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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 963, January 2009 --- Screen Reader Version*

The Taylor Rule and Interval Forecast for Exchange Rates

Jian Wang and Jason J. Wu^*

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

This paper attacks the Meese-Rogoff (exchange rate disconnect) puzzle from a different perspective: out-of-sample interval forecasting. Most studies in the literature focus on point forecasts. In this paper, we apply Robust Semi-parametric (RS) interval forecasting to a group of Taylor rule models. Forecast intervals for twelve OECD exchange rates are generated and modified tests of Giacomini and White (2006) are conducted to compare the performance of Taylor rule models and the random walk. Our contribution is twofold. First, we find that in general, Taylor rule models generate tighter forecast intervals than the random walk, given that their intervals cover out-of-sample exchange rate realizations equally well. This result is more pronounced at longer horizons. Our results suggest a connection between exchange rates and economic fundamentals: economic variables contain information useful in forecasting the distributions of exchange rates. The benchmark Taylor rule model is also found to perform better than the monetary and PPP models. Second, the inference framework proposed in this paper for forecast-interval evaluation, can be applied in a broader context, such as inflation forecasting, not just to the models and interval forecasting methods used in this paper.

Keywords: Exchange rate disconnect puzzle, exchange rate forecast, interval forecasting

JEL classification: F31, C14, C53

1.  Introduction

Recent studies explore the role of monetary policy rules, such as Taylor rules, in exchange rate determination. They find empirical support in these models for the linkage between exchange rates and economic fundamentals. Our paper extends this literature from a different perspective: interval forecasting. We find that the Taylor rule models can outperform the random walk, especially at long horizons, in forecasting twelve OECD exchange rates based on relevant out-of-sample interval forecasting criteria. The benchmark Taylor rule model is also found to perform relatively better than the standard monetary model and the purchasing power parity (PPP) model.

In a seminal paper, Meese and Rogoff (1983) find that economic fundamentals - such as the money supply, trade balance and national income - are of little use in forecasting exchange rates. They show that existing models cannot forecast exchange rates better than the random walk in terms of out-of-sample forecasting accuracy. This finding suggests that exchange rates may be determined by something purely random rather than economic fundamentals. Meese and Rogoff's (1983) finding has been named the Meese-Rogoff puzzle in the literature.

In defending fundamental-based exchange rate models, various combinations of economic variables and econometric methods have been used in attempts to overturn Meese and Rogoff's finding. For instance, Mark (1995) finds greater exchange rate predictability at longer horizons.¹Groen (2000) and Mark and Sul (2001) detect exchange rate predictability by using panel data. Kilian and Taylor (2003) find that exchange rates can be predicted from economic models at horizons of 2 to 3 years after taking into account the possibility of nonlinear exchange rate dynamics. Faust, Rogers, and Wright (2003) find that the economic models consistently perform better using real-time data than revised data, though they do not perform better than the random walk.

Recently, there is a growing strand of literature that uses Taylor rules to model exchange rate determination. Engel and West (2005) derive the exchange rate as present-value asset price from a Taylor rule model. They also find a positive correlation between the model-based exchange rate and the actual real exchange rate between the US dollar and the Deutschmark (Engel and West, 2006). Mark (2007) examines the role of Taylor-rule fundamentals for exchange rate determination in a model with learning. In his model, agents use least-square learning rules to acquire information about the numerical values of the model's coefficients. He finds that the model is able to capture six major swings of the real Deutschemark-Dollar exchange rate from 1973 to 2005. Molodtsova and Papell (2008) find significant short-term out-of-sample predictability of exchange rates with Taylor-rule fundamentals for 11 out of 12 currencies vis-á-vis the U.S. dollar over the post-Bretton Woods period. Molodtsova, Nikolsko-Rzhevskyy, and Papell (2008) find evidence of out-of-sample predictability for the dollar/mark nominal exchange rate with forecasts based on Taylor rule fundamentals using real-time data, but not revised data.

Our paper joins the above literature of Taylor-rule exchange rate models. However, we addresses the Meese and Rogoff puzzle from a different perspective: interval forecasting. A forecast interval captures not only the expected future value of the exchange rate, but a range in which the exchange rate may lie with a certain probability, given a set of predictors available at the time of forecast. Our contribution to the literature is twofold. First, we find that for twelve OECD exchange rates, the Taylor rule models in general generate tighter forecast intervals than the random walk, given that their intervals cover the realized exchange rates (statistically) equally well. This finding suggests an intuitive connection between exchange rates and economic fundamentals beyond point forecasting: the use of economic variables as predictors help narrow down the range in which future exchange rates may lie, compared to random walk forecast intervals which are essentially based on unconditional distributions of exchange rates. Second, we propose an inference framework for cross-model comparison of out-of-sample forecast intervals. The proposed framework can be used for forecast-interval evaluation in a broader context, not just for the models and methods used in this paper. For instance, the framework can also be used to evaluate out-of-sample inflation forecasting.

As we will discuss later, we in fact derive forecast intervals from estimates of the distribution of changes in the exchange rate. Hence, in principle, evaluations across models can be done based on distributions instead of forecast intervals. However, focusing on interval forecasting performance allows us to compare models in two dimensions that are more relevant to practitioners: empirical coverage and length.

While the literature on interval forecasting for exchange rates is sparse, several authors have studied out-of-sample exchange rate density forecasts, from which interval forecasts can be derived. Diebold, Hahn and Tay (1999) use the RiskMetrics model of JP Morgan (1996) to compute half-hour-ahead density forecasts for Deutschmark/Dollar and Yen/Dollar returns. Christoffersen and Mazzotta (2005) provide option-implied one-day-ahead density and interval forecasts for four major exchange rates. Boero and Marrocu (2004) obtain one-day-ahead density forecasts for the Euro nominal effective exchange rate using the self-exciting threshold autoregressive (SETAR) models. Sarno and Valente (2005) evaluate the exchange rate density forecasting performance of the Markov-switching vector equilibrium correction model that is developed by Clarida, Sarno, Taylor and Valente (2003). They find that information from the term structure of forward premia help the model to outperform the random walk in forecasting the out-of-sample densities of the spot exchange rate. More recently, Hong, Li and Zhao (2007) construct half-hour-ahead density forecasts for Euro/Dollar and Yen/Dollar exchange rates using a comprehensive set of univariate time series models that capture fat tails, time-varying volatility and regime switches.

There are several common features across the studies listed above, which make them different from our paper. First, the focus of the above studies is not to make connections between the exchange rate and economic fundamentals. These studies use high frequency data, which are not available for most conventional economic fundamentals. For instance, Diebold, Hahn and Tay (1999) and Hong, Li and Zhao (2007) use intra-day data. With the exception of Sarno and Valente (2005), all the studies focus only on univariate time series models. Second, these studies do not consider multi-horizon-ahead forecasts, perhaps due to the fact that their models are often highly nonlinear. Iterating nonlinear density models multiple horizons ahead is analytically difficult, if not infeasible. Lastly, the above studies assume that the densities are analytically defined for a given model. The semi-parametric method used in this paper does not impose such restrictions.

Our choice of semi-parametric method is motivated by the difficulty of using macroeconomic models in exchange rate interval forecasting: these models typically do not describe the future distributions of exchange rates. For instance, the Taylor rule models considered in this paper do not describe any features of the data beyond the conditional means of future exchange rates. We address this difficulty by applying Robust Semi-parametric forecast intervals (from hereon RS forecast intervals) of Wu (2007).² This method is useful since it does not require the model be correctly specified, or contain parametric assumptions about the future distribution of exchange rates.

We apply RS forecast intervals to a set of Taylor rule models that differ in terms of the assumptions on policy and interest rate smoothing rules. Following Molodtsova and Papell (2008), we include twelve OECD exchange rates (relative to the US dollar) over the post-Bretton Woods period in our dataset. For these twelve exchange rates, the out-of-sample RS forecast intervals at different forecast horizons are generated from the Taylor rule models and then compared with those of the random walk. The empirical coverages and lengths of forecast intervals are used as the evaluation criteria. Our empirical coverage and length tests are modified from Giacomini and White's (2006) predictive accuracy tests in the case of rolling but fixed-size estimation samples.

For a given nominal coverage (probability), the empirical coverage of forecast intervals derived from a forecasting model is the probability that the out-of-sample realizations (exchange rates) lie in the intervals. The length of the intervals is a measure of its tightness: the distance between its upper and lower bound. In general, the empirical coverage is not the same as its nominal coverage. Significantly missing the nominal coverage indicates poor quality of the model and intervals. One certainly wants the forecast intervals to contain out-of-sample realizations as close as possible to the probability they target. Most evaluation methods in the literature focus on comparing empirical coverages across models, following the seminal work of Christoffersen (1998). Following this literature, we first test whether forecast intervals of the Taylor rule models and the random walk have equally accurate empirical coverages. The model with more accurate coverages is considered the better model. In the cases where equal coverage accuracy cannot be rejected, we further test whether the lengths of forecast intervals are the same. The model with tighter forecast intervals provides more useful information about future values of the data, and hence is considered as a more useful forecasting model.

It is also important to establish what this paper is not attempting. First, the inference procedure does not carry the purpose of finding the correct model specification. Rather, inference is on how useful models are in generating forecast intervals, measured in terms of empirical coverages and lengths. Second, this paper does not consider the possibility that there might be alternatives to RS forecast intervals for the exchange rate models we consider. Some models might perform better if parametric distribution assumptions (e.g. the forecast errors are conditionally heteroskedastic and $t-$distributed) or other assumptions (e.g. the forecast errors are independent of the predictors) are added. One could presumably estimate the forecast intervals differently based on the same models, and then compare those with the RS forecast intervals, but this is out of the scope of this paper. As we described, we choose RS method for its robustness and flexibility achieved by the semi-parametric approach.

Our benchmark Taylor rule model is from Engel and West (2005) and Engel, Wang, and Wu (2008). For the purpose of comparison, several alternative Taylor rule models are also considered. These setups have been studied by Molodtsova and Papell (2008) and Engel, Mark, and West (2007). In general, we find that the Taylor rule models perform better than the random walk model, especially at long horizons: the models either have more accurate empirical coverages than the random walk, or in cases of equal coverage accuracy, the models have tighter forecast intervals than the random walk. The evidence of exchange rate predictability is much weaker in coverage tests than in length tests. In most cases, the Taylor rule models and the random walk have statistically equally accurate empirical coverages. So, under the conventional coverage test, the random walk model and the Taylor rule models perform equally well. However, the results of length tests suggest that Taylor rule fundamentals are useful in generating tighter forecasts intervals without losing accuracy in empirical coverages.

We also consider two other popular models in the literature: the monetary model and the model of purchasing power parity (PPP). Based on the same criteria, both models are found to perform better than the random walk in interval forecasting. As with the Taylor rule models, most evidence of exchange rate predictability comes from the length test: economic models have tighter forecast intervals than the random walk given statistically equivalent coverage accuracy. The monetary model performs slightly worse than the benchmark Taylor rule model and the PPP model. The benchmark Taylor rule model performs better than the PPP model at short horizons and equally well at long horizons.

Our findings suggest that exchange rate movements are linked to economic fundamentals. However, we acknowledge that the Meese-Rogoff puzzle remains difficult to understand. Although Taylor rule models offer statistically significant length reductions over the random walk, the reduction of length is quantitatively small. Forecasting exchange rates remains a difficult task in practice. There are some impressive advances in the literature, but most empirical findings remain fragile. As mentioned in Cheung, Chinn, and Pascual (2005), forecasts from economic fundamentals may work well for some currencies during certain sample periods but not for other currencies or sample periods. Engel, Mark, and West (2007) recently show that a relatively robust finding is that exchange rates are more predictable at longer horizons, especially when using panel data. We find greater predictability at longer horizons in our exercise. It would be of interest to investigate connections between our findings and theirs.

Several recent studies have attacked the puzzle from a different angle: there are reasons that economic fundamentals cannot forecast the exchange rate, even if the exchange rate is determined by these fundamentals. Engel and West (2005) show that existing exchange rate models can be written in a present-value asset-pricing format. In these models, exchange rates are determined not only by current fundamentals but also by expectations of what the fundamentals will be in the future. When the discount factor is large (close to one), current fundamentals receive very little weight in determining the exchange rate. Not surprisingly, the fundamentals are not very useful in forecasting. Nason and Rogers (2008) generalize the Engel-West theorem to a class of open-economy dynamic stochastic general equilibrium (DSGE) models. Other factors such as parameter instability and mis-specification (for instance, Rossi 2005) may also play important roles in understanding the puzzle. It is interesting to investigate conditions under which we can reconcile our findings with these studies.

The remainder of this paper is organized as follows. Section two describes the forecasting models we use, as well as the data. In section three, we illustrate how the RS forecast intervals are constructed from a given model. We also propose loss criteria to evaluate the quality of the forecast intervals, and test statistics that are based on Giacomini and White (2006). Section four presents results of out-of-sample forecast evaluation. Finally, section five contains concluding remarks.

2.  Models and Data

Seven models are considered in this paper. Let $m=1,2,...,7$ be the index of these models and the first model be the benchmark model. A general setup of the models takes the form of

$$s_{t+h}-s_{t}=\alpha_{m,h}+\beta^{\prime}_{m,h}\mathbf{X}_{m,t}+\varepsilon _{m,t+h},$$ (1)

where $s_{t+h}-s_{t}$ is $h$-period changes of the (log) exchange rate, and $\mathbf{X}_{m,t}$ contains economic variables that are used in model $m$. Following the literature of long-horizon regressions, both short- and long-horizon forecasts are considered. Models differ in economic variables that are included in matrix $\mathbf{X}_{m,t}$. In the benchmark model,

$$\mathbf{X}_{1,t}\equiv\left[ \begin{array}[c]{ccc} \pi_{t}-\pi_{t}^{\ast} & y_{t}^{gap}-y_{t}^{gap\ast} & q_{t}\\ & & \end{array} \right] ,$$

where $\pi_{t}$ ( $\pi_{t}^{\ast}$) is the inflation rate, and $y_{t}^{gap}$ ( $y_{t}^{gap\ast}$) is output gap in the home (foreign) country. The real exchange rate $q_{t}$ is defined as $q_{t}\equiv s_{t}+p_{t}^{\ast}-p_{t}$, where $p_{t}$ ( $p_{t}^{\ast}$) is the (log) consumer price index in the home (foreign) country. This setup is motivated by the Taylor rule model in Engel and West (2005) and Engel, Wang, and Wu (2008). The next subsection describes this benchmark Taylor rule model in detail.

We also consider the following models that have been studied in the literature:

$$\mathbf{X}_{2,t}\equiv\left[ \begin{array}[c]{cc} \pi_{t}-\pi_{t}^{\ast} & y_{t}^{gap}-y_{t}^{gap\ast}\ & \end{array}\right]$$

$$\mathbf{X}_{3,t}\equiv\left[ \begin{array}[c]{ccc} \pi_{t}-\pi_{t}^{\ast} & y_{t}^{gap}-y_{t}^{gap\ast} & i_{t-1}-i_{t-1}^{\ast }\ & & \end{array}\right] i_{t} i_{t}^{\ast}$$

$$\mathbf{X}_{4,t}\equiv\left[ \begin{array}[c]{cccc} \pi_{t}-\pi_{t}^{\ast} & y_{t}^{gap}-y_{t}^{gap\ast} & q_{t} & i_{t-1} -i_{t-1}^{\ast}\ & & & \end{array}\right]$$

$$\mathbf{X}_{5,t}\equiv q_{t}$$

$$\mathbf{X}_{6,t}\equiv\left[ \begin{array}[c]{c} s_{t}-\left[ (m_{t}-m_{t}^{\ast})-(y_{t}-y_{t}^{\ast})\right] \ \end{array}\right] m_{t} m_{t}^{\ast} y_{t} y_{t}^{\ast}$$

$$\mathbf{X}_{7,t}\equiv0$$

Models 2-4 are the Taylor rule models studied in Molodtsova and Papell (2008). Model 2 can be considered as the constrained benchmark model in which PPP always holds. Molodtsova and Papell (2008) include interest rate lags in models 3 and 4 to take into account potential interest rate smoothing rules of the central bank. Model 5 is the purchasing power parity (PPP) model and model 6 is the monetary model. Both models have been widely used in the literature. See Molodtsova and Papell (2008) for the PPP model and Mark (1995) for the monetary model. Model 7 is the driftless random walk model ( $\alpha _{7,h}\equiv0$).³ Given a date $\tau$ and horizon $h$, the objective is to estimate the forecast distribution of $s_{\tau+h}-s_{\tau}$ conditional on $\mathbf{X}_{m,\tau}$, and subsequently build forecast intervals from the estimated forecast distribution. Before moving to the econometric method, we first describe the Taylor rule model that motivates the setup of our benchmark model.

2.1  Benchmark Taylor Rule Model

Our benchmark model is the Taylor rule model that is derived in Engel and West (2005) and Engel, Wang, and Wu (2008). Following Molodtsova and Papell (2008), we focus on models that depend only on current levels of inflation and output gap.⁴ The Taylor rule in the home country takes the form of

$$\bar{i}_{t}=\bar{i}+\delta_{\pi}(\pi_{t}-\bar{\pi})+\delta_{y}y_{t} ^{gap}+u_{t},$$ (2)

where $\bar{i}_{t}$ is the central bank's target for short-term interest rate at time $t$, $\bar{i}$ is the equilibrium long-run rate, $\pi_{t}$ is the inflation rate, $\bar{\pi}$ is the target inflation rate, and $y_{t}^{gap}$ is output gap. The foreign country is assumed to follow a symmetric Taylor rule. In addition, we follow Engel and West (2005) to assume that the foreign country targets the exchange rate in its Taylor rule:

$$\bar{i}_{t}^{\ast}=\bar{i}+\delta_{\pi}(\pi_{t}^{\ast}-\bar{\pi})+\delta _{y}y_{t}^{gap\ast}+\delta_{s}(s_{t}-\bar{s}_{t})+u_{t}^{\ast} ,$$ (3)

where $\bar{s}_{t}$ is the targeted exchange rate. Assume that the foreign country targets the PPP level of the exchange rate: $\bar{s}_{t}=p_{t} -p_{t}^{\ast}$, where $p_{t}$ and $p_{t}^{\ast}$ are logarithms of the home and foreign aggregate prices. In equation (3), we assume that the policy parameters take the same values in the home and foreign countries. To simplify our presentation, we assume that the home and foreign countries have the same long-run inflation and interest rates. Such restrictions have been relaxed in our econometric model after we include a constant term in estimations.

We do not consider interest rate smoothing in our benchmark model. That is, the actual interest rate ($i_{t}$) is identical to the target rate in the benchmark model:

$$i_{t}=\bar{i}_{t}.$$ (4)

Molodtsova and Papell (2008) consider the following interest rate smoothing rule:

$$i_{t}=(1-\rho)\bar{i}_{t}+\rho i_{t-1}+\nu_{t},$$ (5)

where $\rho$ is the interest rate smoothing parameter. We include these setups in models 3 and 4. Note that our estimation methods do not require the monetary policy shock $u_{t}$ and the interest rate smoothing shock $\nu_{t}$ to satisfy any assumptions, aside from smoothness of their distributions when conditioned on predictors.

Substituting the difference of equations (2) and (3) to Uncovered Interest-rate Parity (UIP), we have

$$s_{t}=E_{t}\left\{ (1-b)\sum_{j=0}^{\infty }b^{j}(p_{t+j}-p^{*}_{t+j})-b\sum_{j=0}^{\infty}b^{j}\left[ \delta_{y} (y^{gap}_{t+j}-y_{t+j}^{gap\ast})+\delta_{\pi}(\pi_{t+j}-\pi^{*}_{t+j})\right] \right\} ,$$ (6)

where the discount factor $b=\frac{1}{1+\delta_{s}}$. Under some conditions, the present value asset pricing format in equation (6) can be written into an error-correction form:⁵

$$s_{t+h}-s_{t}=\alpha_{h}+\beta _{h}z_{t}+\varepsilon_{t+h},$$ (7)

where the deviation of the exchange rate from its equilibrium level is defined as:

$$z_{t}=s_{t}-p_{t}+p_{t}^{\ast}+\frac{b}{1-b}\left[ \delta_{y}(y_{t}^{gap}-y_{t}^{gap\ast})+\delta_{\pi}(\pi_{t}-\pi_{t}^{\ast })\right] .$$ (8)

We use equation (7) as our benchmark setup in calculating h-horizon-ahead out-of-sample forecasting intervals. According to equation (8), the matrix $\mathbf{X} _{1,t}$ in equation (1) includes economic variables $q_{t}\equiv s_{t}+p_{t}^{\ast}-p_{t}$, $y_{t}^{gap}-y_{t}^{gap\ast}$, and $\pi_{t}-\pi_{t}^{\ast}$.

2.2  Data

The forecasting models and the corresponding forecast intervals are estimated using monthly data for twelve OECD countries. The United States is treated as the foreign country in all cases. For each country we synchronize the beginning and end dates of the data across all models estimated. The twelve countries and periods considered are: Australia (73:03-06:6), Canada (75:01-06:6), Denmark (73:03-06:6), France (77:12-98:12), Germany (73:03-98:12), Italy (74:12-98:12), Japan (73:03-06:6), Netherlands (73:03-98:12), Portugal (83:01-98:12), Sweden (73:03-04:11), Switzerland (75:09-06:6), and the United Kingdom (73:03-06:4).

The data is taken from Molodtsova and Papell (2008).⁶ With the exception of interest rates, the data is transformed by taking natural logs and then multiplied by 100. The nominal exchange rates are end-of-month rates taken from the Federal Reserve Bank of St. Louis database. Output data $y_{t}$ are proxied by Industrial Production (IP) from the International Financial Statistics (IFS) database. IP data for Australia and Switzerland are only available at quarterly frequency, and hence are transformed from quarterly to monthly observations using the quadratic-match average option in Eviews 4.0 by Molodtsova and Papell (2008). Following Engel and West (2006), the output gap $y_{t}^{gap}$ is calculated by quadratically de-trending the industrial production for each country.

Prices data $p_{t}$ are proxied by Consumer Price Index (CPI) from the IFS database. Again, CPI for Australia is only available at quarterly frequency and quadratic-match average is used to impute monthly observations. Inflation rates are calculated by taking the first differences of the logs of CPIs. Money market rate from IFS (or "call money rate") is used as a measure of the short-term interest rate set by the central bank. Finally, M1 is used to measure the money supply for most countries. M0 for the UK, and M2 for Italy and Netherlands is used due to the unavailability of M1 data.

3.  Econometric Method

For a given model $m$, the objective is to estimate from equation (1) the distribution of $s_{\tau +h}-s_{\tau}$ conditional on data $\mathbf{X}_{m,\tau}$ that is observed up to time $\tau$. This is the $h$-horizon-ahead forecast distribution of the exchange rate, from which the corresponding forecast interval can be derived. For a given $\alpha$, the forecast interval of coverage $\alpha \in(0,1)$ is an interval in which $s_{\tau+h}-s_{\tau}$ is supposed to lie with a probability of $\alpha$.

Models $m=1,...,7$ in equation (1) provide only point forecasts of $s_{\tau+h}-s_{\tau}$. In order to construct forecast intervals for a given model, we apply robust semi-parametric (RS) forecast intervals to all models. The nominal $\alpha$-coverage forecast interval of $s_{\tau +h}-s_{\tau}$ conditional on $\mathbf{X}_{m,\tau}$ can be obtained by the following three-step procedure:

Step 1.

Estimate model $m$ by OLS and obtain residuals $\widehat{\varepsilon}_{m,t+h}\equiv s_{t+h}-s_{t}-\widehat{\alpha} _{m,h}+\widehat{\beta}^{^{\prime}}_{m,h}\mathbf{X}_{m,t}$ , for $t=1,...,\tau -h$.

Step 2.

For a range of values of $\varepsilon$ (sorted residuals $\{\widehat{\varepsilon}_{m,t+h}\}_{t=1}^{\tau-h}$), estimate the conditional distribution of $\varepsilon_{m,\tau+h}\vert\mathbf{X}_{m,\tau}$ by

$$\widehat{P}(\varepsilon_{m,\tau+h}\le\varepsilon\vert\mathbf{X}_{m,\tau} )\equiv\frac{\sum_{t=1}^{\tau-h}1(\widehat{\varepsilon}_{m,t+h}\le\varepsilon) \mathbf{K}_{b}(\mathbf{X}_{m,t}-\mathbf{X}_{m,\tau})}{\sum_{t=1}^{\tau -h}\mathbf{K}_{b}(\mathbf{X}_{m,t}-\mathbf{X}_{m,\tau})},$$ (9)

where $\mathbf{K}_{b}(\mathbf{X}_{m,t}-\mathbf{X}_{m,\tau})\equiv b^{-d}\mathbf{K}((\mathbf{X}_{m,t}-\mathbf{X}_{m,\tau})/b)$ , $\mathbf{K} (\cdot)$ is a multivariate Gaussian kernel with dimension same as that of $\mathbf{X}_{m,t}$, and $b$ is the smoothing parameter or bandwidth.⁷

Step 3.

Find the $(1-\alpha)/2$ and $(1+\alpha)/2$ quantiles of the estimated distribution, which are denoted by $\widehat{\varepsilon }_{m,h}^{(1-\alpha)/2}$ and $\widehat{\varepsilon}_{m,h}^{(1+\alpha)/2}$. The estimate of the $\alpha$-coverage forecast interval for $s_{\tau+h}-s_{\tau}$ conditional on $\mathbf{X}_{m,\tau}$ is

$$\widehat{I}^{\alpha}_{m,\tau+h}\equiv(\widehat{\beta}_{m,h}^{\prime} \mathbf{X}_{m,\tau}+\widehat{\varepsilon}_{m,h}^{(1-\alpha)/2}, \widehat {\beta}_{m,h}^{\prime}\mathbf{X}_{m,\tau}+\widehat{\varepsilon}_{m,h} ^{(1+\alpha)/2})$$ (10)

For each model $m$, the above method uses the forecast models in equation (1) to estimate the location of the forecast distribution, while nonparametric kernel distribution estimate is used to estimate the shape. As a result, the interval obtained from this method is semi-parametric. Wu (2007) shows that under some weak regularity conditions, this method always consistently estimates the forecast distribution,⁸ and hence the forecast intervals, of $s_{\tau+h}-s_{\tau}$ conditional on $\mathbf{X} _{m,\tau}$, regardless of the quality of model $m$. That is, the forecast intervals are robust. Stationarity of economic variables is one of those regularity conditions. In our models, exchange rate differences, interest rates and inflation rates are well-known to be stationary, while empirical tests for real exchange rates and output gaps generate mixed results. These results may be driven by the difficulty of distinguishing a stationary but persistent variable from a non-stationary one. In this paper, we take the stationarity of these variables as given.

Model seven is the random walk model. The estimator in equation (9) becomes the Empirical Distribution Function (EDF) of the exchange rate innovations. Under regularity conditions, equation (9) consistently estimates the unconditional distribution of $s_{\tau+h}-s_{\tau}$, and can be used to form forecast intervals for $s_{\tau+h}$. The forecast intervals of economic models and the random walk are compared. Our interest is to test whether RS forecast intervals based on economic models are more accurate than those based on the random walk model. We focus on the empirical coverage and the length of forecast intervals in our tests.

Following Christoffersen (1998) and related work, the first standard we use is the empirical coverage. The empirical coverage should be as close as possible to the nominal coverage ($\alpha$). Significantly missing the nominal coverage indicates the inadequacy of the model and predictors for the given sample size. For instance, if 90% forecast intervals calculated from a model contain only 50% of out-of-sample observations, the model can hardly be identified as useful for interval forecasting. This case is called under-coverage. In contrast, over-coverage implies that the intervals could be reduced in length (or improved in tightness), but the forecast interval method and model are unable to do that for the given sample size. An economic model is said to outperform the random walk if its empirical coverage is more accurate than that of the random walk.

On the other hand, the empirical coverage of an economic model may be equally accurate as that of the random walk model, but the economic model has tighter forecast intervals than the random walk. We argue that the lengths of forecast intervals signify the informativeness of the intervals given that these intervals have equally accurate empirical coverages. In this case, the economic model is also considered to outperform the random walk in forecasting exchange rates. The empirical coverage and length tests are conducted at both short and long horizons for six economic models relative to the random walk for each of the twelve OECD exchange rates.

We use tests that are applications of the unconditional predictive accuracy inference framework of Giacomini and White (2006). Unlike the tests of Diebold and Mariano (1995) and West (1996), our forecast evaluation tests do not focus the asymptotic features of the forecasts. Rather, in the spirit of Giacomini and White (2006), we are comparing the population features of forecasts generated by rolling samples of fixed sample size. This contrasts the traditional forecast evaluation methods in that although it uses asymptotic approximations to do the testing, the inference is not on the asymptotic properties of forecasts, but on their population finite sample properties. We acknowledge that the philosophy of this inference framework remains a point of contention, but it does tackle three important evaluation difficulties in this paper. First, it allows for evaluation of forecast intervals that are not parametrically derived. The density evaluation methods developed in well-known studies such as Diebold, Gunther, Tay (1998), Corradi and Swanson (2006a) and references within Corradi and Swanson (2006b) require that the forecast distributions be parametrically specified. Giacomini and White's (2006) method overcomes this challenge by allowing comparisons among parametric, semi-parametric and nonparametric forecasts. As a result, in the cases of semi-parametric and nonparametric forecasts, it also allows comparison of models with predictors of different dimensions, as evident in our exercise. Second, by comparing finite sample properties of RS forecast intervals derived from different models, we avoid rejecting models that are mis-specified,⁹ but are nonetheless good approximations useful for forecasting. Finally, we can individually (though not jointly) test whether the forecast intervals differ in terms of empirical coverages and lengths, for the given estimation sample, and not confined to focus only on empirical coverages or holistic properties of forecast distribution such as probability integral transform.

3.1  Test of Equal Empirical Coverages

Suppose the sample size available to the researcher is $T$ and all data are collected in a vector $\mathbf{W}_{t}$. Our inference procedure is based on a rolling estimation scheme, with the size of the rolling window fixed while $T\rightarrow\infty$. Let $T=R+N$ and $R$ be the size of the rolling window. For each horizon $h$ and model $m$, a sequence of $N(h)=N+1-h$ $\alpha$-coverage forecast intervals are generated using rolling data: $\{\mathbf{W} _{t}\}_{t=1}^{R}$ for forecast for date $R+h$, $\{\mathbf{W}_{t}\}_{t=2} ^{R+1}$ for forecast for date $R+h+1$, and so on, until forecast for date $T$ is generated using $\{\mathbf{W}_{t}\}_{t=N(h)}^{R+N(h)-1}$.

Under this fixed-sample-size rolling scheme, for each finite $h$ we have $N(h)$ observations to compare the empirical coverages and lengths across $m$ models ( $m=1, 2,...,7$). By fixing $R$, we allow the finite sample properties of the forecast intervals to be preserved as $T\rightarrow\infty$. Thus, the forecast intervals and the associated forecast losses are simply functions of a finite and fixed number of random variables. We are interested in approximating the population moments of these objects by taking $N(h)\rightarrow\infty$. A loose analogy would be finding the finite-sample properties of a certain parameter estimator when sample size is fixed at $R$, by a bootstrap with an arbitrarily large number of bootstrap replications.

We conduct individual tests for the empirical coverages and lengths. In each test, we define a corresponding forecast loss, propose a test statistic and derive its asymptotic distribution. As defined in equation (10), let $\widehat{I}^{\alpha}_{m,\tau+h}$ be the $h-$horizon ahead RS forecast interval of model $m$ with a nominal coverage of $\alpha$. For out-of-sample forecast evaluation, we require $\widehat{I}^{\alpha}_{m,\tau+h}$ to be constructed using data from $t=\tau-R+1$ to $t=\tau$. The coverage accuracy loss is defined as

$$CL_{m,h}^{\alpha}=\left[ P(Y_{\tau+h}\in\widehat{I}^{\alpha}_{m,\tau +h})-\alpha\right] ^{2}.$$ (11)

For economic models ($m=1,...,6$), the goal is to compare the coverage accuracy loss of RS forecast intervals of model $m$ with that of the random walk ($m=7$). The null and alternative hypotheses are:

$$H_{0} : \Delta CL_{m,h}^{\alpha}\equiv CL_{7,h}^{\alpha}-CL_{m,h}^{\alpha }=0$$

$$H_{A} : \Delta CL_{m,h}^{\alpha}\ne0.$$

Define the sample analog of the coverage accuracy loss in equation (11):

$$\widehat{CL}_{m,h}^{\alpha}=\left( N(h)^{-1}\sum_{\tau=R}^{T-h}1(Y_{\tau+h} \in\widehat{I}^{\alpha}_{m,\tau+h})-\alpha\right) ^{2},$$

where $1(Y_{\tau+h}\in\widehat{I}^{\alpha}_{m,\tau+h})$ is an index function that equals one when $Y_{\tau+h}\in\widehat{I}^{\alpha}_{m,\tau+h}$, and equals zero otherwise. Applying the asymptotic test of Giacomini and White (2006) to the sequence $\{1(Y_{\tau+h}\in\widehat{I}^{\alpha}_{m,\tau +h})\}_{\tau=R}^{T-h}$ and applying the Delta method, we can show that

$$\sqrt{N(h)}(\Delta\widehat{CL}_{m,h}^{\alpha}-\Delta CL_{m,h}^{\alpha })\overset{d}{\rightarrow}N(0,\Gamma_{m,h}^{^{\prime}}\Omega_{m,h}\Gamma _{m,h}),$$ (12)

where $\overset{d}{\rightarrow}$ denotes convergence in distribution, and $\Omega_{m,h}$ is the long-run covariance matrix between $1(Y_{\tau+h} \in\widehat{I}^{\alpha}_{m,\tau+h})$ and $1(Y_{\tau+h}\in\widehat{I}^{\alpha }_{7,\tau+h})$. The matrix $\Gamma_{m,h}$ is defined as:

$$\Gamma_{m,h}\equiv\left[ \begin{array}[c]{cc} 2\left( P\left( Y_{\tau+h}\in\widehat{I}^{\alpha}_{m,\tau+h}\right) -\alpha\right) & 2\left( P\left( Y_{\tau+h}\in\widehat{I}^{\alpha}_{7,\tau +h}\right) -\alpha\right) \end{array} \right] ^{\prime}.$$

$\Gamma_{m,h}$ can be estimated consistently by its sample analog $\widehat{\Gamma}_{m,h}$, while $\Omega_{m,h}$ can be estimated by some HAC estimator $\widehat{\Omega}_{m,h}$, such as Newey and West (1987).¹⁰The test statistic for coverage test is defined as:

$$Ct_{m,h}^{\alpha}\equiv\frac{\sqrt{N(h)}\Delta\widehat{CL}_{m,h}^{\alpha} }{\sqrt{\widehat{\Gamma}_{m,h}^{^{\prime}}\widehat{\Omega}_{m,h} \widehat{\Gamma}_{m,h}}}\overset{d}{\rightarrow}N(0,1)$$ (13)

3.2  Test of Equal Empirical Lengths

Define the length loss as:

$$LL^{\alpha}_{m,h}\equiv E\left[ leb\left( \widehat{I}^{\alpha}_{m,\tau +h}\right) \right] ,$$ (14)

where $leb(\cdot)$ is the Lesbesgue measure. To compare the length loss of RS forecast intervals of economic models $m=1, 2,...,6$ with that of the random walk ($m=7)$, the null and alternative hypotheses are:

$$H_{0} : \Delta LL_{m,h}^{\alpha}\equiv LL_{7,h}^{\alpha}-LL_{m,h}^{\alpha }=0$$

$$H_{A} : \Delta LL_{m,h}^{\alpha}\ne0.$$

The sample analog of the length loss for model $m$ is defined as:

$$\widehat{LL}_{m,h}^{\alpha}=N(h)^{-1}\sum_{\tau=R}^{T-h}leb(\widehat {I}^{\alpha}_{m,\tau+h}).$$

Directly applying the test of Giacomini and White (2006), we have

$$\sqrt{N(h)}(\Delta\widehat{LL}_{m,h}^{\alpha}-\Delta LL_{m,h}^{\alpha })\overset{d}{\rightarrow}N(0,\Sigma_{m,h}),$$ (15)

where $\Sigma_{m,h}$ is the long-run variance of $leb\left( \widehat {I}^{\alpha}_{7,\tau+h}\right) -leb\left( \widehat{I}^{\alpha}_{m,\tau +h}\right)$ . Let $\widehat{\Sigma}_{m,h}$ be the HAC estimator of $\Sigma_{m,h}$. The test statistic for empirical length is defined as:

$$Lt_{m,h}^{\alpha}\equiv\frac{\sqrt{N(h)}\Delta\widehat{LL}_{m,h}^{\alpha} }{\sqrt{\widehat{\Sigma}_{m,h}}}\overset{d}{\rightarrow}N(0,1).$$ (16)

3.3  Discussion

The coverage accuracy loss function is symmetric in our paper. In practice, an asymmetric loss function may be better when looking for an exchange rate forecast model to help make policy or business decisions. Under-coverage is arguably a more severe problem than over-coverage in practical situations. However, the focus of this paper is the disconnect between economic fundamentals and the exchange rate. Our goal is to investigate which model comes closer to the data: the random walk or fundamental-based models. It is not critical in this case whether coverage inaccuracy comes from the under- or over-coverage. We acknowledge that the use of symmetric coverage loss remains a caveat, especially since we are using the coverage accuracy test as a pre-test for the tests of length. Clearly, there is a tradeoff between the empirical coverage and the length of forecast intervals. Given the same center,¹¹ intervals with under-coverage have shorter lengths than intervals with over-coverage. In this case, the length test is in favor of models that systematically under-cover the targeted nominal coverage when compared to a model that systematically over-covers. This problem cannot be detected by the coverage accuracy test with symmetric loss function because over- and under-coverage are treated equally. However, our results in section 4 show that there is no evidence of systematic under-coverage for the economic models considered in this paper. For instance, in Table 1, one-month-ahead ($h=1$) forecast interval over-covers the nominal coverage (90%) for nine out of twelve exchange rates.¹² Note that under-coverage does not guarantee shorter intervals either in our paper, because forecast intervals of different models usually have different centers.¹³

As we have mentioned, comparisons across models can also be done at the distribution level. We choose interval forecasts for two reasons. First, interval forecasts have been widely used and reported by the practitioners. For instance, the Bank of England calculates forecast intervals of inflation in its inflation reports. Second, compared to evaluation metrics for density forecasts, the empirical coverage and length loss functions of interval forecasts, and the subsequent interpretations of test rejection/acceptance are more intuitive.

4.  Results

We apply RS forecast intervals for each model for a given nominal coverage of $\alpha=0.9$. There is no particular reason why we chose 0.9 as the nominal coverage. Some auxiliary results show that our qualitative findings do not depend on the choice of $\alpha$. Due to different sample sizes across countries, we choose different sizes for the rolling window ($R$) for different countries. Our rule is very simple: for countries with $T\ge300$, we choose $R=200$, otherwise we set $R=150$.¹⁴ Again, from our experience, tampering with $R$ does not change the qualitative results unless $R$ is chosen to be unusually big or small.

For time horizons $h=1,3,6,9,12$ and models $m=1,...,7$, we construct a sequence of $N(h)$ 90% forecast intervals $\{\widehat{I}^{0.9}_{m,\tau +h}\}_{\tau=R}^{T-h}$ for the $h$-horizon change of the exchange rate $s_{t+h}-s_{t}$. Then we compare economic models and the random walk by computing empirical coverages, lengths and test statistics $Ct_{m,h}^{0.9}$ and $Lt_{m,h}^{0.9}$ as described in section 3. We first report the results of our benchmark model. After that, results of alternative models are reported and discussed.

4.1  Results of Benchmark Model

Table 1 shows results of the benchmark Taylor rule model. For each time horizon $h$ and exchange rate, the first column (Cov.) reports the empirical coverage for the given nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals (the distance between upper and lower bounds). The length is multiplied by 100 and therefore expressed in terms of the percentage change of the exchange rate. For instance, the length of the one-month-ahead forecast interval for the Australian dollar is $7.114$. On average, the distance between the upper and lower bound of the one-month-ahead forecast interval for the Australian dollar is $7.114$% change of the Australian dollar against the US dollar. We use superscripts $a$, $b$, and $c$ to denote that the null hypothesis of equal empirical coverage accuracy/length is rejected in favor of the Taylor rule model at a confidence level of 10%, 5%, and 1% respectively. Superscripts $x$, $y$, and $z$ are used for rejections in favor of the random walk analogously.

We summarize our findings in three panels. In the first panel ((1) Coverage Test), the row of "Model Better" reports the number of exchange rates that the Taylor rule model has more accurate empirical coverages than the random walk. The row of "RW Better" reports the number of exchange rates for which the random walk outperforms the Taylor rule model under the same criterion. In the second panel ((2) Length Test Given Equal Coverage Accuracy), a better model is the one with tighter forecast intervals given equal coverage accuracy. In the last panel ((1)+(2)), a better model is the one with either more accurate coverages, or tighter forecast intervals given equal coverage accuracy.

For most exchange rates and time horizons, the Taylor rule model and the random walk model have statistically equally accurate empirical coverages. The null hypothesis of equal coverage accuracy is rejected in only nine out of sixty tests (one rejection at horizon 3, two rejections at horizons 6 and 9, and four at horizon 12). All nine rejections are in favor of the Taylor rule model. That is, the empirical coverage of the Taylor rule model is closer to the nominal coverage than those of the random walk. However, based on the number of rejections (9) in a total of sixty tests, there is no strong evidence that the Taylor rule model can generate more accurate empirical coverages than the random walk. The coverage tests at horizon twelve have more rejections in favor of the Taylor rule model than that at short horizons ($h=1,3$). However, this pattern in coverage tests does not exist in other models that will be discussed in next subsection.

In cases where the Taylor rule model and the random walk have equally accurate empirical coverages, the Taylor rule model generally has equal or significantly tighter forecast intervals than the random walk. In forty-one out of fifty-one cases, the null hypothesis of equally tight forecast intervals is rejected in favor of the Taylor rule model. In contrast, the null hypothesis is rejected in only two cases in favor of the random walk. The evidence of exchange rate predictability is more pronounced at longer horizons. At horizons nine and twelve ($h=9, 12$), for cases where empirical coverage accuracies between the random walk and the Taylor rule model are statistically equivalent, the Taylor rule model has significantly tighter forecast intervals than the random walk.

As for each individual exchange rate, the benchmark Taylor rule model works best for the Canadian dollar, the French Franc, the Deutschmark, and the Swedish Krona: for all time horizons, the model has tighter forecast intervals than the random walk, while their empirical coverages are statistically equally accurate. The Taylor rule model performs better than the random walk in most horizons for remaining exchange rates except the Japanese yen, for which the Taylor rule model outperforms the random walk only at long horizons.

4.2  Results of Alternative Models

Five alternative economic models are also compared with the random walk: three alternative Taylor rule models that are studied in Molodtsova and Papell (2008), the PPP model, and the monetary model. Tables 2-6 report results of these alternative models.

In general, results of coverage tests do not show strong evidence that economic models can generate more accurate coverages than the random walk at either short or long horizons. Though the benchmark Taylor rule model shows a sign of long-horizon predictability based on coverage accuracy tests, there is no clear evidence for such a pattern in any other models. However, after considering length tests, we find that economic models perform better than the random walk, especially at long horizons. The Taylor rule model four (Table 4) and the PPP model (Table 5) perform the best among alternative models. Results of these two models are very similar to that of the benchmark Taylor rule model. At horizon twelve, both models outperform the random walk for all twelve exchange rates under our out-of-sample forecast interval evaluation criteria. The performance of the Taylor rule model two (Table 2) and three (Table 3) is relatively less impressive than other models, but still for about half of exchange rates, the economic models outperform the random walk at several horizons in out-of-sample interval forecasts.

Comparing the benchmark Taylor rule model, the PPP model and the monetary model, the performance of the monetary model (Table 6) is slightly worse than the other two models at long horizons. Compared to the Taylor rule and PPP models, the monetary model outperforms the random walk for a smaller number of exchange rates at horizons 6, 9, and 12. The Taylor rule model and the monetary model perform relatively better than the PPP model at short horizons. Overall, the benchmark Taylor rule model seems to perform slightly better than the monetary and PPP models. Molodtsova and Papell (2008) find similar results in their point forecasts.

4.3  Discussion

After Mark (1995) first documents exchange rate predictability at long horizons, long-horizon exchange rate predictability has become a very active area in the literature. With panel data, Engel, Mark, and West (2007) recently show that the long-horizon predictability of the exchange rate is relatively robust in the exchange rate forecasting literature. We find similar results in our interval forecasts. The evidence of long-horizon predictability seems robust across different models and currencies when both empirical coverage and length tests are used. At horizon twelve, all economic models outperform the random walk for seven exchange rates: the Australia dollar, Canadian dollar, Italian Lira, Japanese yen, Portuguese escudo, Swedish krona, and the British pound in the sense that interval lengths of economic models are smaller than those of the random walk, given equivalent coverage accuracy. This is true only for the Italian Lira at horizon one. We also notice that there is no clear evidence of long-horizon predictability based on the tests of empirical coverage accuracy only.

Molodtsova and Papell (2008) find strong out-of-sample exchange rate predictability for Taylor rule models even at the short horizon. In our paper, the evidence for exchange rate predictability at short horizons is not very strong. This finding may be a result of some assumptions we have used to simplify our computation. For instance, an $\alpha$-coverage forecast interval will always be constructed using the $(1-\alpha)/2$ and $(1+\alpha)/2$ quantiles. Alternatively, we can choose quantiles that minimize the length of intervals, given the nominal coverage.¹⁵ We have also assumed symmetric Taylor rules. Relaxing these assumptions may help us find exchange rate predictability at short horizons. In addition, the development of more powerful testing methods may also be helpful. The evidence of exchange rate predictability in Molodtsova and Papell (2008) is partly driven by the testing method recently developed by Clark and West (2006, 2007). We acknowledge that whether or not short-horizon results can be improved remains an interesting question, but do not pursue this in the current paper. The purpose of this paper is to show the connection between the exchange rate and economic fundamentals from an interval forecasting perspective. Predictability either at short- or long-horizons will serve this purpose.

Though we find that economic fundamentals are helpful for forecasting exchange rates, we acknowledge that exchange rate forecasting in practice is still a difficult task. The forecast intervals from economic models are statistically tighter than those of the random walk, but they remain fairly wide. For instance, the distance between the upper and lower bound of three-month-ahead forecast intervals is usually a 20% change of the exchange rates. Figures 1-3 show forecast intervals generated by the benchmark Taylor rule model and the random walk for the British pound, the Deutschmark, and the Japanese yen at different horizons.¹⁶ To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been re-centered so that they have the same center as the forecast intervals of the Taylor rule model. In these figures, the Taylor rule model has tighter forecast intervals, especially at the horizon of 12 months, than the random walk. However, the difference is quantitatively small.

5.  Conclusion

There is a growing strand of literature that uses Taylor rules to model exchange rate movements. Our paper contributes to the literature by showing that Taylor rule fundamentals are useful in forecasting distribution of exchange rates. We apply Robust Semi-parametric forecast intervals of Wu (2007) to a group of Taylor models for twelve OECD exchange rates. The forecast intervals generated by the Taylor rule models are in general tighter than those of the random walk, given that these intervals cover the realized exchange rates equally well. The evidence of exchange rate predictability is more pronounced at longer horizons, a result that echoes previous long-horizon studies such as Mark (1995). The benchmark Taylor rule model is also found to perform better than the monetary and PPP models based on out-of-sample interval forecasts.

Though we find some empirical support for the connection between the exchange rate and economic fundamentals, we acknowledge that the detected connection is weak. The reductions of the lengths of forecast intervals are quantitatively small, though they are statistically significant. Forecasting exchange rates remains a difficult task in practice. Engel and West (2005) argue that as the discount factor gets closer to one, present value asset pricing models place greater weight on future fundamentals. Consequently, current fundamentals have very weak forecasting power and exchange rates appear to follow approximately a random walk. Under standard assumptions in Engel and West (2005), the Engel-West theorem does not imply that exchange rates are more predictable at longer horizons, or economic models can outperform the random walk in forecasting exchange rates based on out-of-sample interval forecasts. However, modifications to these assumptions may be able to reconcile the Engel-West explanation with empirical findings of exchange rate predictability. For instance, Engel, Wang, and Wu (2008) find that when there exist stationary but persistent unobservable fundamentals, for example risk premium, the Engel-West explanation predicts long-horizon exchange rate predictability in point forecasts, though the exchange rate still approximately follows a random walk at short horizons. It would also be of interest to study conditions under which our findings in interval forecasts can be reconciled with the Engel-West theorem.

We believe other issues, such as parameter instability (Rossi, 2005), nonlinearity (Kilian and Taylor, 2003), real time data (Faust, Rogers, and Wright, 2003, Molodtsova, Nikolsko-Rzhevskyy, and Papell, 2008), are all contributing to the Meese-Rogoff puzzle. Panel data are also found helpful in detecting exchange rate predictability, especially at long horizons. For instance, see Mark and Sul (2001) and Engel, Mark, and West (2007). It would be interesting to incorporate these studies into interval forecasting. We leave these extensions for future research.

Table 1: Panel 1 - Results of Benchmark Taylor Rule Model.

	Cov. h=1	Leng. h=1	Cov. h=3	Leng. h=3	Cov. h=6	Leng. h=6	Cov. h=9	Leng. h=9	Cov. h=12	Leng. h=12
Australian Dollar	0.895	7.114	0.888	14.283c	0.923	20.286c	0.927	24.468c	0.915	28.016c
Canadian Dollar	0.808	3.446c	0.789	6.351c	0.808	8.586c	0.817	10.321c	0.801	12.614c
Danish Kroner	0.920	8.668c	0.939	17.415c	0.949	26.087	0.969	30.776c	0.968	36.962c
French Franc	0.912	8.921c	0.920	17.674c	0.928c	26.007c	0.957	29.924c	0.934	36.883c
Deutschmark	0.927	8.851c	0.879	18.634c	0.894	27.923c	0.960a	33.734c	0.969	38.374c
Italian Lira	0.906	8.754c	0.875	18.305	0.910	26.788c	0.862	34.785c	0.890a	39.958c
Japanese Yen	0.915	9.633z	0.909	19.765	0.902	28.497c	0.932	33.793c	0.883	37.333c
Dutch Guilder	0.917	8.821	0.907	18.649c	0.933	27.649c	0.951a	31.117c	0.959a	40.737c
Portuguese Escudo	0.901	8.205z	0.913a	17.899	0.879c	22.431c	0.825	25.959c	0.883c	32.464c
Swedish Krona	0.844	7.448c	0.860	15.405c	0.874	23.930c	0.861	30.827c	0.834	37.432c
Swiss Franc	0.935	9.759c	0.946	20.036	0.982	27.682c	0.994	32.837c	0.956	38.728c
British Pound	0.919	8.429	0.923	16.570c	0.906	23.623c	0.884	27.849c	0.903c	30.814c

Note: �h denotes forecast horizons for monthly data. �For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. �Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 1: Panel 2 - Results of Benchmark Taylor Rule Model, Coverage Test

	h=1	h=3	h=6	h=9	h=12
Model Better	0	1	2	2	4
RW Better	0	0	0	0	0

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 1: Panel 3 - Results of Benchmark Taylor Rule Model, Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	7	8	9	10	8
RW Better	2	0	0	0	0

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 1: Panel 4 - Results of Benchmark Taylor Rule Model, Coverage Test and Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	7	9	11	12	12
RW Better	2	0	0	0	0

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 2: Panel 1 - Results of Benchmark Taylor Rule Model Two.

	Cov. h=1	Leng. h=1	Cov. h=3	Leng. h=3	Cov. h=6	Leng. h=6	Cov. h=9	Leng. h=9	Cov. h=12	Leng. h=12
Australian Dollar	0.884	7.146y	0.899	15.086c	0.918	20.902a	0.880	26.222c	0.856	30.677c
Canadian Dollar	0.825	3.442c	0.783	6.362c	0.820	8.815c	0.846	10.610c	0.813	13.216c
Danish Kroner	0.925	8.756c	0.934	17.791c	0.959	27.511z	0.963	32.956	0.947	40.387
French Franc	0.922	8.840c	0.920	18.740	0.949c	29.161c	0.936	34.994c	0.868	41.330c
Deutschmark	0.936	9.005	0.879	19.489	0.952	29.658	0.941a	38.006	0.980	44.355z
Italian Lira	0.920	9.095b	0.882	18.558	0.910	27.464c	0.908	37.325c	0.921	43.005c
Japanese Yen	0.915	9.565	0.914	19.752	0.912	29.618	0.937	36.834	0.942	44.455b
Dutch Guilder	0.908	8.645c	0.897	18.983c	0.971	29.391b	0.990	38.867z	0.980	46.650z
Portuguese Escudo	0.916	7.956	0.957	17.924z	0.909c	24.196c	0.889	28.533z	0.883a	35.338c
Swedish Krona	0.861	7.575	0.860	15.679c	0.851	24.916c	0.849	31.108c	0.823	40.458c
Swiss Franc	0.947	10.008	0.928	20.379y	0.976	29.578c	0.988	37.858	0.962	44.675z
British Pound	0.919	8.614z	0.933	17.302c	0.922	26.196c	0.937	31.371a	0.957	37.239c

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 2: Panel 2 - Results of Benchmark Taylor Rule Model Two, Coverage Test

	h=1	h=3	h=6	h=9	h=12
Model Better	0	0	2	1	1
RW Better	0	0	0	0	0

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 2: Panel 3 - Results of Benchmark Taylor Rule Model Two, Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	5	6	7	6	7
RW Better	2	2	1	2	3

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 2: Panel 4 - Results of Benchmark Taylor Rule Model Two, Coverage Test and Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	5	6	9	7	8
RW Better	2	2	1	2	3

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 3: Panel 1 - Results of Benchmark Taylor Rule Model. Three

	Cov. h=1	Leng. h=1	Cov. h=3	Leng. h=3	Cov. h=6	Leng. h=6	Cov. h=9	Leng. h=9	Cov. h=12	Leng. h=12
Australian Dollar	0.884	7.163y	0.893	14.925c	0.881	21.067	0.869	25.590c	0.835	28.714c
Canadian Dollar	0.831	3.453b	0.789	6.402c	0.808	8.764c	0.846	10.644c	0.795	12.592c
Danish Kroner	0.925	8.794b	0.924	17.831c	0.959	27.536z	0.963	33.251	0.942	40.036a
French Franc	0.941	8.880c	0.880	18.389c	0.876c	28.548b	0.915	35.443a	0.813	41.350c
Deutschmark	0.945	9.042	0.897	19.642z	0.914	29.677	0.901c	37.291a	0.878	44.520y
Italian Lira	0.906	8.831c	0.868	18.064c	0.902	27.430c	0.877	37.364c	0.803	41.499c
Japanese Yen	0.905	9.181c	0.873	18.910c	0.881	25.700c	0.927	31.259c	0.894	37.049c
Dutch Guilder	0.927	8.910	0.907	19.204a	0.942	29.637	0.951a	36.896c	0.959	46.321z
Portuguese Escudo	0.930	7.961	0.928	16.808c	0.909c	24.059b	0.873	27.868	0.917b	34.980c
Swedish Krona	0.861	7.375c	0.843	15.096c	0.886	24.770c	0.849	31.044c	0.817	38.468c
Swiss Franc	0.965	9.959	0.934	20.433y	0.957	29.418c	0.926c	37.130	0.911	43.546
British Pound	0.919	8.537	0.939	17.397b	0.927	25.809c	0.926	30.749c	0.968	36.825c

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 3: Panel 2 - Results of Benchmark Taylor Rule Model Three, Coverage Test

	h=1	h=3	h=6	h=9	h=12
Model Better	0	0	2	2	1
RW Better	0	0	0	0	0

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 3: Panel 3 - Results of Benchmark Taylor Rule Model Three, Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	4	5	5	7	8
RW Better	0	2	1	0	2

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 3: Panel 4 - Results of Benchmark Taylor Rule Model Three, Coverage Test and Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	4	5	7	8	9
RW Better	0	2	0	0	2

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 4: Panel 1 - Results of Benchmark Taylor Rule Model Four.

	Cov. h=1	Leng. h=1	Cov. h=3	Leng. h=3	Cov. h=6	Leng. h=6	Cov. h=9	Leng. h=9	Cov. h=12	Leng. h=12
Australian Dollar	0.895	7.119	0.888	14.409c	0.902	20.359c	0.911	24.165c	0.872	27.492c
Canadian Dollar	0.814	3.425c	0.777	6.332c	0.744	8.490c	0.769	10.177c	0.759	12.043c
Danish Kroner	0.920	8.703c	0.929	17.534c	0.964	26.025	0.984x	30.891c	0.963	36.443c
French Franc	0.931	9.065c	0.860	17.422c	0.938c	25.950c	0.883	30.016c	0.791	35.192c
Deutschmark	0.945	8.866c	0.888	18.839c	0.894	26.803c	0.911c	32.839c	0.929	38.699c
Italian Lira	0.891	8.663c	0.838	17.575c	0.865	26.307c	0.777	33.602c	0.756	38.890c
Japanese Yen	0.905	9.160c	0.863	18.708c	0.861	24.386c	0.869	28.730c	0.851	31.697c
Dutch Guilder	0.936	8.797	0.897	18.368c	0.914	26.700c	0.931c	29.974c	0.929c	37.481c
Portuguese Escudo	0.901	8.183y	0.884b	16.237b	0.909c	22.354c	0.905	25.896c	0.917	30.329c
Swedish Krona	0.861	7.382c	0.854	15.095c	0.869	23.340c	0.820	30.370c	0.805	36.487c
Swiss Franc	0.965	9.644c	0.940	19.782a	0.957	27.332c	0.975	31.004c	0.956	35.362c
British Pound	0.904	8.464	0.923	16.287c	0.854	23.394c	0.825	27.333c	0.855	29.796c

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 4: Panel 2 - Results of Benchmark Taylor Rule Model Four, Coverage Test

	h=1	h=3	h=6	h=9	h=12
Model Better	0	0	2	2	2
RW Better	0	0	0	1	0

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 4: Panel 3 - Results of Benchmark Taylor Rule Model Four, Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	8	1	9	9	10
RW Better	1	0	0	0	0

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 4: Panel 4 - Results of Benchmark Taylor Rule Model Four, Coverage Test and Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	8	11	11	11	12
RW Better	1	0	0	1	0

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 5: Panel 1 - Results of Purchasing Power Parity Model.

	Cov. h=1	Leng. h=1	Cov. h=3	Leng. h=3	Cov. h=6	Leng. h=6	Cov. h=9	Leng. h=9	Cov. h=12	Leng. h=12
Australian Dollar	0.895	7.114z	0.883	15.558	0.912	21.311	0.880	26.120c	0.856	30.316c
Canadian Dollar	0.808	3.513	0.811	6.929	0.814	9.669b	0.828	11.912a	0.789	15.303c
Danish Kroner	0.925	8.735b	0.929	18.253	0.938	25.891c	0.969	32.151b	0.952	38.019c
French Franc	0.922	8.918c	0.940	18.137c	0.979	26.944c	0.936	31.597c	0.824	37.655c
Deutschmark	0.936	9.079	0.935	18.797c	0.942	27.588c	1.000	33.585c	0.990	39.821c
Italian Lira	0.913	8.794c	0.875	18.600	0.887	26.619c	0.954	36.347c	0.953	42.926c
Japanese Yen	0.920	9.662z	0.899	19.903	0.912	28.691c	0.932	33.973c	0.899	38.568c
Dutch Guilder	0.936	8.830	0.935	18.904c	0.952	27.902c	1.000	33.468c	0.990	40.045c
Portuguese Escudo	0.901	8.049	0.913a	17.818	0.909c	23.033c	0.841	25.579c	0.900c	32.050c
Swedish Krona	0.861	7.541c	0.876	16.089	0.886	24.345c	0.855	31.744a	0.799	37.943c
Swiss Franc	0.941	9.884c	0.946	19.709c	0.988	28.110c	0.988	33.301c	0.981	39.887c
British Pound	0.934	8.643z	0.939	17.317c	0.938	25.283c	0.952	29.418c	0.930	32.964c

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 5: Panel 2 - Results of Purchasing Power Parity Model, Coverage Test

	h=1	h=3	h=6	h=9	h=12
Model Better	0	1	1	0	1
RW Better	0	0	0	0	0

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 5: Panel 3 - Results of Purchasing Power Parity Model, Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	5	5	10	12	11
RW Better	3	0	0	0	0

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 5: Panel 4 - Results of Purchasing Power Parity Model, Coverage Test and Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	5	6	11	12	12
RW Better	3	0	0	0	0

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Table 6: Panel 1 - Results of Monetary Model.

	Cov. h=1	Leng. h=1	Cov. h=3	Leng. h=3	Cov. h=6	Leng. h=6	Cov. h=9	Leng. h=9	Cov. h=12	Leng. h=12
Australian Dollar	0.869	7.089	0.853	15.357	0.840	21.353	0.822	25.707c	0.766	30.099c
Canadian Dollar	0.791	3.529	0.783	6.854	0.779	9.903	0.787	12.336	0.747	15.176c
Danish Kroner	0.905	8.771b	0.893	18.049	0.871	26.413	0.864	31.655b	0.856	38.564c
French Franc	0.922	8.830c	0.910	18.346c	0.949b	26.794c	0.957	32.389c	0.956	38.113c
Deutschmark	0.936	8.944a	0.897	18.615c	0.875	27.610c	0.901c	33.223c	0.908	40.409c
Italian Lira	0.913	9.001c	0.882	18.445b	0.925	26.613c	0.954	34.968c	0.945	41.395c
Japanese Yen	0.920	9.542	0.919	19.374c	0.871	28.312c	0.864	33.401c	0.814	38.149c
Dutch Guilder	0.917	8.753a	0.916	19.384a	0.962	29.149b	0.970	38.173	0.908c	43.896c
Portuguese Escudo	0.916	8.073	0.957	17.811	0.985	24.971z	0.968	28.026	1.000	34.598c
Swedish Krona	0.856	7.460c	0.837	15.587c	0.823	22.536c	0.826	28.641c	0.781	33.112c
Swiss Franc	0.929	9.910	0.868	19.539c	0.793	26.827c	0.745	31.797c	0.722	36.189c
British Pound	0.929	8.398c	0.928	17.355c	0.896	25.383c	0.884	30.600c	0.850	34.251c

Note: –h denotes forecast horizons for monthly data. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengths are averages across N(h) out-of-sample trials. –Superscripts a, b, c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x, y, z are defined analogously for rejections in favor of the random walk.

Table 6: Panel 2 - Results of Monetary Model, Coverage Test

	h=1	h=3	h=6	h=9	h=12
Model Better	0	0	1	1	1
RW Better	0	0	0	0	0

Note: In this panel, a better model is the one with more accurate empirical coverages. RW is the abbreviation of Random Walk.

Table 6: Panel 3 - Results of Monetary Model, Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	7	8	8	8	11
RW Better	0	0	1	0	0

Note: In this panel, a better model is the one with tighter forecast intervals given equal coverage accuracy.

Table 6: Panel 4 - Results of Monetary Model, Coverage Test and Length Test Given Equal Coverage Accuracy

	h=1	h=3	h=6	h=9	h=12
Model Better	7	8	8	9	12
RW Better	0	0	1	0	0

Note: In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy.

Figure 1a.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (British Pound), 1-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1989M12	143.69	0.6099	0.6629	0.6103	0.6638
1990M1	144.98	0.601	0.6531	0.6012	0.6538
1990M2	145.69	0.5812	0.6314	0.5812	0.6322
1990M3	153.31	0.5664	0.615	0.5659	0.6155
1990M4	158.46	0.5909	0.6418	0.5908	0.6427
1990M5	154.04	0.5878	0.6382	0.5862	0.6377
1990M6	153.70	0.5727	0.622	0.5722	0.6224
1990M7	149.04	0.5609	0.6094	0.5611	0.6104
1990M8	147.46	0.5285	0.5764	0.5303	0.5769
1990M9	138.44	0.5032	0.5481	0.5045	0.5491
1990M10	129.59	0.5091	0.5545	0.5103	0.5555
1990M11	129.22	0.491	0.5349	0.493	0.5366
1990M12	133.89	0.4851	0.5286	0.4883	0.5315
1991M1	133.70	0.4964	0.5407	0.4991	0.5432
1991M2	130.54	0.4926	0.5367	0.4958	0.5397
1991M3	137.39	0.4855	0.529	0.4884	0.5316
1991M4	137.11	0.5244	0.5717	0.5266	0.5736
1991M5	138.22	0.5479	0.597	0.5482	0.5971
1991M6	139.75	0.5543	0.604	0.5564	0.6061
1991M7	137.83	0.5796	0.6322	0.5814	0.6334
1991M8	136.82	0.5787	0.6313	0.5808	0.6327
1991M9	134.30	0.5683	0.62	0.5696	0.6205
1991M10	130.77	0.5548	0.6053	0.5555	0.6052
1991M11	129.63	0.5556	0.6063	0.5567	0.6064
1991M12	128.04	0.5375	0.586	0.539	0.5871
1992M1	125.46	0.5233	0.5708	0.5249	0.5718
1992M2	127.70	0.5281	0.5761	0.5302	0.5776
1992M3	132.86	0.5374	0.5862	0.5395	0.5877
1992M4	133.54	0.5546	0.6048	0.5564	0.6062
1992M5	130.77	0.545	0.5944	0.546	0.5948
1992M6	126.84	0.5302	0.5784	0.53	0.5774
1992M7	125.88	0.5164	0.5635	0.5171	0.5633
1992M8	126.23	0.4993	0.5448	0.5002	0.5449
1992M9	122.60	0.4926	0.5373	0.4936	0.5377
1992M10	121.17	0.5181	0.5655	0.5194	0.5661
1992M11	123.88	0.5782	0.6315	0.5803	0.6338
1992M12	124.04	0.6264	0.684	0.6282	0.6861
1993M1	124.99	0.6169	0.6736	0.6184	0.6754
1993M2	120.76	0.6252	0.6826	0.6259	0.6835
1993M3	117.02	0.6623	0.7232	0.6663	0.7294
1993M4	112.41	0.6535	0.7136	0.6562	0.7183
1993M5	110.34	0.6175	0.6747	0.6204	0.6798
1993M6	107.41	0.6162	0.6731	0.6192	0.6768
1993M7	107.69	0.6337	0.6919	0.6354	0.6945
1993M8	103.77	0.6385	0.6971	0.6408	0.7004
1993M9	105.57	0.6391	0.6977	0.6426	0.7024
1993M10	107.02	0.6254	0.6826	0.6284	0.6869
1993M11	107.88	0.6365	0.6946	0.6379	0.6973
1993M12	109.91	0.6443	0.7031	0.6471	0.7074
1994M1	111.44	0.639	0.6972	0.6426	0.7024
1994M2	106.30	0.641	0.6994	0.6422	0.7019
1994M3	105.10	0.6435	0.7022	0.6478	0.7081
1994M4	103.48	0.6392	0.6975	0.6423	0.7021
1994M5	103.75	0.6394	0.6977	0.6465	0.7067
1994M6	102.53	0.632	0.6897	0.6371	0.6964
1994M7	98.45	0.6262	0.6834	0.6279	0.6863
1994M8	99.94	0.6189	0.6755	0.6196	0.6773
1994M9	98.77	0.6181	0.6746	0.6219	0.6792
1994M10	98.35	0.6085	0.6642	0.6124	0.6688
1994M11	98.04	0.5936	0.648	0.5971	0.6521
1994M12	100.18	0.6007	0.6557	0.6036	0.6592
1995M1	99.77	0.6106	0.6664	0.6153	0.672
1995M2	98.24	0.6086	0.6643	0.6091	0.6652
1995M3	90.52	0.6077	0.6633	0.6102	0.6664
1995M4	83.69	0.5975	0.6522	0.5994	0.6546
1995M5	85.11	0.5922	0.6464	0.5967	0.6517
1995M6	84.64	0.6011	0.6561	0.6042	0.6599
1995M7	87.40	0.5998	0.6548	0.6015	0.6568
1995M8	94.74	0.6	0.655	0.6013	0.6567
1995M9	100.55	0.609	0.6648	0.6122	0.6686
1995M10	100.84	0.612	0.6681	0.6153	0.6719
1995M11	101.94	0.6086	0.6643	0.6079	0.6638
1995M12	101.85	0.611	0.6669	0.6139	0.6704
1996M1	105.75	0.6172	0.6736	0.6226	0.68
1996M2	105.79	0.6271	0.6844	0.6274	0.6852
1996M3	105.94	0.6222	0.6775	0.6248	0.6819
1996M4	107.20	0.6269	0.682	0.6284	0.6859
1996M5	106.34	0.631	0.6865	0.633	0.6909
1996M6	108.96	0.6326	0.6884	0.6334	0.6914
1996M7	109.19	0.623	0.6779	0.6225	0.6795
1996M8	107.87	0.6206	0.6752	0.6179	0.6745
1996M9	109.93	0.618	0.6725	0.6192	0.6758
1996M10	112.41	0.6147	0.669	0.6155	0.6718
1996M11	112.30	0.6065	0.6599	0.605	0.6603
1996M12	113.98	0.5773	0.6299	0.577	0.6301
1997M1	117.91	0.5756	0.628	0.5764	0.6295
1997M2	122.96	0.579	0.6316	0.5783	0.6316
1997M3	122.77	0.5907	0.6441	0.59	0.6444
1997M4	125.64	0.5974	0.6512	0.5959	0.6508
1997M5	119.19	0.5892	0.6419	0.5887	0.6429
1997M6	114.29	0.5887	0.6412	0.5877	0.6418
1997M7	115.38	0.585	0.6373	0.5831	0.6368
1997M8	117.93	0.577	0.6286	0.5745	0.6274
1997M9	120.89	0.6007	0.6548	0.5982	0.6532
1997M10	121.06	0.6034	0.6576	0.599	0.6528
1997M11	125.38	0.5936	0.6469	0.5874	0.6401
1997M12	129.73	0.5744	0.6263	0.5679	0.6189
1998M1	129.55	0.5827	0.6357	0.5779	0.6298
1998M2	125.85	0.5941	0.6476	0.5867	0.6391
1998M3	129.08	0.5882	0.6405	0.5846	0.6367
1998M4	131.75	0.5809	0.6323	0.5771	0.6286
1998M5	134.90	0.5748	0.6257	0.5736	0.6248
1998M6	140.33	0.5901	0.6431	0.5855	0.6378
1998M7	140.79	0.5875	0.6399	0.5812	0.6331
1998M8	144.68	0.5895	0.6419	0.5836	0.6357
1998M9	134.48	0.5929	0.6456	0.5869	0.6393
1998M10	121.05	0.5751	0.6264	0.5701	0.621
1998M11	120.29	0.5733	0.624	0.5661	0.6166
1998M12	117.07	0.5838	0.6355	0.5774	0.629
1999M1	113.29	0.5801	0.6314	0.5741	0.6253
1999M2	116.67	0.5904	0.6425	0.5814	0.6332
1999M3	119.47	0.5952	0.6477	0.5893	0.6419
1999M4	119.77	0.5974	0.65	0.5916	0.6444
1999M5	122.00	0.6023	0.6553	0.5961	0.6493
1999M6	120.72	0.5989	0.6514	0.5937	0.6467
1999M7	119.33	0.6067	0.6601	0.6013	0.655
1999M8	113.23	0.6162	0.6706	0.609	0.6633
1999M9	106.88	0.6021	0.655	0.5973	0.6506
1999M10	105.97	0.595	0.6473	0.5904	0.6431
1999M11	104.65	0.5843	0.6358	0.5788	0.6305
1999M12	102.58	0.5968	0.6494	0.5919	0.6447
2000M1	105.30	0.5997	0.6524	0.5946	0.6477
2000M2	109.39	0.5932	0.6453	0.5847	0.6369
2000M3	106.31	0.6047	0.658	0.5995	0.653
2000M4	105.63	0.613	0.667	0.6071	0.6613
2000M5	108.32	0.6088	0.6625	0.6062	0.6603
2000M6	106.13	0.6403	0.6967	0.6357	0.6925
2000M7	108.21	0.641	0.6973	0.6355	0.6923
2000M8	108.08	0.6427	0.6995	0.6362	0.6931
2000M9	106.84	0.6476	0.7052	0.6442	0.7018
2000M10	108.44	0.6722	0.7316	0.669	0.7288
2000M11	109.01	0.6652	0.724	0.6612	0.7203
2000M12	112.21	0.6738	0.7339	0.6727	0.7329
2001M1	116.67	0.6569	0.7159	0.6557	0.7143
2001M2	116.23	0.6543	0.7134	0.6491	0.7072
2001M3	121.51	0.6591	0.7195	0.6603	0.7194
2001M4	123.77	0.6633	0.7242	0.664	0.7233
2001M5	121.77	0.6665	0.7282	0.6685	0.7282
2001M6	122.35	0.6693	0.7311	0.6724	0.7324
2001M7	124.50	0.6814	0.7444	0.6841	0.7452
2001M8	121.37	0.6759	0.7359	0.6779	0.7379
2001M9	118.61	0.6614	0.7186	0.6674	0.7257
2001M10	121.45	0.6516	0.7081	0.6553	0.7126
2001M11	122.41	0.6564	0.7133	0.6615	0.7193
2001M12	127.59	0.6654	0.7217	0.6684	0.7265
2002M1	132.68	0.6611	0.7171	0.6658	0.7237
2002M2	133.64	0.6672	0.7237	0.6701	0.7283
2002M3	131.06	0.6742	0.7279	0.6746	0.7331
2002M4	130.77	0.6742	0.7279	0.6744	0.7329
2002M5	126.38	0.6638	0.7165	0.6651	0.7228
2002M6	123.29	0.6562	0.7084	0.6586	0.7145
2002M7	117.90	0.6481	0.6997	0.648	0.703
2002M8	118.99	0.6169	0.6687	0.6166	0.6701
2002M9	121.08	0.6233	0.6758	0.6244	0.6787
2002M10	123.91	0.6138	0.6656	0.6166	0.6702
2002M11	121.61	0.6154	0.6673	0.6161	0.6697
2002M12	121.89	0.6098	0.6611	0.6108	0.6638
2003M1	118.81	0.6036	0.6544	0.605	0.6575
2003M2	119.34	0.5958	0.6461	0.5933	0.6448
2003M3	118.69	0.5982	0.6487	0.5968	0.6487
2003M4	119.90	0.6071	0.6587	0.6064	0.6591
2003M5	117.37	0.606	0.6576	0.6097	0.6627
2003M6	118.33	0.5917	0.6406	0.5915	0.6429
2003M7	118.70	0.5813	0.6292	0.5778	0.6279
2003M8	118.66	0.594	0.6426	0.5916	0.643
2003M9	114.80	0.6038	0.6536	0.6032	0.6544
2003M10	109.50	0.5946	0.6437	0.5952	0.6456
2003M11	109.18	0.5733	0.621	0.5729	0.6211
2003M12	107.74	0.5691	0.6166	0.5693	0.6172
2004M1	106.27	0.5492	0.5951	0.5493	0.5955
2004M2	106.71	0.5328	0.5776	0.5267	0.5713
2004M3	108.52	0.5186	0.5626	0.5149	0.5586
2004M4	107.66	0.5302	0.5749	0.5265	0.5712
2004M5	112.20	0.5346	0.5799	0.5332	0.5785
2004M6	109.43	0.5411	0.5869	0.5383	0.584
2004M7	109.49	0.529	0.5739	0.5263	0.5706
2004M8	110.23	0.5237	0.5681	0.5218	0.5657
2004M9	110.09	0.5298	0.5748	0.5285	0.573
2004M10	108.78	0.5377	0.5833	0.5363	0.5815
2004M11	104.70	0.5365	0.5806	0.5337	0.5769
2004M12	103.81	0.5202	0.5631	0.5185	0.5605
2005M1	103.34	0.4995	0.5415	0.5002	0.5408
2005M2	104.94	0.5169	0.5597	0.5132	0.5544
2005M3	105.25	0.5132	0.5553	0.5112	0.5517
2005M4	107.19
2005M5	106.60
2005M6	108.75
2005M7	111.95
2005M8	110.61

 

Figure 1b.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (British Pound), 6-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1990M5	0.5962	0.572	0.7404	0.5638	0.738
1990M6	0.5847	0.5606	0.7257	0.5553	0.7269
1990M7	0.5525	0.5406	0.6994	0.5369	0.7029
1990M8	0.5260	0.5325	0.6903	0.5227	0.6843
1990M9	0.5321	0.5549	0.7167	0.5457	0.7144
1990M10	0.5140	0.58	0.7477	0.5415	0.7089
1990M11	0.5091	0.5448	0.7012	0.5286	0.6919
1990M12	0.5203	0.5255	0.6759	0.5184	0.6785
1991M1	0.5169	0.5004	0.6433	0.4899	0.6413
1991M2	0.5091	0.4809	0.6209	0.4663	0.6104
1991M3	0.5490	0.4846	0.6262	0.4708	0.6175
1991M4	0.5715	0.468	0.6073	0.4542	0.5965
1991M5	0.5801	0.452	0.5925	0.4452	0.5908
1991M6	0.6062	0.4622	0.6056	0.455	0.6039
1991M7	0.6056	0.4555	0.5967	0.452	0.5999
1991M8	0.5938	0.4541	0.5948	0.4452	0.5909
1991M9	0.5792	0.4852	0.6357	0.4801	0.6372
1991M10	0.5803	0.5159	0.6762	0.4998	0.6633
1991M11	0.5619	0.512	0.6715	0.5073	0.6733
1991M12	0.5473	0.5354	0.7025	0.53	0.7037
1992M1	0.5528	0.5325	0.6998	0.5295	0.7055
1992M2	0.5625	0.528	0.6941	0.5192	0.6919
1992M3	0.5801	0.5188	0.6817	0.5065	0.6749
1992M4	0.5693	0.5191	0.6806	0.5075	0.6762
1992M5	0.5526	0.5027	0.6566	0.4913	0.6548
1992M6	0.5391	0.4893	0.643	0.4785	0.6377
1992M7	0.5215	0.4905	0.6439	0.4834	0.6441
1992M8	0.5146	0.4975	0.6537	0.4918	0.6554
1992M9	0.5416	0.5118	0.6725	0.5073	0.676
1992M10	0.6050	0.5069	0.6666	0.4978	0.6633
1992M11	0.6550	0.4979	0.6555	0.4832	0.6439
1992M12	0.6447	0.4827	0.6355	0.4714	0.6281
1993M1	0.6525	0.4686	0.6157	0.456	0.6076
1993M2	0.6947	0.4631	0.6079	0.45	0.5996
1993M3	0.6841	0.4823	0.6304	0.4735	0.631
1993M4	0.6474	0.529	0.6913	0.529	0.705
1993M5	0.6461	0.5721	0.7448	0.5727	0.7638
1993M6	0.6630	0.5658	0.7385	0.5638	0.7551
1993M7	0.6687	0.5765	0.7501	0.5706	0.7644
1993M8	0.6705	0.5939	0.7743	0.6074	0.8167
1993M9	0.6558	0.5919	0.7706	0.5982	0.8086
1993M10	0.6656	0.5641	0.7332	0.5661	0.7651
1993M11	0.6753	0.5619	0.7298	0.565	0.7636
1993M12	0.6706	0.578	0.7501	0.5798	0.7836
1994M1	0.6701	0.5794	0.7512	0.5847	0.7903
1994M2	0.6760	0.5765	0.747	0.5863	0.7925
1994M3	0.6703	0.5681	0.7332	0.5734	0.7751
1994M4	0.6746	0.5806	0.7489	0.582	0.7867
1994M5	0.6648	0.5823	0.7507	0.5905	0.7981
1994M6	0.6552	0.5767	0.7437	0.5864	0.7925
1994M7	0.6465	0.583	0.7518	0.586	0.792
1994M8	0.6484	0.5778	0.7454	0.5911	0.799
1994M9	0.6385	0.5773	0.7456	0.5861	0.7922
1994M10	0.6225	0.5683	0.7346	0.5899	0.7973
1994M11	0.6292	0.5663	0.7323	0.5813	0.7857
1994M12	0.6416	0.5711	0.7387	0.5729	0.7744
1995M1	0.6351	0.566	0.7321	0.5654	0.7641
1995M2	0.6361	0.5598	0.7241	0.567	0.7664
1995M3	0.6249	0.551	0.7127	0.5583	0.7547
1995M4	0.6222	0.5408	0.6998	0.5443	0.7357
1995M5	0.6300	0.5482	0.7097	0.5502	0.7437
1995M6	0.6270	0.552	0.7145	0.561	0.7582
1995M7	0.6269	0.5621	0.7284	0.5553	0.7506
1995M8	0.6382	0.5566	0.7221	0.5563	0.7519
1995M9	0.6414	0.5491	0.713	0.5465	0.7386
1995M10	0.6338	0.5382	0.6988	0.544	0.7353
1995M11	0.6400	0.5486	0.7124	0.5508	0.7445
1995M12	0.6491	0.5521	0.717	0.5483	0.7411
1996M1	0.6541	0.5513	0.7151	0.5482	0.7409
1996M2	0.6510	0.5555	0.7205	0.5581	0.7543
1996M3	0.6548	0.5595	0.7204	0.5609	0.7581
1996M4	0.6596	0.5658	0.7265	0.5542	0.749
1996M5	0.6600	0.5595	0.7186	0.5596	0.7564
1996M6	0.6487	0.5593	0.7185	0.5676	0.7672
1996M7	0.6439	0.576	0.7402	0.572	0.7731
1996M8	0.6452	0.5656	0.7272	0.5693	0.7694
1996M9	0.6413	0.5689	0.7314	0.5726	0.7739
1996M10	0.6304	0.5755	0.7396	0.5768	0.7796
1996M11	0.6016	0.5788	0.7446	0.5771	0.78
1996M12	0.6010	0.5757	0.7414	0.5672	0.7667
1997M1	0.6030	0.5773	0.7438	0.563	0.761
1997M2	0.6152	0.5674	0.7312	0.5642	0.7625
1997M3	0.6213	0.5649	0.7279	0.5608	0.758
1997M4	0.6138	0.5633	0.724	0.5512	0.7451
1997M5	0.6127	0.5432	0.6979	0.526	0.711
1997M6	0.6079	0.5404	0.6911	0.5255	0.7103
1997M7	0.5990	0.5463	0.6965	0.5272	0.7126
1997M8	0.6236	0.5588	0.7119	0.5379	0.727
1997M9	0.6245	0.5678	0.7223	0.5432	0.7343
1997M10	0.6124	0.5597	0.7115	0.5367	0.7254
1997M11	0.5921	0.5604	0.7119	0.5357	0.7241
1997M12	0.6025	0.5593	0.7115	0.5316	0.7185
1998M1	0.6116	0.5524	0.7026	0.5238	0.7079
1998M2	0.6095	0.5778	0.7343	0.5453	0.7332
1998M3	0.6017	0.5843	0.743	0.5461	0.7316
1998M4	0.5980	0.5785	0.736	0.5355	0.7172
1998M5	0.6104	0.5705	0.7249	0.5177	0.6905
1998M6	0.6059	0.5761	0.7316	0.5269	0.7021
1998M7	0.6084	0.5895	0.7497	0.5348	0.7127
1998M8	0.6119	0.5804	0.7381	0.5329	0.7101
1998M9	0.5944	0.5766	0.7308	0.5261	0.7011
1998M10	0.5902	0.5711	0.7225	0.5229	0.6968
1998M11	0.6020	0.6005	0.7587	0.5338	0.7113
1998M12	0.5985	0.5959	0.7522	0.5298	0.706
1999M1	0.6061	0.5906	0.7448	0.532	0.7089
1999M2	0.6144	0.6039	0.7613	0.535	0.713
1999M3	0.6168	0.5853	0.7375	0.5198	0.6926
1999M4	0.6215	0.5925	0.7468	0.5161	0.6877
1999M5	0.6190	0.5955	0.7507	0.5264	0.7015
1999M6	0.6270	0.5884	0.7426	0.5234	0.6974
1999M7	0.6349	0.605	0.7629	0.53	0.7062
1999M8	0.6227	0.5977	0.7555	0.5372	0.7159
1999M9	0.6155	0.5955	0.7541	0.5393	0.7187
1999M10	0.6034	0.6022	0.7638	0.5435	0.7242
1999M11	0.6171	0.5964	0.7496	0.5413	0.7213
1999M12	0.6199	0.6004	0.7542	0.5482	0.7305
2000M1	0.6096	0.6119	0.7686	0.5552	0.7398
2000M2	0.6250	0.5911	0.7428	0.5445	0.7256
2000M3	0.6330	0.583	0.7328	0.5382	0.7172
2000M4	0.6320	0.5799	0.7291	0.5277	0.7031
2000M5	0.6627	0.5862	0.7372	0.5396	0.7191
2000M6	0.6626	0.5926	0.7455	0.542	0.7223
2000M7	0.6633	0.5975	0.7527	0.5331	0.7103
2000M8	0.6716	0.5935	0.7494	0.5465	0.7283
2000M9	0.6975	0.6025	0.7612	0.5534	0.7375
2000M10	0.6894	0.5941	0.7513	0.5526	0.7364
2000M11	0.7014	0.6213	0.7855	0.5795	0.7722
2000M12	0.6836	0.6232	0.7876	0.5794	0.7721
2001M1	0.6768	0.6293	0.795	0.58	0.7729
2001M2	0.6885	0.6215	0.7849	0.5873	0.7826
2001M3	0.6923	0.6402	0.81	0.6099	0.8128
2001M4	0.6970	0.6345	0.7998	0.6028	0.8032
2001M5	0.7010	0.6315	0.795	0.6133	0.8173
2001M6	0.7133	0.6204	0.7749	0.5977	0.7964
2001M7	0.7068	0.6215	0.7763	0.5918	0.7884
2001M8	0.6958	0.6105	0.7606	0.602	0.802
2001M9	0.6832	0.612	0.7552	0.6053	0.7888
2001M10	0.6896	0.6103	0.7497	0.6094	0.7782
2001M11	0.6966	0.6076	0.7459	0.613	0.7796
2001M12	0.6938	0.6158	0.7519	0.6237	0.7932
2002M1	0.6982	0.6129	0.7483	0.618	0.786
2002M2	0.7029	0.5942	0.7257	0.6084	0.7738
2002M3	0.7027	0.5943	0.7255	0.6037	0.7597
2002M4	0.6930	0.5982	0.7299	0.6102	0.7669
2002M5	0.6850	0.606	0.7389	0.6175	0.7746
2002M6	0.6740	0.6002	0.7317	0.6175	0.7716
2002M7	0.6425	0.6079	0.7412	0.6218	0.7765
2002M8	0.6507	0.6081	0.7416	0.627	0.7816
2002M9	0.6425	0.6091	0.7424	0.6269	0.7815
2002M10	0.6421	0.5991	0.7308	0.6182	0.7707
2002M11	0.6365	0.5916	0.7217	0.6111	0.7618
2002M12	0.6304	0.601	0.7332	0.6013	0.7495
2003M1	0.6182	0.5794	0.707	0.5732	0.7145
2003M2	0.6219	0.5792	0.7066	0.5805	0.7236
2003M3	0.6319	0.5691	0.6946	0.5732	0.7146
2003M4	0.6354	0.5723	0.6997	0.5728	0.714
2003M5	0.6164	0.5673	0.6947	0.5678	0.7078
2003M6	0.6021	0.5627	0.685	0.5624	0.701
2003M7	0.6165	0.5684	0.6895	0.5515	0.6875
2003M8	0.6274	0.5631	0.6811	0.5548	0.6916
2003M9	0.6190	0.5664	0.6864	0.5637	0.7027
2003M10	0.5955	0.5552	0.6725	0.5668	0.7065
2003M11	0.5918	0.5549	0.6721	0.5499	0.6854
2003M12	0.5709	0.5542	0.6707	0.5379	0.6695
2004M1	0.5478	0.5566	0.6758	0.5531	0.6856
2004M2	0.5355	0.5635	0.684	0.5664	0.6977
2004M3	0.5476	0.5553	0.6741	0.5589	0.6884
2004M4	0.5546	0.5433	0.6586	0.5377	0.6623
2004M5	0.5599	0.5436	0.6584	0.5343	0.6581
2004M6	0.5471	0.5306	0.6439	0.5155	0.6349
2004M7	0.5424	0.5306	0.6459	0.4914	0.6091
2004M8	0.5494	0.5178	0.6309	0.4805	0.5955
2004M9	0.5575	0.5209	0.6361	0.4913	0.609
2004M10	0.5532	0.5199	0.636	0.4976	0.6167
2004M11	0.5374	0.5298	0.6475	0.5055	0.6226
2004M12	0.5185	0.5191	0.638	0.4986	0.6084
2005M1	0.5320	0.5142	0.6329	0.4944	0.6031
2005M2	0.5299	0.5202	0.6375	0.5007	0.6109
2005M3	0.5251	0.5252	0.6436	0.5081	0.62
2005M4	0.5274	0.5282	0.6473	0.5042	0.6151
2005M5	0.5388	0.5108	0.6278	0.4898	0.5976
2005M6	0.5501	0.4935	0.6064	0.4726	0.5766
2005M7	0.5712	0.5183	0.637	0.4849	0.5916
2005M8	0.5573	0.5107	0.6279	0.483	0.5893
2005M9	0.5536	0.51	0.6273	0.4786	0.584

 

Figure 1c.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (British Pound), 12-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1990M11	0.5091	0.564	0.8106	0.5316	0.8054
1990M12	0.5203	0.5528	0.7946	0.5236	0.7933
1991M1	0.5169	0.5349	0.7677	0.5063	0.7671
1991M2	0.5091	0.5283	0.7569	0.4929	0.7468
1991M3	0.5490	0.5481	0.7848	0.5146	0.7797
1991M4	0.5715	0.5761	0.8252	0.5106	0.7736
1991M5	0.5801	0.5445	0.7806	0.4984	0.7551
1991M6	0.6062	0.53	0.7608	0.4888	0.7405
1991M7	0.6056	0.5052	0.7276	0.4619	0.6998
1991M8	0.5938	0.492	0.7109	0.4397	0.6662
1991M9	0.5792	0.493	0.7159	0.4448	0.6739
1991M10	0.5803	0.4784	0.692	0.4245	0.651
1991M11	0.5619	0.4655	0.675	0.418	0.6448
1991M12	0.5473	0.4712	0.6969	0.4272	0.659
1992M1	0.5528	0.4628	0.6836	0.4244	0.6547
1992M2	0.5625	0.4667	0.6828	0.418	0.6449
1992M3	0.5801	0.4894	0.7198	0.4508	0.6954
1992M4	0.5693	0.5182	0.7619	0.4693	0.7239
1992M5	0.5526	0.504	0.7424	0.4763	0.7348
1992M6	0.5391	0.5229	0.7703	0.4977	0.7677
1992M7	0.5215	0.5159	0.7599	0.4972	0.767
1992M8	0.5146	0.5155	0.7585	0.4875	0.7521
1992M9	0.5416	0.5077	0.7502	0.4755	0.7336
1992M10	0.6050	0.5084	0.7521	0.4765	0.7351
1992M11	0.6550	0.4941	0.7301	0.4613	0.7117
1992M12	0.6447	0.4873	0.7165	0.4493	0.6887
1993M1	0.6525	0.4876	0.7186	0.4538	0.6943
1993M2	0.6947	0.4937	0.7226	0.4618	0.7034
1993M3	0.6841	0.5059	0.7384	0.4763	0.7255
1993M4	0.6474	0.505	0.7359	0.4674	0.7119
1993M5	0.6461	0.5008	0.7172	0.4537	0.6911
1993M6	0.6630	0.4858	0.6928	0.4426	0.6733
1993M7	0.6687	0.4721	0.6649	0.4281	0.6477
1993M8	0.6705	0.466	0.6565	0.4225	0.6391
1993M9	0.6558	0.4796	0.6731	0.4446	0.6726
1993M10	0.6656	0.5145	0.7227	0.4967	0.7514
1993M11	0.6753	0.55	0.768	0.5377	0.8134
1993M12	0.6706	0.5437	0.7612	0.5294	0.8008
1994M1	0.6701	0.5504	0.778	0.5357	0.8104
1994M2	0.6760	0.5587	0.7935	0.5704	0.8628
1994M3	0.6703	0.5579	0.7916	0.5617	0.8497
1994M4	0.6746	0.5382	0.7625	0.5315	0.804
1994M5	0.6648	0.5341	0.7565	0.5305	0.8024
1994M6	0.6552	0.5467	0.7779	0.5444	0.8235
1994M7	0.6465	0.5464	0.7773	0.549	0.8351
1994M8	0.6484	0.5422	0.7728	0.5505	0.8437
1994M9	0.6385	0.5363	0.764	0.5384	0.8252
1994M10	0.6225	0.5463	0.7779	0.5465	0.8376
1994M11	0.6292	0.5455	0.7766	0.5544	0.8498
1994M12	0.6416	0.5416	0.7709	0.5506	0.8438
1995M1	0.6351	0.5467	0.7781	0.5502	0.8432
1995M2	0.6361	0.5405	0.7692	0.555	0.8507
1995M3	0.6249	0.5412	0.7701	0.5503	0.8434
1995M4	0.6222	0.533	0.7581	0.5539	0.8489
1995M5	0.6300	0.5321	0.7567	0.5458	0.8365
1995M6	0.6270	0.5391	0.7622	0.5379	0.8245
1995M7	0.6269	0.5345	0.7555	0.5308	0.8136
1995M8	0.6382	0.5286	0.7471	0.5324	0.816
1995M9	0.6414	0.521	0.7364	0.5242	0.8035
1995M10	0.6338	0.5146	0.7274	0.5111	0.7833
1995M11	0.6400	0.5205	0.7364	0.5166	0.7918
1995M12	0.6491	0.5229	0.7394	0.5267	0.8073
1996M1	0.6541	0.5343	0.7562	0.5214	0.7992
1996M2	0.6510	0.5294	0.7495	0.5223	0.8005
1996M3	0.6548	0.5268	0.7422	0.5131	0.7864
1996M4	0.6596	0.5164	0.7272	0.5108	0.7829
1996M5	0.6600	0.5259	0.74	0.5172	0.7927
1996M6	0.6487	0.5311	0.7468	0.5148	0.7891
1996M7	0.6439	0.5288	0.7432	0.5147	0.7888
1996M8	0.6452	0.532	0.7478	0.524	0.8032
1996M9	0.6413	0.5348	0.7503	0.5267	0.8072
1996M10	0.6304	0.5426	0.7587	0.5203	0.7975
1996M11	0.6016	0.5345	0.7474	0.5255	0.8053
1996M12	0.6010	0.5326	0.7449	0.533	0.8168
1997M1	0.6030	0.5469	0.765	0.537	0.8231
1997M2	0.6152	0.5374	0.752	0.5345	0.8192
1997M3	0.6213	0.5381	0.7536	0.5376	0.824
1997M4	0.6138	0.5509	0.7715	0.5416	0.83
1997M5	0.6127	0.5543	0.7756	0.5419	0.8305
1997M6	0.6079	0.5558	0.7735	0.5326	0.8163
1997M7	0.5990	0.5577	0.7746	0.5287	0.8103
1997M8	0.6236	0.5499	0.7642	0.5297	0.8119
1997M9	0.6245	0.5497	0.7629	0.5266	0.807
1997M10	0.6124	0.5479	0.761	0.5176	0.7933
1997M11	0.5921	0.5353	0.744	0.4939	0.757
1997M12	0.6025	0.5346	0.7413	0.4934	0.7562
1998M1	0.6116	0.5377	0.7454	0.495	0.7587
1998M2	0.6095	0.5507	0.7633	0.505	0.7741
1998M3	0.6017	0.5593	0.7752	0.5101	0.7759
1998M4	0.5980	0.5531	0.7662	0.5039	0.7622
1998M5	0.6104	0.5542	0.7673	0.503	0.7575
1998M6	0.6059	0.5544	0.7675	0.4991	0.7508
1998M7	0.6084	0.5462	0.7573	0.4918	0.7383
1998M8	0.6119	0.5712	0.7931	0.512	0.7671
1998M9	0.5944	0.5807	0.8065	0.5127	0.7667
1998M10	0.5902	0.5787	0.8036	0.5028	0.7516
1998M11	0.6020	0.5792	0.8042	0.4861	0.7173
1998M12	0.5985	0.5861	0.8135	0.4947	0.7296
1999M1	0.6061	0.5955	0.8264	0.5022	0.7407
1999M2	0.6144	0.5918	0.8212	0.5004	0.738
1999M3	0.6168	0.5841	0.8139	0.494	0.7286
1999M4	0.6215	0.5837	0.8165	0.491	0.7241
1999M5	0.6190	0.6145	0.8594	0.5012	0.7392
1999M6	0.6270	0.605	0.8458	0.4975	0.7337
1999M7	0.6349	0.5948	0.8318	0.4995	0.7368
1999M8	0.6227	0.6181	0.8638	0.5024	0.741
1999M9	0.6155	0.5984	0.833	0.488	0.7198
1999M10	0.6034	0.6026	0.834	0.4846	0.7147
1999M11	0.6171	0.598	0.8302	0.4943	0.7277
1999M12	0.6199	0.5881	0.8158	0.4914	0.7235
2000M1	0.6096	0.5958	0.8267	0.4976	0.7326
2000M2	0.6250	0.5881	0.8161	0.5044	0.7427
2000M3	0.6330	0.5833	0.8087	0.5064	0.7456
2000M4	0.6320	0.5892	0.8168	0.5103	0.7513
2000M5	0.6627	0.5834	0.8084	0.5082	0.7483
2000M6	0.6626	0.5846	0.8101	0.5147	0.7578
2000M7	0.6633	0.593	0.822	0.5213	0.7674
2000M8	0.6716	0.5756	0.7946	0.5113	0.7528
2000M9	0.6975	0.5689	0.7826	0.5054	0.744
2000M10	0.6894	0.5701	0.784	0.4954	0.7294
2000M11	0.7014	0.5742	0.789	0.5067	0.7459
2000M12	0.6836	0.5828	0.8008	0.509	0.7493
2001M1	0.6768	0.5877	0.8054	0.5005	0.7369
2001M2	0.6885	0.5811	0.7956	0.5131	0.7555
2001M3	0.6923	0.5878	0.7968	0.5196	0.7651
2001M4	0.6970	0.5838	0.775	0.5189	0.7639
2001M5	0.7010	0.6042	0.7987	0.5441	0.8011
2001M6	0.7133	0.6051	0.7856	0.544	0.7895
2001M7	0.7068	0.6107	0.7873	0.5446	0.783
2001M8	0.6958	0.6034	0.7748	0.5515	0.7921
2001M9	0.6832	0.6202	0.7902	0.5727	0.8218
2001M10	0.6896	0.6111	0.7737	0.566	0.806
2001M11	0.6966	0.6064	0.7687	0.5759	0.8175
2001M12	0.6938	0.5969	0.7524	0.5612	0.7968
2002M1	0.6982	0.5911	0.7442	0.5557	0.7888
2002M2	0.7029	0.5824	0.7322	0.5652	0.8024
2002M3	0.7027	0.584	0.7331	0.5684	0.8069
2002M4	0.6930	0.5864	0.7345	0.5722	0.8124
2002M5	0.6850	0.5846	0.7303	0.5756	0.8171
2002M6	0.6740	0.5897	0.7355	0.5856	0.8314
2002M7	0.6425	0.5882	0.7327	0.5803	0.8238
2002M8	0.6507	0.574	0.7148	0.5747	0.811
2002M9	0.6425	0.5733	0.7146	0.5675	0.7963
2002M10	0.6421	0.5764	0.7185	0.5765	0.8038
2002M11	0.6365	0.58	0.7229	0.5824	0.8119
2002M12	0.6304	0.5773	0.7191	0.582	0.8087
2003M1	0.6182	0.5808	0.7232	0.5929	0.8138
2003M2	0.6219	0.5798	0.7218	0.5999	0.8192
2003M3	0.6319	0.5811	0.7232	0.5998	0.8191
2003M4	0.6354	0.573	0.7136	0.5915	0.8078
2003M5	0.6164	0.5657	0.7054	0.5847	0.7984
2003M6	0.6021	0.586	0.729	0.5753	0.7856
2003M7	0.6165	0.5655	0.7024	0.5484	0.7489
2003M8	0.6274	0.5616	0.6953	0.5554	0.7584
2003M9	0.6190	0.5559	0.6885	0.5484	0.7489
2003M10	0.5955	0.5574	0.6919	0.548	0.7483
2003M11	0.5918	0.556	0.6898	0.5432	0.7419
2003M12	0.5709	0.5511	0.6851	0.5381	0.7348
2004M1	0.5478	0.5566	0.692	0.5277	0.7206
2004M2	0.5355	0.5509	0.6848	0.5308	0.7249
2004M3	0.5476	0.552	0.6841	0.5393	0.7365
2004M4	0.5546	0.5452	0.6763	0.5423	0.7405
2004M5	0.5599	0.5478	0.6811	0.5261	0.7184
2004M6	0.5471	0.5474	0.6812	0.5139	0.7017
2004M7	0.5424	0.5472	0.6802	0.5309	0.7186
2004M8	0.5494	0.5515	0.6853	0.5418	0.7312
2004M9	0.5575	0.5478	0.6807	0.5364	0.7215
2004M10	0.5532	0.5397	0.6706	0.5161	0.6941
2004M11	0.5374	0.5446	0.6766	0.5128	0.6898
2004M12	0.5185	0.5396	0.6699	0.4948	0.6654
2005M1	0.5320	0.5365	0.6687	0.4747	0.6385
2005M2	0.5299	0.5299	0.6633	0.4625	0.6242
2005M3	0.5251	0.5274	0.6611	0.4729	0.6383
2005M4	0.5274	0.5288	0.6632	0.4789	0.6464
2005M5	0.5388	0.5343	0.6704	0.4835	0.6526
2005M6	0.5501	0.5279	0.663	0.4724	0.6376
2005M7	0.5712	0.5287	0.664	0.4684	0.6322
2005M8	0.5573	0.5327	0.6688	0.4744	0.6403
2005M9	0.5536	0.5343	0.671	0.4814	0.6498
2005M10	0.5665	0.5368	0.6743	0.4777	0.6447
2005M11	0.5764	0.5267	0.6624	0.4641	0.6264
2005M12	0.5728	0.5179	0.6532	0.4478	0.6044
2006M1	0.5654	0.5299	0.6693	0.4594	0.6201
2006M2	0.5721	0.5239	0.662	0.4576	0.6177
2006M3	0.5733	0.5249	0.6637	0.4535	0.6121
2006M4	0.5656	0.5192	0.6577	0.4554	0.6147

 

Figure 2a.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (Deutschmark), 1-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1989M12	1.7378	1.7399	1.8989	1.7372	1.9062
1990M1	1.6914	1.6522	1.8051	1.6497	1.8101
1990M2	1.6758	1.5999	1.746	1.6062	1.7619
1990M3	1.7053	1.5983	1.7426	1.5987	1.7456
1990M4	1.6863	1.6166	1.7628	1.6268	1.7762
1990M5	1.6630	1.6108	1.7579	1.6087	1.7565
1990M6	1.6832	1.5829	1.7274	1.5865	1.7322
1990M7	1.6375	1.5939	1.7378	1.6058	1.7529
1990M8	1.5702	1.5492	1.6898	1.5622	1.7054
1990M9	1.5701	1.4808	1.6153	1.498	1.6343
1990M10	1.5238	1.4812	1.6161	1.4978	1.6341
1990M11	1.4857	1.4441	1.5756	1.4537	1.586
1990M12	1.4982	1.3991	1.5262	1.4174	1.5464
1991M1	1.5091	1.4163	1.5453	1.4293	1.5594
1991M2	1.4805	1.4206	1.5502	1.4396	1.5707
1991M3	1.6122	1.3997	1.5272	1.4124	1.541
1991M4	1.7027	1.5249	1.662	1.538	1.679
1991M5	1.7199	1.6234	1.7741	1.6243	1.7736
1991M6	1.7828	1.6393	1.7915	1.6408	1.7916
1991M7	1.7852	1.7029	1.8614	1.7008	1.8571
1991M8	1.7435	1.7233	1.8838	1.703	1.8595
1991M9	1.6933	1.6578	1.811	1.6633	1.8161
1991M10	1.6893	1.609	1.7571	1.6154	1.7639
1991M11	1.6208	1.6331	1.7833	1.6116	1.7596
1991M12	1.5630	1.5431	1.6854	1.5462	1.6883
1992M1	1.5788	1.4857	1.6227	1.4911	1.6281
1992M2	1.6186	1.5047	1.6432	1.5062	1.6446
1992M3	1.6616	1.5404	1.6825	1.5442	1.6857
1992M4	1.6493	1.5797	1.722	1.5852	1.7305
1992M5	1.6225	1.5752	1.7165	1.5735	1.7177
1992M6	1.5726	1.5469	1.6857	1.5479	1.6898
1992M7	1.4914	1.497	1.6321	1.5002	1.6377
1992M8	1.4475	1.421	1.5545	1.4162	1.5532
1992M9	1.4514	1.3736	1.5016	1.3745	1.5074
1992M10	1.4851	1.3755	1.5048	1.3782	1.5115
1992M11	1.5875	1.4106	1.5431	1.4103	1.5467
1992M12	1.5822	1.5147	1.6585	1.5076	1.6537
1993M1	1.6144	1.511	1.6539	1.5025	1.6481
1993M2	1.6414	1.5578	1.7061	1.5331	1.6817
1993M3	1.6466	1.573	1.723	1.5586	1.7097
1993M4	1.5964	1.5691	1.7184	1.5636	1.7151
1993M5	1.6071	1.5256	1.6706	1.516	1.663
1993M6	1.6547	1.5332	1.6791	1.5261	1.674
1993M7	1.7157	1.5822	1.7333	1.5713	1.7236
1993M8	1.6944	1.6458	1.803	1.6292	1.7871
1993M9	1.6219	1.6096	1.7636	1.609	1.7649
1993M10	1.6405	1.5422	1.6896	1.5402	1.6894
1993M11	1.7005	1.558	1.707	1.5578	1.7088
1993M12	1.7105	1.6227	1.7773	1.6148	1.7713
1994M1	1.7426	1.634	1.7898	1.6243	1.7818
1994M2	1.7355	1.6687	1.8283	1.6548	1.8152
1994M3	1.6909	1.6598	1.8182	1.6481	1.8078
1994M4	1.6984	1.6108	1.7638	1.6058	1.7614
1994M5	1.6565	1.6197	1.7736	1.6128	1.7692
1994M6	1.6271	1.5856	1.7358	1.573	1.7255
1994M7	1.5674	1.5521	1.6982	1.5451	1.6949
1994M8	1.5646	1.4999	1.6408	1.4884	1.6326
1994M9	1.5491	1.4948	1.6355	1.4857	1.6297
1994M10	1.5195	1.4747	1.6145	1.4711	1.6137
1994M11	1.5396	1.4496	1.5872	1.443	1.5828
1994M12	1.5716	1.4709	1.6109	1.462	1.6037
1995M1	1.5302	1.5033	1.6469	1.4924	1.6371
1995M2	1.5022	1.465	1.6033	1.4531	1.5939
1995M3	1.4061	1.4405	1.5757	1.4265	1.5647
1995M4	1.3812	1.3414	1.4708	1.3348	1.4646
1995M5	1.4096	1.3186	1.4458	1.3113	1.4388
1995M6	1.4012	1.3494	1.4768	1.3386	1.4683
1995M7	1.3886	1.3421	1.4683	1.3306	1.4595
1995M8	1.4456	1.3321	1.4573	1.3186	1.4464
1995M9	1.4601	1.3852	1.5151	1.3727	1.5058
1995M10	1.4143	1.3981	1.5296	1.3865	1.5209
1995M11	1.4173	1.3541	1.481	1.343	1.4731
1995M12	1.4406	1.3608	1.4882	1.3459	1.4764
1996M1	1.4635	1.3859	1.5154	1.3681	1.5007
1996M2	1.4669	1.4	1.5312	1.3897	1.5244
1996M3	1.4776	1.4097	1.5432	1.3931	1.5281
1996M4	1.5044	1.4136	1.5474	1.4031	1.5391
1996M5	1.5324	1.4398	1.5776	1.4286	1.5671
1996M6	1.5282	1.4681	1.6095	1.4551	1.5962
1996M7	1.5025	1.465	1.6065	1.4512	1.5919
1996M8	1.4826	1.4404	1.5794	1.4268	1.565
1996M9	1.5080	1.4197	1.5565	1.4079	1.5443
1996M10	1.5277	1.4425	1.5815	1.432	1.5708
1996M11	1.5118	1.4615	1.6012	1.4508	1.5911
1996M12	1.5525	1.4465	1.5839	1.4356	1.5744
1997M1	1.6047	1.4959	1.6302	1.4743	1.6169
1997M2	1.6747	1.5476	1.6862	1.5238	1.6711
1997M3	1.6946	1.6117	1.7556	1.5903	1.7444
1997M4	1.7119	1.6281	1.7734	1.6091	1.7651
1997M5	1.7048	1.6442	1.791	1.6256	1.7832
1997M6	1.7277	1.6455	1.7917	1.6188	1.7757
1997M7	1.7939	1.6606	1.8075	1.6407	1.7997
1997M8	1.8400	1.7289	1.8824	1.7035	1.8686
1997M9	1.7862	1.7674	1.9258	1.7473	1.9167
1997M10	1.7575	1.7117	1.8637	1.6962	1.8595
1997M11	1.7323	1.6837	1.8338	1.669	1.8296
1997M12	1.7788	1.6641	1.8128	1.6451	1.8034
1998M1	1.8165	1.71	1.8612	1.6891	1.8513
1998M2	1.8123	1.7372	1.8932	1.725	1.8906
1998M3	1.8272	1.7339	1.8919	1.721	1.8863
1998M4	1.8132	1.7407	1.9018	1.7352	1.9018
1998M5	1.7753	1.7395	1.8903	1.7319	1.8872
1998M6	1.7928	1.7057	1.8538	1.6957	1.8478
1998M7	1.7976	1.7203	1.8697	1.7124	1.866
1998M8	1.7869	1.7251	1.8749	1.717	1.8711
1998M9	1.6990	1.7119	1.8605	1.7068	1.8599
1998M10	1.6381	1.6276	1.7716	1.6153	1.7683
1998M11	1.6827	1.5666	1.7073	1.5574	1.7049
1998M12	1.6698	1.616	1.7591	1.5998	1.7514

 

Figure 2b.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (Deutschmark), 6-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1990M5	1.6630	1.6049	2.1331	1.573	2.106
1990M6	1.6832	1.5484	2.0562	1.4953	1.9999
1990M7	1.6375	1.4571	1.9356	1.4554	1.9466
1990M8	1.5702	1.4817	1.9695	1.442	1.9286
1990M9	1.5701	1.4596	1.9408	1.4673	1.9517
1990M10	1.5238	1.4952	1.9612	1.4509	1.93
1990M11	1.4857	1.452	1.9005	1.4309	1.9033
1990M12	1.4982	1.4259	1.8641	1.4483	1.9265
1991M1	1.5091	1.3895	1.8218	1.409	1.8742
1991M2	1.4805	1.3131	1.7223	1.3511	1.7971
1991M3	1.6122	1.3159	1.7267	1.3509	1.797
1991M4	1.7027	1.3343	1.7486	1.3111	1.744
1991M5	1.7199	1.241	1.6286	1.2784	1.7005
1991M6	1.7828	1.2898	1.6939	1.2892	1.7148
1991M7	1.7852	1.2618	1.6596	1.2985	1.7272
1991M8	1.7435	1.2813	1.6852	1.2739	1.6945
1991M9	1.6933	1.3359	1.7528	1.3872	1.8452
1991M10	1.6893	1.456	1.9108	1.4651	1.9488
1991M11	1.6208	1.4804	1.9671	1.4799	1.9794
1991M12	1.5630	1.5521	2.0652	1.534	2.0639
1992M1	1.5788	1.6847	2.2388	1.536	2.0687
1992M2	1.6186	1.5142	2.011	1.5002	2.0348
1992M3	1.6616	1.468	1.9492	1.457	1.9762
1992M4	1.6493	1.6527	2.1945	1.4535	1.9715
1992M5	1.6225	1.4623	1.941	1.3946	1.8916
1992M6	1.5726	1.3909	1.853	1.3449	1.8241
1992M7	1.4914	1.4184	1.8947	1.3585	1.8426
1992M8	1.4475	1.4282	1.9096	1.3928	1.8891
1992M9	1.4514	1.4377	1.921	1.4298	1.9392
1992M10	1.4851	1.4633	1.9574	1.4192	1.925
1992M11	1.5875	1.4294	1.9117	1.3961	1.8936
1992M12	1.5822	1.3808	1.8463	1.3531	1.8353
1993M1	1.6144	1.3402	1.791	1.2833	1.7405
1993M2	1.6414	1.2676	1.6943	1.2455	1.6893
1993M3	1.6466	1.263	1.6855	1.2488	1.6938
1993M4	1.5964	1.3009	1.7374	1.2779	1.7333
1993M5	1.6071	1.4234	1.9016	1.366	1.8528
1993M6	1.6547	1.4263	1.9035	1.3614	1.8465
1993M7	1.7157	1.5509	2.0696	1.3892	1.8842
1993M8	1.6944	1.5132	2.0229	1.4123	1.9155
1993M9	1.6219	1.4714	1.9675	1.4168	1.9217
1993M10	1.6405	1.4499	1.9386	1.3737	1.8632
1993M11	1.7005	1.4478	1.9358	1.3828	1.8755
1993M12	1.7105	1.5079	2.016	1.4238	1.9311
1994M1	1.7426	1.583	2.1161	1.4762	2.0023
1994M2	1.7355	1.4937	1.9966	1.4579	1.9774
1994M3	1.6909	1.4384	1.9223	1.3956	1.8929
1994M4	1.6984	1.4444	1.9304	1.4116	1.9146
1994M5	1.6565	1.5306	2.0454	1.4632	1.9846
1994M6	1.6271	1.5498	2.0709	1.4718	1.9963
1994M7	1.5674	1.5981	2.1352	1.4995	2.0338
1994M8	1.5646	1.5813	2.1116	1.4933	2.0255
1994M9	1.5491	1.5065	2.0093	1.455	1.9735
1994M10	1.5195	1.5251	2.0314	1.4614	1.9822
1994M11	1.5396	1.5211	2.0232	1.4253	1.9332
1994M12	1.5716	1.4658	1.9426	1.4	1.8989
1995M1	1.5302	1.4334	1.9054	1.3486	1.8292
1995M2	1.5022	1.4162	1.884	1.3462	1.8259
1995M3	1.4061	1.3818	1.837	1.333	1.808
1995M4	1.3812	1.3695	1.8239	1.3075	1.7734
1995M5	1.4096	1.396	1.8588	1.3247	1.7968
1995M6	1.4012	1.439	1.91	1.3523	1.8342
1995M7	1.3886	1.4048	1.8659	1.3167	1.7859
1995M8	1.4456	1.3902	1.8416	1.2925	1.7531
1995M9	1.4601	1.2929	1.7118	1.2099	1.641
1995M10	1.4143	1.2773	1.6907	1.1885	1.612
1995M11	1.4173	1.3012	1.7211	1.2129	1.6451
1995M12	1.4406	1.2965	1.7169	1.2056	1.6353
1996M1	1.4635	1.2955	1.7167	1.1948	1.6206
1996M2	1.4669	1.3304	1.7631	1.2438	1.6871
1996M3	1.4776	1.338	1.7729	1.2563	1.704
1996M4	1.5044	1.3005	1.7234	1.2169	1.6505
1996M5	1.5324	1.3227	1.7529	1.2196	1.6542
1996M6	1.5282	1.3542	1.794	1.2396	1.6814
1996M7	1.5025	1.3394	1.772	1.2592	1.708
1996M8	1.4826	1.3749	1.8167	1.2623	1.7121
1996M9	1.5080	1.3566	1.791	1.2714	1.7244
1996M10	1.5277	1.3862	1.8273	1.2945	1.7558
1996M11	1.5118	1.416	1.8668	1.3185	1.7884
1996M12	1.5525	1.418	1.8695	1.315	1.7835
1997M1	1.6047	1.3976	1.8425	1.2928	1.7535
1997M2	1.6747	1.3723	1.8077	1.2757	1.7303
1997M3	1.6946	1.3863	1.8257	1.2976	1.76
1997M4	1.7119	1.4072	1.8535	1.3146	1.783
1997M5	1.7048	1.3945	1.8348	1.3008	1.7644
1997M6	1.7277	1.44	1.895	1.3359	1.8119
1997M7	1.7939	1.4898	1.9602	1.3807	1.8727
1997M8	1.8400	1.5313	2.0168	1.4409	1.9544
1997M9	1.7862	1.5372	2.0225	1.4581	1.9776
1997M10	1.7575	1.5464	2.0376	1.473	1.9838
1997M11	1.7323	1.5796	2.085	1.4668	1.9735
1997M12	1.7788	1.5638	2.062	1.4866	1.9774
1998M1	1.8165	1.6468	2.1538	1.5436	2.0352
1998M2	1.8123	1.6658	2.1632	1.5833	2.0865
1998M3	1.8272	1.6017	2.0692	1.537	2.024
1998M4	1.8132	1.577	2.0337	1.5123	1.9852
1998M5	1.7753	1.5769	2.0334	1.4906	1.9568
1998M6	1.7928	1.6242	2.0941	1.5305	2.0091
1998M7	1.7976	1.6237	2.0938	1.563	2.0518
1998M8	1.7869	1.627	2.0987	1.5594	2.0471
1998M9	1.6990	1.6167	2.0856	1.5722	2.0639
1998M10	1.6381	1.6208	2.0911	1.5602	2.0481
1998M11	1.6827	1.6031	2.0684	1.5276	2.0053
1998M12	1.6698	1.6079	2.0727	1.5426	2.0251

 

Figure 2c.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (Deutschmark), 12-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1990M11	1.4857	1.446	2.1626	1.3878	2.1567
1990M12	1.4982	1.3932	2.0833	1.3178	2.0481
1991M1	1.5091	1.3401	2.0037	1.2827	1.9935
1991M2	1.4805	1.3508	2.0198	1.2709	1.9751
1991M3	1.6122	1.348	2.0158	1.2932	2.0097
1991M4	1.7027	1.3593	2.0384	1.2788	1.9873
1991M5	1.7199	1.3284	1.9932	1.2611	1.9599
1991M6	1.7828	1.3184	1.9773	1.2765	1.9838
1991M7	1.7852	1.2856	1.9265	1.2418	1.93
1991M8	1.7435	1.2291	1.8388	1.1908	1.8506
1991M9	1.6933	1.2275	1.8335	1.1907	1.8504
1991M10	1.6893	1.2258	1.8284	1.1556	1.7959
1991M11	1.6208	1.1707	1.7341	1.1267	1.751
1991M12	1.5630	1.1848	1.7641	1.1362	1.7658
1992M1	1.5788	1.1621	1.7296	1.1444	1.7786
1992M2	1.6186	1.1711	1.7338	1.1228	1.7449
1992M3	1.6616	1.2187	1.816	1.2226	1.9001
1992M4	1.6493	1.296	1.9286	1.2912	2.0067
1992M5	1.6225	1.314	1.9551	1.3043	2.0271
1992M6	1.5726	1.3533	2.0135	1.352	2.1012
1992M7	1.4914	1.4182	2.1116	1.3538	2.1039
1992M8	1.4475	1.3383	1.9954	1.3222	2.0548
1992M9	1.4514	1.3172	1.9697	1.2842	1.9957
1992M10	1.4851	1.3762	2.0574	1.2811	1.9909
1992M11	1.5875	1.2915	1.9311	1.2291	1.9102
1992M12	1.5822	1.2538	1.8747	1.1853	1.8421
1993M1	1.6144	1.2745	1.906	1.1973	1.8608
1993M2	1.6414	1.301	1.945	1.2275	1.9077
1993M3	1.6466	1.3265	1.9812	1.2601	1.9583
1993M4	1.5964	1.3368	1.9964	1.2508	1.9439
1993M5	1.6071	1.3151	1.9635	1.2305	1.9123
1993M6	1.6547	1.2772	1.9088	1.1926	1.8534
1993M7	1.7157	1.235	1.8474	1.131	1.7577
1993M8	1.6944	1.1906	1.7809	1.0977	1.7059
1993M9	1.6219	1.19	1.7762	1.1006	1.7105
1993M10	1.6405	1.2168	1.8205	1.1263	1.7503
1993M11	1.7005	1.3	1.946	1.2039	1.8711
1993M12	1.7105	1.3045	1.9518	1.1999	1.8647
1994M1	1.7426	1.3618	2.0372	1.2243	1.9027
1994M2	1.7355	1.367	2.0454	1.2447	1.9344
1994M3	1.6909	1.3541	2.0294	1.2487	1.9406
1994M4	1.6984	1.3287	1.9914	1.2107	1.8816
1994M5	1.6565	1.3306	1.9942	1.2187	1.894
1994M6	1.6271	1.3705	2.054	1.2548	1.9501
1994M7	1.5674	1.426	2.1321	1.3011	2.022
1994M8	1.5646	1.3874	2.0716	1.2849	1.9969
1994M9	1.5491	1.3472	2.009	1.23	1.9115
1994M10	1.5195	1.3576	2.0256	1.2441	1.9334
1994M11	1.5396	1.4092	2.1029	1.2896	2.0041
1994M12	1.5716	1.4219	2.1206	1.2972	2.016
1995M1	1.5302	1.4596	2.1747	1.3215	2.0538
1995M2	1.5022	1.4537	2.1644	1.3161	2.0454
1995M3	1.4061	1.4118	2.1011	1.2824	1.9929
1995M4	1.3812	1.4212	2.1145	1.288	2.0017
1995M5	1.4096	1.4121	2.0988	1.2562	1.9523
1995M6	1.4012	1.3796	2.051	1.2339	1.9176
1995M7	1.3886	1.3517	2.0075	1.1886	1.8473
1995M8	1.4456	1.3484	2.0037	1.1865	1.8439
1995M9	1.4601	1.3251	1.9648	1.1748	1.8258
1995M10	1.4143	1.3134	1.9488	1.1523	1.7909
1995M11	1.4173	1.3353	1.9767	1.1675	1.8145
1995M12	1.4406	1.365	2.0198	1.1918	1.8522
1996M1	1.4635	1.3514	1.9952	1.1604	1.8034
1996M2	1.4669	1.329	1.9642	1.1392	1.7704
1996M3	1.4776	1.2669	1.8738	1.0663	1.6572
1996M4	1.5044	1.2425	1.8547	1.0475	1.6279
1996M5	1.5324	1.2582	1.8731	1.069	1.6613
1996M6	1.5282	1.2591	1.8743	1.0626	1.6514
1996M7	1.5025	1.2498	1.859	1.0531	1.6366
1996M8	1.4826	1.2979	1.9326	1.0963	1.7037
1996M9	1.5080	1.303	1.9402	1.1073	1.7208
1996M10	1.5277	1.2944	1.9242	1.0725	1.6668
1996M11	1.5118	1.2929	1.9201	1.0749	1.6705
1996M12	1.5525	1.3028	1.9352	1.0925	1.6979
1997M1	1.6047	1.315	1.9538	1.1098	1.7248
1997M2	1.6747	1.322	1.964	1.1125	1.7289
1997M3	1.6946	1.3341	1.9804	1.1205	1.7414
1997M4	1.7119	1.3595	2.0171	1.1409	1.773
1997M5	1.7048	1.3628	2.02	1.1621	1.806
1997M6	1.7277	1.3679	2.028	1.1589	1.8011
1997M7	1.7939	1.3628	2.0212	1.1394	1.7707
1997M8	1.8400	1.3397	1.9886	1.1243	1.7473
1997M9	1.7862	1.3505	2.0044	1.1436	1.7773
1997M10	1.7575	1.3665	2.0262	1.1586	1.7973
1997M11	1.7323	1.351	2.0014	1.1465	1.7786
1997M12	1.7788	1.359	2.0131	1.1774	1.8265
1998M1	1.8165	1.3946	2.0644	1.2169	1.8784
1998M2	1.8123	1.4399	2.1279	1.27	1.9576
1998M3	1.8272	1.4563	2.1342	1.2851	1.9682
1998M4	1.8132	1.4591	2.1298	1.2982	1.9693
1998M5	1.7753	1.4649	2.1268	1.2928	1.9589
1998M6	1.7928	1.4524	2.1021	1.3102	1.9814
1998M7	1.7976	1.4973	2.1671	1.3604	2.0573
1998M8	1.7869	1.5417	2.254	1.3954	2.1145
1998M9	1.6990	1.5063	2.1867	1.3546	2.0548
1998M10	1.6381	1.48	2.15	1.3328	2.0218
1998M11	1.6827	1.4717	2.1404	1.3138	1.9929
1998M12	1.6698	1.5016	2.1792	1.3489	2.0462

 

Figure 3a.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (Japanese Yen), 1-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1989M12	143.69	137.1448	150.3042	135.8702	149.366
1990M1	144.98	137.0572	150.1959	136.0333	149.5454
1990M2	145.69	138.3936	151.6314	137.2494	150.8823
1990M3	153.31	139.0866	152.3841	137.9236	151.6234
1990M4	158.46	146.6827	160.6354	145.1402	159.9882
1990M5	154.04	151.6278	165.9187	150.0097	165.3559
1990M6	153.70	147.417	161.4377	145.8239	160.7419
1990M7	149.04	146.9849	161.1507	145.5035	159.9562
1990M8	147.46	142.2518	156.0701	141.0903	155.1046
1990M9	138.44	140.3027	154.2165	139.6026	153.4539
1990M10	129.59	131.1932	144.3835	130.8171	144.0557
1990M11	129.22	122.5274	134.9261	122.4619	134.855
1990M12	133.89	122.3886	134.7416	122.1073	134.4645
1991M1	133.70	126.9561	139.8379	126.5199	139.3237
1991M2	130.54	126.7481	139.5959	126.3429	139.1288
1991M3	137.39	123.6171	136.2016	123.3591	135.843
1991M4	137.11	130.1833	143.4535	129.8266	142.9793
1991M5	138.22	129.8952	143.1449	129.5672	142.6936
1991M6	139.75	131.0981	144.4562	130.6079	143.8398
1991M7	137.83	132.6852	146.2326	132.0658	145.4453
1991M8	136.82	130.782	144.1128	130.2427	143.4376
1991M9	134.30	129.7122	142.9256	129.2954	142.3943
1991M10	130.77	127.3166	140.3002	126.9128	139.7702
1991M11	129.63	123.5467	136.1673	123.569	136.0878
1991M12	128.04	122.755	135.2936	122.4986	134.9089
1992M1	125.46	121.3312	133.7332	120.989	133.2464
1992M2	127.70	118.7918	130.9355	118.5577	130.5688
1992M3	132.86	120.9751	133.3118	120.6749	132.9004
1992M4	133.54	125.7367	138.524	125.5495	138.2689
1992M5	130.77	126.1439	138.9549	126.1914	138.9758
1992M6	126.84	123.6608	136.3031	123.569	136.0878
1992M7	125.88	120.1263	132.4361	119.8571	131.9997
1992M8	126.23	119.3691	131.5722	118.9496	131.0004
1992M9	122.60	119.3371	131.5382	119.2831	131.3677
1992M10	121.17	115.9207	127.7959	115.8504	127.5872
1992M11	123.88	114.63	126.3828	114.5029	126.1031
1992M12	124.04	117.1965	129.206	117.0616	128.921
1993M1	124.99	117.1537	129.2048	117.2138	129.0887
1993M2	120.76	118.3556	130.3984	118.1081	130.0735
1993M3	117.02	114.2326	125.9636	114.1142	125.6751
1993M4	112.41	110.5448	121.905	110.5757	121.778
1993M5	110.34	106.04	116.9426	106.2293	116.9913
1993M6	107.41	104.3046	115.0288	104.2717	114.8354
1993M7	107.69	101.5958	112.0406	101.5042	111.7875
1993M8	103.77	101.609	112.0558	101.7684	112.0786
1993M9	105.57	98.0503	108.129	98.062	107.9966
1993M10	107.02	99.7681	110.0253	99.7633	109.8702
1993M11	107.88	101.3172	111.7358	101.1293	111.3747
1993M12	109.91	102.1469	112.6526	101.9416	112.2692
1994M1	111.44	103.7283	114.2849	103.8658	114.3884
1994M2	106.30	105.3095	116.0551	105.3091	115.9779
1994M3	105.10	100.5524	110.7573	100.454	110.631
1994M4	103.48	99.1887	109.2225	99.3153	109.3769
1994M5	103.75	97.6774	107.5405	97.7878	107.6947
1994M6	102.53	97.9322	107.8845	98.0424	107.975
1994M7	98.45	97.1121	107.0364	96.8923	106.7084
1994M8	99.94	93.1883	102.7285	93.028	102.4526
1994M9	98.77	94.3563	103.9987	94.4433	104.0113
1994M10	98.35	93.2214	102.7473	93.3355	102.7912
1994M11	98.04	92.6616	102.1547	92.935	102.3502
1994M12	100.18	92.7087	102.2432	92.6473	102.0334
1995M1	99.77	94.7109	104.4119	94.6703	104.2613
1995M2	98.24	94.3687	104.0684	94.2829	103.8347
1995M3	90.52	93.1143	102.6935	93	102.2376
1995M4	83.69	85.5857	94.4056	85.6955	94.2075
1995M5	85.11	78.8412	86.9637	79.0594	87.095
1995M6	84.64	80.2507	88.5377	80.3989	88.5706
1995M7	87.40	79.8708	88.0313	79.9579	88.076
1995M8	94.74	82.5905	91.0398	82.5662	90.9491
1995M9	100.55	89.5286	98.7561	89.4965	98.593
1995M10	100.84	94.8839	104.681	94.9927	104.9307
1995M11	101.94	95.4478	105.3138	95.259	105.2249
1995M12	101.85	96.3677	106.2827	96.3031	106.0913
1996M1	105.75	96.1693	106.0504	96.2164	105.9959
1996M2	105.79	100.1183	110.4129	99.903	110.0572
1996M3	105.94	100.1107	110.4005	99.943	110.1012
1996M4	107.20	100.1748	110.4614	100.083	110.2555
1996M5	106.34	101.2044	111.6215	101.271	111.5642
1996M6	108.96	100.4388	110.7321	100.454	110.6642
1996M7	109.19	103.0411	113.5438	102.9352	113.3862
1996M8	107.87	103.2287	113.739	103.1516	113.6246
1996M9	109.93	101.9811	112.3662	101.9008	112.2468
1996M10	112.41	103.7504	114.3077	103.8451	114.3884
1996M11	112.30	106.203	117.0035	106.1974	116.9797
1996M12	113.98	106.268	117.0627	106.0913	116.8627
1997M1	117.91	107.7368	118.645	107.8995	118.6052
1997M2	122.96	111.5738	122.8637	111.62	122.6948
1997M3	122.77	116.3477	128.1381	116.4079	127.9705
1997M4	125.64	116.0634	127.8241	116.2217	127.7659
1997M5	119.19	118.0081	129.999	118.9377	130.7517
1997M6	114.29	112.3734	123.9277	112.832	124.0395
1997M7	115.38	107.9781	119.0616	108.1912	118.9377
1997M8	117.93	109.1976	120.4083	109.2239	120.073
1997M9	120.89	111.3451	122.7849	111.6423	122.7316
1997M10	121.06	113.7762	125.4698	114.4456	125.8134
1997M11	125.38	114.2197	125.9511	114.606	125.9897
1997M12	129.73	118.6781	130.869	118.6882	130.4774
1998M1	129.55	122.415	135.0145	122.8176	135.0169
1998M2	125.85	122.3367	134.922	122.6457	134.828
1998M3	129.08	118.8284	131.018	119.1401	130.9742
1998M4	131.75	121.5706	134.0612	122.1928	134.3301
1998M5	134.90	124.2904	137.0916	124.7236	137.1123
1998M6	140.33	127.2105	140.3396	127.7021	140.3866
1998M7	140.79	132.9272	146.6644	132.8473	146.0428
1998M8	144.68	133.49	147.289	133.2864	146.5256
1998M9	134.48	136.9805	151.1832	136.9615	150.5658
1998M10	121.05	126.39	139.4667	127.0397	139.9381
1998M11	120.29	113.383	125.1976	114.2284	125.9645
1998M12	117.07	113.0988	124.8832	113.511	125.1734
1999M1	113.29	110.2031	121.6818	110.4762	121.8268
1999M2	116.67	106.9306	117.9395	106.9114	117.8721
1999M3	119.47	110.014	121.3704	110.0902	121.3768
1999M4	119.77	112.554	124.1653	112.7418	124.3002
1999M5	122.00	112.8164	124.4674	113.024	124.6114
1999M6	120.72	114.6104	126.2066	115.1229	126.9255
1999M7	119.33	113.6874	125.1962	113.9204	125.5997
1999M8	113.23	112.8048	124.2021	112.7305	124.1512
1999M9	106.88	106.3352	117.0174	106.9648	117.8014
1999M10	105.97	100.6024	110.7098	100.8869	111.1966
1999M11	104.65	99.5617	109.6013	100.033	110.2555
1999M12	102.58	98.9207	108.9049	98.7805	108.875
2000M1	105.30	96.6317	106.3703	96.8245	106.7191
2000M2	109.39	99.2742	109.3461	99.3948	109.5521
2000M3	106.31	103.6896	114.2491	103.2548	113.8065
2000M4	105.63	100.4624	110.6755	100.3536	110.6088
2000M5	108.32	99.0885	109.1598	99.7034	109.8922
2000M6	106.13	101.8785	112.156	102.2479	112.6967
2000M7	108.21	100.4251	110.6095	100.1831	110.421
2000M8	108.08	102.268	112.6512	102.1457	112.584
2000M9	106.84	101.3201	111.6058	102.0232	112.449
2000M10	108.44	101.0619	111.3277	100.8465	111.1522
2000M11	109.01	102.1351	112.5001	102.3604	112.8207
2000M12	112.21	102.8055	113.245	102.8941	113.4089
2001M1	116.67	105.3585	116.0631	105.9217	116.7459
2001M2	116.23	110.0807	121.49	110.1232	121.3768
2001M3	121.51	109.802	121.0614	109.7165	120.9165
2001M4	123.77	114.3424	126.172	114.6977	126.4188
2001M5	121.77	116.2528	128.27	116.8277	128.7664
2001M6	122.35	114.4249	126.5201	114.9388	126.6845
2001M7	124.50	115.037	127.2639	115.4919	127.2941
2001M8	121.37	116.5541	128.9739	117.519	129.5284
2001M9	118.61	113.2041	125.2614	114.5601	126.2672
2001M10	121.45	111.7232	123.1354	111.9553	123.3962
2001M11	122.41	113.2307	124.8044	114.6404	126.3556
2001M12	127.59	114.8505	126.5979	115.5496	127.3577
2002M1	132.68	118.8042	131.4102	120.4338	132.7808
2002M2	133.64	124.2441	137.4541	125.236	138.0754
2002M3	131.06	125.6902	139.0484	126.141	139.0731
2002M4	130.77	122.6819	135.7181	123.7174	136.4011
2002M5	126.38	122.2512	135.2525	123.4332	136.0878
2002M6	123.29	117.9296	130.4507	119.3905	131.5254
2002M7	117.90	115.6443	127.8922	116.4661	128.3037
2002M8	118.99	111.0714	122.8514	111.3747	122.6948
2002M9	121.08	111.6028	123.4421	112.4041	123.8288
2002M10	123.91	114.0112	125.7751	114.6289	126.0149
2002M11	121.61	116.862	128.9033	117.3077	128.9597
2002M12	121.89	114.3845	126.1665	115.1229	126.5579
2003M1	118.81	114.5082	126.3267	115.388	126.8493
2003M2	119.34	112.4318	124.0381	112.4715	123.6432
2003M3	118.69	113.413	125.0666	113.0353	124.2008
2003M4	119.90	112.0637	123.5696	112.4153	123.5196
2003M5	117.37	112.4222	123.9644	113.5678	124.786
2003M6	118.33	110.19	121.5344	111.1633	122.1439
2003M7	118.70	111.8546	123.2355	112.0786	123.1127
2003M8	118.66	112.0936	123.4951	112.4265	123.4949
2003M9	114.80	111.9399	123.316	112.3928	123.4579
2003M10	109.50	108.3948	119.4097	108.7335	119.4383
2003M11	109.18	103.1588	113.6451	103.7102	113.9204
2003M12	107.74	103.2173	113.6914	103.5754	113.5905
2004M1	106.27	101.4557	111.7814	102.207	112.0898
2004M2	106.71	100.9333	111.1857	100.8163	110.5646
2004M3	108.52	101.1693	111.4677	101.2305	111.0188
2004M4	107.66	102.8155	113.2199	102.9455	112.8997
2004M5	112.20	101.8971	112.202	102.1355	112.0113
2004M6	109.43	106.357	117.0372	106.442	116.9329
2004M7	109.49	103.8684	113.785	103.8347	114.0458
2004M8	110.23	103.9626	113.8367	104.4282	114.1028
2004M9	110.09	104.4655	114.3732	105.1408	114.8814
2004M10	108.78	104.3049	114.1881	105.0042	114.7321
2004M11	104.70	103.1904	112.9815	103.7516	113.3636
2004M12	103.81	99.5597	109.0679	99.8631	109.1147
2005M1	103.34	98.6076	108.0353	99.0178	108.1912
2005M2	104.94	98.4622	107.8653	98.5634	107.6947
2005M3	105.25	100.4329	109.9926	100.093	109.1802
2005M4	107.19	100.3011	109.8516	100.3837	109.4973
2005M5	106.60	102.2425	111.9461	102.2376	111.5196
2005M6	108.75	100.8962	110.4973	101.6769	110.9079
2005M7	111.95	103.5451	113.4482	103.7309	113.1484
2005M8	110.61	106.7007	116.8631	106.7831	116.4777
2005M9	111.24	105.264	115.3071	105.4988	115.0768
2005M10	114.87	106.3474	116.4835	106.1019	115.7346
2005M11	118.45	109.1451	119.4848	109.563	119.51
2005M12	118.46	112.01	122.5664	112.9788	123.2359
2006M1	115.48	111.7955	122.1451	112.9901	123.2359
2006M2	117.86	110.413	120.5558	110.1453	120.085
2006M3	117.28	112.5427	122.8268	112.4153	122.5599
2006M4	117.07	111.7563	121.9831	111.8658	121.9608
2006M5	111.73	111.9335	122.1922	111.6646	121.7415
2006M6	114.63	106.1331	115.9514	106.5698	116.1869

 

Figure 3b.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (Japanese Yen), 6-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1990M5	154.04	126.5472	171.9341	118.5696	160.5973
1990M6	153.70	125.2612	170.1406	118.712	160.7901
1990M7	149.04	126.4843	171.7585	119.7732	162.2276
1990M8	147.46	126.815	171.2492	120.3615	163.0244
1990M9	138.44	133.277	180.0206	126.6592	171.2972
1990M10	129.59	136.6261	185.0265	130.9087	177.0443
1990M11	129.22	133.7383	180.8796	127.2559	172.1042
1990M12	133.89	134.3002	181.6167	126.9762	171.726
1991M1	133.70	130.809	176.8975	123.125	166.5174
1991M2	130.54	129.4392	174.9506	121.8268	164.7617
1991M3	137.39	120.7606	163.0848	114.3656	154.671
1991M4	137.11	112.4895	151.8927	106.155	144.7923
1991M5	138.22	111.7187	150.9232	105.8476	144.373
1991M6	139.75	114.3646	153.4259	109.6726	149.5903
1991M7	137.83	113.0787	151.925	109.5192	149.381
1991M8	136.82	110.3666	148.0935	106.9327	145.8531
1991M9	134.30	114.8958	154.0108	112.539	153.4999
1991M10	130.77	115.1056	154.4195	112.3142	153.1932
1991M11	129.63	116.8139	156.7761	113.2163	154.4237
1991M12	128.04	118.9457	159.6412	114.48	156.1473
1992M1	125.46	117.3254	157.502	112.8997	153.9919
1992M2	127.70	116.4048	156.4145	112.0786	152.8719
1992M3	132.86	114.6656	153.9692	110.0132	150.0547
1992M4	133.54	110.3883	148.0732	107.1147	146.1013
1992M5	130.77	110.5288	148.4431	106.1868	144.8357
1992M6	126.84	109.4923	147.1243	104.8782	143.0508
1992M7	125.88	106.8164	143.6438	102.7707	140.1762
1992M8	126.23	108.481	145.9819	104.6059	142.6794
1992M9	122.60	111.8873	150.6733	108.8314	148.4428
1992M10	121.17	111.6999	150.5394	109.3879	149.2018
1992M11	123.88	110.6138	149.2283	107.1147	146.1013
1992M12	124.04	108.06	145.8709	103.897	141.7124
1993M1	124.99	107.8631	145.6424	103.1104	140.6395
1993M2	120.76	106.9894	144.4981	103.3995	141.0339
1993M3	117.02	103.8831	140.3355	100.4239	136.9752
1993M4	112.41	103.2573	139.4883	99.2558	135.3819
1993M5	110.34	105.4319	142.3876	101.4737	138.4072
1993M6	107.41	104.9019	141.5134	101.6057	138.5872
1993M7	107.69	106.6669	143.9181	102.3809	139.6445
1993M8	103.77	102.793	138.6643	98.9189	134.9224
1993M9	105.57	99.7553	134.4642	95.8515	130.7386
1993M10	107.02	95.4726	128.7032	92.0839	125.5997
1993M11	107.88	94.5428	127.0667	90.387	123.2852
1993M12	109.91	92.5148	124.1955	87.988	120.013
1994M1	111.44	91.6733	123.1441	88.217	120.3254
1994M2	106.30	89.0754	119.514	85.0041	115.9432
1994M3	105.10	90.3351	121.2212	86.4789	117.9546
1994M4	103.48	92.1319	123.5695	87.663	119.5698
1994M5	103.75	92.8297	124.492	88.3671	120.5302
1994M6	102.53	93.4006	125.2466	90.0351	122.8053
1994M7	98.45	95.1951	127.5781	91.2862	124.5117
1994M8	99.94	91.5529	122.4255	87.0776	118.7713
1994M9	98.77	90.0621	120.3844	86.0906	117.425
1994M10	98.35	88.6525	118.5669	84.7665	115.619
1994M11	98.04	88.9276	119.1328	84.9871	115.92
1994M12	100.18	88.7033	118.907	83.9902	114.5601
1995M1	99.77	85.0085	113.9959	80.6404	109.9912
1995M2	98.24	85.1437	114.1868	81.8673	111.6646
1995M3	90.52	84.181	112.749	80.907	110.3547
1995M4	83.69	83.4181	111.5872	81.2475	109.8812
1995M5	85.11	84.178	112.5195	81.1501	109.5411
1995M6	84.64	85.8576	114.644	83.0464	111.9329
1995M7	87.40	86.7068	115.2053	82.7066	111.475
1995M8	94.74	85.4649	113.3406	81.4346	109.7604
1995M9	100.55	79.1169	104.1444	75.0384	101.1394
1995M10	100.84	72.036	95.4649	69.3732	93.5036
1995M11	101.94	73.3869	96.6591	70.5485	95.0877
1995M12	101.85	72.4823	95.6661	70.1615	94.415
1996M1	105.75	74.5471	98.8398	72.4502	97.4072
1996M2	105.79	80.0589	106.4486	78.5315	105.5622
1996M3	105.94	84.2908	112.2547	83.3543	112.0449
1996M4	107.20	85.4273	114.3314	83.588	112.3816
1996M5	106.34	86.0601	115.8653	84.5041	113.8976
1996M6	108.96	86.0144	115.838	84.4281	113.966
1996M7	109.19	90.2576	121.5061	87.663	118.3327
1996M8	107.87	90.2467	121.6686	87.6981	118.38
1996M9	109.93	90.1027	121.6576	87.8209	118.5459
1996M10	112.41	90.9335	123.274	88.8634	119.7732
1996M11	112.30	90.343	122.3428	88.1465	118.7476
1996M12	113.98	92.7838	125.4779	90.3237	121.6807
1997M1	117.91	92.8799	125.4488	90.5136	121.9364
1997M2	122.96	91.8122	123.9509	89.416	120.4579
1997M3	122.77	93.1062	125.623	91.1221	122.7562
1997M4	125.64	95.2381	128.4716	93.1862	125.5369
1997M5	119.19	95.481	128.5838	93.0931	125.4115
1997M6	114.29	96.1425	129.8174	94.4811	127.2814
1997M7	115.38	99.5392	134.3288	97.7389	131.6702
1997M8	117.93	103.7282	139.7992	101.9314	137.593
1997M9	120.89	103.2951	139.2933	101.7684	137.373
1997M10	121.06	104.1555	140.3126	104.1466	140.5833
1997M11	125.38	100.3373	135.0831	98.8002	133.3664
1997M12	129.73	96.9513	130.5874	94.7366	127.881
1998M1	129.55	98.0285	132.1465	95.6408	129.1016
1998M2	125.85	99.8793	134.9306	97.7585	131.9602
1998M3	129.08	101.9423	136.3365	100.2132	135.1115
1998M4	131.75	102.4976	136.8599	100.3536	135.2061
1998M5	134.90	106.1825	141.7347	103.9282	140.0221
1998M6	140.33	109.4348	146.1606	107.544	144.9951
1998M7	140.79	109.3473	146.128	107.3935	144.9661
1998M8	144.68	106.6326	142.5158	104.3238	140.8225
1998M9	134.48	109.1187	145.7877	106.9969	144.4307
1998M10	121.05	111.4806	148.8975	109.213	147.4221
1998M11	120.29	113.9107	152.4394	111.8211	150.9427
1998M12	117.07	118.3852	158.6411	116.3264	157.0242
1999M1	113.29	118.8181	159.341	116.7109	157.5432
1999M2	116.67	122.3459	164.2871	119.929	161.8872
1999M3	119.47	114.2203	152.7245	111.475	150.295
1999M4	119.77	103.9364	138.6279	100.3436	135.2872
1999M5	122.00	103.0983	136.8734	99.7134	134.4376
1999M6	120.72	100.099	132.9064	97.0474	130.8432
1999M7	119.33	96.687	127.8758	93.5972	126.6212
1999M8	113.23	98.1602	130.2217	95.4879	130.3861
1999M9	106.88	99.9288	132.7275	97.7878	133.5265
1999M10	105.97	100.0689	132.8695	98.0326	133.8607
1999M11	104.65	101.9217	135.233	99.8531	136.3466
1999M12	102.58	101.2602	134.2276	98.8101	134.9224
2000M1	105.30	100.3604	133.1777	97.6705	133.3664
2000M2	109.39	95.6659	126.8824	92.6751	126.5453
2000M3	106.31	91.0664	120.669	87.4791	119.4503
2000M4	105.63	90.1092	119.5899	86.7387	118.4392
2000M5	108.32	89.1164	118.5726	85.6526	116.9563
2000M6	106.13	87.0437	115.9217	83.9566	114.6404
2000M7	108.21	88.8145	118.3677	86.1853	117.6836
2000M8	108.08	91.7752	122.2908	89.5323	122.2539
2000M9	106.84	89.5391	119.5438	87.0167	118.8188
2000M10	108.44	89.0509	118.998	86.4529	118.049
2000M11	109.01	91.1302	121.5668	88.6592	121.0617
2000M12	112.21	89.8777	119.952	86.8689	118.617
2001M1	116.67	91.1593	121.7181	88.5706	120.9407
2001M2	116.23	90.9983	121.4626	88.4644	120.7956
2001M3	121.51	90.0906	119.9845	87.4441	119.4025
2001M4	123.77	91.2234	121.4928	88.7568	121.1949
2001M5	121.77	91.9352	122.034	89.2195	121.8268
2001M6	122.35	94.5181	124.8467	91.8448	125.4115
2001M7	124.50	97.3	128.035	95.4879	130.3861
2001M8	121.37	96.9839	127.0561	95.1353	129.9046
2001M9	118.61	99.9774	131.118	99.4545	135.9653
2001M10	121.45	101.3019	132.5279	101.3014	138.9758
2001M11	122.41	99.8105	130.6722	99.6635	136.7289
2001M12	127.59	100.155	130.5393	100.1431	137.3868
2002M1	132.68	100.8215	131.0986	101.9008	139.7982
2002M2	133.64	98.6228	128.0512	99.3352	136.2784
2002M3	131.06	97.6201	126.2849	97.0766	133.1798
2002M4	130.77	98.7388	127.338	99.4048	136.3738
2002M5	126.38	99.5688	128.3102	100.1932	137.4555
2002M6	123.29	101.8414	131.4407	104.4282	143.2656
2002M7	117.90	105.0004	137.1861	109.6069	148.9782
2002M8	118.99	105.613	138.1444	110.7749	150.0547
2002M9	121.08	104.3139	136.3466	109.3223	147.1717
2002M10	123.91	104.5984	136.6671	109.0929	146.8336
2002M11	121.61	102.2895	133.1046	106.0171	141.911
2002M12	121.89	101.0627	131.3248	104.1571	138.4349
2003M1	118.81	97.7554	127.1707	100.1631	132.3831
2003M2	119.34	98.1064	127.7883	101.099	133.6066
2003M3	118.69	99.7366	129.8777	103.0382	135.9653
2003M4	119.90	101.7145	132.5327	106.2931	139.1427
2003M5	117.37	100.3467	130.782	104.3134	136.5512
2003M6	118.33	100.7549	131.3324	104.5536	136.8657
2003M7	118.70	98.6095	128.5949	101.911	133.4064
2003M8	118.66	98.7963	128.8288	102.3706	134.0081
2003M9	114.80	98.2868	128.1687	101.8092	133.273
2003M10	109.50	99.1609	129.3144	102.8529	134.6394
2003M11	109.18	97.5554	127.2198	100.6753	131.7887
2003M12	107.74	98.1913	128.0256	101.5042	132.8738
2004M1	106.27	98.3694	128.577	101.8193	133.2864
2004M2	106.71	98.3625	128.7925	101.7888	133.2464
2004M3	108.52	95.68	125.3265	98.4747	128.9081
2004M4	107.66	91.9363	120.3391	93.9253	122.9527
2004M5	112.20	91.7817	120.0967	93.6533	122.5967
2004M6	109.43	90.657	118.4952	92.416	120.9769
2004M7	109.49	89.77	117.1685	91.1585	119.3309
2004M8	110.23	89.999	117.4514	91.533	119.8211
2004M9	110.09	91.0782	118.9395	93.2515	121.8511
2004M10	108.78	90.296	118.0491	92.5177	120.8923
2004M11	104.70	93.439	122.1081	96.4187	125.9897
2004M12	103.81	91.3371	119.3785	94.0381	122.879
2005M1	103.34	91.3434	119.4222	94.0851	122.9404
2005M2	104.94	91.7328	119.9194	94.7271	123.7793
2005M3	105.25	91.6662	119.8852	94.604	123.6185
2005M4	107.19	90.951	119.0036	93.4756	122.1439
2005M5	106.60	88.247	115.4753	89.9721	117.566
2005M6	108.75	87.5134	114.5226	89.2106	116.5709
2005M7	111.95	87.2688	114.1256	88.8012	116.0359
2005M8	110.61	88.625	115.8657	90.1793	117.8367
2005M9	111.24	88.4136	115.5687	90.4412	118.1789
2005M10	114.87	89.6973	117.2266	92.1115	120.3615
2005M11	118.45	88.7022	115.8948	91.6063	119.7014
2005M12	118.46	90.761	118.5663	93.4569	122.1195
2006M1	115.48	93.0582	121.7595	96.2068	125.7128
2006M2	117.86	91.9929	120.5355	95.0497	123.6185
2006M3	117.28	92.7904	121.7512	95.593	124.3251
2006M4	117.07	94.7149	124.2568	98.7114	128.3807
2006M5	111.73	96.896	127.4209	101.7888	132.3831
2006M6	114.63	96.7553	127.3269	101.799	132.3963

 

Figure 3c.  Forecast Intervals of Benchmark Taylor Rule and Random Walk (Japanese Yen), 12-month-ahead forecast

Note: To facilitate graphical comparisons, the 6- and 12-month-ahead forecast intervals of the random walk have been relocated such that they have the same center as the intervals of the Taylor rule model.

Date	Realization	Taylor Rule 5%	Taylor Rule 95%	RW 5%	RW 95%
1990M11	129.22	119.9269	189.1412	101.8499	167.2016
1990M12	133.89	119.0975	187.9188	101.9722	167.4023
1991M1	133.70	120.0614	189.4829	102.8838	168.8988
1991M2	130.54	120.2957	189.7437	103.3891	169.7285
1991M3	137.39	126.1044	198.7747	108.7988	179.5404
1991M4	137.11	128.6518	202.5936	112.449	185.8983
1991M5	138.22	126.0977	199.0752	109.3113	180.7112
1991M6	139.75	125.5027	198.5642	109.0711	180.314
1991M7	137.83	122.6752	193.6962	105.7629	174.845
1991M8	136.82	121.1749	191.3707	104.6478	173.0015
1991M9	134.30	113.7143	179.1286	98.2387	162.4061
1991M10	130.77	107.5264	168.7311	91.9643	152.0334
1991M11	129.63	106.8016	167.6006	91.6979	151.5931
1991M12	128.04	108.534	169.6307	95.0117	157.0713
1992M1	125.46	106.8953	167.7761	94.8788	156.8516
1992M2	127.70	104.6625	164.229	92.6381	153.1473
1992M3	132.86	107.2349	168.2753	97.4949	161.1765
1992M4	133.54	106.3819	166.2427	97.3001	160.8545
1992M5	130.77	107.1213	166.6683	98.0816	162.1465
1992M6	126.84	108.8759	168.663	99.1764	163.9563
1992M7	125.88	106.8214	165.353	97.8074	161.6931
1992M8	126.23	105.8252	163.6627	97.096	160.517
1992M9	122.60	104.8871	162.0557	95.3067	157.559
1992M10	121.17	101.345	156.5641	92.7957	153.4078
1992M11	123.88	102.4156	158.1051	91.9919	152.079
1992M12	124.04	102.2723	157.3334	90.8582	150.2048
1993M1	124.99	100.1645	154.0562	89.0324	147.1864
1993M2	120.76	101.6537	156.0648	90.6223	149.8148
1993M3	117.02	104.1865	159.6661	94.2829	155.8665
1993M4	112.41	103.6534	158.3628	94.765	156.6635
1993M5	110.34	103.7239	158.5979	92.7957	153.4078
1993M6	107.41	101.5174	155.4807	90.0081	148.7995
1993M7	107.69	101.6095	155.4439	89.3267	147.6729
1993M8	103.77	100.4867	153.7348	89.5771	148.087
1993M9	105.57	97.1502	149.3427	86.9993	143.8254
1993M10	107.02	97.4055	150.0489	85.9873	142.1524
1993M11	107.88	99.2957	153.0368	87.9088	145.329
1993M12	109.91	98.493	152.0059	88.0232	145.518
1994M1	111.44	100.2269	154.6441	88.6947	146.6282
1994M2	106.30	96.6196	149.1789	85.6955	141.6699
1994M3	105.10	94.4201	145.8637	83.0381	137.2769
1994M4	103.48	90.1753	139.1901	79.7742	131.881
1994M5	103.75	89.3975	137.71	78.3041	129.4507
1994M6	102.53	88.0407	135.9688	76.2258	126.0149
1994M7	98.45	86.6753	134.3791	76.4242	126.3429
1994M8	99.94	84.8642	131.8639	73.6409	121.7415
1994M9	98.77	85.7114	133.3108	74.9184	123.8536
1994M10	98.35	87.4345	136.1068	75.9443	125.5495
1994M11	98.04	87.7933	136.7141	76.5543	126.5579
1994M12	100.18	88.0434	136.7725	77.9993	128.9468
1995M1	99.77	89.9991	139.6722	79.0832	130.7386
1995M2	98.24	85.8894	133.6655	75.4372	124.7111
1995M3	90.52	85.0029	131.0134	74.5821	123.2975
1995M4	83.69	84.2589	128.8576	73.6703	121.4011
1995M5	85.11	86.5461	129.4535	74.7912	121.7172
1995M6	84.64	86.7195	128.3936	74.7388	120.2893
1995M7	87.40	83.5581	123.5857	71.758	115.4919
1995M8	94.74	84.8298	122.9645	72.8498	117.249
1995M9	100.55	84.1623	121.9483	71.9952	115.8736
1995M10	100.84	83.2353	120.4482	71.6863	115.3764
1995M11	101.94	84.0288	121.4351	71.4644	115.0193
1995M12	101.85	85.3146	123.1535	73.0249	117.5307
1996M1	105.75	86.3606	124.5625	72.7261	117.0499
1996M2	105.79	84.2589	121.4521	71.6075	115.2496
1996M3	105.94	77.4991	111.8336	65.9832	106.1974
1996M4	107.20	69.7317	102.6051	61.0016	98.1798
1996M5	106.34	70.8602	104.6528	62.0351	99.8431
1996M6	108.96	70.238	103.7775	61.6948	98.6028
1996M7	109.19	72.4342	106.1658	63.7074	101.6057
1996M8	107.87	77.3663	113.0743	69.0548	109.6288
1996M9	109.93	81.2467	118.3925	73.2956	116.1869
1996M10	112.41	81.9867	119.5704	73.5011	115.6999
1996M11	112.30	82.4019	119.8586	74.3066	116.7109
1996M12	113.98	81.893	118.8233	74.2398	116.501
1997M1	117.91	85.7119	124.2632	77.0843	120.9648
1997M2	122.96	85.6593	124.2164	77.1152	121.0132
1997M3	122.77	85.4524	124.012	77.2232	121.2919
1997M4	125.64	85.7739	125.6751	78.1398	123.0019
1997M5	119.19	85.2217	126.2286	77.5095	122.8667
1997M6	114.29	87.3307	131.1966	79.424	126.6719
1997M7	115.38	87.1891	131.2292	79.5909	127.205
1997M8	117.93	86.2233	129.7163	78.6258	125.6625
1997M9	120.89	87.0336	130.9527	80.126	128.0601
1997M10	121.06	88.7809	133.4524	81.941	130.9611
1997M11	125.38	89.2273	133.9948	81.8591	130.8302
1997M12	129.73	89.4459	134.4356	83.0797	132.7808
1998M1	129.55	92.1231	138.4039	85.9444	137.3593
1998M2	125.85	95.63	143.9001	89.6309	143.2512
1998M3	129.08	95.1319	142.8929	89.4876	143.0222
1998M4	131.75	95.835	144.2101	91.5788	147.0393
1998M5	134.90	92.7928	139.354	86.8776	139.491
1998M6	140.33	90.3624	135.8651	83.3043	133.7537
1998M7	140.79	91.2778	137.3715	84.0994	135.0304
1998M8	144.68	92.9861	140.1152	85.9615	138.0202
1998M9	134.48	95.057	143.145	88.12	141.4859
1998M10	121.05	95.8515	144.9603	88.2435	141.6841
1998M11	120.29	99.6553	149.7976	91.3867	146.7309
1998M12	117.07	102.443	153.7653	94.5662	151.8358
1999M1	113.29	101.9087	153.2802	94.4339	151.6234
1999M2	116.67	100.073	150.524	91.7346	147.2895
1999M3	119.47	102.2854	153.7302	94.0851	151.0635
1999M4	119.77	103.7301	157.4173	96.0338	154.1922
1999M5	122.00	106.1261	159.6394	98.3271	157.8744
1999M6	120.72	109.0931	164.0652	102.2888	164.2353
1999M7	119.33	108.7053	163.8753	102.6269	164.7781
1999M8	113.23	111.8497	169.238	105.4567	169.5588
1999M9	106.88	106.2428	160.5039	98.0228	157.6063
1999M10	105.97	98.3217	148.3382	88.2347	141.8684
1999M11	104.65	97.5761	146.8559	87.6805	140.9775
1999M12	102.58	95.3679	143.041	85.3363	137.2083
2000M1	105.30	92.3543	138.0084	82.5827	132.7808
2000M2	109.39	94.3502	140.5712	85.0382	136.7289
2000M3	106.31	95.9133	142.6704	87.0863	140.0221
2000M4	105.63	96.3669	143.1689	87.3043	140.3726
2000M5	108.32	97.6358	145.7462	88.9256	142.9793
2000M6	106.13	96.5183	143.1834	87.9968	141.4859
2000M7	108.21	94.9051	140.6468	86.9819	139.8541
2000M8	108.08	91.6146	135.1967	82.5331	132.7012
2000M9	106.84	87.6588	129.1037	77.9058	125.2611
2000M10	108.44	87.1514	128.4722	77.2464	124.2008
2000M11	109.01	86.1511	127.0927	76.2792	122.6457
2000M12	112.21	84.7047	125.0664	74.7687	120.2172
2001M1	116.67	86.38	127.5995	76.7536	123.4085
2001M2	116.23	88.3445	130.3393	79.7343	128.2011
2001M3	121.51	86.3646	127.5877	77.494	124.5989
2001M4	123.77	86.4851	128.2428	76.9919	123.7917
2001M5	121.77	87.5732	130.7866	78.9567	126.9508
2001M6	122.35	86.1428	127.4977	77.3623	124.3873
2001M7	124.50	86.7698	125.9311	78.8778	126.824
2001M8	121.37	87.6861	127.3138	78.7832	126.6719
2001M9	118.61	86.064	123.2552	77.8746	125.211
2001M10	121.45	87.4853	125.3195	79.0436	127.0906
2001M11	122.41	88.4658	123.3907	79.4557	127.7532
2001M12	127.59	91.0381	124.2886	81.7937	131.5122
2002M1	132.68	92.2676	125.6516	85.0382	136.7289
2002M2	133.64	91.7911	124.2697	84.7241	136.2239
2002M3	131.06	93.7488	126.9106	88.5706	142.4085
2002M4	130.77	94.441	126.6749	90.2154	145.0531
2002M5	126.38	93.4343	125.6215	88.7568	142.7079
2002M6	123.29	93.0479	125.9778	89.1839	143.3946
2002M7	117.90	93.7099	126.8544	90.7492	145.9115
2002M8	118.99	92.9194	125.6962	88.4644	142.2378
2002M9	121.08	91.3919	124.0641	86.4529	139.0036
2002M10	123.91	92.4352	125.4883	94.0663	142.3374
2002M11	121.61	92.6413	126.1507	95.8036	143.4663
2002M12	121.89	94.5935	128.9616	100.9676	149.5304
2003M1	118.81	96.4432	131.7778	105.4461	155.4929
2003M2	119.34	97.1343	133.1306	106.6337	156.6165
2003M3	118.69	96.8017	132.4609	104.7315	153.6074
2003M4	119.90	97.668	133.3781	105.7523	153.2545
2003M5	117.37	96.668	131.3118	103.1413	148.1166
2003M6	118.33	96.3946	130.3061	101.1293	144.4885
2003M7	118.70	94.2765	127.18	96.9214	138.1721
2003M8	118.66	94.2317	127.4704	98.1307	139.4491
2003M9	114.80	95.4371	129.0806	99.953	141.911
2003M10	109.50	97.0536	130.9072	102.2888	145.2273
2003M11	109.18	95.7842	129.4644	100.3837	142.5225
2003M12	107.74	96.3846	130.1341	101.3419	142.8507
2004M1	106.27	94.5969	127.7515	98.7805	139.2401
2004M2	106.71	94.633	127.9591	99.226	139.8681
2004M3	108.52	93.9993	127.109	98.6817	139.101
2004M4	107.66	94.5641	127.8824	99.6935	140.5271
2004M5	112.20	93.3974	126.292	97.5827	137.5517
2004M6	109.43	93.9791	127.0752	98.3861	138.6843
2004M7	109.49	93.9196	127.1953	99.6436	139.1149
2004M8	110.23	94.7317	127.3732	99.893	139.0731
2004M9	110.09	92.8407	124.4549	96.66	134.5452
2004M10	108.78	89.5846	120.1628	92.3975	128.3293
2004M11	104.70	89.5739	120.0481	92.4622	127.9577
2004M12	103.81	88.5943	118.6547	91.2406	126.2672
2005M1	103.34	87.7321	117.4317	89.9991	124.5491
2005M2	104.94	88.1639	117.9983	90.3689	125.0608
2005M3	105.25	88.8312	118.9476	91.8999	127.1796
2005M4	107.19	87.9718	117.8578	91.1768	126.1788
2005M5	106.60	90.4506	121.1065	95.0212	131.4991
2005M6	108.75	88.5691	118.5889	92.6751	128.2524
2005M7	111.95	88.5522	118.6342	92.7215	128.3165
2005M8	110.61	88.8437	118.9948	93.3541	129.192
2005M9	111.24	88.7335	118.9091	93.2328	129.0242
2005M10	114.87	88.2904	118.3191	92.1207	127.4852
2005M11	118.45	86.0486	115.3249	88.6681	122.7071
2005M12	118.46	85.5127	114.6032	87.9176	121.6685
2006M1	115.48	85.147	114.1141	87.5141	121.1101
2006M2	117.86	86.1565	115.3738	88.8723	122.9896
2006M3	117.28	85.9235	115.0455	89.1304	123.3468
2006M4	117.07	86.7376	116.1665	90.7765	125.6248
2006M5	111.73	86.0627	115.2782	90.2786	124.9358
2006M6	114.63	87.6359	117.6383	92.1023	127.4597

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Rossi, B. (2005) "Testing Long-horizon Predictive Ability with High Persistence, and the Meese-Rogoff Puzzle," International Economic Review 46(1), 61-92.

Sarno, L. and G. Valente (2005) "Empirical Exchange Rate Models and Currency Risk: Some Evidence from Density Forecasts," Journal of International, Money and Finance 24(2), 363-385.

West, K. D. (1996) "Asymptotic Inference About Predictive Ability," Econometrica 64, 1067-1084.

Wu, J. J. (2007) "Robust Semiparametric Forecast Intervals," Manuscript, the Federal Reserve Board.

APPENDIX

A.1  Monetary and Taylor Rule Models

In this section, we describe the monetary and Taylor rule models used in the paper.

A.1.1  Monetary Model

Assume the money market clearing condition in the home country is

$$m_t=p_t+\gamma y_t-\alpha i_t+v_t,$$

where $m_t$ is the log of money supply, $p_t$ is the log of aggregate price, $i_t$ is the nominal interest rate, $y_t$ is the log of output, and $v_t$ is money demand shock. A symmetric condition holds in the foreign country and we use an asterisk in subscript to denote variables in the foreign country. Subtracting foreign money market clearing condition from the home, we have

$$i_t-i_t^\ast=\frac{1}{\alpha}\left[-(m_t-m_t^\ast)+(p_t-p_t^\ast)+\gamma(y_t-y_t^\ast)+(v_t-v_t^\ast)\right].$$ (A.1.1)

The nominal exchange rate is equal to its purchasing power value plus the real exchange rate:

$$s_t=p_t-p_t^\ast+q_t.$$ (A.1.2)

The uncovered interest rate parity in financial market takes the form

$$E_ts_{t+1}-s_t=i_t-i_t^\ast+\rho_t,$$ (A.1.3)

where $\rho_t$ is the uncovered interest rate parity shock. Substituting equations (A.1.1) and (A.1.2) into (A.1.3), we have

$$s_t=(1-b)\left[m_t-m_t^\ast-\gamma(y_t-y_t^\ast)+q_t-(v_t-v_t^\ast)\right]-b\rho_t+bE_ts_{t+1},$$ (A.1.4)

where $b=\alpha/(1+\alpha)$. Solving $s_t$ recursively and applying "no-bubbles" condition, we have

$$s_t=E_t\left\{(1-b)\sum_{j=0}^\infty b^j\left[m_{t+j}-m_{t+1}^\ast-\gamma(y_{t+j}-y_{t+1}^\ast)+q_{t+1}-(v_{t+j}-v_{t+j}^\ast)\right]-b\sum_{j=1}^\infty b^{j}\rho_{t+j}\right\}.$$ (A.1.5)

In the standard monetary model, such as Mark (1995), purchasing power parity ($q_t=0$) and uncovered interest rate parity hold ($\rho_t=0$). Furthermore, it is assumed that the money demand shock is zero ( $v_t=v_t^\ast=0$) and $\gamma=1$. Equation (A.1.5) reduces to

$$s_{t}=E_{t}\left\{(1-b)\sum_{j=0}^{\infty}b^{j}\left(m_{t+j}-m^{*}_{t+j}-(y_{t+j}-y^{*}_{t+j})\right)\right\}.$$

A.1.2  Taylor Rule Model

We follow Engel and West (2005) to assume that both countries follow the Taylor rule and the foreign country targets the exchange rate in its Taylor rule. The interest rate differential is

$$i_t-i_t^\ast=\delta_s(s_t-\bar{s}_t^\ast)+\delta_y(y_t^{gap}-y_t^{gap\ast })+\delta_\pi(\pi_t-\pi_t^{\ast})+v_t-v_t^\ast,$$ (A.1.6)

where $\bar{s}_t^\ast$ is the targeted exchange rate. Assume that monetary authorities target the PPP level of the exchange rate: $\bar{s}_t^\ast=p_t-p_t^\ast$. Substituting this condition and the interest rate differential to the UIP condition, we have

$$s_t=(1-b)(p_t-p_t^\ast)-b\left[\delta_y(y_t^{gap}-y_t^{gap\ast })+\delta_\pi(\pi_t-\pi_t^{\ast})+v_t-v_t^\ast\right]-b\rho_t+bE_ts_{t+1},$$ (A.1.7)

where $b=\frac{1}{1+\delta_s}$. Assuming that uncovered interest rate parity hold ($\rho_t=0$) and monetary shocks are zero, equation (A.1.7)) reduces to the benchmark Taylor rule model in our paper:

$$s_{t}=E_{t}\left\{(1-b)\sum_{j=0}^{\infty}b^{j}(p_{t+j}-p^{*}_{t+j})-b\sum_{j=0}^{\infty}b^{j}(\delta_{y}(y^{gap}_{t+j}-y_{t+j}^{gap*})+\delta_{\pi}(\pi_{t+j}-\pi^{*}_{t+j}))\right\}.$$

 

A.2  Long-horizon Regressions

In this section, we derive long-horizon regressions for the monetary model and the benchmark Taylor rule model.

A.2.1  Monetary Model

In the monetary model,

$$s_{t}=E_{t}\left\{(1-b)\sum_{j=0}^{\infty}b^{j}\left(m_{t+j}-m^{*}_{t+j}-(y_{t+j}-y^{*}_{t+j})\right)\right\},$$

where $m_{t}$ and $y_{t}$ are logarithms of domestic money stock and output, respectively. The superscript $*$ denotes the foreign country. Money supplies ($m_t$ and $m_t^*$) and total outputs ($y_t$ and $y_t^*$) are usually I(1) variables. The general form considered in Engel, Wang, and Wu(2008) is:

$$s_{t} = (1-b)\sum_{j=0}^{\infty}b^{j}E_{t}\alpha^{'}\mathbf{D}_{t}$$

$$(I_{n}-\Phi(L))\Delta\mathbf{D}_{t} = \varepsilon_{t}$$ (A.2.1)

$$E(\varepsilon_{t+j}\vert\varepsilon_{t},\varepsilon_{t-1},...) \equiv E_{t}(\varepsilon_{t+j})=0, \forall j\ge 1,$$

where $n$ is the dimension of $\mathbf{D}_{t}$ and $I_n$ is an $n\times n$ identity matrix. $L$ is the lag operator and $\Phi(L)=\phi_{1}L+\phi_{2}L^{2}+...+\phi_{p}L^{p}$. Assume $\Phi(1)$ is non-diagonal and the covariance matrix of $\varepsilon_t$ is given by $\Omega=E_t[\varepsilon_t\varepsilon_t']$. We assume that the change of fundamentals follows a VAR(p) process in our setup. From proposition 1 of Engel, Wang, Wu (2008), we know that for a fixed discount factor $b$ and $p\ge 2$,

$$s_{t+h}-s_{t}=\beta_{h}z_{t}+\delta^{'}_{0,h}\Delta\mathbf{D}_{t}+...+\delta^{'}_{p-2,h}\Delta\mathbf{D}_{t-p+2}+\zeta_{t+h}$$

is a correctly specified regression where the regressors and errors do not correlate. In the case of $p=1$, the long-horizon regressions reduces to

$$s_{t+h}-s_{t}=\beta_{h}z_{t}+\zeta_{t+h}.$$

Following the literature, for instance Mark (1995), we do not include $\Delta\mathbf{D}_{t}$ and its lags in our long-horizon regressions. The monetary model can be written in the form of (A.2.1) by setting $\mathbf{D}_{t}=[m_{t}\quad m^{*}_{t}\quad y_{t}\quad y^{*}_{t}]'$ , $\alpha=[1\quad -1\quad -1\quad 1]'$. By definition, $z_{t}=s_{t}-(m_{t}-m^{*}_{t})+(y_{t}-y^{*}_{t})$. This corresponds to $\beta_{m,h}=1$, $\mathbf{X}_{m,t} =s_{t}-(m_{t}-m^{*}_{t})+(y_{t}-y^{*}_{t})$ in equation (1) of section 3.

A.2.2  Taylor Rule Model

In the Taylor rule model,

$$s_{t}=E_{t}\left\{(1-b)\sum_{j=0}^{\infty}b^{j}(p_{t+j}-p^{*}_{t+j})-b\sum_{j=0}^{\infty}b^{j}(\delta_{y}(y^{gap}_{t+j}-y_{t+j}^{gap*})+\delta_{\pi}(\pi_{t+j}-\pi^{*}_{t+j}))\right\},$$

where $p_{t}$, $y^{gap}_{t}$ and $\pi_{t}$ are domestic aggregate price, output gap and inflation rate, respectively. $\delta_{y}$ and $\delta_{\pi}$ are coefficients of the Taylor rule model. The aggregate prices $p_t$ and $p_t^\ast$ are usually I(1) variables. Inflation and output gap are more likely to be I(0). Engel, Wang, and Wu (2008) consider a setup which includes both stationary and non-stationary variables:

$$s_t=(1-b)\sum_{j=0}^\infty b^jE_t\left[f_{1t+j}\right] +b\sum_{j=0}^\infty b^jE_t\left[f_{2t+j}+u_{2t+j}\right]$$

$$f_{1t} =\alpha_1'\mathbf{D_t}\sim I(1)$$

$$f_{2t} =\alpha_2'\mathbf{\Delta D_t}\sim I(0)$$

$$u_{2t} =\alpha_3'\mathbf{\Delta D_t}\sim I(0)$$

$$(I_{n} -\Phi(L))\Delta\mathbf{X}_{t}=\varepsilon_{t},$$ (A.2.2)

where $f_{1t}$ and $f_{2t}$ ($u_{2t}$) are observable (unobservable) fundamentals. $\Delta\mathbf{D}_{t}$ is the first difference of $\mathbf{D}_{t}$, which contains I(1) economic variables.¹⁷ From proposition 2 of Engel, Wang, and Wu (2008), we know that for a fixed discount factor $b$ and $h\ge 2$,

$$s_{t+h}-s_{t}=\tilde{\beta}_{h}z_{t}+\sum_{k=0}^{p-1}\tilde{\delta}^{'}_{k,h}\Delta\mathbf{D}_{t-k}+\tilde{\zeta}_{t+h}$$ (A.2.3)

is a correctly specified regression, where the regressors and errors do not correlate. In the case of $p=1$, the long-horizon regressions reduces to

$$s_{t+h}-s_{t}=\tilde{\beta}_{h}z_{t}+\tilde{\zeta}_{t+h}.$$

The Taylor rule model can be written into the form of (A.2.2) by setting

$$\mathbf{D}_{t}=\left[p_{t}\quad p^{*}_{t}\quad \sum_{s=-\infty}^t y^{gap}_{s}\quad \sum_{s=-\infty}^ty^{gap*}_s,\sum_{s=-\infty}^t\pi_{s}\quad \sum_{s=-\infty}^t\pi^{*}_{s}\right]'.$$

By definition, $z_{t}=s_t-p_{t}+p^{*}_{t}+\frac{b}{1-b}(\delta_{y}(y^{gap}_{t}-y^{gap*}_{t})+\delta_{\pi}(\pi_{t}-\pi^{*}_{t}))$ . This corresponds to $\beta_{m,h}=[1\quad \frac{b}{1-b}\delta_y\quad \frac{b}{1-b}\delta_\pi]$ and $\mathbf{X}_{m,t}=[q_t\quad y_t^{gap}-y_t^{gap*}\quad \pi_t-\pi_t^*]$ , where $q_t=s_t-p_{t}+p^{*}_{t}$ is the real exchange rate. $\beta_{m,h}$ and $\mathbf{X}_{m,t}$ can be defined differently. For instance, $\beta_{m,h}=1$ and $\mathbf{X}_{m,t}=s_t-p_{t}+p^{*}_{t}+\frac{b}{1-b}(\delta_{y}(y^{gap}_{t}-y^{gap*}_{t})+\delta_{\pi}(\pi_{t}-\pi^{*}_{t}))$ . Our results do not change qualitatively under this alternative setup.

Footnotes

*  We thank Menzie Chinn, Charles Engel, Bruce Hansen, Mark Wynne, and Ken West for invaluable discussions. We would also like to thank seminar participants at the Dallas Fed for helpful comments. All views are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas, the Board of Governors or the Federal Reserve System. All GAUSS programs are available upon request. Email: [email protected] Address: Research Department, Federal Reserve Bank of Dallas, 2200 N. Pearl Street, Dallas, TX 75201 Phone: (214) 922-6471. Email: [email protected] Address: Board of Governors of the Federal Reserve System, 20th and C Streets, Washington, D.C. 20551 Phone: (202) 452-2556.  Return to text

1.  Chinn and Meese (1995), and MacDonald and Taylor (1994) find similar results. However, the long-horizon exchange rate predictability in Mark (1995) has been challenged by Kilian (1999) and Berkowitz and Giorgianni (2001) in succeeding studies. Return to text

2.  For brevity, we omit RS and simply say forecast intervals when we believe that it causes no confusion. Return to text

3.  We also tried the random walk with a drift. It does not change our results. Return to text

4.  Clarida, Gali, and Gertler (1998) find empirical support for forward looking Taylor rules. Forward looking Taylor rules are ruled out because they require forecasts of predictors, which creates additional complications in out-of-sample forecasting. Return to text

5.  See appendix for more detail. While the long-horizon regression format of the benchmark Taylor model is derived directly from the underlying Taylor rule model, this is not the case for the models with interest rate smoothing (models 3 and 4). Molodtsova and Papell (2007) only consider the short-horizon regression for the Taylor rule models. We include long-horizon regressions of these models only for the purpose of comparison. Return to text

6.  We thank the authors for the data, which we downloaded from David Papell's website. For the exact line numbers and sources of the data, see the data appendix of Molodtsova and Papell (2008). Return to text

7.  We choose $b$ using the method of Hall, Wolff, and Yao (1999). Return to text

8.  It is consistent in the sense of convergence in probability as the estimation sample size goes to infinity. Return to text

9.  While RS intervals remedy mis-specifications asymptotically, it does not guarantee such corrections in a given finite sample. Return to text

10.  We use Newey and West (1987) for our empirical work, with a window width of 12. Return to text

11.  Center here means the half way point between the upper and lower bound for a given interval. Return to text

12.  These nine exchange rates are the Danish Kroner, the French Franc, the Deutschmark, the Italian Lira, the Japanese Yen, the Dutch Guilder, the Portuguese Escudo, the Swiss Franc, and the British pound. Similar results hold at other horizons. Return to text

13.  When comparing the intervals for $S_{\tau+h} - S_{\tau}$, the random walk model builds the forecast interval around 0, while economic model $m$ build it around $\widehat{\beta}^{^{\prime}}_{m,h}\mathbf{X}_{m,\tau}$.  Return to text

14.  The only exception is Portugal, where only 192 data points were available. In this case, we choose R = 120. Return to text

15.  See Wu (2007) for more discussions. Return to text

16.  Figures in other countries show similar patterns. Results are available upon request. Return to text

17.  To incorporate I(0) economic variables, $\mathbf{D}_{t}$ contains the levels of I(1) variables and the summation of I(0) variables from negative infinity to time $t$. Return to text

This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to text

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