section 4.4 considers the effect of rolling windows. Section 4.5 reports the results of combining model forecasts.

4.1 Forecast Accuracy

Time-Series Models

In Table 4, we report RMSE statistics, in annual percentage terms, for the ARIMA model out-of-sample forecasts over the the post-1985 and post-1995 periods. The ARIMA RMSEs generally range from around 0.4-0.7% for PUXX to around 1.4-2.2% for PUXHS. For the post-1985 sample, the ARMA (1,1) model generates the lowest RMSE among all ARIMA models in forecasting PUNEW and PUXHS, but the annual Atkeson-Ohanian (2001) random walk is superior in forecasting core inflation (PUXX) and PCE. As the best quarterly ARIMA model, we select the ARMA(1,1) model for the remainder of the paper.⁸ In the post-1995 period, it beats both the quarterly RW and AR models in forecasting the PUXHS and PCE measure, but the AR model has a lower RMSE in forecasting PUNEW and PUXX, whereas the quarterly RW generates a lower RMSE in forecasting PUXX . Yet, the improvements are minor and the ARMA(1,1) model remains overall best among the three quarterly ARIMA models. However, the annual random walk is the best forecasting model for PUXX and PCE. It beats the ARMA(1,1) model for three of the four inflation measures and generates a much lower RMSE for forecasting core inflation (PUXX).

Table 4 also reports the RMSEs of the non-linear regime-switching model, RGM. Over the post-1985 period, RGM generally performs in line with, and slightly worse than, a standard ARMA model. There is some evidence that non-linearities are important for forecasting in the post-1995 sample, where the regime-switching model outperforms all the ARIMA models in forecasting PUNEW and PUXHS. Both these inflation series become much less persistent post-1995, and the RGM model captures this by transitioning to a regime of less persistent inflation. However, the Hamilton (1989) RGM model performs worse than a linear ARMA model for forecasting PUXX and PCE.

OLS Phillips Curve Forecasts

Table 5 reports the out-of-sample RMSEs and the model comparison regression estimates (equation (17)) for the Phillips curve models described in Section 3.2, relative to the benchmark of the ARMA(1,1) model. The overall picture in Table 5 is that the ARMA(1,1) model typically outperform the Phillips curve forecasts. Of the 80 comparisons (10 models, 2 out-samples, and 4 inflation measures), the model comparison regression coefficient $(1-\lambda)$ is not significantly positive at the 95% level in any of 80 cases using West (1996) standard errors! It must be said that the coefficients are sometimes positive and far away from zero, but the standard errors are generally rather large. When we compute Hansen-Hodrick (1980) standard errors, we still only obtain 14 cases of significant $(1-\lambda)$ coefficients with p-values less than 5%, and of these 14 cases, only nine are positive.

The OLS Phillips curve regressions are most successful in forecasting core inflation, PUXX. Of the nine cases where the Phillips curve produces lower RMSEs than the ARMA(1,1) model, five occur for PUXX. The best model forecasting PUXX inflation uses the composite Bernanke-Boivin-Eliasz aggregate real activity factor (PC8). While the $(1-\lambda)$ coefficients are large for PC8, their West (1996) standard errors are also large, so they are insignificant for both samples. Another relatively successful Phillips curve specification is the PC7 model that uses the Stock-Watson nonfinancial Experimental Leading Index-2. This index does not embed asset pricing information. PC7 for PUXHS post-1985 is the only case, out of 80 cases, that generates a positive $(1-\lambda)$ coefficient which is significant at a level higher than the 90% level using West standard errors, but its performance deteriorates for the post-1995 sample. All of the RMSEs of PC7 are also higher than the RMSE of an ARMA(1,1) model. In contrast, the PC1 model, which simply uses past inflation and past GDP growth, delivers five of the nine relative RMSEs below one and beats PC7 in all but one case.

Among the various Phillips curve models, it is also striking that the PC4 model consistently beats the PC2 and PC3 models, sometimes by a wide margin in terms of RMSE. The PC2 and PC3 models use detrended measures of output that are often used to proxy for the output gap. PC4 uses the labor share as a real activity measure, which is sometimes used as a proxy for the marginal cost concept in New Keynesian models. This is interesting because the recent Phillips curve literature (see Gali and Gertler, 1999) stresses that marginal cost measures provide a better characterization of (in-sample) inflation dynamics than detrended output measures. Our results suggest that the use of marginal cost measures also leads to better out-of-sample predictive power. However, the use of GDP growth leads to significantly better forecasts than the labor share measure, but GDP growth remains, so far, conspicuously absent in the recent Phillips curve literature.

Finally, using Table 4 together with Table 5, it is easy to verify whether the Atkeson-Ohanian (2001) results hold up for our models and data. Essentially, they do: the annual random walk beats the Phillips curve models in 72 out of 80 cases. All the cases where a Phillips curve model beats the annual random walk occur in forecasting the PUNEW or PUXHS measures.

Term Structure Forecasts

In Table 6, we report the out-of-sample forecasting results for the various term structure models (see Section 3.3). Generally, the term structure based forecasts perform worse than the Phillips-curve based forecasts. Over a total of 120 statistics (15 models, 4 inflation measures, 2 sample periods), term structure based-models beat the ARMA(1,1) model in only eight cases in terms of producing smaller RMSE statistics. The $(1-\lambda)$ coefficients are usually positive for forecasting PUXX in the post-1985 period, but half are negative in the post-1995 sample. Unfortunately, the use of West (1996) standard errors turns 10 cases of significantly positive $(1-\lambda)$ coefficients using Hansen-Hodrick (1980) standard errors into insignificant coefficients. The performance of the term structure forecasts is so poor that using West (1996) standard errors, in none of the 120 cases is the $(1-\lambda)$ parameters significant at the 95% level. This may be caused by many of the term structure models, especially the no-arbitrage models, having relatively large numbers of parameters.

The term structure models most successfully forecast core inflation, PUXX, which delivers six of the eight cases with smaller RMSEs than an ARMA(1,1) model. In particular, the TS1 model that includes inflation, GDP growth, and the short rate beats an ARMA(1,1) model and has a positive $(1-\lambda)$, but insignificant, coefficient in both the post-1985 and post-1995 samples. The other models with term structure information that are successful at forecasting PUXX are TS6 and TS8, both of which also include short rate information.

The finance literature has typically used term spreads, not short rates, to predict future inflation changes (see, for example, Mishkin, 1990, 1991). In contrast to the relative success of the models with short rate information, models TS9-TS11, which incorporate information from the term spread, perform badly. They produce higher RMSE statistics than the benchmark ARMA(1,1) model for all four inflation measures. This is consistent with Estrella and Mishkin (1997) and Kozicki (1997), who find that the forecasting ability of the term spread is diminished after controlling for lagged inflation. However, we show that the short rate still contains modest predictive power even after controlling for lagged inflation. Thus, the short rate, not the term spread, contains the most predictive power in simple forecasting regressions.

Table 6 shows that the performance of iterated VAR forecasts is mixed. VARs produce lower RMSEs than ARMA(1,1) models. The relatively poor performance of long-horizon VAR forecasts for inflation contrasts with the good performance for VARs in forecasting GDP (see Ang, Piazzesi and Wei, 2004) and for forecasting other macroeconomic time series (see Marcellino, Stock and Watson, 2006). The non-linear empirical regime-switching VAR (RGMVAR) generally fares worse than the VAR. This result stands in contrast to the relatively strong performance of the univariate regime-switching model using only inflation data (RGM in Table 4) for forecasting PUNEW and PUXX. This implies that the non-linearities in term structure data have no marginal value for forecasting inflation above the non-linearities already present in inflation itself.

The last two lines of each panel in Table 6 shows that there is some evidence that no-arbitrage forecasts (MDL1-2) are useful for forecasting PUXX in the post-1985 sample. While the $(1-\lambda)$ coefficients are significant using Hansen-Hodrick (1980) standard errors, they are not significant with West (1996) standard errors. Moreover, both no-arbitrage term structure models always fail to beat the ARMA(1,1) forecasts in terms of RMSE. While the finance literature shows that inflation is a very important determinant of yield curve movements, our results show that the no-arbitrage cross-section of yields appears to provide little marginal forecasting ability for the dynamics of future inflation over simple time-series models.

Surveys

Table 7 reports the results for the survey forecasts and reveals several notable results. First, surveys perform very well in forecasting PUNEW, PUXHS, and PUXX. With only one exception, the raw survey forecasts SPF1, LIV1 and MICH1 have lower RMSEs than ARMA(1,1) forecasts over both the post-1985 and the post-1995 samples (the exception is MICH1 for PUXX over the post-1985 sample). For example, for the post-1985 (post-1995) sample, the RMSE ratio of the raw SPF forecasts relative to an ARMA(1,1) is 0.779 (0.861) when predicting PUNEW. The horse races always assign large, positive $(1-\lambda)$ weights to the pure survey forecasts (the lowest one is 0.383) in both out-of-sample periods. Ignoring parameter uncertainty, the coefficients are significantly different from zero in every case, but taking into account parameter uncertainty, statistical significance disappears for the post-1995 samples, and in the case of the PUXX measure, even for the post-1985 sample. This is true for all three surveys.

Second, while the SPF and Livingston surveys do a good job at forecasting all three measures of CPI inflation (PUNEW, PUXHS, and PUXX) out-of-sample, the Michigan survey is relatively unsuccessful at forecasting core inflation, PUXX. It is not surprising that consumers in the Michigan survey fail to forecast PUXX, since PUXX excludes food and energy which are integral components of the consumer's basket of goods. Note that while the annual PUNEW and PUXHS measures have the highest correlations with each other (99% in both out-samples), core inflation is less correlated with the other CPI measures. In particular, post-1995, the correlation of annual PUXX with annual PUNEW (PUXHS) is only 33% (21%). Surveys do less well at forecasting PCE inflation, always producing worse forecasts in terms of RMSE than an ARMA(1,1). This result is expected because the survey participants are asked to forecast CPI inflation, rather than the consumption deflator PCE.

Third, the raw survey forecasts outperform the linear or non-linear bias adjusted forecasts (with the only notable exception being the bias-adjusted forecasts for PCE). As a specific example, for PUNEW, the relative RMSE ratios are always higher for the models with suffix 2 (linear bias adjustment) or the models with suffix 3 (non-linear bias adjustment) compared to the raw survey forecasts across all three surveys. This result is perhaps not surprising given the mixed evidence regarding biases in the survey data (see Table 3). While there are some significant biases, these biases must be small, relative to the total amount of forecast error in predicting inflation.

Finally, we might expect that the Livingston and SPF surveys produce good forecasts because they are conducted among professionals. In contrast, participants in the Michigan survey are consumers, not professionals. It is indeed the case that the professionals uniformly beat the consumers in forecasting inflation. Nevertheless, in most cases, the Michigan forecasts are of the same order of magnitude as the Livingston and SPF surveys. For example, for PUNEW over the post-1995 sample, the Michigan RMSE ratio is 0.862, just slightly above the RMSE ratio of 0.861 for the SPF survey. It is striking that information aggregated over non-professionals also produces accurate forecasts that beat ARIMA time-series models.

It is conceivable that consumers simply extrapolate past information to the future and that the Michigan survey forecasts are simply random walk forecasts, similar to the Atkeson and Ohanian (2001) (AORW) random walk forecasts. Indeed, Table 3 demonstrated the relatively good forecasting performance of the annual random walk model, which beats the ARMA(1,1) model in a number of cases. Nevertheless, comparing the performance of the survey forecasts relative to the AORW model, we find that the random walk model produces smaller RMSEs than the Michigan survey only for PUXX and PCE inflation, which consumers are not directly asked to forecast. The AORW also outperforms the SPF survey for PUXX inflation over the post-1995 period, but the AORW model always performs worse than the Livingston survey for the CPI inflation measures. Looking at PUNEW, the inflation measure which the survey participants are actually asked to forecast, the AORW model performs worse than all the surveys, including the Michigan surveys. Thus, survey forecasts clearly are not simply random walk forecasts!

4.2 Summary

Let us summarize the results so far. First, among ARIMA time-series models, the ARMA (1,1) model is the best overall quarterly model, but the annual random walk also performs very well. Nevertheless, some models that incorporate real activity information, term structure information, or, especially, survey information, beat the ARMA(1,1) model, even when ARMA(1,1) forecasts are used as the benchmark in a forecast comparison regression. Second, the simplest Phillips curve model using only past inflation and GDP growth is a good predictor. Third, adding term structure information occasionally leads to an improvement in inflation forecasts, but generally only for core inflation. No-arbitrage restrictions do not improve forecasting performance. Fourth, the survey forecasts perform very well in forecasting all inflation measures except PCE inflation.

To get an overall picture of the relative forecasting power of the various models, Table 8 reports the relative RMSE ratios of the best models from each of the first three categories (pure time-series, Phillips-curve, and term structure models) and of each raw survey forecast. The most remarkable result in Table 8 is that for CPI inflation (PUNEW, PUXHS, and PUXX), the survey forecasts completely dominate the Phillips curve or term structure models in both out-of-sample periods. For the post-1985 sample, the RMSEs are around 20% smaller for the survey forecasts compared to forecasts from Phillips-curve or term structure models. The natural exception is PCE inflation, where the best model in both samples is just the annual random walk model!

For the post-1985 sample, a survey forecast delivers the overall lowest RMSE for all CPI inflation measures. The performance of the survey forecasts remains impressive in the post-1995 sample, but the Hamilton (1989) regime-switching model (RGM) has a slightly lower RMSE for PUNEW and PUXHS. Impressively, the Livingston survey continues to deliver the most accurate forecast of PUXX post-1995.

For the Phillips curve forecasts, the simple PC1 regression using only past inflation and GDP growth frequently outperforms more complicated models for both PUNEW and PUXHS. Other measures of economic growth are more successful at forecasting PUXX and PCE. For PUXX inflation, PC8 produces forecasts that beat an ARMA(1,1) model for both the post-1985 and post-1995 sample. The PC8 forecasting model uses the Bernanke et al. (2005) composite indicator. For the PCE measure, models combining multiple time series (PC6 through 8) continue to do well, and the PC6 measure, which uses the Stock and Watson experimental leading index (XLI), produces the lowest RMSE for the post-1995 sample. For the post-1985 sample, PC4, which uses the labor share performs best. However, all the Phillips curve models are always beaten by time-series models or surveys.

Among the term structure models, models incorporating past inflation, the short rate, and one of the combination real activity measures (TS6 through TS8) perform relatively well. TS7 (using XLI-2) is best for the PUNEW and PCE measure for the post-1985 sample, whereas TS8 (using the Bernanke et al., 2005, composite indicator) is best for all measures except PUXX in the post-1995 sample. For PUXX, the TS6 model (which uses XLI as the real activity measure) produces the lowest RMSE. Like the Phillips curve models, all the term structure forecasts are also soundly beaten by time-series models or survey forecasts.

4.3 Stability of the Best Forecasting Models

One requirement for a good forecasting model is that it must consistently perform well. In Table 9, we report the ex-ante best models within each category (time-series, Phillips curve, term structure, and surveys) and across all models over the post-1995 sample. Since we record the best models at the end of each quarter, we include only the SPF and Michigan survey forecasts because the Livingston survey is only available semi-annually. This understates the performance of the surveys as the Livingston survey sometimes outperforms the other two survey measures, especially for PUXX (see Table 8). The best models are evaluated recursively, so at each point in time, we select the model within each group that yields the lowest forecast RMSEs over the sample from 1985:Q4 to the present. Naturally, as we roll through the sample, the best ex-ante models up to the end of each quarter converge to the best models reported for the post-1985 period in Table 8. If the best ex-ante models for 2002:Q4 were reported, these would be identical to the best models in the post-1985 sample in Table 8, with the exception that the Livingston survey is excluded.

Table 9 shows that for PUNEW and PUXHS, the ARMA(1,1) model is consistently the best time-series model, whereas for PUXX and PCE, the Atkeson-Ohanian (2001) model is always best. Given the good forecasting performance of these time-series models, this implies that the time-series models represent extremely good benchmarks. In contrast, there is little stability for the best ex-ante Phillips curve model, which is also stressed by Brave and Fisher (2004). For PUNEW, the best Phillips curve models alternate between PC1 (using GDP growth) and PC5 (using unemployment). For PUXHS, the best Phillips curve is PC7 (using XLI-2) at the beginning of the period, but transitions to PC1 at the end of the sample. For core inflation, PUXX, PC8 (using the composite Bernanke, Boivin and Eliasz, 2005, factor) alternates with PC1. This instability further reduces the usefulness of the Phillips curve forecasts and hence, the knowledge that sometimes these Phillips curve forecasts may beat an ARMA(1,1) model is hard to translate into consistent, accurate forecasts.

The best term structure models are also generally unstable over time for PUNEW and PUXX. While the VAR model is consistently the best performer for PUXHS and TS7 (using XLI-2 with the short rate) is always the best term structure model for PCE, this consistent performance is less useful because both of these models cannot beat an ARMA(1,1). A sharp contrast to the unstable Phillips curve and term structure models are the survey results. For all three CPI measures (PUNEW, PUXHS, and PUXX), professionals always forecast better than consumers, with the SPF beating the Michigan survey. A remarkable result is that the raw SPF survey always dominates all other models throughout the period for the CPI measures. Surveys consistently deliver superior inflation forecasts!

4.4 Rolling Window Forecasts

McConnell and Perez-Quiros (2000) and Stock and Watson (2002b), among others, document that there has been a structural break since the mid-1980s. This has been called the "Great Moderation" because it is characterized by lower volatility of many macro variables. It is conceivable that professional forecasters fast adapt to structural changes. In contrast, the models use relatively long windows (necessary to retain some estimation efficiency and power) to estimate parameters. These model parameters would respond only slowly to a structural break as new data points are added. If changes in the time series properties of inflation play a role in the relative forecasting prowess of models versus the surveys, allowing the model parameters to change more quickly through rolling windows should generate superior model performance.

In Table 10, we use a constant 10-year rolling window to estimate all the linear time-series, Phillips curve, and term structure models. We do not consider the regime-switching models (RGM, RGMVAR) and the no-arbitrage term structure models, (MDL1, which is an affine model, and MDL2, which is a regime-switching model). The regime-switching data generating processes in the RGM, RGMVAR, and MDL2 models produce forecasts that may already potentially account for structural breaks. We report the relative RMSEs of the ex-post best models in each category together with the raw survey forecasts results, using the same recursively estimated ARMA(1,1) model as the benchmark.

Table 10 shows that over both the post-1985 and post-1995 samples, surveys still provide the best forecasts for all CPI inflation measures. Note that with a 10-year rolling window, the post-1995 sample results involve models estimated only on the post-Great Moderation sample. Thus, surveys still out-perform even when the models are estimated only with data from the Great Moderation regime. But, estimating the models with only post-1985 data does improve their performance, as a comparison between the RMSE ratios between Tables 8 and 10 reveals, especially for the PUXX and PCE measures. This implies that the model parameters may indeed only have adjusted to the new situation by 1995 and raises the possibility that the out-performance of the surveys may not last. In fact, it is striking that an older literature, summarized by Croushore (1998), stressed that the surveys performed relatively poorly in forecasting compared to models.

To investigate this, we use the Livingston survey, which is the only survey available over our full sample, from 1952-2002. We compute the RMSE ratio of the out-of-sample forecasts for the Livingston survey relative to an ARMA(1,1) model for 1960-1985 and 1986-2002, where the first eight years are used as an in-sample estimation period for the ARMA(1,1) model. Over the pre-1985 sample, the Livingston RMSE ratio is 1.046 (with a RMSE level of 2.324), while over the post-1985 sample, the RMSE ratio is 0.789 (with a RMSE level of 0.896). Consequently, professionals are more adept at forecasting inflation in the post-1985 period.⁹

4.5 Combining Model Forecasts

Surveys may be averaging information from many different sources, whereas our models implicitly always constrain the information set to a limited number of variables. If this is the source of the out-performance of the surveys, the model combination techniques should perform better than any individual model by itself.

Table 11 investigates whether we can improve the forecasting performance by combining different models. We first combine models within each of the four categories (time-series, Phillips curve, term structure, and survey models), then combine the four ex-ante best models from each category in the column labelled "Best Models," and finally combine across all the models in the last column labelled "All Models." The models in the survey category comprise only the SPF and Michigan surveys because the Livingston survey is conducted at a semiannual frequency. Table 7 shows that the Livingston forecasts are very similar to the SPF and Michigan surveys for PUNEW and PUXHS, and that the Livingston survey is the best single forecaster for PUXX. Thus, excluding the Livingston survey places a conservative higher bound on the RMSEs for the forecast combinations involving surveys.

We use five methods of model combination: means or medians over all the models, linear combinations using weights recursively computed by OLS, and linear combinations using weights recursively computed by mixed combination regressions either with an equal-weight prior or a prior that places a unit weight on the ex-ante best model. We start the model combination regressions at 1995:Q4 using realized inflation and the out-of-sample forecasts over 1985:Q4 to 1995:Q4. At each subsequent period, we advance the data sample by one quarter and re-run the model combination regression to obtain the slope coefficient estimates. For comparison, the last row in each panel reports the RMSE ratio, relative to an ARMA(1,1) forecast, of the recursively-updated ex-ante best performing individual model, as reported in Table 9.¹⁰

There are three main findings in Table 11. First, using mean or median forecasts mostly does not improve the forecast performance relative to the best individual ex-ante model. There are 24 cases to consider: four inflation measures and six different sets of model combinations. Combining forecasts by taking their means only improves out-of-sample forecasts in six out of 24 cases. Taking medians produces the same results, improving forecasts for exactly the same cases as taking means. The mean or median combination methods work best for PUNEW and PUXHS using time-series models. However, when these forecasting improvements occur for model combinations, the improvements are small. Thus, simple methods of combining forecasts provide little additional predictive power relative to the best model.

Second, updating the model weights based on previous model performance does not always lead to superior performance. For the Phillips Curve models, OLS model combinations outperform means and medians for all inflation measures. However, when OLS model combinations are taken across all models, using an OLS combination is never better than the best individual model.

Finally, the performance of the equal-weight prior and the unit prior that places weight only the best ex-ante model are generally close to the OLS forecast combination method. Across all models, the unit weight prior produces lower RMSE ratios than the OLS or equal-weight priors. However, it is only for PUXX that the various regression-based model combination methods produce better forecasts than the best individual forecasts. For PUNEW, PUXHS, and PCE, the best individual models beat the model combinations, and for PUNEW and PUXHS, the best individual ex-ante forecasts are surveys.

To help interpret the results, we investigate the ex-ante OLS weights on some selected models. In Figure 2, we plot the OLS slope estimates of regression (18) for various inflation measures over the period of 1995:Q4 to 2002:Q4. For clarity, we restrict the regression to combinations of the ex-ante best model within each category (time-series, Phillips Curve, term structure) together with the SPF survey. Note that by choosing the best model in each category, we handicap the survey forecasts. We compute the weights in the regression recursively like the forecasts in Table 11; that is, we start in 1995:Q4, and recursively compute forecasts from 1985:Q4 to 1995:Q4.

Figure 2 shows that when forecasting all the CPI inflation measures (PUNEW, PUXHS, and PUXX), the data consistently place the largest ex-ante weights on survey forecasts and very little weight on the other models. The weights on the SPF survey forecast are generally constant and lie around 0.8 for PUNEW, PUXX, and PUXHS. There is no consistent, best model that dominates for the remaining 0.1-0.2 weights. The weights on the time-series models are always zero for PUNEW, but temporarily spike upward in the middle of the sample to around 0.15 for PUXHS and 0.20 for PUXX. For PUNEW and PUXHS, the Phillips curves fare best at the beginning of the sample, but the regressions place very little weight on Phillips curve forecasts at the end of the sample. For PCE inflation, surveys contain little information. The weight on the best survey stays close to zero until late 1999, then rises to 0.2. For forecasting PCE among the other categories of models, the Phillips Curve forecast stands out, with weights ranging from 0.2 to 0.6. Term structure models receive the highest weight at the end of the sample. We conclude that combining model forecasts, at least using the techniques here, is not a very useful forecasting tool, especially compared to using just survey data for forecasting CPI inflation.

5 Robustness to Non-Stationary Inflation

5.1 Definition and Models

In this section we investigate the robustness of our results to the alternative assumption that quarterly inflation is difference stationary. Our exercise is now to forecast four-quarter ahead inflation changes:

$$\mathrm{E}_{t}\left( \pi_{t+4,4}-\pi_{t,4}\right) =\mathrm{E}_{t}\left[ \sum_{i=-3}^{3} (4-\vert i\vert)\Delta\pi_{t+1+i}\right]$$

$$= \mathrm{E}_{t} \left[ \sum_{i=0}^{3} (4-i) \Delta\pi_{t+1+i} \right] + 4 \pi_{t} - \pi_{t,4},$$ (A-21)

where $\pi_{t+4,4}$ is annual inflation defined in equation (2).

We now replace quarterly inflation, $\pi_{t}$, by quarterly inflation changes, $\Delta\pi_{t+1}=\pi_{t+1}-\pi_{t}$ in all the models considered in Sections 3.1 to 3.3. For example, we estimate an ARMA(1,1) on first differences of inflation:

$$\Delta\pi_{t+1} =\mu+\phi\Delta\pi_{t}+\psi\varepsilon_{t}+\varepsilon_{t+1}$$

and an AR($p$) on first differences of inflation:

$$\Delta\pi_{t+1} =\mu+\phi_{1}\Delta\pi_{t}+\phi_{2}\Delta\pi_{t-1} +\ldots+\phi_{p}\Delta\pi_{t-p+1}+\varepsilon_{t+1}.$$

The OLS Phillips Curve and term structure regressions include quarterly inflation changes as one of the regressors, rather than quarterly inflation. From the models estimated on $\Delta\pi_{t}$, we compute forecasts of inflation changes over the next year, $\mathrm{E}_{t}(\pi _{t+4,4}-\pi_{t,4})$.

There are three models for which we do not estimate a counterpart using quarterly inflation differences. We do not consider a random walk model for inflation changes and do not specify the no-arbitrage term structure models (MLD1 and MLD2) to have non-stationary inflation dynamics, although we still consider the forecasts of annual inflation changes implied by the original stationary models. In all other cases, we examine the forecasts of both the original stationary models and the new non-stationary models that use first differences of inflation.

The original models estimated on inflation levels generate RMSEs for forecasting annual inflation changes that are identical to the RMSEs for forecasting annual inflation levels. Hence, the question is whether models estimated on differences provide superior forecasts to models estimated on levels. By including a new set of models estimated on inflation changes, we also enrich the set of forecasts which we can combine. We maintain the ARMA(1,1) model estimated on inflation rate levels as a benchmark.

5.2 Performance of Individual Models

Table 12 reports the RMSE ratios of the best performing models estimated on levels or differences within each model category. Time-series models estimated on levels always provide lower RMSEs than time-series models estimated on differences. For both Phillips curve and term structure models, using inflation differences or levels produces similar forecasting performance for both the PUNEW and PUXHS measures. For these inflation measures, the Phillips curve models are slightly better estimated on levels, but for term structure models, there is no clear overall winner. However, for the PUXX and PCE measures, Phillips curve and term structure regressions using past inflation changes are more accurate than regressions with past inflation levels.

Our major finding that surveys generally outperform other model forecasts is robust to specifying the models in inflation differences. For the CPI inflation measures (PUNEW, PUXHS, PUXX) over the post-1985 sample, surveys deliver lower RMSEs than the best time-series, Phillips curve, and term structure forecasts. First difference models are most helpful for lowering RMSEs for core inflation (PUXX) over the post-1995 sample, where the best time-series model estimated on differences (ARMA) produces a relative RMSE ratio of 0.649. This is still beaten by the raw Livingston survey, with a RMSE ratio of 0.557.¹¹

5.3 Performance of Combining Models

In this section, we run forecast combination regressions to determine the best combination of models to forecast inflation changes (similar to Section 3.6 for inflation levels). The model weights are computed from the regression:

$$\pi_{s+4,4}-\pi_{s,4}=\sum_{i=1}^{n}\omega_{s} ^{i}f_{s}^{i}+\varepsilon_{s,s+4}, s=1,\ldots,t.$$ (A-22)

We repeat the exercise of Table 11 and compute ex-ante recursive weights over 1995:Q4-2002:Q4 using the best ex-ante forecasting models in each category and across all models. In unreported results available upon request, we find that our original results for forecasting inflation levels also extends to forecasting inflation changes. Specifically, there is generally no improvement in combining model forecasts, or when model combinations result in out-performance, the improvement is small. Specifically, for PUNEW and PUXHS, using means, medians, OLS, or an equal-weight prior produces higher RMSEs than the best individual model. For these inflation measures, all model combinations produce RMSEs that are higher than the survey forecasts. This result is robust to both combining models in levels and also combining models in differences. There are some improvements for forecasting PCE inflation using models in differences, but the forecasting gains are very small.

In Figures 3 and 4, we plot the OLS coefficient estimates of equation (22) for the models specified in differences and the models specified in levels, respectively, together with the best survey forecast. We consider only the SPF and the Michigan surveys at the end of each quarter, and the SPF survey always dominates the Michigan survey. Similar to Figure 2, we choose the best ex-ante performing time-series, Phillips Curve, and term structure models at each time, and compute the OLS ex-ante weights recursively over 1995:Q4 to 2004:Q4. Both Figures 3 and 4 confirm that the surveys produce superior forecasts of inflation changes.

In Figure 3, the weight on the SPF survey for PUNEW and PUXHS changes is above or around 0.8. The surveys clearly dominate the I(1) time-series, Phillips Curve, and term structure models. For PUXX changes, the regressions still place the largest weight on the survey, but the weight is around 0.5. In contrast, for forecasting PUXX inflation levels, the weights on the survey range from 0.6 to above 0.9. Thus, there is now additional information in the other models for forecasting PUXX changes, most particularly the Phillips Curve PC1 model, which has a weight around 0.4. Nevertheless, surveys still receive the highest weight. Consistent with the results for forecasting inflation levels, surveys provide little information to forecast PCE changes. For PCE changes, the largest ex-ante weight in the forecast combination regression is for the ARMA(1,1) estimated on inflation differences.

Figure 4 combines the surveys with stationary models. While Table 12 reveals that the RGM model estimated on inflation levels yields the lowest RMSE over the post-1995 sample in forecasting PUNEW and PUXHS differences, there appears to be little additional value in the RGM forecast once surveys are included. Figure 4 shows that the forecast combination regression places almost zero ex-ante weight on the RGM model. The weights on the other I(0) models are also low, whereas the survey weights are around 0.8 or higher. Compared to the other stationary model categories, surveys also have an edge at forecasting PUXX inflation. Again, surveys do not perform well relative to I(0) models for forecasting PCE changes.

6 Conclusions

We conduct a comprehensive analysis of different inflation forecasting methods using four inflation measures and two different out-of-sample periods (post-1985 and post-1995). We investigate forecasts based on time-series models; Phillips curve inspired forecasts; and forecasts embedding information from the term structure. Our analysis of term structure models includes linear regressions, non-linear regime switching models, and arbitrage-free term structure models. We compare these model forecasts with the forecasting performance of three different survey measures (the SPF, Livingston, and Michigan surveys), examining both raw and bias-adjusted survey measures.

Our results can be summarized as follows. First, the best time series model is mostly a simple ARMA(1,1) model, which can be motivated by thinking of inflation comprising stochastic expected inflation following an AR(1) process, and shocks to inflation. Post-1995, the annual random walk used by Atkeson and Ohanian (2001) is a serious competitor. Second, while the ARMA(1,1) model is hard to beat in terms of RMSE forecast accuracy, it is never the best model. For CPI measures, the survey measures consistently deliver better forecasts than ARMA(1,1) models, and in fact, much better forecasts than Phillips curve-based regressions, term structure models based on OLS regressions, non-linear models, iterated VAR forecasts, and even no-arbitrage term structure models that use information from the entire cross-section of yields. Naturally, surveys do a relatively poor job at forecasting PCE inflation, which they are not designed to forecast.

Some of our results shed light on the validity of some simple explanations of the superior performance of survey forecasts. One possibility is that the surveys simply aggregate information from many different sources, not captured by a single model. The superior information in median survey forecasts may be due to an effect similar to Bayesian Model Averaging, or averaging across potentially hundreds of different individual forecasts and extracting common components (see Stock and Watson, 2002a; Timmermann, 2004). For example, it is striking that the Michigan survey, which is conducted among relatively unsophisticated consumers, beats time-series, Phillips curve, and term structure forecasts. The Livingston and SPF surveys, conducted among professionals, do even better.

If there is information in surveys not included in a single model, combining model forecasts may lead to superior forecasts. However, when we examine forecasts that combine information across models or from various data sources (like the Bernanke et al., 2005, index of real activity that uses 65 macro factors measuring real activity), we find that the surveys still outperform. Across all models, combination methods of simple means or medians, or forecast combination regressions which use prior information never outperform survey forecasts. In ex-ante model combination exercises for forecasting CPI inflation, almost all the weight is placed on survey forecasts. One avenue for future research is to investigate whether alternative techniques for combining forecasts perform better (see Inoue and Killian, 2005, for a survey and study of one promising technique).

Another potential reason why surveys outperform is because survey information is not captured in any of the variables or models that we use. If this is the case, our results strongly suggest that there would be additional information to include survey forecasts in the large datasets used to construct a small number of composite factors, which are designed to summarize aggregate macroeconomic dynamics (see, among others, Bernanke et al., 2005; Stock and Watson, 2005).

Our results also have important implications for term structure modelling. Extant sophisticated no-arbitrage term structure models, while performing well in sample, seem to provide relatively poor forecasts relative to simpler term structure or Phillips curve models out-of-sample. A potential solution is to introduce the information present in the surveys as additional state variables in the term structure models. Pennacchi (1991) was an early attempt in that direction and Kim (2004) is a recent attempt to build survey expectations into a no-arbitrage quadratic term structure model. Brennan, Wang and Xia (2004) also recently use the Livingston survey to estimate an affine asset pricing model.

Finally, surveys may forecast well because they quickly react to changes in the data generating process for inflation in the post-1985 sample. In particular, since the mid-1980s, the volatility of many macroeconomic series, including inflation, has declined. This "Great Moderation" may also explain why a univariate regime-switching model for inflation provides relatively good forecasts over this sample period. Nevertheless, when we re-do our forecasting exercises using a 10-year rolling window, the surveys forecasts remain superior.

We conjecture that the surveys likely perform well for all of these reasons: the pooling of large amounts of information; the efficient aggregation of that information; and the ability to quickly adapt to major changes in the economic environment such as the Great Moderation. While our analysis shows that surveys provide superior forecasts of CPI inflation, the PCE deflator is often the Federal Reserve's preferred inflation indicator for the conduct of monetary policy. Since existing surveys target only the CPI index, professional surveys designed to forecast the PCE deflator may also deliver superior forecasts of PCE inflation.

Appendix: Computation of West (1996) Standard Errors

By subtracting $f_{t}^{ARMA}$ from both sides of equation (17) and letting $e_{t,t+4}^{ARMA}$ denote the forecast residuals of the ARMA(1,1) model and $e_{t,t+4}^x$ denote the forecast residuals of candidate model $x$, we can write:

$$e_{t,t+4}^{ARMA}= (1-\lambda) ( e_{t,t+4}^{ARMA} -e_{t,t+4}^{x}) +\varepsilon_{t+4,4}.$$ (A-1)

The estimated slope coefficient $\hat{\lambda}$ has the asymptotic distribution:

$$\sqrt{P}( \hat{\lambda}-\lambda) \overset{d}{\rightarrow }\mathcal{N}\left( 0, \ensuremath{{\rm E}}(d_{t+4}d_{t+4}')^{-1} \Omega_{ff} \ensuremath{{\rm E}}(d_{t+4}d_{t+4}')^{-1}\right),$$ (A-2)

where $P$ is the length of the out-sample, $\Omega_{ff}= \ensuremath{\text{var}}(f_{t,t+4})$, $f_{t,t+4}=e_{t,t+4}^{ARMA}( e_{t,t+4}^{ARMA}-e_{t,t+4}^{x})$ and $d_{t,t+4} = e_{t,t+4}^{ARMA}-e_{t,t+4}^{x}$. West (1996) derives the long-run asymptotic variance $\Omega_{ff}$ after taking into account parameter uncertainty.

We use the notation based on West (2006). The forecast horizon is four quarters ahead. For each model $i$ there are $P$ out-of-sample forecasts in all, which rely on estimates of a $k_{i}\times1$ unknown parameter vector $\theta_{i}$. The first forecast uses data from a sample of length $R$ to predict a time $t = (R+4)$ variable, while the last forecast uses data from time $t = R+P-1\equiv T$ to forecast a time $t = T+4$ variable. The total sample size is $R+P-1+4=T+4.$

For the $i$th candidate model, $\hat{\theta}_i$, the small-sample estimate of the parameters $\theta_i$ satisfies:

$$\hat{\theta}_{i}(t)-\theta_{i}= B_{i}(t) H_{i}(t),$$ (A-3)

where $B_{i}(t)$ is a $k_{i}\times q_{i}$ matrix and $H_{i}(t)$ is a $q_{i}\times 1$ vector. The vector $H_i(t)$ represents orthogonality conditions of the model and the matrix $B_i(t)$ is a linear combination of the orthogonality conditions to recover the parameters. We assume that $B_{i}(t) \overset{p}{\rightarrow}B_{i}$, where $B_{i}$ is a matrix with rank $k_{i}$. The moment conditions $H_{i}(t)$ are given by:¹²

$$H_{i}(t) = \frac{1}{t}\sum_{s=1}^{t}h_{s}^{i}(\theta_i),$$ (A-4)

for the recursive forecast case which we investigate, where $h_{s}^{i}(\theta_i)$ are $q_{i}\times 1$ orthogonality conditions. For models estimated by maximum likelihood, the matrix $B_{i}(t)$ is the inverse of the Hessian and $h_{t}^{i}(\theta_i)$ is the score. For linear models in the form of $y_{t}=X_{t}^{i \prime} \theta^{i}+\varepsilon_{t}$, $B_{i}(t) = \ensuremath{{\rm E}}(X_t^{i} X_t^{i \prime})^{-1}$ and $h_{t}^{i}(\theta_i) = X_{t}^{i \prime}( y_{t}-X_{t}^{i \prime }\theta_i)$.

We stack the parameters of the ARMA(1,1) benchmark model and the parameters of the $i$th candidate model in the vector $\theta= (\theta_{ARMA}, \theta_i )$. Then, we can write $\hat{\theta}(t) =B(t) H(t)$, where $H(t) = \frac{1}{t} \sum_{s=1}^t h_s(\theta)$, where:

$$B\left( t\right) = \left[ \begin{array}[c]{cc} B_{ARMA}(t) & 0\ 0 & B_i(t) \end{array}\right],$$

$$h_{t}(\theta) = \left[ \begin{array}[c]{c} h_{t}^{ARMA}(\theta_{ARMA}) \ h_{t}^{i}(\theta_{i}) \end{array}\right],$$ (A-5)

and $B(t) \overset{p}{\rightarrow}B$, where

$$B =\left[ \begin{array}[c]{cc} B_{ARMA} & 0\\ 0 & B_i \end{array} \right].$$ (A-6)

We define the derivative $F$ of the moment conditions with respect to $\theta$ as:

$$F= \ensuremath{{\rm E}}\left[ \frac{\partial f_{t,t+4}(\theta)} {\partial \theta} \right] = \left[ \begin{array}[c]{c} F_{1} \\ F_{2} \end{array} \right],$$ (A-7)

where $F_1$ and $F_2$ are given by:

$$F_{1} = \ensuremath{{\rm E}}\left[ \frac{\partial f_{t,t+4}\left( \theta\right) } {\partial\theta_{ARMA}}\right] = \ensuremath{{\rm E}}\left[ \left( 2e_{t,t+4}^{ARMA} -e_{t,t+4}^{x}\right) \frac{\partial e_{t,t+4}^{ARMA}}{\partial\theta_{ARMA} }\right]$$

$$F_{2} = \ensuremath{{\rm E}}\left[ \frac{\partial f_{t,t+4}\left( \theta\right) } {\partial\theta_{i}}\right] =-\ensuremath{{\rm E}}\left[ e_{t,t+4}^{ARMA}\frac{\partial e_{t,t+4}^{x}}{\partial\theta_{i}}\right].$$ (A-8)

Finally, for the asymptotic results, we need $P \rightarrow \infty$ and $R \rightarrow \infty$ with

$$\rho =\lim_{T\rightarrow\infty}\frac{P}{R}<\infty.$$ (A-9)

Following West (2006), we define the constants $\lambda_{fh}$ and $\lambda_{hh}$:

$$\lambda_{fh} = 1-\rho^{-1} \ln ( 1+\rho ),$$

$$\lambda_{hh} = 2[ 1-\rho^{-1}\ln( 1+\rho)].$$ (A-10)

Under these assumptions, West (1996) derives that the asymptotic variance $\Omega_{ff}$ is given by:

$$\Omega_{ff}=S_{ff}+\lambda_{fh}\left( FBS_{fh}^{\prime}+S_{fh}B^{\prime }F^{\prime}\right) +\lambda_{hh}FBV_{hh}B^{\prime}F^{\prime}$$ (A-11)

where

$$S_{ff} = \sum_{j=-\infty}^{\infty} \ensuremath{{\rm E}}\left[ ( f_{t,t+4}-\ensuremath{{\rm E}} f_{t,t+4})(f_{t-j,t-j+4}-\ensuremath{{\rm E}}f_{t,t+4})^{\prime} \right],$$

$$S_{fh} = \sum_{j=-\infty}^{\infty}\ensuremath{{\rm E}}\left[( f_{t,t+4}-\ensuremath{{\rm E}} f_{t,t+4}) h_{t-j}^{\prime} \right],$$

$$S_{hh} = \sum_{j=-\infty}^{\infty}\ensuremath{{\rm E}} \left[h_{t}h_{t-j}^{\prime} \right].$$ (A-12)

Note that the estimate without parameter uncertainty is simply $S_{ff}$, and taking into account parameter uncertainty can increase or decrease the long-run variance of $\hat{\lambda}$ depending on the covariances of $f_{t,t+4}$ with $h_{t+4}$.

A consistent estimator can be constructed using the small-sample counterparts. In particular, we compute $\hat{\lambda}_{fh}$ and $\hat{\lambda}_{hh}$ setting $\hat{\rho} =P/R$,

$$\hat{F} = \frac{1}{P}\sum_{t=R}^{T}\frac{\partial f(\theta) } {\partial\theta} \Big\vert _{\theta = \hat{\theta}},$$

$$\hat{B} \equiv B(T) \overset{p}{\rightarrow} B,$$ (A-13)

and construct $\hat{f}_{t,t+4} = f_{t,t+4}(\hat{\theta}(t))$ and $\hat{h}_{t} = h_{t}(\hat{\theta}(t))$ using the estimates $\hat{\theta}(t)$, which are recursively updated each time using data up to time $t$. The sample covariances, $\hat{S}_{ff}$, $\hat{S}_{fh}$ and $S_{hh}$ converge to their population equivalents in equation (A-12). To estimate these, we define the vector of moments:

$$\hat{g}_{t}=\left[ \begin{array}[c]{cc} \hat{f}_{t,t+4} & \hat{F}\hat{B}\hat{h}_{t} \end{array} \right].$$ (A-14)

To construct a non-singular estimate for the covariance of $\hat{g}_t$, which we denote as $\hat{\Omega}$, we use a Newey-West (1987) covariance estimator with three lags. We partition $\hat{\Omega}$ as the $2 \times 2$ matrix:

$$\hat{\Omega} = \left[ \begin{array}[c]{cc} \hat{\Omega}_{11} & \hat{\Omega}_{12} \\ \hat{\Omega}_{21} & \hat{\Omega}_{22} \end{array} \right].$$ (A-15)

Then, a consistent estimate of $\Omega_{ff}$ is given by:

$$\hat{\Omega}_{ff}=\hat{\Omega}_{11}+\hat{\lambda}_{fh}( \hat{\Omega}_{12}+\hat{\Omega}_{21}) +\hat{\lambda}_{hh}\hat{\Omega}_{22}.$$ (A-16)

Bibliography

Atkeson, A., Ohanian, L.E., 2001

Are Phillips Curves useful for forecasting inflation? Federal Reserve Bank of Minneapolis Quarterly Review 25, 2-11.

Ang, A., Bekaert, G., 2002

Regime switches in interest rates. Journal of Business and Economic Statistics 20, 163-182.

Ang, A., Bekaert, G., Wei, M., 2006

The term structure of real rates and expected inflation. Working paper, Columbia University.

Ang, A., Piazzesi, M., Wei, M., 2004

What does the yield curve tell us about GDP growth? Journal of Econometrics, forthcoming.

Bai, J., Ng, S., 2004

A panic attack on unit roots and cointegration. Econometrica 72, 1127-1177.

Bates, J. M., Granger, C.W.J., 1969

The combination of forecasts. Operations Research Quarterly 20, 451-468.

Bekaert, G., Cho, S., Moreno, A., 2005

New Keynesian macroeconomics and the term structure. Working paper, Columbia University.

Bekaert, G., Hodrick, R.J., Marshall, D., 2001

Peso problem explanations for term structure anomalies. Journal of Monetary Economics 48, 241-270.

Bernanke, B.S., Boivin, J., 2003

Monetary policy in a data-rich environment. Journal of Monetary Economics 50, 525-546.

Bernanke, B.S., Boivin, J., Eliasz, P., 2005

Measuring the effects of monetary policy: A factor-augmented vector autoregressive (FAVAR) approach. Quarterly Journal of Economics 120, 387-422.

Boivin, J., Ng, S., 2006

Are more data always better for factor analysis? Journal of Econometrics, forthcoming.

Brave, S., Fisher, J.D.M., 2004

In search of a robust inflation forecast. Federal Reserve Bank of Chicago Economic Perspectives 28, 12-30.

Brennan, M.J., Wang, A.W., Xia, Y., 2004

Estimation and test of a simple model of intertemporal capital asset pricing. Journal of Finance 59, 1743-1775.

Bryan, M.F., Cecchetti, S.G., 1993

The consumer price index as a measure of inflation. Economic Review of the Federal Reserve Bank of Cleveland 29, 15-24.

Campbell, S.D., 2004

Volatility, predictability and uncertainty in the great moderation: Evidence from the survey of professional forecasters. Working paper, Federal Reserve Board of Governors.

Carlson, J.A., 1977

A study of price forecasts. Annals of Economic and Social Measurement 1, 27-56.

Clark, T.E., 1999

A comparison of the CPI and the PCE price index. Federal Reserve Bank of Kansas City Economic Review 3, 15-29.

Clark, T.E., McCracken, M.W., 2006

The predictive content of the output gap for inflation: Resolving in-sample and out-of-sample evidence. Journal of Money, Credit and Banking, forthcoming.

Clemen, R.T., 1989

Combining forecasts: A review and annotated bibliography. International Journal of Forecasting 5, 559-581.

Chen, R.R., Scott, L., 1993

Maximum likelihood estimation for a multi-factor equilibrium model of the term structure of interest rates. Journal of Fixed Income 3, 14-31.

Cecchetti, S., Chu, R., Steindel, C., 2000

The unrealiability of inflation indicators. Federal Reserve Bank of New York Current Issues in Economics and Finance 6, 1-6.

Cochrane, J., Piazzesi, M., 2005

Bond risk premia. American Economic Review 95, 1, 138-160.

Cogley, T., Sargent T.J., 2005

Drifts and volatilities: Monetary policies and outcomes in the post WWII U.S. Review of Economic Dynamics 8, 262-302.

Croushore, D., 1998

Evaluating inflation forecasts. Working Paper 98-14, Federal Reserve Bank of St. Louis.

Curtin, R.T., 1996

Procedure to estimate price expectations. Mimeo, University of Michigan Survey Research Center.

Dai, Q., Singleton, K.J., 2002

Expectation puzzles, time-varying risk premia, and affine models of the term structure. Journal of Financial Economics 63, 415-41.

Diebold, F.X., 1989

Forecast combination and encompassing: Reconciling two divergent literatures. International Journal of Forecasting 5, 589-92.

Diebold, F.X., Lopez, J.A., 1996

Forecasting evaluation and combination, in G.S. Maddala and C.R. Rao, eds., Handbook of statistics (Elsevier, Amsterdam) 241-268.

Duffee, G.R., 2002

Term premia and the interest rate forecasts in affine models. Journal of Finance 57, 405-443.

Duffie, D., Kan, R., 1996

A yield-factor model of interest rates. Mathematical Finance 6, 379-406.

Estrella, A., Mishkin, F.S., 1997

The predictive power of the term structure of interest rates in Europe and the United States: Implications for the European Central Bank. European Economic Review 41, 1375-401.

Evans, M.D.D., Lewis, K.K., 1995

Do expected shifts in inflation affect estimates of the long-run Fisher relation? Journal of Finance 50, 225-253.

Evans, M.D.D., Wachtel, P., 1993

Inflation regimes and the sources of inflation uncertainty. Journal of Money, Credit and Banking 25, 475-511.

Fama, E.F., 1975

Short-term interest rates as predictors of inflation. American Economic Review 65, 269-282.

Fama, E.F., Gibbons, M.R., 1984

A comparison of inflation forecasts. Journal of Monetary Economics 13, 327-348.

Fisher, J.D.M., Liu, C.T., Zhou, R., 2002

When can we forecast inflation? Federal Reserve Bank of Chicago Economic Perspectives 1, 30-42.

Frankel, J.A., Lown, C.S., 1994

An indicator of future inflation extracted from the steepness of the interest rate yield curve along its entire length. Quarterly Journal of Economics 59, 517-530.

Fuhrer, J., Moore, G., 1995

Inflation persistence. Quarterly Journal of Economics 110, 127-159.

Gali, J., and M. Gertler, 1999

"Inflation Dynamics: A Structural Econometrics Analysis," textit">Journal of Monetary Economics, 44, 2, 195-222.

Grant, A.P., Thomas, L.B., 1999

Inflation expectations and rationality revisited. Economics Letters 62, 331-338.

Gray, S.F., 1996

Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42, 27-62.

Hamilton, J.D., 1985

Uncovering financial market expectations of inflation. Journal of Political Economy 93, 1224-1241.

Hamilton, J., 1988

Rational-expectations econometric analysis of changes in regime: An investigation of the term structure of interest rates. Journal of Economic Dynamics and Control 12, 385-423.

Hamilton, J., 1989

A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357-384.

Hansen, L.P., Hodrick, R.J., 1980

Forward exchange rates as optimal predictors of future spot rates: an econometric analysis. Journal of Political Economy 88, 829-853.

Hodrick, R.J., Prescott, E.C., 1997

Postwar U.S. business cycles: An empirical investigation. Journal of Money, Credit and Banking 29, 1-16.

Holden, S., Driscoll, J.C., 2003

Inflation persistence and relative contracting. American Economic Review 93, 1369-1372.

Inoue, A., Kilian, L., 2005

How useful is bagging in forecasting economic time series? A case study of U.S. CPI inflation. Working paper, University of Michigan.

Jorion, P., Mishkin, F.S., 1991

A multi-country comparison of term structure forecasts at long horizons. Journal of Financial Economics 29, 59-80.

Kim, D.H., 2004

Inflation and the real term structure. Working paper, Federal Reserve Board of Governors.

Kim, C.J., Nelson, C.R., 1999

Has the U.S. economy become more stable? A Bayesian approach based on a Markov switching model of the business cycle. Review of Economics and Statistics 81, 608-616.

Kozicki, S., 1997

Predicting real growth and inflation with the yield spread. Federal Reserve Bank of Kansas City Economic Review 82, 39-57.

Marcellino, M., Stock, J.H., Watson, M.W., 2006

A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series. Journal of Econometrics, forthcoming.

Mehra, Y.P., 2002

Survey measures of expected inflation: Revisiting the issues of predictive content and rationality. Federal Reserve Bank of Richmond Economic Quarterly 88, 17-36.

McConnell, M.M., Perez-Quiros, G., 2000

Output fluctuations in the United States: What has changed since the early 1950's. American Economic Review 90, 1464-1476.

Mishkin, F.S., 1990

What does the term structure tell us about future inflation? Journal of Monetary Economics 25, 77-95.

Mishkin, F.S., 1991

A multi-country study of the information in the term structure about future inflation. Journal of International Money and Finance 19, 2-22.

Nelson, C.R., Schwert, G.W., 1977

On testing the hypothesis that the real rate of interest is constant. American Economic Review 67, 478-486.

Newey, W.K., West K.D., 1987

A simple positive, semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703-708.

Ng, S., Perron, P., 2001

Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519-1554.

Orphanides, A., van Norden, S., 2003

The reliability of inflation forecasts based on output gap estimates in real time. Working paper, CIRANO.

Pennacchi, G.G., 1991

Identifying the dynamics of real interest rates and inflation: Evidence using survey data. Review of Financial Studies 4, 53-86.

Plosser, C.I., Schwert, G.W., 1978

Money, income, and sunspots: Measuring the economic relationships and the effects of difference. Journal of Monetary Economics 4, 637-660.

Quah, D., Vahey, S.P., 1995

Measuring core inflation. Economic Journal 105, 1130-1144.

Schorfheide, F., 2005

VAR forecasting under misspecification. Journal of Econometrics 128, 99-136.

Sims, C.A., 2002

The role of models and probabilities in the monetary policy process. Brookings Papers on Economic Activity 2, 1-40.

Souleles, N.S., 2004

Expectations, heterogeneous forecast errors and consumption:Micro evidence from the Michigan consumer sentiment surveys. Journal of Money, Credit and Banking 36, 39-72.

Stock, J.H., Watson, M.W., 1989

New indexes of coincident and leading economic indicators, in O.J. Blanchard and S. Fischer, eds., NBER Macroeconomics Annual (MIT Press, Boston) 351-394.

Stock, J.H., Watson, M.W., 1999

Forecasting inflation. Journal of Monetary Economics 44, 293-335.

Stock, J.H., Watson, M.W., 2002

a. Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97, 1167-1179.

Stock, J.H., Watson, M.W., 2002

b. Has the business cycle changed and why? in M. Gertler M. and K. Rogoff, eds., NBER Macroeconomics Annual 2002 (MIT Press, Boston) 159-218.

Stock, J.H., Watson, M.W., 2003

Forecasting output and inflation: The role of asset prices. Journal of Economic Literature 41, 788-829.

Stock J.H., Watson, M.W., 2005

An empirical comparison of methods for forecasting using many predictors. Working paper, Harvard University.

Stockton, D., Glassman, J., 1987

An evaluation of the forecast performance of alternative models of inflation. Review of Economics and Statistics 69, 108-117.

Theil, H., 1963

On the use of incomplete prior information in regression analysis. Journal of the American Statistical Association 58, 401-414.

Theil, H., Goldberger, A.S., 1961

On pure and mixed estimation in economics. International Economic Review 2, 65-78.

Thomas, L.B., 1999

Survey measures of expected U.S. inflation. Journal of Economic Perspectives 13, 125-144.

Timmermann, A., 2006

Forecast combinations, in G. Elliot, C.W.J. Granger and A. Timmermann, eds., Handbook of Economic Forecasting (Elsevier, Amsterdam), in press.

West, K.D., 1996

Asymptotic inference about predictive ability. Econometrica 64, 1067-1084.

West, K.D., 2006

Forecast evaluation, in G. Elliott, C.W.J. Granger, and A. Timmermann, eds., Handbook of Economic Forecasting (Elsevier, Amsterdam), in press.

West, K.D., McCracken, M.W., 1998

Regression-based tests of predictive ability. International Economic Review 39, 817-840.

Wright, J.H., 2004

Forecasting U.S. inflation by Bayesian model averaging. Working paper, Federal Reserve Board of Governors.

Table 1: Summary Statistics -
Panel A: 1952:Q2 - 2002:Q4$^\ast$

	PUNEW	PUXHS	PUXX	PCE
Mean	3.84	3.60	4.24	3.84
Standard Error: Mean	(0.20)	(0.20)	(0.19)	(0.19)
Standard Deviation	2.86	2.78	2.56	2.45
Standard Error: Standard Deviation	(0.14)	(0.14)	(0.14)	(0.13)
Autocorrelation	0.78	0.74	0.77	0.79
Standard Error: Autocorrelation	(0.08)	(0.09)	(0.11)	0.09)
Correlations: PUXHS	0.99
Correlations: PUXX	0.94	0.91
Correlations: PCE	0.98	0.98	0.93

Table 1 continued: Summary Statistics -
Panel B: 1986:Q1-2002:Q4

	PUNEW	PUXHS	PUXX	PCE
Mean	3.09	2.87	3.21	2.58
Standard Error: Mean	(0.14)	(0.17)	0.12)	(0.14)
Standard Deviation	1.12	1.37	0.97	1.08
Standard Error: Standard Deviation	(0.10)	(0.12)	(0.09)	(0.10)
Autocorrelation	0.47	0.37	0.77	0.69
Standard Error: Autocorrelation	(0.07)	(0.10)	(0.08)	(0.07)
Correlations PUXHS	0.99
Correlations PUXX	0.85	0.79
Correlations PCE	0.95	0.93	0.90

Table 1 continued: Summary Statistics -
Panel C: 1996:Q1-2002:Q4

	PUNEW	PUXHS	PUXX	PCE
Mean	2.27	1.84	2.32	1.70
Standard Error:	(0.17)	(0.25)	(0.05)	(0.13)
Standard Deviation	0.81	1.19	0.24	0.62
Standard Error: Standard Deviation	(0.12)	(0.17)	(0.03)	(0.09)
Autocorrelation	-0.13	-0.19	-0.38	0.05
Standard Error: Autocorrelation	(0.23)	(0.23)	(0.14)	(0.18)
Correlations PUXHS	0.99
Correlations PUXX	0.33	0.21
Correlations PCE	0.89	0.88	0.19

This table reports various moments of different measures of annual inflation sampled at a quarterly frequency for different sample periods. PUNEW is CPI-U All Items; PUXHS is CPI-U Less Shelter; PUXX is CPI-U All Items Less Food and Energy, also called core CPI; and PCE is the Personal Consumption Expenditure deflator. All measures are in annual percentage terms. The autocorrelation reported is the fourth order autocorrelation with the quarterly inflation data, representing the first-order autocorrelation of annual inflation. Standard errors reported in parentheses are computed by GMM.

** For PUXX, the start date is 1958:Q2 and for PCE, the start date is 1960:Q2.

Table 2: Forecasting Models

Abbreviation	Specification	Type of Model
ARMA	ARMA(1,1)	Time-Series Model
AR	Autoregressive model	Time-Series Model
RW	Random walk on quarterly inflation	Time-Series Model
AORW	Random walk on annual inflation	Time-Series Model
RGM	Univariate regime-switching model	Time-Series Model
PC1	INFL + GDPG	Phillips Curve (OLS)
PC2	INFL + GAP1	Phillips Curve (OLS)
PC3	INFL + GAP2	Phillips Curve (OLS)
PC4	INFL + LSHR	Phillips Curve (OLS)
PC5	INFL + UNEMP	Phillips Curve (OLS)
PC6	INFL + XLI	Phillips Curve (OLS)
PC7	INFL + XLI-2	Phillips Curve (OLS)
PC8	INFL + FAC	Phillips Curve (OLS)
PC9	INFL + GAP1 + LSHR	Phillips Curve (OLS)
PC10	INFL + GAP2 + LSHR	Phillips Curve (OLS)
TS1	INFL + GDPG + RATE	OLS Term Structure Model
TS2	INFL + GAP1 + RATE	OLS Term Structure Model
TS3	INFL + GAP2 + RATE	OLS Term Structure Model
TS4	INFL + LSHR + RATE	OLS Term Structure Model
TS5	INFL + UNEMP + RATE	OLS Term Structure Model
TS6	INFL + XLI + RATE	OLS Term Structure Model
TS7	INFL + XLI-2 + RATE	OLS Term Structure Model
TS8	INFL + FAC + RATE	OLS Term Structure Model
TS9	INFL + SPD	OLS Term Structure Model
TS10	INFL + RATE + SPD	OLS Term Structure Model
TS11	INFL + GDPG + RATE + SPD	OLS Term Structure Models
VAR	VAR(1) on RATE, SPD, INFL, GDPG	Empirical Term Structure Model
RGMVAR	Regime-switching model on RATE, SPD, INFL	Empirical Term Structure Model
MDL1	Three-factor affine model	No-Arbitrage Term Structure Model
MDL2	General three-factor regime-switching model	No-Arbitrage Term Structure Model
SPF1	Survey of Professional Forecasters	Inflation Surveys
SPF2	Linear bias-corrected SPF	Inflation Surveys
SPF3	Non-linear bias-corrected SPF	Inflation Surveys
LIV1	Livingston Survey	Inflation Surveys
LIV2	Linear bias-corrected Livingston	Inflation Surveys
LIV3	Non-linear bias-corrected Livingston	Inflation Surveys
MICH1	Michigan Survey	Inflation Surveys
MICH2	Linear bias-corrected Michigan	Inflation Surveys
MICH3	Non-linear bias-corrected Michigan	Inflation Surveys

INFL refers to the inflation rate over the previous quarter; GDPG to GDP growth; GAP1 to detrended log real GDP using a quadratic trend; GAP2 to detrended log real GDP using the Hodrick-Prescott filter; LSHR to the labor income share; UNEMP to the unemployment rate; XLI to the Stock-Watson Experimental Leading Index; XLI-2 to the Stock-Watson Experimental Leading Index-2; FAC to an aggregate composite real activity factor constructed by Bernanke, Boivin and Eliasz (2005); RATE to the one-quarter yield; and SPD to the difference between the 20-quarter and the one-quarter yield.

Table 3: Bias of Survey Forecasts - PUNEW

	$\alpha_{1}$	$\alpha_{2}$	$\beta_{1}$	$\beta_{2}$
SPF	1.321		0.482
Standard Error: SPF	(0.694)		(0.190)
Livingston	0.637		0.993
Standard Error: Livingston	(0.375)		(0.161)
Michigan	-0.823		1.276
Standard Error: Michigan	(0.658)		(0.205)
SPF	1.437*	-0.188	0.414**	0.128
Standard Error: SPF	(0.671)	(0.585)	(0.180)	(0.140)
Livingston	0.589**	-0.295	0.806**	0.461**
Standard Error: Livingston	(0.184)	(0.506)	(0.068)	(0.160)
Michigan	0.039	-1.261	0.959	0.482
Standard Error: Michigan	(0.429)	(0.822)	(0.099)	(0.249)

Table 3 continued: Bias of Survey Forecasts - PUXHS

	$\alpha_{1}$	$\alpha_{2}$	$\beta_{1}$	$\beta_{2}$
SPF	0.638		0.601*
Standard Error: SPF	(0.803)		(0.199)
Livingston	0.561		0.942
Standard Error: Livingston	(0.337)		(0.130)
Michigan	-0.741		1.167
Standard Error: Michigan	(0.621)		(0.166)
SPF	0.612	-0.269	0.580*	0.147
Standard Error: SPF	(0.717)	(1.085)	(0.164)	(0.279)
Livingston	0.568**	-0.191	0.765**	0.389**
Standard Error: Livingston	(0.202)	(0.576)	(0.070)	(0.129)
Michigan	-0.267	-0.723	1.002	0.262*
Standard Error: Michigan	(0.613)	(0.571)	(0.143)	(0.132)

Table 3 continued: Bias of Survey Forecasts - PUXX

	$\alpha_{1}$	$\alpha_{2}$	$\beta_{1}$	$\beta_{2}$
SPF	0.852		0.694
Standard Error: SPF	(0.612)		(0.179)
Livingston	0.381		1.055
Standard Error: Livingston	(0.429)		(0.133)
Michigan	-0.279		1.194
Standard Error: Michigan	(0.466)		(0.124)
SPF	0.966	-0.201	0.643	0.100
Standard Error: SPF	(0.662)	(0.495)	(0.192)	(0.123)
Livingston	0.433	0.124	0.931	0.165
Standard Error:	(0.303)	(0.558)	(0.104)	(0.136)
Michigan	-0.160	-0.042	1.137	0.059
Standard Error: Michigan	(0.579)	(0.842)	(0.146)	(0.245)

Table 3 continued: Bias of Survey Forecasts - PCE

	$\alpha_{1}$	$\alpha_{2}$	$\beta_{1}$	$\beta_{2}$
SPF	0.041		0.728*
Standard Error: SPF	(0.500)		(0.125)
Livingston	0.234		0.949
Standard Error: Livingston	(0.479)		(0.136)
Michigan	-0.547		1.058
Standard Error: Michigan	(0.521)		(0.139)
SPF	0.122	-0.571	0.689**	0.213
Standard Error: SPF	(0.482)	(0.751)	(0.108)	(0.187)
Livingston	0.278	-0.094	0.785*	0.399**
Standard Error: Livingston	(0.453)	(0.480)	(0.087)	(0.085)
Michigan	-0.061	-0.688	0.900	0.228
Standard Error: Michigan	(0.581)	(0.559)	(0.145)	(0.117)

This table reports the coefficient estimates in equations (15) and (16). We denote standard errors of $\alpha_{1}$, $\alpha_{2}$ and$\beta_{2}$ that reject the hypothesis that the coefficients are different to zero and standard errors of $\beta_{1}$ that reject that $\beta_{1}$ = 1 at the 95% and 99% level by * and **, respectively, based on Hansen and Hodrick (1980) standard errors (reported in parentheses). For the SPF survey, the sample is 1981:Q3 to 2002:Q4; for the Livingston survey, the sample is 1952:Q2 to 2002:Q4 for PUNEW and PUXHS, 1958:Q2 to 2002:Q4 for PUXX, and 1960:Q2 to 2002:Q4 for PCE; and for the Michigan survey, the sample is 1978:Q1 to 2002:Q4.

Table 4: Time-Series Forecasts of Annual Inflation - PUNEW

	Post-1985 Sample RMSE	Post-1985 Sample ARMA=1	Post-1995 Sample RMSE	Post-1995 Sample ARMA=1
ARMA	1.136	1.000	1.144	1.000
AR	1.140	1.003	1.130	0.988
RGM	1.420	1.250	0.873	0.764
AORW	1.177	1.036	1.128	0.986
RW	1.626	1.431	1.529	1.337

Table 4 continued: Time-Series Forecasts of Annual Inflation - PUXHS

	Post-1985 Sample RMSE	Post-1985 Sample ARMA=1	Post-1995 Sample RMSE	Post-1995 Sample ARMA=1
ARMA	1.490	1.000	1.626	1.000
AR	1.515	1.017	1.634	1.005
RGM	1.591	1.068	1.355	0.833
AORW	1.580	1.061	1.670	1.027
RW	2.172	1.458	2.146	1.320

Table 4 continued: Time-Series Forecasts of Annual Inflation - PUXX

	Post-1985 Sample RMSE	Post-1985 Sample ARMA=1	Post-1995 Sample RMSE	Post-1995 Sample ARMA=1
ARMA	0.630	1.000	0.600	1.000
AR	0.644	1.023	0.593	0.988
RGM	0.677	1.075	0.727	1.211
AORW	0.516	0.819	0.372	0.620
RW	0.675	1.072	0.549	0.915

Table 4 continued: Time-Series Forecasts of Annual Inflation - PCE

	Post-1985 Sample RMSE	Post-1985 Sample ARMA=1	Post-1995 Sample RMSE	Post-1995 Sample ARMA=1
ARMA	0.878	1.000	0.944	1.000
AR	0.942	1.073	1.014	1.074
RGM	0.945	1.077	1.081	1.145
AORW	0.829	0.945	0.869	0.921
RW	1.140	1.298	1.215	1.288

We forecast annual inflation out-of-sample from 1985:Q4 to 2002:Q4 and from 1995:Q4 to 2002:Q4 at a quarterly frequency. Table 2 contains full details of the time-series models. Numbers in the RMSE columns are reported in annual percentage terms. The column labeled ARMA = 1 reports the ratio of the RMSE relative to the ARMA(1,1) specification.

Table 5: OLS Phillips Curve Forecasts of Annual Inflation - PUNEW

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
PC1	0.979	0.639	0.392	0.596	0.977	0.673	0.624	0.984
PC2	1.472	0.066	0.145	0.155	1.956	-0.117	0.199	0.169
PC3	1.166	0.269	0.233	0.258	1.295	0.171	0.349	0.344
PC4	1.078	-1.043	0.632	1.266	1.025	0.046	0.890	1.389
PC5	1.032	0.354	0.288	0.372	1.115	-0.174	0.222	0.458
PC6	1.103	-0.303	0.575	0.634	1.086	-0.633	0.488	1.054
PC7	1.022	0.460	0.161**	0.283	1.040	0.367	0.406	0.531
PC8	1.039	0.319	0.477	0.515	0.993	0.468	0.793	0.901
PC9	1.576	0.006	0.119	0.144	1.994	-0.121	0.174	0.159
PC10	1.264	0.146	0.205	0.235	1.426	0.119	0.246	0.287

Table 5 continued: OLS Phillips Curve Forecasts of Annual Inflation - PUXHS

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
PC1	1.000	0.498	0.458	0.758	0.992	0.618	0.814	1.182
PC2	1.328	-0.022	0.218	0.239	1.586	-0.192	0.317	0.266
PC3	1.113	0.200	0.310	0.329	1.105	0.239	0.522	0.519
PC4	1.096	-0.988	0.497*	1.064	1.029	0.008	0.745	1.229
PC5	1.083	-0.080	0.299	0.491	1.076	-0.411	0.358	0.708
PC6	1.131	-1.074	0.519*	0.822	1.061	-1.316	0.512**	1.463
PC7	1.001	0.498	0.186**	0.301	1.070	0.085	0.529	0.590
PC8	1.094	-0.325	0.466	0.713	1.007	0.101	1.259	1.337
PC9	1.394	-0.055	0.186	0.224	1.624	-0.204	0.290	0.254
PC10	1.165	0.125	0.273	0.308	1.202	0.150	0.340	0.392

Table 5 continued: OLS Phillips Curve Forecasts of Annual Inflation - PUXX

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
PC1	0.866	1.432	0.340**	1.632	0.825	1.182	0.120**	1.384
PC2	2.463	-0.120	0.072	0.100	3.257	-0.227	0.093*	0.119
PC3	1.664	0.054	0.213	0.190	2.076	-0.063	0.275	0.226
PC4	1.234	0.126	0.143	0.261	1.330	0.187	0.214	0.230
PC5	1.024	0.460	0.207*	0.370	1.185	0.134	0.445	0.551
PC6	1.005	0.479	0.477	1.053	0.916	1.009	0.277**	1.935
PC7	1.074	0.381	0.277	0.426	1.089	0.293	0.500	0.731
PC8	0.862	0.809	0.297**	0.751	0.767	1.127	0.275**	1.340
PC9	2.485	-0.076	0.069	0.100	3.262	-0.168	0.069*	0.120
PC10	1.873	0.079	0.136	0.153	2.562	0.038	0.150	0.151

Table 5 continued: OLS Phillips Curve Forecasts of Annual Inflation - PCE

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
PC1	1.053	0.029	0.469	0.972	1.088	-0.240	0.434	1.119
PC2	1.698	-0.136	0.141	0.178	1.997	-0.240	0.223	0.218
PC3	1.274	-0.031	0.280	0.252	1.407	-0.239	0.354	0.340
PC4	1.027	0.343	0.392	1.004	1.031	0.339	0.535	1.138
PC5	1.125	-0.080	0.327	0.434	1.214	-0.635	0.389	0.629
PC6	1.053	0.036	0.484	1.233	1.020	0.273	0.509	1.795
PC7	1.033	0.436	0.175*	0.359	1.116	0.034	0.334	0.651
PC8	1.040	0.269	0.476	0.807	1.044	0.044	1.101	2.018
PC9	1.518	-0.100	0.166	0.193	1.786	-0.282	0.258	0.258
PC10	1.247	0.120	0.201	0.297	1.432	-0.068	0.235	0.322

We forecast annual inflation out-of-sample over 1985:Q4 to 2002:Q4 and over 1995:Q4 to 2002:Q4 at a quarterly frequency. Table 2 contains full details of the Phillips Curve models. The column labelled "Relative RMSE" reports the ratio of the RMSE relative to the ARMA(1,1) specification. The column titled "$1-\lambda$" reports the coefficient ( $1-\lambda$) from equation (17). Standard errors computed using the Hansen-Hodrick (1980) method and the West (1996) method are reported in the columns titled "HH SE" and "West SE," respectively. We denote standard errors that reject the hypothesis of ( $1-\lambda$) equal to zero at the 95% (99%) level by * (**).

Table 6: Term Structure Forecasts of Annual Inflation - PUNEW

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
TS1	1.096	0.137	0.332	0.393	1.030	0.362	0.410	0.653
TS2	1.444	0.019	0.145	0.148	1.826	-0.147	0.229	0.182
TS3	1.176	0.193	0.229	0.259	1.226	0.156	0.335	0.358
TS4	1.166	-0.108	0.249	0.321	1.018	0.370	0.474	0.959
TS5	1.134	0.088	0.186	0.278	1.122	0.006	0.187	0.429
TS6	1.194	-0.241	0.326	0.371	1.112	-0.162	0.406	0.578
TS7	1.091	0.309	0.252	0.290	1.039	0.373	0.434	0.523
TS8	1.119	0.116	0.332	0.365	1.010	0.380	0.816	0.864
TS9	1.363	0.086	0.085	0.129	1.229	-0.008	0.083	0.305
TS10	1.196	-0.024	0.143	0.220	1.043	0.132	0.639	0.685
TS11	1.198	-0.124	0.431	0.414	1.052	0.286	0.318	0.611
VAR	1.106	0.307	0.187	0.225	1.328	-0.101	0.259	0.270
RGMVAR	1.647	0.050	0.050	0.090	1.518	-0.170	0.198	0.226
MDL1	1.323	0.161	0.064*	0.356	1.345	-0.088	0.192	0.247
MDL2	1.192	0.225	0.117	0.392	1.329	-0.118	0.251	0.278

Table 6 continued: Term Structure Forecasts of Annual Inflation - PUXHS

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
TS1	1.080	-0.025	0.413	0.508	1.014	0.373	0.553	0.824
TS2	1.345	-0.017	0.205	0.216	1.584	-0.197	0.329	0.265
TS3	1.116	0.186	0.278	0.309	1.118	0.195	0.435	0.463
TS4	1.085	-0.275	0.499	0.670	0.996	0.542	0.592	1.077
TS5	1.113	-0.082	0.214	0.358	1.094	-0.191	0.265	0.557
TS6	1.140	-0.566	0.342	0.534	1.069	-0.360	0.419	0.776
TS7	1.081	0.161	0.298	0.342	1.070	0.089	0.410	0.564
TS8	1.083	-0.054	0.411	0.497	0.975	0.559	1.057	1.055
TS9	1.173	0.114	0.105	0.201	1.130	-0.123	0.211	0.478
TS10	1.140	-0.594	0.468	0.658	1.032	-0.034	0.090	0.855
TS11	1.102	-0.121	0.423	0.482	1.049	0.093	0.164	0.667
VAR	1.001	0.496	0.264	0.354	1.137	0.041	0.426	0.433
RGMVAR	1.363	0.070	0.085	0.159	1.285	-0.149	0.366	0.383
MDL1	1.225	0.127	0.081	0.263	1.186	-0.048	0.266	0.320
MDL2	1.047	0.395	0.203	0.702	1.156	0.000	0.406	0.386

Table 6 continued: Term Structure Forecasts of Annual Inflation - PUXX

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
TS1	0.945	0.667	0.322*	0.655	0.945	0.665	0.317*	0.924
TS2	2.262	-0.092	0.084	0.100	2.982	-0.225	0.099*	0.117
TS3	1.399	0.121	0.260	0.249	1.698	-0.057	0.344	0.288
TS4	1.232	0.260	0.156	0.229	1.268	0.319	0.225	0.248
TS5	1.081	0.392	0.203	0.299	1.258	0.085	0.407	0.454
TS6	0.969	0.567	0.294	0.601	0.866	0.788	0.078**	0.882
TS7	1.068	0.419	0.203*	0.354	1.118	0.342	0.289	0.505
TS8	0.948	0.568	0.197**	0.459	0.958	0.520	0.253*	0.832
TS9	1.372	0.050	0.239	0.247	1.282	-0.101	0.457	0.504
TS10	1.034	0.433	0.284	0.467	1.208	-0.048	0.548	0.737
TS11	1.017	0.474	0.246	0.439	1.192	0.099	0.502	0.686
VAR	1.651	0.041	0.178	0.154	2.238	-0.276	0.151	0.183
RGMVAR	1.572	0.120	0.138	0.147	1.622	-0.211	0.340	0.278
MDL1	1.506	0.253	0.091**	0.381	1.593	-0.004	0.280	0.303
MDL2	1.834	0.262	0.039**	0.443	1.329	0.355	0.069**	0.298

Table 6 continued: Term Structure Forecasts of Annual Inflation - PCE

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
TS1	1.075	-0.073	0.453	0.847	1.078	-0.207	0.433	1.192
TS2	1.670	-0.149	0.145	0.181	1.966	-0.247	0.226	0.221
TS3	1.279	-0.053	0.288	0.259	1.373	-0.245	0.376	0.360
TS4	1.075	0.018	0.372	0.864	1.059	0.234	0.442	0.816
TS5	1.126	-0.115	0.331	0.456	1.202	-0.645	0.383	0.663
TS6	1.094	-0.149	0.428	0.896	1.100	-0.358	0.397	1.322
TS7	1.018	0.443	0.271	0.481	1.106	0.033	0.303	0.673
TS8	1.027	0.374	0.414	0.720	1.025	0.346	1.058	1.855
TS9	1.141	-0.024	0.192	0.304	1.121	-0.825	0.584	0.939
TS10	1.087	-0.569	0.549	0.992	1.110	-0.850	0.638	1.177
TS11	1.086	0.006	0.418	0.665	1.132	-0.396	0.288	0.878
VAR	1.286	-0.179	0.274	0.298	1.511	-0.337	0.392	0.327
RGMVAR	1.507	-0.242	0.131	0.237	1.461	-0.356	0.233	0.424
MDL1	1.169	0.144	0.235	0.432	1.271	-0.374	0.284	0.481
MDL2	1.314	-0.205	0.159	1.220	1.339	-0.331	0.120**	0.589

We forecast annual inflation out-of-sample over 1985:Q4 to 2002:Q4 and over 1995:Q4 to 2002:Q4 at a quarterly frequency. Table 2 contains full details of the term structure models. The column labelled "Relative RMSE" reports the ratio of the RMSE relative to the ARMA(1,1) specification. The column titled "$1-\lambda$" reports the coefficient $1-\lambda$ from equation (17). Standard errors computed using the Hansen-Hodrick (1980) method and the West (1996) method are reported in the columns titled "HH SE" and "West SE," respectively. We denote standard errors that reject the hypothesis of $1-\lambda$ equal to zero at the 95% (99%) level by * (**).

Table 7: Survey Forecasts of Annual Inflation - PUNEW

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
SPF1	0.779	1.051	0.177**	0.439*	0.861	0.869	0.407*	0.554
SPF2	0.964	0.564	0.216**	0.308	0.902	0.745	0.377*	0.484
SPF3	0.976	0.541	0.207**	0.302	0.915	0.728	0.414	0.479
LIV1	0.789	1.164	0.102**	0.585	0.792	1.140	0.203**	0.913
LIV2	1.180	0.335	0.177	0.281	1.092	0.403	0.437	0.550
LIV3	1.299	0.251	0.163	0.226	1.152	0.275	0.517	0.549
MICH1	0.902	0.771	0.324*	0.379*	0.862	1.113	0.520*	0.684
MICH2	0.961	0.675	0.327*	0.370	0.930	0.861	0.644	0.609
MICH3	0.968	0.655	0.347	0.375	0.947	0.776	0.653	0.567

Table 7 continued: Survey Forecasts of Annual Inflation - PUXHS

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
SPF1	0.819	0.939	0.171**	0.430*	0.914	0.773	0.394*	0.546
SPF2	0.924	0.666	0.227**	0.312*	0.888	0.825	0.357*	0.504
SPF3	1.348	0.103	0.183	0.193	0.958	0.582	0.323	0.362
LIV1	0.844	1.098	0.099**	0.573	0.856	1.072	0.214**	0.878
LIV2	1.054	0.554	0.176**	0.386	1.031	0.550	0.366	0.615
LIV3	1.199	0.327	0.156*	0.299	1.053	0.502	0.443	0.605
MICH1	0.881	0.876	0.273**	0.398*	0.937	0.750	0.434	0.476
MICH2	0.918	0.815	0.290**	0.395*	0.932	0.814	0.515	0.528
MICH3	0.970	0.608	0.251*	0.347	0.953	0.684	0.492	0.474

Table 7 continued: Survey Forecasts of Annual Inflation - PUXX

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
SPF1	0.691	0.968	0.140**	0.654	0.699	1.260	0.225**	1.437
SPF2	1.145	0.125	0.362	0.555	1.104	0.091	0.852	1.177
SPF3	1.179	0.035	0.373	0.555	1.180	-0.358	0.956	1.390
LIV1	0.655	0.803	0.192**	0.730	0.557	1.227	0.134**	1.453
LIV2	1.355	-0.185	0.177	0.185	1.387	-0.423	0.415	0.557
LIV3	1.289	-0.095	0.259	0.262	1.278	-0.496	0.735	0.850
MICH1	1.185	0.383	0.159*	0.301	0.822	1.041	0.208**	2.124
MICH2	1.343	-0.153	0.248	0.272	1.566	-0.385	0.286	0.356
MICH3	1.360	-0.242	0.253	0.285	1.617	-0.493	0.273	0.363

Table 7 continued: Survey Forecasts of Annual Inflation - PCE

	Post-1985 Sample Relative RMSE	Post-1985 Sample $1-\lambda$	Post-1985 Sample HH SE	Post-1985 Sample West SE	Post-1995 Sample Relative RMSE	Post-1995 Sample $1-\lambda$	Post-1995 Sample HH SE	Post-1995 Sample West SE
SPF1	1.199	0.147	0.267	0.241	1.250	0.090	0.395	0.349
SPF2	0.980	0.537	0.206**	0.375	0.924	0.655	0.325*	0.570
SPF3	1.034	0.454	0.180*	0.306	1.040	0.453	0.234	0.362
LIV1	1.082	0.175	0.325	0.300	1.101	0.132	0.412	0.400
LIV2	1.397	-0.050	0.189	0.234	1.303	-0.026	0.265	0.358
LIV3	1.380	-0.123	0.149	0.212	1.341	-0.191	0.272	0.375
MICH1	1.217	0.108	0.216	0.192	1.338	-0.030	0.327	0.283
MICH2	1.194	0.039	0.253	0.216	1.205	0.056	0.415	0.350
MICH3	1.248	-0.022	0.239	0.200	1.255	-0.003	0.399	0.334

We forecast annual inflation out-of-sample over 1985:Q4 to 2002:Q4 and from 1995:Q4 to 2002:Q4 at a quarterly frequency for the SPF survey (SPF1-3) and the Michigan survey (MICH1-3). The frequency of the Livingston survey (LIV1-3) is biannual and forecasts are made at the end of the second and end of the fourth quarter. Table 2 contains full details of the survey models. The column labelled "Relative RMSE" reports the ratio of the RMSE relative to the ARMA(1,1) specification. The column titled " $1-\lambda$" reports the coefficient ($1-\lambda$ from equation (17). Standard errors computed using the Hansen-Hodrick (1980) method and the West (1996) method are reported in the columns titled "HH SE" and "West SE," respectively. We denote standard errors that reject the hypothesis of ($1-\lambda$) equal to zero at the 95% (99%) level by * (**).

Table 8: Best Models in Forecasting Annual Inflation - Panel A: Post-1985 Sample

	PUNEW	PUNEW	PUXHS	PUXHS	PUXX	PUXX	PCE	PCE
Best Time-Series Model	ARMA	1.000	ARMA	1.000	AORW	0.819	AORW	0.945*
Best Phillips-Curve Model	PC1	0.979	PC1	1.000	PC8	0.862	PC4	1.027
Best Term-Structure Model	TS7	1.091	VAR	1.001	TS1	0.945	TS7	1.018
Raw Survey Forecasts	SPF1	0.779*	SPF1	0.819*	SPF1	0.691	SPF1	1.199
Raw Survey Forecasts	LIV1	0.789	LIV1	0.844	LIV1	0.655*	LIV1	1.082
Raw Survey Forecasts	MICH1	0.902	MICH1	0.881	MICH1	1.185	MICH1	1.217

Table 8 continued: Best Models in Forecasting Annual Inflation - Panel B: Post-1995 Sample

	PUNEW	PUNEW	PUXHS	PUXHS	PUXX	PUXX	PCE	PCE
Best Time-Series Model	RGM	0.764*	RGM	0.833*	AORW	0.620	AORW	0.921*
Best Phillips-Curve Model	PC1	0.977	PC1	0.992	PC8	0.767	PC6	1.020
Best Term-Structure Model	TS8	1.010	TS8	0.975	TS6	0.866	TS8	1.025
Raw Survey Forecasts	SPF1	0.861	SPF1	0.914	SPF1	0.699	SPF1	1.250
Raw Survey Forecasts	LIV1	0.792	LIV1	0.856	LIV1	0.557*	LIV1	1.101
Raw Survey Forecasts	MICH1	0.862	MICH1	0.937	MICH1	0.822	MICH1	1.338

The table reports the best time-series model, the best OLS Phillips Curve model, the best model using term structure data, along with SPF1, LIV1, and MCH1 forecasts for out-of-sample forecasting of annual inflation at a quarterly frequency. Each entry reports the ratio of the model RMSE to the RMSE of an ARMA(1,1) forecast. The smallest RMSEs for each inflation measure are marked with an asterisk.

Table 9: Ex-Ante Best Models in Forecasting Annual Inflation

Date	PUNEW Time Series	PUNEW Phillips Curve	PUNEW Term Structure	PUNEW Surveys	PUNEW All Models	PUXHS Time Series	PUXHS Phillips Curve	PUXHS Term Structure	PUXHSSurveys	PUXHS All Models
1995Q4	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1996Q1	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1996Q2	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1996Q3	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1996Q4	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1997Q1	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1997Q2	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1997Q3	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1997Q4	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1998Q1	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1998Q2	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1998Q3	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1998Q4	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1999Q1	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1999Q2	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1999Q3	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
1999Q4	ARMA	PC5	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
2000Q1	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
2000Q2	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
2000Q3	ARMA	PC1	VAR	SPF1	SPF1	ARMA	PC7	VAR	SPF1	SPF1
2000Q4	ARMA	PC1	TS1	SPF1	SPF1	ARMA	PC1	VAR	SPF1	SPF1
2001Q1	ARMA	PC1	TS1	SPF1	SPF1	ARMA	PC1	VAR	SPF1	SPF1
2001Q2	ARMA	PC1	TS1	SPF1	SPF1	ARMA	PC1	VAR	SPF1	SPF1
2001Q3	ARMA	PC1	TS1	SPF1	SPF1	ARMA	PC1	VAR	SPF1	SPF1
2001Q4	ARMA	PC1	TS7	SPF1	SPF1	ARMA	PC1	VAR	SPF1	SPF1

Table 9 Continued

Date	PUXX Time Series	PUXX Phillips Curve	PUXX Term Structure	PUXX Surveys	PUXX All Models	PCE Time Series	PCE Phillips Curve	PCE Term Structure	PCESurveys	PCE All Models
1995Q4	AORW	PC1	TS11	SPF1	SPF1	AORW	PC7	TS7	MICH1	TS7
1996Q1	AORW	PC1	TS11	SPF1	SPF1	AORW	PC7	TS7	MICH1	TS7
1996Q2	AORW	PC1	TS11	SPF1	SPF1	AORW	PC7	TS7	MICH1	TS7
1996Q3	AORW	PC1	TS11	SPF1	SPF1	AORW	PC7	TS7	MICH1	TS7
1996Q4	AORW	PC8	TS11	SPF1	SPF1	AORW	PC7	TS7	MICH1	AORW
1997Q1	AORW	PC1	TS11	SPF1	SPF1	AORW	PC7	TS7	MICH1	AORW
1997Q2	AORW	PC8	TS11	SPF1	SPF1	AORW	PC7	TS7	MICH1	AORW
1997Q3	AORW	PC8	TS11	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
1997Q4	AORW	PC8	TS11	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
1998Q1	AORW	PC8	TS1	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
1998Q2	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
1998Q3	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
1998Q4	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
1999Q1	AORW	PC8	TS8	SPF1	SPF1	AORW	PC7	TS7	MICH1	TS7
1999Q2	AORW	PC8	TS8	SPF1	SPF1	AORW	PC7	TS7	MICH1	TS7
1999Q3	AORW	PC8	TS8	SPF1	SPF1	AORW	PC7	TS7	MICH1	TS7
1999Q4	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	TS7
2000Q1	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
2000Q2	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
2000Q3	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
2000Q4	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	MICH1	AORW
2001Q1	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	SPF1	AORW
2001Q2	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	SPF1	AORW
2001Q3	AORW	PC8	TS8	SPF1	SPF1	AORW	PC4	TS7	SPF1	AORW
2001Q4	AORW	PC8	TS1	SPF1	SPF1	AORW	PC4	TS7	SPF1	AORW

The table reports the ex-ante best model within each category of time-series, Phillips curve, and term structure models, together with the SPF and Michigan surveys. We also report the best ex-ante model across all models. The best models within each category, and across all models, yield the lowest out-of-sample RMSE for forecasting annual inflation at a quarterly frequency during the post-1985 sample period. The ex-ante best models are evaluated recursively through the sample starting with the first forecast in 1985:Q4 and the last forecast ending on the date given in the first column.

Table 10: Best Models in Forecasting Annual Inflation: Rolling Estimation - Panel A: Post-1985 Sample

	PUNEW	PUNEW	PUXHS	PUXHS	PUXX	PUXX	PCE	PCE
Best Time-Series Model	AR	0.967	AR	1.002	AORW	0.819	AORW	0.945*
Best Phillips-Curve Model	PC7	1.070	PC1	1.068	PC8	1.179	PC8	1.082
Best Term-Structure Model	TS1	1.199	TS9	1.073	TS6	1.350	TS6	1.182
Raw Survey Forecasts	SPF1	0.779*	SPF1	0.819*	SPF1	0.691	SPF1	1.199
Raw Survey Forecasts	LIV1	0.789	LIV1	0.844	LIV1	0.655*	LIV1	1.082
Raw Survey Forecasts	MICH1	0.902	MICH1	0.881	MICH1	1.185	MICH1	1.217

Table 10 continued: Best Models in Forecasting Annual Inflation: Rolling Estimation - Panel B: Post-1995 Sample

	PUNEW	PUNEW	PUXHS	PUXHS	PUXX	PUXX	PCE	PCE
Best Time-Series Model	AR	0.879	AR	0.914	ARMA	0.635	ARMA	0.730*
Best Phillips-Curve Model	PC6	0.951	PC6	0.955	PC7	0.560	PC6	0.799
Best Term-Structure Model	VAR	0.987	VAR	0.998	TS5	0.881	TS3	0.990
Raw Survey Forecasts	SPF1	0.861*	SPF1	0.914	SPF1	0.699	SPF1	1.250
Raw Survey Forecasts	LIV1	0.792	LIV1	0.856*	LIV1	0.557*	LIV1	1.101
Raw Survey Forecasts	MICH1	0.862	MICH1	0.937	MICH1	0.822	MICH1	1.338

The table reports the ex-post best ARIMA and random walk time-series models, the best OLS Phillips Curve model, the best linear model using term structure data, along with SPF1, LIV1, and MCH1 forecasts for out-of-sample forecasting of annual inflation at a quarterly frequency. All models are estimated using a rolling window of 10 years. We do not consider the regime-switching models (RGM and RGMVAR) and the no-arbitrage term structure models (MDL1 and MLD2). Each entry reports the ratio of the model RMSE to the RMSE of a recursively estimated ARMA(1,1) model. Models with the smallest RMSEs are marked with an asterisk.

Table 11: Combined Forecasts of Annual Inflation - PUNEW

Model Combination Method	Time-Series	Phillips Curve	Term Structure	Surveys	Best Models	All Models
Mean	0.898	1.123	1.057	0.851	0.992	0.998
Median	0.934	1.093	1.079	0.851	1.016	1.045
OLS	0.970	1.007	1.116	0.858	0.867	0.876
Equal Weight Prior	0.955	1.007	1.102	0.858	0.861	0.879
Unit Weight Prior	0.977	0.951	1.115	0.859	0.862	0.873
Best Individual Model	1.000	0.960	1.207	0.861	0.861	0.861

Table 11 continued: Combined Forecasts of Annual Inflation - PUXHS

Model Combination Method	Time-Series	Phillips Curve	Term Structure	Surveys	Best Models	All Models
Mean	0.954	1.065	1.012	0.921	0.975	0.992
Median	0.953	1.082	1.053	0.921	1.009	1.039
OLS	0.963	1.001	1.069	0.917	0.919	0.924
Equal Weight Prior	0.950	1.008	1.058	0.918	0.920	0.935
Unit Weight Prior	0.977	0.992	1.085	0.916	0.914	0.914
Best Individual Model	1.000	1.029	1.137	0.914	0.914	0.914

Table 11 continued: Combined Forecasts of Annual Inflation - PUXX

Model Combination Method	Time-Series	Phillips Curve	Term Structure	Surveys	Best Models	All Models
Mean	0.835	1.547	1.322	0.719	0.727	1.235
Median	0.940	1.167	1.211	0.719	0.735	1.052
OLS	0.631	0.885	0.964	0.699	0.665	0.706
Equal Weight Prior	0.687	0.878	0.956	0.699	0.652	0.661
Unit Weight Prior	0.650	0.836	0.947	0.699	0.658	0.658
Best Individual Model	0.620	0.779	0.977	0.699	0.699	0.699

Table 11 continued: Combined Forecasts of Annual Inflation - PCE

Model Combination Method	Time-Series	Phillips Curve	Term Structure	Surveys	Best Models	All Models
Mean	0.968	1.160	1.127	1.285	0.999	1.105
Median	0.979	1.136	1.130	1.285	0.999	1.118
OLS	0.935	0.974	1.019	1.288	0.921	0.964
Equal Weight Prior	0.938	0.984	1.017	1.287	0.922	0.968
Unit Weight Prior	0.917	0.967	1.010	1.287	0.911	0.948
Best Individual Model	0.921	1.057	1.106	1.289	0.887	0.887

The table reports the RMSEs relative to the ARMA(1,1) model for forecasting annual inflation at a quarterly frequency out-of-sample from 1995:Q4 to 2002:Q4 by combining models within each category (time-series, Phillips curve, term structure, surveys), using the ex-ante best models in each category, or over all models. Forecasts reported include the mean and median forecasts, and linear combinations of forecasts using recursively-computed weights computed from OLS, or model combination regressions with various priors. We investigate an equal weight prior and a prior that places only a unit weight on the best ex-ante model. We consider only unadjusted SPF and Michigan survey forecasts in the survey category. For comparison, the last row in each panel reports the relative RMSE of using the ex-ante best performing single forecast model at each period (as reported in Table 9).

PUNEW

Table 12: Best Models in Forecasting Annual Inflation Changes - PUNEW

	Post-1985 Sample Estimated on Levels Model	Post-1985 Sample Estimated on Levels RMSE	Post-1985 Estimated on Differences Sample Model	Post-1985 Estimated on Differences Sample RMSE	Post-1995 Sample Estimated on Levels Model	Post-1995 Sample Estimated on Levels RMSE	Post-1995 Estimated on Differences Sample Model	Post-1995 Estimated on Differences Sample RMSE
Best Time-Series Model	ARMA	1.000	ARMA	1.071	RGM	0.764*	ARMA	1.025
Best Phillips-Curve Model	PC1	0.979	PC7	1.005	PC1	0.977	PC7	0.976
Best Term-Structure Model	TS7	1.091	TS7	1.023	TS8	1.010	TS1	0.968
Raw Survey Forecasts	SPF1	0.779*			SPF1	0.861
Raw Survey Forecasts	LIV1	0.789			LIV1	0.792
Raw Survey Forecasts	MICH1	0.902			MICH1	0.862

Table 12 continued: Best Models in Forecasting Annual Inflation Changes - PUXHS

	Post-1985 Sample Estimated on Levels Model	Post-1985 Sample Estimated on Levels RMSE	Post-1985 Estimated on Differences Sample Model	Post-1985 Estimated on Differences Sample RMSE	Post-1995 Sample Estimated on Levels Model	Post-1995 Sample Estimated on Levels RMSE	Post-1995 Estimated on Differences Sample Model	Post-1995 Estimated on Differences Sample RMSE
Best Time-Series Model	ARMA	1.000	ARMA	1.098	RGM	0.833*	ARMA	1.046
Best Phillips-Curve Model	PC1	1.000	PC7	1.027	PC1	0.992	PC1	1.023
Best Term-Structure Model	VAR	1.001	TS7	1.004	TS8	0.975	TS7	0.987
Raw Survey Forecasts	SPF1	0.819*			SPF1	0.914
Raw Survey Forecasts	LIV1	0.844			LIV1	0.856
Raw Survey Forecasts	MICH1	0.881			MICH1	0.937

Table 12: Best Models in Forecasting Annual Inflation Changes - PUXX

	Post-1985 Sample Estimated on Levels Model	Post-1985 Sample Estimated on Levels RMSE	Post-1985 Estimated on Differences Sample Model	Post-1985 Estimated on Differences Sample RMSE	Post-1995 Sample Estimated on Levels Model	Post-1995 Sample Estimated on Levels RMSE	Post-1995 Estimated on Differences Sample Model	Post-1995 Estimated on Differences Sample RMSE
Best Time-Series Model	AORW	0.819	ARMA	0.837	AORW	0.620	ARMA	0.649
Best Phillips-Curve Model	PC8	0.862	PC1	0.722	PC8	0.767	PC1	0.652
Best Term-Structure Model	TS1	0.945	TS8	0.861	TS6	0.866	TS6	0.655
Raw Survey Forecasts	SPF1	0.691			SPF1	0.699
Raw Survey Forecasts	LIV1	0.655*			LIV1	0.557*
Raw Survey Forecasts	MICH1	1.185			MICH1	0.822

Table 12: Best Models in Forecasting Annual Inflation Changes - PCE

	Post-1985 Sample Estimated on Levels Model	Post-1985 Sample Estimated on Levels RMSE	Post-1985 Estimated on Differences Sample Model	Post-1985 Estimated on Differences Sample RMSE	Post-1995 Sample Estimated on Levels Model	Post-1995 Sample Estimated on Levels RMSE	Post-1995 Estimated on Differences Sample Model	Post-1995 Estimated on Differences Sample RMSE
Best Time-Series Model	AORW	0.945	ARMA	1.029	AORW	0.921	ARMA	1.004
Best Phillips-Curve Model	PC4	1.027	PC8	0.978	PC6	1.020	PC6	1.018
Best Term-Structure Model	TS7	1.018	TS8	0.945*	TS8	1.025	TS4	0.951*
Raw Survey Forecasts	SPF1	1.199			SPF1	1.250
Raw Survey Forecasts	LIV1	1.082			LIV1	1.101
Raw Survey Forecasts	MICH1	1.217			MICH1	1.338

This table reports the relative RMSE for forecasting annual inflation changes of the best performing out-of-sample forecasting model in each model category (time-series, Phillips Curve, and term structure models) and those of the raw survey forecasts. The models are estimated in either inflation levels or inflation differences. Table 2 contains full details of all the forecasting models. We report the RMSE ratios relative to an ARMA(1,1) specification estimated on levels. Models with the smallest RMSEs are marked with an asterisk.

Figure 1: Annual Inflation and Survey Forecasts

In the top panel, we graph the four inflation measures: CPI-U All Items, PUNEW; CPI-U Less Shelter, PUXHS; CPI-U All Items Less Food and Energy, or core CPI, PUXX; and the Personal Consumption Expenditure deflator, PCE. We also plot the Livingston survey forecast. The survey forecast is lagged one year, so that in December 1990, we plot inflation from December 1989 to December 1990 together with the survey forecasts of December 1989. In the bottom panel, we plot all three survey forecasts (SPF, Livingston, and the Michigan surveys), together with PUNEW inflation. The survey forecasts are also lagged one year for comparison.

Figure 2: Ex-Ante Weights on Best Models for Forecasting Annual Inflation

We graph the ex-ante OLS weights on models from regression (18) over the period 1995:Q4 to 2002:Q4. We combine the ex-ante best model within each category (time-series, Phillips Curve, and term structure) from Table 11 with the raw SPF survey. The weights are computed recursively through the sample.

Figure 3: Ex-Ante Weights on Best I(1) Models for Forecasting Annual Inflation Changes

We graph the ex-ante OLS weights on models from regression (22) over the period 1995:Q4 to 2002:Q4. We combine the ex-ante best non-stationary model within each category (time-series, Phillips Curve, and term structure) together with the raw SPF survey. The weights are computed recursively through the sample.

Figure 4: Ex-Ante Weights on Best I(0) Models for Forecasting Annual Inflation Changes

We graph the ex-ante OLS weights on models from regression (22) over the period 1995:Q4 to 2002:Q4. We combine the ex-ante best stationary model within each category (time-series, Phillips Curve, and term structure) together with the raw SPF survey. The weights are computed recursively through the sample.

Footnotes

We thank Jean Boivin for kindly providing data. Andrew Ang acknowledges support from the National Science Foundation. We have benefitted from the comments of Todd Clark, Dean Croushore, Bob Hodrick, Jonas Fisher, Robin Lumsdaine, Michael McCracken, Antonio Moreno, Serena Ng, and Tom Stark, and seminar participants at Columbia University and Goldman Sachs Asset Management. We especially thank the editor, Charles Plosser, and an anonymous referee for excellent comments. The opinions expressed in this paper do not necessarily reflect those of the Federal Reserve Board or the Federal Reserve system. Return to Text

Columbia Business School, 805 Uris Hall, 3022 Broadway, New York, NY 10027; ph: (212) 854-9154; fax: (212) 662-8474; email: [email protected]; WWW: http://www.columbia.edu/~aa610 Return to Text

Columbia Business School, 802 Uris Hall, 3022 Broadway, New York, NY 10027; ph: (212) 854-9156; fax: (212) 662-8474; email: [email protected]; WWW: http://www.gsb.columbia.edu/faculty/gbekaert Return to Text

Federal Reserve Board of Governors, Division of Monetary Affairs, Washington, DC 20551; ph: (202) 736-5619; fax: (202) 452-2301; email: [email protected]; WWW: www.federalreserve.gov/research/staff/weiminx.htm Return to Text

¹ To achieve stationarity of the underlying individual macro series, various transformations are employed by Bernanke, Boivin and Eliasz (2005). In particular, many series are first differenced at a monthly frequency. Better forecasting results might be potentially obtained by taking a long 12-month difference to forecast annual inflation (see comments by, among others, Plosser and Schwert, 1978), or pre-screening the variables to be used in the construction of the composite factor (see Boivin and Ng, 2006). We do not consider these adjustments and use the original Bernanke-Boivin-Eliasz series. Return to Text

² We obtain data for the Livingston survey and SPF data from the Philadelphia Fed website (http://www.phil.frb.org/econ/liv and http://www.phil.frb.org/econ/spf, respectively). We take the Michigan survey data from the St. Louis Federal Reserve FRED database (http://research.stlouisfed.org/fred2/series/MICH/). Median Michigan survey data is also available from the University of Michigan's website (http://www.sca.isr.umich.edu/main.php. However, there are small discrepancies between the two sources before September 1996. We choose to use data from FRED because it is consistent with the values reported in Curtin (1996). Return to Text

³ In the data, the correlation between log CPI changes, $\log(P_{t+4}/P_{t})$ and simple inflation, $P_{t+4}/P_{t}-1$ is 1.000 for all four measures of inflation across our full sample period. The correlation between end-of-quarter log CPI changes and quarterly average CPI changes is above 0.994. The differences in log CPI changes, simple inflation, and changes in quarterly average CPI are very small, and an order of magnitude smaller than the forecast RMSEs. As an illustration, for PUNEW, the means of $\log(P_{t+4}/P_{t})$, $P_{t+4}/P_{t}-1$, and changes in quarterly average CPI-U are 3.83%, 3.82%, and 3.86%, respectively, while the volatilities are 2.87%, 2.86%, and 2.91%, respectively. Return to Text

⁴ We also examined bias adjustments using the change in annual inflation, using

$$\pi_{t+4,4} - \pi_{t,4} = \alpha_1 + \beta_1(f_t^S - \pi_{t,4}) + \varepsilon_{t+4,4}$$

in place of equation (15) and

$$\pi_{t+4,4} - \pi_{t,4} = \alpha_1 + \alpha_2 D_t + \beta_1 (f_t^S - \pi_{t,4}) + \beta_2 D_t (f_t^S - \pi_{t,4}) + \varepsilon_{t+4,4}$$

in place of equation (16). Like the bias adjustments in equations (15) and (16), these bias adjustments also do not outperform the raw survey forecasts and generally perform worse than the bias adjustments using inflation levels. Return to Text

⁵ While the RMSEs for the Livingston survey represent a different sample than those of all other models and surveys, we also produced forecasts for a common semi-annual sample. The results are robust and we do not further comment on them. Return to Text

⁶ See the literature reviews by, among others, Clemen (1989), Diebold and Lopez (1996), and more recently Timmermann (2006). Return to Text

⁷ Diebold (1989) shows that when the target is persistent, as in the case of inflation, the forecast error from the combination regression will typically be serially correlated and hence predictable, unless the constraint that the weights sum to one is imposed. Return to Text

⁸ The estimated ARMA models contain large autoregressive roots with negative MA roots. As Ng and Perron (2001) comment, the negative MA components lead unit root tests to over-reject the null of non-stationarity. Return to Text

⁹ In contrast to the superior performance of surveys relative to models for forecasting inflation, Campbell (2004) finds that for forecasting GDP post-1985, surveys perform worse relative to a simple AR(1). However, Campbell shows that for forecasting GDP, surveys outperform an AR(1) benchmark prior to 1985. Return to Text

¹⁰ We also ask the question whether ex-post, a particular combination of models would have performed better than individual forecasts. This ex-post analysis cannot be used for actual forecasting, but indicates which models would have been most successful forecasting inflation out-of-sample ex-post. For the ex-post combinations, we find that the improvement generated by the combined forecasts is also relatively minor, even for the unit-weight prior model, which uses forward-looking information to find the best performing model over the whole sample. These results are available upon request. Return to Text

¹¹ We also ran model comparison regressions as in equation (17), but with inflation changes on the left hand side, and keeping the stationary ARMA(1,1) model as the benchmark model. These results are available upon request. We find that while generally the models specified in differences do not fare any better than the models specified in levels in terms of beating the RMSE of a stationary ARMA(1,1), there are more I(1) models with significant $(1-\lambda)$ coefficients using Hansen-Hodrick (1980) standard errors. The largest increase occurs for PUXX inflation. Like the model comparisons for forecasting inflation levels, surveys consistently provide significant improvement in forecasting CPI inflation changes above an ARMA(1,1) model on levels, especially for the post-1985 sample period. Return to Text

¹² West and McCracken (1998) derive similar forms for $\Omega_{ff}$ under the cases of rolling and fixed out-of-sample forecasts. Return to Text

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