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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 918, January 2008---Screen Reader Version*

International Competition and Inflation: A New Keynesian Perspective^*

Luca Guerrieri, Christopher Gust^† and David López-Salido

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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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:

We develop and estimate an open economy New Keynesian Phillips curve (NKPC) in which variable demand elasticities give rise to changes in desired markups in response to changes in competitive pressure from abroad. A parametric restriction on our specification yields the standard NKPC, in which the elasticity is constant, and there is no role for foreign competition to influence domestic inflation. By comparing the unrestricted and restricted specifications, we provide evidence that foreign competition plays an important role in accounting for the behavior of inflation in the traded goods sector. Our estimates suggest that foreign competition has lowered domestic goods inflation about 1 percentage point over the 2000-2006 period. Our results also provide evidence against demand curves with a constant elasticity in the context of models of monopolistic competition.

Keywords: Inflation, New Keynesian Phillips curve, variable markups

JEL classification: F41, E31, E32

1  Introduction

An important question in macroeconomics is the extent to which global factors influence the behavior of aggregate prices. While it is widely recognized that import prices have a direct effect on consumer prices, there is less agreement about the extent to which global factors influence domestic prices. One prominent view is that the prices of U.S. domestic producers mainly depend on domestic variables, with international factors having only a limited impact. Recent work has challenged this view, arguing that the intensifying trend of global economic integration has changed the behavior of inflation, and international considerations have become an important determinant of inflation dynamics.¹

We address this question in the context of a structural model of inflation in the spirit of Dornbusch and Fischer (1984) and Dornbusch (1985), who emphasized how variations in the desired markups of domestic firms could arise in response to changes in competitive pressures from abroad. These competitiveness effects arise in our model, because a firm faces an elasticity of demand as in Kimball (1995), which depends on its price relative to its competitors. As a result, a reduction in the prices of foreign competitors can induce domestic firms to lower their desired markups. We embed these non-constant elasticity preferences into a short-run model of inflation in which firms only infrequently re-optimize their prices due to the presence of Calvo (1983) contracts.

We derive a specification for domestic inflation that depends not only on real marginal cost, but on the prices of imported or foreign goods relative to domestic prices.² A parametric restriction on our specification yields the standard New Keynesian Phillips curve (NKPC) in which the elasticity of demand is constant, and there is no role for competition abroad to directly influence inflation.³ By comparing the unrestricted and restricted versions of our model, we are able to evaluate the extent to which foreign competition influences the behavior of inflation. In addition, we empirically assess the hypothesis of a constant elasticity of substitution (CES), which is often used by researchers due to its analytical convenience rather than its empirical validity.

Our methodology for estimating inflation closely parallels the present-value approach used in the empirical finance literature. To estimate our model, we use data on the prices of U.S. domestic tradable goods rather than a broader price measure. While this choice represents a departure from most of the empirical literature on inflation, it is motivated by two considerations. First, tradable prices are appropriate given our theoretical model, which focuses on the interactions between foreign and domestic producers of tradable products. Second, the behavior of domestic tradable prices should be particularly revealing regarding the influence of global factors on the domestic economy. We view substantiating that domestic tradeable prices are influenced by global factors as an important first step in building a similar case for broader measures of domestic inflation that include non-tradables.

Our results provide evidence that foreign competition has played an important role in explaining the behavior of traded goods inflation. For instance, we estimate that foreign competition, by reducing the desired markups of domestic producers, has lowered the annual inflation rate for domestic goods about 1 percentage point, on average, over the 2000-2006 period. In addition, movements in relative import prices associated with changes in foreign competition account for over $\frac{1}{3}$ of the volatility of goods price inflation over the 1983-2006 sample period.

Our benchmark estimate for the degree of nominal rigidities are consistent with firms that re-optimize prices, on average, once every 3 to 4 quarters.⁴ We also find that once we account for the endogenous changes in desired markups, there is a limited, if not negligible, role for backward-looking price setting behavior in explaining the dynamics of traded goods inflation. In contrast, much of the NKPC literature including Galí and Gertler (1999) and Eichenbaum and Fischer (2007) estimate degrees of backward-looking behavior that are significantly different from zero. The difference in our results with these earlier papers reflects our focus on inflation for tradeable goods, which inherits a considerable degree of persistence from movements in relative import prices.

Another contribution of our work is that we show that in an open economy the variability in desired markups can be separately identified from changes in markups arising from nominal rigidities. As demonstrated by Eichenbaum and Fischer (2007), in a closed economy, it is not possible to separately identify the frequency of price re-optimization from the real rigidity associated with changes in desired markups. To estimate the frequency of price adjustment in closed economy models, researchers frequently resort to calibrating the parameter governing the variation in the demand elasticity with little empirical guidance. In an open economy, relative import prices are informative about the competitive interaction between foreign and domestic firms, and can shed light on the nature of the demand curve. In this context, our estimates provide evidence against CES demand curves. In particular, we find a large and statistically significant departure from a constant elasticity of substitution, and our estimates for the demand curve are consistent with the calibrated values used in closed economy contexts by Eichenbaum and Fischer (2007), Coenen, Levin, and Christoffel (2007), and Dotsey and King (2005).

The rest of this paper proceeds as follows. Section 2 describes our open economy model with a variable demand elasticity and discusses the issue of identification. Section 3 and 4 describe our data and empirical methodology. Section 5 discusses our estimation results, while Section 6 concludes.

2  An Open Economy Model with a Variable Demand Elasticity

In this section, we describe the analytical framework that leads to the open economy New Keynesian Phillips curve that we estimate. Our framework can be viewed as part of a general equilibrium model which also includes households and the producers of non-tradable goods and services. However, in order to help minimize model misspecification, we employ a limited information approach in estimating traded goods inflation, and only describe the part of the model that is relevant for our estimation approach. In doing so, we emphasize how the international dimension of our model allows us to separately identify the degree to which markups vary due to nominal rigidities from variation in desired markups arising from changes in competition.

2.1  Final Good Producers

At time $t$, an aggregate final good, $A_{t}$, is produced by perfectly competitive firms. The representative firm combines a continuum of intermediate goods produced at home and another continuum produced abroad. The firm chooses domestically-produced goods, $A_{Dt}(i)$, $i\in\lbrack0,1]$, imported goods, $A_{Mt}(i)$, $i\in\lbrack0,1]$, and $A_{t}$ to maximize profits:

$$\max P_{At}A_{t}-\left[ \int_{0}^{1}P_{Dt}(i)A_{Dt}(i)di-\int_{0}^{1} P_{Mt}(i)A_{Mt}(i)di\right] ,$$ (1)

subject to $\int_{0}^{1} D\left( \frac{A_{Dt}(i)}{A_{t}},\frac{A_{Mt} (i)}{A_{t}}\right) di \ge1$ . In maximizing profits, the firm takes the prices of the domestic, $P_{dt}(i)$, imported goods, $P_{Mt}(i)$, and the final good, $P_{At}$, as given.

For $\int_{0}^{1} D\left( \frac{A_{Dt}(i)}{A_{t}},\frac {A_{Mt}(i)}{A_{t}}\right) di$ , we adopt the aggregator used by Gust, Leduc, and Vigfusson (2006), who extend the one discussed in Dotsey and King (2005) to an international environment. This aggregator is given by:

$$\int^{1}_{0} D\left( \frac{A_{Dt}(i)}{A_{t}},\frac{A_{Mt}(i)}{A_{t}}\right) di = \left[ V_{Dt}^{1/\rho}+V_{Mt}^{1/\rho}\right] ^{\rho}-\frac{1} {(1-\nu)\gamma_{t}}+1 .$$ (2)

In turn, $V_{Dt}$ is an aggregator of domestically-produced goods given by

$$V_{Dt}=\int_{0}^{1}\frac{(1-\omega)^{\rho}}{(1-\nu)\gamma_{t}}\left[ \frac{1-\nu}{1-\omega}\frac{A_{Dt}(i)}{A_{t}}+\nu\right] ^{\gamma_{t} }di,$$ (3)

and $V_{Mt}$ is an aggregator of imported goods given by

$$V_{Mt}=\int_{0}^{1}\frac{\omega^{\rho}}{(1-\nu)\gamma_{t}}\left[ \frac {(1-\nu)}{\omega}\frac{A_{Mt}(i)}{A_{t}}+\nu\right] ^{\gamma_{t} }di.$$ (4)

In the above, $\rho$ influences the substitutability between domestic and foreign goods. The share parameter $\omega$ is related to the degree of home bias in preferences and can be thought of as indexing the degree of trade openness.

Our estimation strategy explicitly requires us to model an error to our structural equation for inflation. We let $\gamma_{t}$ be an exogenous shock influencing the elasticity of substitution between varieties produced within a given country, which, as we discuss later, introduces exogenous variations in markups and in aggregate inflation. We specify that $\gamma_{t}$ evolves according to:

$$\gamma_{t} = \gamma\exp(\epsilon_{\gamma t}),$$ (5)

where $\epsilon_{\gamma t}$ is an identically and independently distributed (iid) process with zero-mean and standard deviation, $\sigma_{\gamma}$. Later, we verify that once you take into account endogenous variations of the markup, this error is in fact white noise and thus makes no contribution to inflation persistence. In contrast, recent empirical applications such as Ireland (2004) and Smets and Wouters (2007) have generally assumed that the exogenous movements in the markup are serially autocorrelated.

To understand our aggregator, it is useful to abstract from the iid markup shock. In that case, when $\nu>0$ and $\gamma_{t} = \gamma$, the elasticity of demand is variable (VES) and the (absolute value of the) demand elasticity can be expressed as an increasing function of a firm's relative price. When $\nu=0$ and $\gamma_{t} = \gamma$, the demand aggregator has a constant elasticity of substitution (CES) and can be thought of as the combination of a Dixit-Stiglitz and Armington (1969) aggregators. In particular, in this case, our aggregator can be rewritten as:

$$A_{t}=\left[ (1-\omega)A_{Dt}^{\frac{\gamma}{\rho}}+\omega A_{Mt} ^{\frac{\gamma}{\rho}}\right] ^{\frac{\rho}{\gamma}},$$

where $A_{Dt}=\left( \int_{0}^{1}A_{Dt}(i)^{\gamma}di\right) ^{\frac {1}{\gamma}}$ and $A_{Mt}=\left( \int_{0}^{1}A_{Mt}(i)^{\gamma}di\right) ^{\frac{1}{\gamma}}$ .

As shown in Appendix A, profit maximization by the representative final good producer implies that its demand for domestic good $i$ is given by:

$$A_{Dt}(i)=(1-\omega)\left[ \frac{1}{1-\nu}\left( \frac{P_{Dt}(i)}{P_{Dt} }\right) ^{\frac{1}{\gamma_{t}-1}}\left( \frac{P_{Dt}}{P_{Ft}}\right) ^{\frac{\rho}{\gamma_{t}-\rho}}-\frac{\nu}{1-\nu}\right] A_{t} .$$ (6)

In these demand curves, $P_{Mt}$, and $P_{Dt}$ are price indices of domestic and imported goods given by:

$$P_{Dt}=\left( \int_{0}^{1}P_{Dt}\left( i\right) ^{\frac{\gamma_{t}} {\gamma_{t}-1}}di\right) ^{\frac{\gamma_{t}-1}{\gamma_{t}}}\hspace {0.5cm} \hspace{0.5cm}P_{Mt}=\left( \int_{0}^{1}P_{Mt}\left( i\right) ^{\frac{\gamma_{t}}{\gamma_{t}-1}}di\right) ^{\frac{\gamma_{t} -1}{\gamma_{t}}},$$ (7)

while $P_{Ft}$ is a price index consisting of all the prices of a firm's competitors:

$$P_{Ft}=\left[ (1-\omega)P_{Dt}^{\frac{\gamma_{t}}{\gamma_{t}-\rho}}+\omega P_{Mt}^{\frac{\gamma_{t}}{\gamma_{t}-\rho}}\right] ^{\frac{\gamma_{t}-\rho }{\gamma_{t}}}.$$ (8)

As in Dotsey and King (2005), when $\nu\neq0$, these demand curves have a linear term which implies that the elasticity of demand depends on a firm's price relative to the prices of its competitors, $P_{Ft}$.

2.2  Intermediate Good Producers

Intermediate good $i$ is produced by a monopolistically competitive firm, whose technology is Cobb-Douglas over capital and labor. Intermediate goods producers face perfectly competitive factor input markets within a country. Capital and labor are assumed to be immobile across countries but completely mobile within a country. Thus, within a country, all firms have the same marginal cost, $MC_{t}$.

Intermediate goods producers sell their products to the consumption goods distributors, and we assume that markets are segmented so that firms can charge different prices at home and abroad (i.e., price to market). The domestic price is determined according to Calvo-style contracts. In particular, firm $i$ faces a constant probability $1-\theta$ of being able to re-optimize its price. This probability is assumed to be independent across time, firms, and countries. If firm $i$ can not re-optimize its price at time $t$, the firms resets its price based on lagged inflation as in Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2003). In particular, $P_{Dt}\left( i\right) =\pi^{1-\delta_{D}}\pi_{t-1}^{\delta_{D}}P_{Dt-1}\left( i\right)$ , where $\pi_{t-1}=P_{Dt}/P_{Dt-1}$, and the parameter $0\leq\delta_{D}\leq1$ captures the degree of indexation to past inflation. In this specification $\delta_{D}=0$ corresponds to indexation to steady state inflation ($\pi$), and $\delta_{D}=1$ implies full indexation to past inflation. When firm $i$ can re-optimize in period $t$, it maximizes:

$$E_{t}\sum_{j=0}^{\infty} \xi_{t+j} \theta^{j} \left[ I_{Dt+j}P_{Dt}\left( i\right) -MC_{t+j}\right] A_{Dt+j}(i),$$ (9)

taking $MC_{t+j}$, its demand schedule, and the indexing scheme, $I_{Dt+j} =\Pi_{h=1}^{j}\pi^{1-\delta_{D}}\pi_{t+h-1}^{\delta_{D}}$ as given. In the above, $\xi_{t+j}$ is the stochastic discount factor with steady state value, $\beta\in(0,1)$, and $E_{t}$ denotes the conditional expectations operator at date $t$. The first-order condition from this problem is:

$$E_{t}\sum_{j=0}^{\infty} \xi_{t+j} \theta^{j}\left[ 1-\left( 1-\frac {MC_{t+j} }{I_{Dt+j}P_{Dt}(i)}\right) \epsilon_{t+j}(i)\right] A_{Dt+j}(i)=0,$$ (10)

where the elasticity of demand for good $i$ in the domestic market is:

$$\epsilon_{t}(i)=\frac{1}{1-\gamma_{t}}\left[ 1-\nu\left( \frac{P_{Dt} (i)}{P_{Dt}}\right) ^{\frac{1}{1-\gamma_{t}}}\left( \frac{P_{Dt}}{P_{Ft} }\right) ^{\frac{\rho}{\rho-\gamma_{t}}}\right] ^{-1}.$$ (11)

This elasticity results in a time-varying markup of the form:

$$\mu_{t}(i)=\frac{\epsilon_{t}(i)}{\epsilon_{t}(i)-1}=\left[ \gamma_{t} +\nu(1-\gamma_{t})p_{Dt}(i)^{\frac{1}{1-\gamma_{t}}}p_{Ft}^{\frac{\rho} {\gamma_{t}-\rho}}\right] ^{-1},$$ (12)

where the lower case variables denote relative prices (i.e., $p_{Dt} (i)=\frac{P_{Dt}(i)}{P_{Dt}}$ and $p_{Ft}=\frac{P_{Ft}}{P_{Dt}}$).

To understand variations in the desired markup (i.e., the markup in the absence of price rigidities and the exogenous shock $\gamma_{t}$), it is useful to log-linearize this expression around a steady state in which relative prices are equal to one and write it as:

$$\hat{\mu}_{t}(i)= \hat{\mu}_{Dt}(i) - \varphi_{\mu} \hat{\gamma} _{t}.$$ (13)

where $\hat{\mu}_{Dt}(i)$ is the log-linearized desired markup and $\varphi_{\mu} =(\mu-1)\frac{\gamma}{1-\gamma}$. The desired markup is given by:

$$\hat{\mu}_{Dt}(i)=-\left[ \frac{\partial\epsilon(i)}{\partial p_{D}(i)} \frac{1}{\epsilon}\right] (\mu-1)\hat{p}_{Dt}(i)+\left[ \frac{\partial \epsilon(i)}{\partial p_{M}}\frac{1}{\epsilon}\right] (\mu-1)\hat{p} _{Mt}.$$ (14)

The steady state markup of an intermediate good producer is given by

$$\mu=\frac{1}{\gamma+(1-\gamma)\nu}>1,$$ (15)

and $\epsilon=\frac{1}{(1-\gamma)(1-\nu)}$ is the steady state demand elasticity.

According to equation (14), there are two sources of variations in desired markups. The first reflects variations arising from deviations in a firm's price relative to the prices of its domestic competitors. Variations in desired markups arising from this source depend on $\frac{\partial\epsilon(i)}{\partial p_{D}(i)}\frac{1}{\epsilon} = \nu\epsilon$ , which is the elasticity of the elasticity with respect to a firm's relative price. For $\nu> 0$, this elasticity measures how much $\epsilon_{t}(i)$ rises when a firm raises its price above the prices of its domestic competitors. In that case, a firm will lower its desired markup so that its desired price does not deviate too far from those of its domestic competitors. If $\nu= 0$, then the demand curves are CES absent the markup shock, and $\frac{\partial\epsilon(i)}{\partial p_{D}(i)}\frac{1}{\epsilon} = 0$ .

The second source of variation in a firm's desired markup arises from foreign competition. This effect depends on $\frac{\partial\epsilon (i)}{\partial p_{M}}\frac{1}{\epsilon}=\nu\epsilon_{A}\omega$ , where

$$\epsilon_{A} =\frac{\rho}{(\rho-\gamma)(1-\nu)}>0$$ (16)

is the elasticity of substitution between home and foreign goods. The elasticity of the elasticity with respect to the relative import price, $\frac{\partial\epsilon(i)}{\partial p_{M}}\frac{1}{\epsilon}$, measures how much $\epsilon_{t}(i)$ rises when relative import prices fall. In that case, a firm faces stiffer competition from abroad and will lower its desired markup. For $\nu=0$, the CES case, there is no effect of foreign competitiveness on domestic markups and $\frac{\partial\epsilon(i)}{\partial p_{M}}\frac {1}{\epsilon}=0$ . The importance of foreign competitiveness on the desired markups of domestic firms depends on the degree of trade openness ($\omega$) and the elasticity of substitution between home and foreign goods. International competition has a larger influence on desired markups when an economy is more open or its goods are closer substitutes with foreign goods.

Substituting out $\frac{\partial\epsilon(i)}{\partial p_{D} (i)}\frac{1}{\epsilon}$ and $\frac{\partial\epsilon(i)}{\partial p_{M}} \frac{1}{\epsilon}$, the desired markup can be expressed as:

$$\hat{\mu}_{Dt}(i) = - \frac{\Psi}{1-\Psi} \hat{p}_{Dt}(i) + \frac{\Psi} {1-\Psi} \frac{\epsilon_{A}}{\epsilon} \omega\hat{p}_{Mt} ,$$ (17)

where the parameter $\Psi$ reflects the variations in the desired markup associated with competition from other firms and is given by:

$$\Psi= \frac{(\mu-1) \frac{\partial\epsilon(i)}{\partial p_{D}(i)}\frac {1}{\epsilon}}{1+(\mu-1) \frac{\partial\epsilon(i)}{\partial p_{D}(i)}\frac {1}{\epsilon}} = \frac{\nu\mu}{1+\nu\mu}.$$ (18)

In our empirical work, we focus on estimating $\Psi$ while calibrating the values of $\mu$ and $\epsilon_{A}$. These three parameters uniquely determine the demand curve parameters discussed earlier - $\rho$, $\gamma$, and $\nu$ - via equations (15), (16), and (18).

2.3  Inflation Dynamics

To understand the role of variations in desired markups for inflation, we log-linearize the firm's first-order condition for price re-optimization, equation (10). As detailed in Appendix A, after some algebraic manipulation, a first-order approximation to this equation yields:

$$\hat{\pi}_{t}-\delta_{D}\hat{\pi}_{t-1}=\beta E_{t}[\hat{\pi}_{t+1} -\delta_{D}\hat{\pi}_{t}]+ \kappa\left[ (1-\Psi) \hat{s}_{t}+ \Psi\omega \frac{\epsilon_{A}}{\epsilon}\hat{p}_{Mt}+\varphi\widehat{\gamma}_{t}\right] ,$$ (19)

$$\hspace{0.25cm} \kappa= \frac{(1-\beta\theta)(1-\theta )}{\theta},$$

and where $\hat{\pi}_{t}$ is domestic price inflation expressed as a log deviation from steady state, $s_{t}$ represents real marginal cost (defined using $P_{Dt}$), and the composite parameter, $\varphi$, influences the sensitivity of inflation to exogenous variations in the markup and is given by $\varphi= 2 \Psi-1$.⁵

Since we allow for partial indexation to lagged inflation, current inflation is affected by inflation in the previous period. Similar to a standard new Keynesian Phillips curve (e.g., Galí and Gertler (1999)), the Calvo price setting parameter, $\theta$, affects the responsiveness of inflation to real marginal cost through its effect on $\kappa$. However, equation (19) differs from the standard specification, since relative import prices also affect inflation. In an open economy, a domestic firm must take into account the prices of its foreign competitors on its desired markup.⁶ If foreign goods become relatively less expensive, then domestic firms will respond by lowering their desired markups in order to maintain a competitive price; hence, this puts downward pressure on $\pi_{t}$.

The importance of this foreign competitiveness effect on domestic inflation depends on the degree of trade openness ($\omega$) and the import price elasticity ( $\epsilon_{A}$) as well as $\Psi$. We use $\Psi$ to gauge the extent of the real rigidity associated with pricing complementarities between firms. A higher value of $\Psi$ reduces the sensitivity of inflation to real marginal cost and raises the sensitivity of inflation to relative import prices.

Identifying the Real Rigidity.  Equation (19) nests two important cases. With $\Psi=0$, the CES case, there is no direct effect of international competition on domestic prices. Equation (19>) is observationally-equivalent to the specification estimated by Galí and Gertler (1999) among others. Another interesting case is the one considered by Eichenbaum and Fischer (2007) in which $\omega= 0$. In this case, the domestic economy does not import foreign goods, and a domestic firm, while willing to vary its desired markup in response to domestic competition, need not be concerned with foreign competition. Accordingly, relative import prices do not affect domestic price inflation.

As discussed by Eichenbaum and Fischer (2007), one can not separately identify $\Psi$ and $\theta$ in the closed economy ($\omega=0$) using equation (19). As a result, many researchers opt to calibrate the value of $\Psi$ with little empirical guidance. However, when $\omega>0$, relative import prices are informative about the extent to which firms vary their desired markups, and it is clear from equation (19) that it is possible to jointly identify both $\Psi$ and $\theta$.

Coenen, Levin, and Christoffel (2007) alter the standard Calvo framework and show how one can separately identify real and nominal rigidities in a closed economy framework in which there are nominal pricing contracts of different durations. Their approach exploits the more complex dynamics between inflation and real marginal cost induced by their contracting structure and they use simulated methods of moments to estimate the parameters. Instead, we use the baseline Calvo model and exploit variation in relative import prices to provide information regarding the nature of demand curves and endogenous changes in desired markups.

2.4  Firm-Specific Capital

We now extend the analysis to incorporate firm-specific capital. To do so, we assume that the production function for intermediate good $i$ is given by:

$$Y_{t}(i)=\bar{K}^{\alpha}(Z_{t}L_{t}(i))^{1-\alpha},$$ (20)

where $L_{t}(i)$ is a firm's demand for labor and $Z_{t}$ is a common technological factor. Finally, $\bar{K}$ denotes each firm's fixed stock of capital. As discussed in Coenen, Levin, and Christoffel (2007), the firm specific level of capital can be interpreted more broadly as production factors that remain fixed in the short run (such as land and overhead labor), while $L_{t}(i)$ can be interpreted as those factors which are variable in the short run.

Under these assumptions, firm $i$'s marginal cost is given by:

$$MC_{t}(i)=\frac{Q}{1-\alpha} \frac{W_{t}}{Z_{t}}Y_{t}(i)^{\frac{\alpha }{1-\alpha}},$$ (21)

where $Q=\bar{K}^{\frac{\alpha}{\alpha-1}}$ and $\frac{\alpha}{1-\alpha}>0$ can be interpreted as the short-run elasticity of a firm's marginal cost to output. Because capital specificity implies that a firm's marginal cost is an increasing function of its output, it acts as another source of real rigidity. In particular, following an increase in nominal demand, a firm with the opportunity to raise its price will have a weaker incentive to do so, since the fall in the relative demand for its good reduces its marginal cost.

In the benchmark economy, a domestic producer may set different prices at home and abroad, and its pricing decision in its home market is completely independent of its pricing decision in its foreign market. With firm-specific capital, this is no longer true. A firm's export price affects a firm's domestic price through its effect on the demand for its product, $Y_{t}(i)$, which alters its marginal cost. To keep our analysis tractable, we abstract from these effects and assume that the domestic firms who compete with foreign firms in the domestic market are distinct from those firms which export. While this assumption simplifies our analysis, it is also more in line with the empirical evidence discussed in Bernard, Eaton, Jensen, and Kortum (2003) than the standard assumption in which all domestic goods producers export their goods. In particular, Bernard, Eaton, Jensen, and Kortum (2003) document that in 1992 only 21 percent of manufacturing establishments were exporters.

With a firm's production equal to its domestic demand (i.e., $Y_{t}(i) = A_{Dt}(i)$ $\forall i$), the first order condition for a firm that re-optimizes its price at date $t$ is:

$$E_{t}\sum_{j=0}^{\infty} \xi_{t+j} \theta^{j} \left[ 1-\left( 1-\frac {MC_{t+j}(i)}{V_{Dt+j}P_{Dt}(i)}\right) \epsilon_{Dt+j}(i)\right] A_{Dt+j}(i)=0.$$ (22)

The log-linearized expression for domestic inflation in this case is given by:

$$\hat{\pi}_{t}-\delta_{D}\hat{\pi}_{t-1}=\beta E_{t}[\hat{\pi}_{t+1}-\delta _{D}\hat{\pi}_{t}]+ \kappa_{D} \left[ (1-\Psi) \hat{s}_{t}+ \Psi\omega \frac{\epsilon_{A}}{\epsilon}\hat{p}_{Mt}+\varphi\widehat{\gamma}_{t}\right] ,$$ (23)

where $\kappa_{D} = \frac{\kappa}{1+\epsilon\frac{\alpha}{1-\alpha} (1-\Psi)}$ , and $\Psi$ and $\kappa$ are defined as before. Comparing equation (23) with equation (19), it is clear that capital specificity does not alter the form of the NKPC but lowers the reduced-form slope coefficient since $\kappa_{D} < \kappa$ with $\alpha> 0$. An implication of this result is that we can not separately identify the real rigidity associated with firm-specific capital from the Calvo-price setting parameter. However, the real rigidity associated with variations in desired markups can still be separately identified and estimated provided information on either $\alpha$ or $\theta$.

3  Data

We use quarterly data on inflation, marginal cost, and relative import prices from 1983-2006 to estimate our model. We focus on this sample period to help abstract from changes in monetary policy regimes. Since our theoretical analysis is for the prices of tradables, we construct on inflation measure based on goods prices (from NIPA Table 1.2.4). We also net out the prices of exported goods, reflecting that prices at home and abroad can differ.⁷ The upper panel of Figure 1 plots goods inflation and inflation in the non-farm business sector from 1979-2006. The two series are positively correlated with each other (the correlation is 0.5). Goods price inflation, however, has been lower, on average, than overall inflation, as well as more volatile, particularly over the past 15 years.

To measure real marginal cost, $s_{t}$, we use data on the labor share in the non-farm business sector defined as nominal labor compensation divided by nominal output. This measure is the standard one used by Galí and Gertler (1999), Sbordone (2002), and Eichenbaum and Fischer (2007) among others.⁸ The lower panel of Figure 1 plots the labor share in the non-farm business sector along with GDP goods inflation. The labor share declined throughout the first half of the 1990s, rose noticeably at the end of the 1990s, and then dropped sharply from 2001-2005.

We measure relative import prices by dividing the NIPA price deflator for non-oil imported goods by the deflator for domestic goods prices.⁹ This series is shown in Figure 2 along with domestic goods inflation. Relative import prices are positively correlated with goods inflation, rising and falling with inflation in the 1980s and moving lower in the 1990s before trending upward in the past five years.

The top panel of Figure 3 shows the correlations between the current value of inflation and the leads of relative import prices. The correlation between traded goods inflation and relative import prices is above 0.3 for the first twelve leads of import prices. These correlations are consistent with our theoretical model, which links inflation to expected future values of relative import prices. The bottom panel of Figure 3 also shows these dynamic correlations between inflation and real unit labor costs. Inflation is also positively correlated with leads of real unit labor costs, as suggested by our theoretical model.

4  Empirical Methodology

Our methodology closely parallels the present-value approach used in the empirical finance literature.¹⁰ In particular, we rewrite equation (19) as a relationship between inflation and the expected discounted value of the future values of real marginal cost and relative import prices:

$$\widehat{\pi}_{t}=\delta_{D}\widehat{\pi}_{t-1}+ \kappa_{D} \sum_{k=0} ^{\infty}\beta^{k} E_{t}\left[ (1-\Psi) \hat{s}_{t+k}+ \Psi\omega \frac{\epsilon_{A}}{\epsilon} \hat{p}_{Mt+k}+\varphi\hat{\gamma}_{t+k} \right] ,$$ (24)

where $\kappa_{D} = \kappa$ if capital is not firm specific. As discussed in Appendix B, we choose to estimate our model using equation (24) rather than applying a generalized method of moments' (GMM) estimator to equation (23), because we found that the small-sample properties of our approach were superior.

To estimate the parameters of interest using (24), we need forecasts of real marginal cost and relative import prices, which we obtain through a VAR. Defining $X_{t}$ as a vector of variables that includes $s_{t}$ and $p_{Mt}$, our VAR in companion form can be written as:

$$X_{t}=AX_{t-1}+u_{t},$$ (25)

where $A$ is a matrix of VAR coefficients, and $u_{t}$ is a vector of iid innovations that may be correlated with each other. With the VAR expressed in this way, we compute the forecasts of $X_{t}$ using the relationship: $E_{t}\{X_{t+k}\}=A^{k}$ $X_{t}$.

For our benchmark specification of the VAR, we include only measures of real unit labor costs and relative import prices in $X_{t}$. Furthermore, we used the Box-Jenkins methodology to test down from an unrestricted VAR with longer lag length. We choose an AR(1) process for real unit labor costs and an AR(2) process for relative import prices. Later, we conduct sensitivity analysis in which we allow for feedback between unit labor costs and import prices in our VAR. For our benchmark specification of the VAR, the equation for inflation that we estimate is:

$$\hat{\pi}_{t}=\delta_{D}\hat{\pi}_{t-1}+\kappa_{D} \left[ \frac{1-\Psi} {1-\beta\rho_{s}} \hat{s}_{t}+ \omega\frac{\epsilon_{A}}{\epsilon} \frac {\Psi(1+\beta\rho_{M2}L)}{1-\beta\rho_{M1}-\beta^{2} \rho_{M2}} \hat{p} _{Mt}\right] + \epsilon_{\pi t},$$ (26)

where $L$ is the lag operator, and $\rho_{s}$ is the autoregressive coefficient for unit labor costs, and $\rho_{M1}$, and $\rho_{M2}$ are the autoregressive coefficients for import prices. We jointly estimate the VAR, equation (25), along with the process for inflation, equation (26).

Our estimation strategy explicitly requires us to model an error to equation (26). In our model, this error reflects iid shocks to the markup and is given by $\epsilon_{\pi t} = \kappa\varphi \hat{\gamma}_{t}$. Since the exogenous variation in markups may be correlated with unit labor costs and import prices, we use lagged variables as instruments.¹¹ Our benchmark set of instruments includes one lag of traded goods inflation, one lag of real unit labor costs, two lags of relative import prices, two lags of a measure of commodity price inflation, two lags of quadratically-detrended output in the goods sector, and four lags of the spread between the 10-year Treasury note and the 3-month Treasury bill.¹²

Since it is possible that our instruments are only weakly correlated with the endogenous variables in our model, we follow Stock, Wright, and Yogo (2002) and Stock and Yogo (2004) and check for the presence of weak instruments based on the $g_{min}$ statistic of Cragg and Donald (1993). We compare this statistic against the critical values for the null hypothesis of weak instruments compiled by Stock and Yogo (2004). Finally, as robustness, we also use maximum likelihood estimation as an alternative to GMM.

Identification and Calibration.  We estimate $\delta$, $\theta$, $\Psi$, as well as $A$, the coefficients from the VAR used to forecast unit labor costs and import prices (for our benchmark specification, the relevant elements of $A$ are $\rho_{s}$, $\rho_{M1}$, and $\rho_{M2}$). We calibrate $\mu$, $\omega$, and $\epsilon _{A}$. Given considerable uncertainty about the values of these parameters, we report results for alternative calibrations in our sensitivity analysis. Throughout our analysis, we set $\beta= 0.99$.

For our benchmark calibration, we choose $\mu= 1.2$, which is at the midpoint of the estimates surveyed by Rotemberg and Woodford (1995) but higher than the estimate of Basu and Fernald (1997). This value of $\mu$ implies $\epsilon=6$. We choose $\epsilon_{A}$, the elasticity of substitution between home and foreign goods, to be 1.5. This estimate is toward the higher end of estimates using macroeconomic data, which are typically below unity in the short run and near unity in the long run (e.g., Hooper, Johnson, and Marquez (2000)). Nevertheless, estimates of this elasticity following a tariff change are typically much higher.¹³

We choose $\omega$ based on the ratio of non-oil imported goods to total goods production. Because of a secular rise in the share of imports, it is difficult to determine an appropriate value for $\omega$, which in our model corresponds to the steady state import share. For our benchmark calibration, we choose $\omega=0.26$, which is the sample average for the 1983:Q1-2006:Q4 period that we use throughout our analysis. For the version of the model with firm-specific capital, following Coenen, Levin, and Christoffel (2004), we set $\alpha= 0.4$.

5  Estimation Results

Table 1 reports our estimates of $\theta$, $\Psi$, and $\delta_{D}$ for the version of the model in which capital moves instantaneously across domestic firms. Table 2 shows these results for the version of the model in which capital is firm-specific. (For our estimates of the auxiliary VAR, see Appendix C.) Comparing Tables 1 and 2, it is apparent that with the exception of the parameter $\theta$, the estimates are very similar. With firm-specific capital, the second column of Table 2 shows that our estimate of $\theta$ is 0.75 for the model with a variable elasticity (VES), which implies that a firm, on average, re-optimizes its price every four quarters. In contrast, without capital specificity, $\theta= 0.82$, implying an average contract duration of over 5 quarters.¹⁴ Since our estimate of $\theta$ is the only difference in results between these two specifications, and it is quite reasonable to believe that some production factors are firm-specific, we now proceed to focus exclusively on the model in which capital is firm-specific.

Table 2 shows that our estimate of $\Psi$ implies a demand elasticity that is far from constant, as the value of $\Psi$ is 0.73 and also statistically significant. To understand what this estimate implies for an individual firm's demand, the upper left panel of Figure 4 plots the demand curve of good $i$ for different values of $\frac{P_{D}(i)}{P_{D}}$ and compares it to the CES demand curve (i.e., $\Psi= 0$). As shown there, because the elasticity increases as a firm raises its price, demand falls more for the VES demand curve than the CES demand curve. With a rising elasticity of demand, the upper right panel shows that a firm will reduce its desired markup as its price rises above those of its domestic competitors.

Our estimate of $\Psi$ implies that demand for good $i$ falls about 14 percent in response to a 2 percent increase in a firm's price above its steady-state value and about 45 percent in response to a 5 percent increase. Correspondingly, these relative price movements are associated with 6 and 12 percentage point falls in desired markups, respectively. These estimates seem quite reasonable in contrast to the values discussed in Chari, Kehoe, and McGrattan (2000). They criticize the calibration of the demand curve in Kimball (1995), because 2 percent and 2.3 percent increases in a firm's price induce a 78 percent and 100 percent fall in demand.

The lower right panel of Figure 4 shows that a decrease in foreign prices relative to domestic prices induces a domestic firm to lower its desired markup. However, a firm's desired markup varies much less in response to a change in foreign prices than in response to its own price, reflecting home bias in tradable consumption (i.e., the calibrated value of $\omega$) and the lower elasticity between home and foreign goods ( $\epsilon_{A}$) than between home goods ($\epsilon$). A 10 percent fall in the relative import price from its steady state value induces a firm to lower its desired markup only about 2 percentage points. Still, as discussed below, such movements in relative import prices and desired markups of firms are enough to have substantial effects on domestic price inflation.

The results in Table 2 also suggest that there is upward bias in the degree of indexation for the CES demand curves. In particular, there is a relatively large and significant coefficient on lagged inflation in this case. In contrast, in the unrestricted VES specification, the coefficient on lagged inflation is smaller and not statistically significant. Intuitively, with the VES demand curves, inflation is inheriting persistence from movements in relative import prices, and as a result, one does not need the partial indexation scheme to compensate. Later, we report results from a Monte Carlo exercise that substantiate this interpretation.

Table 2 reports the Ljung-Box Q-statistic at lags 1 and 4. For the VES specification with indexation, we can reject the presence of serially correlated markup shocks. For the CES specification without indexation, there is strong evidence that the markup shocks are serially correlated, suggesting that the model is misspecified. Although there appears to be less serial correlation for the CES specification with indexation, the Q-statistic at lag 4 still suggests model misspecification. In contrast, even if we omit indexation from the VES specification, the Q-statistics are consistent with no serial correlation.

Table 2 computes the $g_{min}$ statistic, which can be used to test for the presence of weak instruments using the critical values from Stock and Yogo (2004). Based on both definitions of weak instruments discussed there, we can reject that the instrument set is weak for all four specifications shown in Table 2.¹⁵

To assess the fit of our model without indexation (shown in the third column of Table 2), Figure 5 plots predicted inflation, $\hat{\pi}^{F}_{t}$, defined as:

$$\hat{\pi}^{F}_{t}= \kappa\left[ \frac{1-\Psi} {1-\beta\rho_{s}} \hat{s}_{t}+ \omega\frac{\epsilon_{A}}{\epsilon} \frac{\Psi(1+\beta\rho_{M2}L)}{1-\beta \rho_{M1}-\beta^{2} \rho_{M2}} \hat{p}_{Mt}\right] ,$$ (27)

using our estimates for $\theta$ and $\Psi$. The dashed red line in the figure shows a four-quarter moving-average of $\hat{\pi}^{F}_{t}$, while the solid black line shows a four-quarter moving average of actual inflation. Predicted inflation tracks the broad contours of observed inflation. In particular, the predicted series rises in the mid to late 1980s, trends downward with inflation in the 1990s, and rises and falls with actual inflation in the first half of this decade.

An important implication of our estimate of $\Psi$ is that international competition plays an important role in influencing domestic inflation. To assess this role, the dashed blue line in Figure 5 plots predicted inflation for the CES specification in which $\Psi= 0$ and foreign prices do not influence the desired markups of domestic firms. As shown there, without this foreign competitiveness channel, the model fails to account for the increase in inflation in the late 1980s and its subsequent reversal in the early 1990s. More disconcerting, the CES specification overstates the level of inflation for the last seven years of our sample: the model predicts an average, annualized inflation rate of 0.5 percent from 2000-2006 compared to a slight deflation of 0.4 percent. In contrast, the average value of predicted inflation for the VES specification is very close to the observed value over this period. Since the difference between these two specifications reflects the influence of foreign competition on desired markups, our estimates suggest that foreign competition has lowered domestic goods inflation nearly 1 percentage point over the last seven years.

We can also assess the role of foreign competition for inflation dynamics by computing its contribution to the volatility of the four-quarter change in domestic goods prices. For the VES specification, as shown in Table 2 in the row labelled "$\frac{\sigma_{\pi^{F}}} {\sigma_{\pi}}$", predicted inflation accounts for nearly 75 percent of the volatility of observed inflation, with movements in relative import prices accounting for about a $\frac{1}{3}$ of actual inflation volatility. In comparison, the CES specification that allows for lagged indexation only accounts for 35 percent of the volatility of inflation. Accordingly, these variance decompositions offer additional evidence that foreign competition has played an important role in accounting for movements in domestic goods prices.

Model Misspecification and Indexation.  The results shown in Table 2 suggested that the CES model, by excluding import prices is misspecified. In particular, this specification appears to generate upward biased estimates of $\delta_{D}$, the degree of indexation. We investigate this hypothesis by considering the following Monte Carlo experiment. We use the VES specification with pseudo-true values of $\Psi= 0.67$, $\delta_{D} = 0.16$, $\theta= 0.75$ to bootstrap 10,000 repetitions of artificial data, each with 96 observations (i.e., the length of 1983Q1-2006Q4 sample period).¹⁶ For each Monte Carlo sample, we re-estimated the VES and CES specifications with indexation. We also repeated this exercise by generating bootstrapped data with an alternative parameterization of $\Psi$.

The top panels of Figure 6 plot the sampling distributions of our estimates for $\delta_{D}$ and $\theta$ for the first Monte Carlo experiment in which the pseudo-true value of $\Psi$ is $0.67$. The estimate of $\delta_{D}$ from the VES specification appears to be unbiased with the mass of the distribution narrowly concentrated around its pseudo-true value, while the estimate of $\theta$ displays some small sample bias and a bit wider distribution than implied by the asymptotic standard errors provided in Table 2. Still, these results suggest that our GMM estimator fares well in small samples.¹⁷

Figure 6 also shows that the misspecification bias of the CES formulation leads to estimates of $\delta_{D}$ and $\theta$ above their pseudo-true values. As shown in the bottom panels, the bias for $\delta_{D}$ and $\theta$ becomes more severe, when we increase the value of $\Psi$ from 0.67 to 0.9 and lower $\theta$ from 0.75 to 0.67.¹⁸ In particular, the mean estimate of $\delta_{D}$ is 0.47 compared to its pseudo-true value of $0.16$ when $\Psi= 0.9$. This upward bias arises, because the misspecification associated with the omitted import price variable gives rise to serially correlated markup shocks. As a result, the estimate of $\delta_{D}$ rises above its pseudo-true value to help soak up this residual autocorrelation. Thus, an econometrician, who ignored the influence of foreign competition on inflation, may mistakenly conclude that lagged indexation plays an important role in explaining inflation.

Comparison with the Literature.  As discussed earlier, $\Psi$ can be used to gauge the degree of real rigidities associated with variations in desired markups arising from domestic competition. From equation (18), we can see that $\Psi$ depends on both the steady state demand elasticity or markup, and the elasticity of the demand elasticity with respect to a firm's price, $\frac{\partial\epsilon(i)}{\partial p_{D}(i)}\frac{1}{\epsilon}$ . It is therefore a useful metric to compare our estimates with calibrated values of the Kimball (1995) preferences used in the literature.

Table 3 shows our estimated value for $\Psi$ as well as the elasticity of the elasticity with respect to a firm's price. Although our estimates suggest that those discussed in Chari, Kehoe, and McGrattan (2000) are high, a number of researchers use calibrations that are validated by our results. In contrast, Dossche, Heylen, and den Poel (2007) use scanner data from a euro-area supermarket chain to argue that most calibrations of the Kimball (1995) aggregator impose too high a value of $\frac{\partial\epsilon(i)}{\partial p_{D}(i)}\frac{1}{\epsilon}$ , as the median estimate for the goods they consider is only 0.8. However, given that they estimate a demand elasticity with a (net) markup of 250%, their implied estimate of $\Psi$ is 0.67, quite close to our estimate. In our view, $\Psi$ is the relevant metric for comparing results, since $\frac{\partial\epsilon(i)}{\partial p_{D}(i)} \frac{1}{\epsilon}$ is not a sufficient statistic for describing the demand curve or the degree of variation in desired markups.¹⁹

Our results are also related to Batini, Jackson, and Nickell (2005), who estimate an open economy NKPC for the United Kingdom in which foreign prices affect inflation due to both variations in desired markups and the presence of imported intermediate goods. In contrast to our results, they find that their measure of external competitiveness does not have a statistically significant role in explaining the variation in inflation. However, there are a number of important differences in their paper from ours. Most notably, they adopt an ad hoc specification for variations in desired markups.

In our purely forward-looking model, we estimate a value of $\theta$, which implies an average contract duration of four quarters. This estimate is broadly consistent with the micro evidence of Nakamura and Steinsson (2007), who find a median duration of non-sale prices of 8-11 months using prices for both consumers and producer's finished goods.²⁰ Our estimates are also broadly in line though slightly higher than those of Coenen, Levin, and Christoffel (2004) and Eichenbaum and Fischer (2007), who incorporate both VES demand curves and firm-specific capital into New Keynesian Phillips curves.

Our estimate of an insignificant degree of indexation are in line with two recent papers by Ireland (2004) and Coenen, Levin, and Christoffel (2004). Ireland (2004) finds no role for indexation in a closed economy model when he allows for serially autocorrelated markup shocks. In contrast, we use iid markup shocks to show that once we allow for endogenous variations in markups, lagged indexation is not significant. Coenen, Levin, and Christoffel (2004) estimate a closed economy Phillips curve and argue that backward-looking price-setting is not needed to explain aggregate inflation in the context of a stable monetary policy regime. Contrary to their analysis, our results do not hinge on the use of a dummy variable to account for a change in the U.S. monetary policy regime occurring in 1991.²¹

5.1  Alternative Model Calibrations

Table 4 considers the sensitivity of our estimates to the calibrated values of $\omega$, $\epsilon_{A}$, and $\mu$. As a point of comparison, Table 4 also reports our estimates from the VES specification with firm-specific capital and lagged indexation using the benchmark calibration of these parameters.

The parameter $\omega$ determines the share of imports in goods production. For our benchmark calibration, we choose $\omega= 0.26$, the sample average of the ratio of nonfuel goods imports to goods production from 1983-2006. As an alternative, we set $\omega= 0.35$, its value in 2004. With $\omega= 0.35$, our estimate of $\Psi$ declines slightly from 0.73 in our benchmark calibration to 0.66.

The third column of Table 4 shows the effect of lowering the import price elasticity, $\epsilon_{A}$, from its benchmark value of 1.5 to 0.5, a value consistent with short-run estimates. In this case, the estimate of $\Psi$ rises to 0.84, still within the 90% confidence interval of the benchmark model. Alternatively, an increase in $\epsilon_{A}$ to 2 lowers our estimate of $\Psi$ to 0.56. This fall in $\Psi$, however, does not necessarily imply that foreign competition has a smaller effect on the desired markups of domestic firms. In particular, for a given value of $\Psi$, a higher import price elasticity raises the responsiveness of domestic firms' desired markups to foreign prices. The final column of Table 4 shows the estimation results using a markup of 10%, a value in line with the estimates of Basu and Fernald (1997). In this case, our benchmark estimate for $\Psi$ rises from 0.73 to 0.82.

Although the point estimates for the parameters governing the nominal and real rigidities are somewhat dependent on our calibration choices, the qualitative results are not. For a wide set of parameter choices, as evidenced in Table 4, we find that nominal contracts last on average between 3 and 4 quarters, and that foreign competition by inducing changes in the desired markups plays a significant role in explaining the dynamic of inflation.

5.2  Alternative Instruments, Data, and Estimation Procedures

Table 5 compares the structural estimates for the VES model assuming firm-specific capital with a number of alternatives. The first alternative examines the estimation results when we use a smaller instrument set. In particular, in this case, we include two lags of relative import prices, two lags of inflation, and one lag of real unit labor costs. While the point estimate for $\theta$ does not change, we find that the degree of real rigidities is modestly lower in this case.

In our benchmark specification for forecasting unit labor costs and import prices, we ignored any feedback between these variables by considering separate AR processes for these variables. In the third column, we consider an alternative forecasting process in which these variables are modeled as a VAR(2): $X_{t} = AX_{t-1} + u_{t}$ with $X_{t}^{\prime}=[\hat {s}_{t}$ $\hat{p}_{Mt}$ $\hat{s}_{t-1}$ $\hat{p}_{Mt-1}]$ and $u_{t}^{\prime }=[u_{st}$ $u_{p_{M}t}$ 0 $0]^{\prime}$ .²² For the VAR, it is useful to define the vectors $e_{1}^{\prime}=[1$ 0 0 $0]$ and $e_{2}^{\prime}=[0$ $1$ 0 $0]$ to pick out the the first element, $\hat {s}_{t}$, and the second element, $\hat{p}_{Mt}$, from the vector $X_{t}$. Using the previous definitions, we can solve equation (23), the second order difference equation governing inflation dynamics and rewrite it as:

$$\widehat{\pi}_{t}=\delta_{D}\widehat{\pi}_{t-1}+\kappa_{D}[(1-\Psi )e_{1}^{\prime}(I-\beta A)^{-1}X_{t}+\omega\frac{\epsilon_{A}}{ \epsilon}\Psi e_{2}^{\prime}(I-\beta A)^{-1}X_{t}]+\epsilon_{\pi t}.$$ (28)

The third column of Table 5 reports the estimation results under this alternative forecasting model, and Appendix C reports our estimate of $A$. As shown in Table 5, the estimate of $\Psi$ is somewhat larger; however, overall, the restrictions we place on the forecasting model do not appreciably alter the estimates vis-à-vis our benchmark model.

To be consistent with our theoretical model, it would be ideal to have data on a basket of imported goods that matches the basket of domestically-produced goods. However, no such series are available, and instead our benchmark measure of imported prices excludes oil prices, reflecting that oil's share of imports is much larger than its share of domestic goods production. Since the same argument is applicable to other commodities, the third column of Table 5 presents results based on a measure of import prices that excludes oil, materials, and industrial goods. As shown there, our estimates only change slightly from the benchmark case; the most noticeable difference is for the indexation parameter, which equals 0.18 instead of the benchmark estimate of 0.1.

The last column of Table 5 presents results from estimating our system of equations (i.e., the structural inflation equation and the two AR processes for unit labor costs and import prices) using maximum likelihood estimation (MLE). Despite this different estimation strategy, the results are remarkably similar to our GMM estimates.

Overall, we conclude that our results are robust to some broad changes in the instrument set, the forecasting process, the import price series, and the estimation method. In particular, in all cases, these estimates suggest that movements in relative import prices have been an important factor in accounting for traded goods inflation; once we account for endogenous markup variation, there is a limited, if not negligible, contribution of indexation to traded goods inflation.

6  Conclusions

In this paper, we developed a structural model and showed that foreign competition has played an important role in accounting for the behavior of goods inflation through changes in desired markups of domestic firms. In particular, we found that foreign competition has lowered domestic goods inflation by nearly 1 percentage point over the 2000-2006 period. In addition, our results also provided evidence in favor of demand curves which lead to endogenous variations in markups. In contrast to previous work, we found that an inflation specification without backward-looking behavior performed reasonably well in explaining movements in traded goods inflation due to endogenous changes in desired markups.

Although we view this as an important step in understanding how international factors influence domestic prices, goods production is about $\frac{1}{3}$ of overall GDP. A rough estimate would suggest that foreign competition has lowered overall GDP inflation about $\frac{1}{3}$ of 1 percentage point over the 2000-2006 period. However, this estimate does not take into account any interaction between the traded and non-traded sectors, which may magnify these effects. We leave the exploration of this issue to future research.

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Table 1:  Estimates of Open Economy Calvo Model (Capital Not Firm Specific) 1983:Q1 - 2006Q4 ^a,b

Parameter	VES with indexation	VES without indexation	CES with indexation	CES without indexation
θ	0.82 (0.03)	0.82 (0.03)	0.92 (0.02)	0.91 (0.02)
ψ	0.73 (0.11)	0.72 (0.12)	0 (-)	0 (-)
δD	0.10 (0.08)	0 (-)	0.34 (0.08)	0 (-)
$\frac{\sigma_{\pi^{F}}}{\sigma_{\pi}}$	0.73	0.76	0.34	0.28
$\frac{\sigma^{p_{m}}_{\pi}}{\sigma_{\pi}}$	0.33	0.40	0.00	0.00
Q-Statistic(1)	0.20 [0.65]	2.22 [0.14]	0.38 [0.54]	11.43 [0.00]
Q-Statistic(4)	3.93 [0.42]	6.87 [0.14]	11.34 [0.02]	44.33 [0.00]
gmin	71.64	62.03	72.30	62.33

^a  Standard errors are reported in parentheses. A dash in lieu of a standard error indicates that we restricted the corresponding parameter. Q-statistic refers to the Ljung-Box test for serial correlation of $\epsilon_{\pi t}$ at lags 1 and 4. Probability values of Q-statistics are reported in brackets. $\frac{\sigma_{\pi^{F}}}{\sigma_{\pi}}$ refers to the ratio of the volatility of predicted inflation to the volatility of actual inflation, and $\frac{\sigma_{\pi^{P_{m}}}}{\sigma_{\pi}}$ refers to the contribution of the relative import price to inflation volatility.

^b  The estimated inflation equation from Section 4 is:

$$\hat{\pi}_{t}=\delta_{D}\hat{\pi}_{t-1}+\kappa\left[ \frac{1-\Psi} {1-\beta\rho_{s}} \hat{s}_{t}+ \omega\frac{\epsilon_{A}}{\epsilon} \frac {\Psi(1+\beta\rho_{M2}L)}{1-\beta\rho_{M1}-\beta^{2} \rho_{M2}} \hat{p} _{Mt}\right] + \epsilon_{\pi t},$$

where $\kappa= \frac{(1-\beta\theta)(1-\theta)}{\theta}$.

Table 2:  Estimates of Open Economy Calvo Model (Firm-Specific Capital) 1983Q1 - 2006Q4 ^a,b

Parameter	VES with indexation	VES without indexation	CES with indexation	CES without indexation
θ	0.75 (0.03)	0.74 (0.03)	0.83 (0.03)	0.80 (0.04)
ψ	0.73 0.11)	0.72 (0.12)	0 (-)	0 (-)
δD	0.10 (0.08)	0 (-)	0.34 (0.08)	0 (-)
$\frac{\sigma_{\pi^{F}}}{\sigma_{\pi}}$	0.73	0.76	0.34	0.28
$\frac{\sigma^{p_{m}}_{\pi}}{\sigma_{\pi}}$	0.33	0.40	0.00	0.00
Q-Statistic(1)	0.20 [0.65]	2.22 [0.14]	0.38 [0.54]	11.43 [0.00]
Q-Statistic(4)	3.93 [0.42]	6.87 [0.14]	11.34 [0.02]	44.33 [0.00]
gmin	71.64	62.03	72.30	62.33

^a  Standard errors are reported in parentheses. A dash in lieu of a standard error indicates that we restricted the corresponding parameter. Q-statistic refers to the Ljung-Box test for serial correlation of $\epsilon_{\pi t}$ at lags 1 and 4. Probability values of Q-statistics are reported in brackets. $\frac{\sigma_{\pi^{F}}}{\sigma_{\pi}}$ refers to the ratio of the volatility of predicted inflation to the volatility of actual inflation, and $\frac{\sigma_{\pi^{P_{m}}}}{\sigma_{\pi}}$ refers to the contribution of the relative import price to inflation volatility.

^b  The estimated inflation equation is:

$$\hat{\pi}_{t}=\delta_{D}\hat{\pi}_{t-1}+\kappa_{D} \left[ \frac{1-\Psi} {1-\beta\rho_{s}} \hat{s}_{t}+ \omega\frac{\epsilon_{A}}{\epsilon} \frac {\Psi(1+\beta\rho_{M2}L)}{1-\beta\rho_{M1}-\beta^{2} \rho_{M2}} \hat{p} _{Mt}\right] + \epsilon_{\pi t},$$

where $\kappa_{D} = \frac{(1-\beta\theta)(1-\theta)}{\theta\left[ 1+\epsilon\frac{1-\alpha}{\alpha} (1-\Psi)\right] }$ .

Table 3:  Comparison of Benchmark Estimates and Calibrated Demand Curves in the Literature

Study	$\epsilon$	$\mu$	$\frac{\partial\epsilon(i)}{\partial p_{D}(i)}\frac {1}{\epsilon}$	$\Psi$
Benchmark Estimates	6	1.2	13.67	0.73
Chari, Kehoe, and McGrattan (2000)	10	0.11	300	0.97
Coenen, Levin, and Christoffel (2007)	5-20	0.05-0.25	10-33	0.47-0.89
Eichenbaum and Fischer (2007)	11	0.1	10-33	0.5-0.77
Dossche, Heylen, and den Poel (2007)b	1.4	2.5	0.8	0.67
Dotsey and King (2005)	10	0.11	60	0.87
Gust, Leduc, and Vigfusson (2006)	6	0.2	18.30	0.78

^b  Median estimated demand elasticity and curvature from Table 5.

Table 4:  Estimates of VES Specification Under Alternative Calibrations

Parameter	Benchmark*	ω = 0.35	εA = 0.5	εA = 2	μ = 0.1
θ	0.75 (0.03)	0.77 (0.03)	0.72 (0.04)	0.77 (0.03)	0.69 (0.04)
ψ	0.73 (0.11)	0.66 (0.14)	0.84 (0.08)	0.56 (0.14)	0.82 (0.08)
δD	0.10 (0.08)	0.10 (0.08)	0.10 (0.08)	0.10 (0.08)	0.10 (0.08)

^*  The benchmark column refers to the model including firm-specific capital. The estimated inflation equation is:

$$\hat{\pi}_{t}=\delta_{D}\hat{\pi}_{t-1}+\kappa_{D} \left[ \frac{1-\Psi} {1-\beta\rho_{s}} \hat{s}_{t}+ \omega\frac{\epsilon_{A}}{\epsilon} \frac {\Psi(1+\beta\rho_{M2}L)}{1-\beta\rho_{M1}-\beta^{2} \rho_{M2}} \hat{p} _{Mt}\right] + \epsilon_{\pi t},$$

where $\kappa_{D} = \frac{(1-\beta\theta)(1-\theta)}{\theta\left[ 1+\epsilon\frac{1-\alpha}{\alpha} (1-\Psi)\right] }$ .

Table 5:  Estimates of VES Specification Under Alternative Assumptions ^a

Parameter	Benchmark VESb	Smaller Instrument Setc	VAR(2) Forecasting Modeld	Alternative Import Price Seriese	Maximum Likelihood
θ	0.75 (0.03)	0.75 (0.04)	0.72 (0.05)	0.75 (0.03)	0.77 (0.05)
ψ	0.73 (0.11)	0.67 (0.17)	0.82 (0.16)	0.76 (0.12)	0.71 (0.18)
δD	0.10 (0.08)	0.16 (0.09)	0.11 (0.08)	0.18 (0.09)	0.11 (0.08)
$\frac{\sigma_{\pi^{F}}}{\sigma_{\pi}}$	0.73	0.77	0.69	0.74	0.74
Q-Statistic(1)	0.20 [0.65]	0.00 [0.99]	0.29 [0.59]	0.12 [0.72]	0.15 [0.70]
Q-Statistic(4)	3.93 [0.42]	3.43 [0.49]	3.82 [0.43]	5.56 [0.23]	3.37 [0.50]
gmin	71.64	156.81	71.03	84.07	-

^a  Standard errors are reported in parentheses. Q-statistic refers to the Ljung-Box test for serial correlation of $\epsilon_{\pi t}$ at lags 1 and 4, respectively. Probability values of the Q-statistics are reported in brackets. $\frac{\sigma_{\pi^{F}}}{\sigma_{\pi}}$ refers to the ratio of the volatility of predicted inflation to the volatility of actual inflation.

^b  The benchmark column refers to the model including firm-specific capital. The estimated inflation equation is:

$$\hat{\pi}_{t}=\delta_{D}\hat{\pi}_{t-1}+\kappa_{D} \left[ \frac{1-\Psi} {1-\beta\rho_{s}} \hat{s}_{t}+ \omega\frac{\epsilon_{A}}{\epsilon} \frac {\Psi(1+\beta\rho_{M2}L)}{1-\beta\rho_{M1}-\beta^{2} \rho_{M2}} \hat{p} _{Mt}\right] + \epsilon_{\pi t},$$

where $\kappa_{D} = \frac{(1-\beta\theta)(1-\theta)}{\theta\left[ 1+\epsilon\frac{1-\alpha}{\alpha} (1-\Psi)\right] }$ .

^c  The benchmark instrument set includes two lags of relative import prices, commodity price inflation, and quadratically-detrended output in the traded goods sector, 4 lags of the interest rate spread, and one lag of real unit labor costs and inflation. The smaller instrument set includes two lags of relative import prices, two lags of inflation, and one lag of real unit labor costs.

^d  The benchmark system includes an AR(1) process for real unit labor costs and an AR(2) for relative import prices. The VAR(2) model refers to replacing these part of the benchmark system with an unrestricted VAR(2) model for real unit labor costs and relative import prices.

^e  The benchmark relative import price series is the NIPA price deflator for non-oil imported goods divided by domestic goods prices. The alternative series excludes import prices of industrial goods and materials in addition to fuel prices.

Figure 1:  Tradable Goods Inflation and Unit Labor Costs, 1983-2006

Data for Figure1

Dates	Domestic Goods Price Inflation	Non-Farm Business Sector Inflation	Unit Labor Costs
1983	2.8390	1.8553	3.3911
1983.25	3.9175	1.2721	1.5894
1983.5	1.9786	3.7874	0.2259
1983.75	2.0320	2.3771	0.2612
1984	3.3539	2.7976	0.9329
1984.25	0.6681	3.4265	0.4333
1984.5	2.0522	3.1707	0.6021
1984.75	1.7985	1.9937	0.7526
1985	4.8667	4.4514	0.7372
1985.25	1.9664	2.3090	0.6983
1985.5	1.8678	2.4543	0.1822
1985.75	0.5355	1.2463	1.1360
1986	-0.3250	1.9776	0.7175
1986.25	0.2367	0.8868	0.5865
1986.5	1.9943	1.2382	0.8277
1986.75	0.0566	2.9350	2.0034
1987	2.7829	1.9729	2.1472
1987.25	-0.3525	2.0822	1.8502
1987.5	1.3277	3.4704	2.0007
1987.75	0.5075	1.5697	1.9133
1988	-0.5784	3.1992	2.4675
1988.25	1.3514	3.2521	2.4381
1988.5	5.2921	4.0455	2.2711
1988.75	2.6042	3.1522	1.5122
1989	2.0836	3.9879	1.2261
1989.25	4.0562	4.3763	0.1547
1989.5	3.6807	2.7329	-0.2027
1989.75	1.4782	1.9931	0.4384
1990	5.3471	4.4436	0.2994
1990.25	4.7997	4.5670	0.6130
1990.5	2.6994	3.4917	0.8619
1990.75	0.2351	3.4966	1.5854
1991	5.7227	4.4916	1.0451
1991.25	3.5588	2.1808	0.9132
1991.5	2.5889	3.0999	0.5089
1991.75	0.2515	1.2932	0.6331
1992	1.0400	1.8833	0.4089
1992.25	0.5457	1.5246	0.1443
1992.5	-0.0294	1.6933	0.2372
1992.75	0.9115	2.2596	-0.7603
1993	3.3976	2.9483	-0.5355
1993.25	-0.4375	1.7784	-0.0315
1993.5	0.6595	1.6005	-0.4020
1993.75	1.9460	1.5836	-1.1852
1994	1.2021	1.6559	-1.1418
1994.25	-0.3595	1.7437	-1.6586
1994.5	0.5197	3.1456	-1.8575
1994.75	-0.6822	1.4510	-2.6545
1995	-0.8997	1.9080	-2.0581
1995.25	-1.1882	1.4568	-1.9789
1995.5	0.8094	1.2324	-1.8100
1995.75	1.0045	0.8248	-1.9503
1996	2.3045	1.6015	-2.3052
1996.25	1.0560	1.3863	-2.7168
1996.5	1.1070	1.5432	-2.6384
1996.75	0.8429	1.9244	-2.8660
1997	1.3992	2.4015	-2.5176
1997.25	0.5054	1.4543	-3.3062
1997.5	-0.9303	1.0973	-3.4545
1997.75	-0.4448	1.0181	-2.3995
1998	-0.2187	0.4408	-1.4227
1998.25	-0.4110	0.1393	-0.3860
1998.5	1.2764	1.1461	-0.1596
1998.75	-0.7948	0.3709	-0.2606
1999	0.5943	1.2786	0.3754
1999.25	-0.0240	1.5594	0.1563
1999.5	-1.3982	1.4016	-0.0304
1999.75	-1.3468	1.3491	-0.0477
2000	-0.2779	3.2462	3.0463
2000.25	1.5568	1.7692	1.0945
2000.5	-0.2134	1.4300	2.9124
2000.75	-0.3365	1.3068	2.1898
2001	0.6231	2.5208	3.3219
2001.25	3.9925	2.7078	1.8680
2001.5	0.4587	0.7057	1.7860
2001.75	0.4751	1.4847	0.6883
2002	-0.5486	-0.0137	0.5147
2002.25	-1.5601	1.9666	0.9178
2002.5	-1.7272	0.9129	0.1306
2002.75	-0.3690	1.5506	-0.1971
2003	-3.1754	1.9408	-0.1774
2003.25	-1.0376	0.3836	-0.0365
2003.5	0.4889	1.2930	-1.5192
2003.75	-1.0384	1.1261	-0.8640
2004	-0.4198	3.5424	-1.8877
2004.25	1.1612	3.0094	-2.9529
2004.5	-3.0142	2.3606	-2.5238
2004.75	0.0885	3.6080	-2.1693
2005	0.3693	3.2671	-2.8145
2005.25	-1.9514	2.4909	-3.5902
2005.5	-0.3220	3.4302	-3.7031
2005.75	-3.6163	3.2297	-3.7721
2006	1.6565	2.8366	-2.3027
2006.25	0.7587	3.3286	-3.7783
2006.5	-2.1775	0.8359	-3.7241
2006.75	-0.7866	0.5262	-2.2646

Figure 2:  Relative Import Prices, 1983-2006

Data for Figure 2

Dates	Domestic Goods Price Inflation	Relative Import Prices
1983	2.8390	5.5375
1983.25	3.9175	4.1483
1983.5	1.9786	4.4951
1983.75	2.0320	4.3155
1984	3.3539	3.8445
1984.25	0.6681	5.4790
1984.5	2.0522	4.4800
1984.75	1.7985	3.1225
1985	4.8667	-0.6729
1985.25	1.9664	-1.5823
1985.5	1.8678	-2.4255
1985.75	0.5355	0.5466
1986	-0.3250	3.8426
1986.25	0.2367	5.4281
1986.5	1.9943	7.0653
1986.75	0.0566	7.8847
1987	2.7829	8.6348
1987.25	-0.3525	11.3002
1987.5	1.3277	12.1361
1987.75	0.5075	14.6368
1988	-0.5784	17.2440
1988.25	1.3514	19.3260
1988.5	5.2921	17.3124
1988.75	2.6042	18.2270
1989	2.0836	17.9804
1989.25	4.0562	17.4079
1989.5	3.6807	15.8572
1989.75	1.4782	15.9689
1990	5.3471	14.7110
1990.25	4.7997	13.1674
1990.5	2.6994	13.0817
1990.75	0.2351	14.8954
1991	5.7227	14.6210
1991.25	3.5588	13.0388
1991.5	2.5889	11.0222
1991.75	0.2515	11.4805
1992	1.0400	12.2291
1992.25	0.5457	11.8010
1992.5	-0.0294	12.9427
1992.75	0.9115	12.9829
1993	3.3976	10.8832
1993.25	-0.4375	12.0155
1993.5	0.6595	12.4006
1993.75	1.9460	12.4081
1994	1.2021	12.4506
1994.25	-0.3595	12.9318
1994.5	0.5197	13.6667
1994.75	-0.6822	14.5583
1995	-0.8997	14.9775
1995.25	-1.1882	16.5707
1995.5	0.8094	16.6688
1995.75	1.0045	15.6307
1996	2.3045	13.5245
1996.25	1.0560	11.4663
1996.5	1.1070	9.5401
1996.75	0.8429	8.0172
1997	1.3992	6.4488
1997.25	0.5054	5.7724
1997.5	-0.9303	5.5154
1997.75	-0.4448	4.7928
1998	-0.2187	3.7051
1998.25	-0.4110	2.8980
1998.5	1.2764	1.6000
1998.75	-0.7948	1.6680
1999	0.5943	1.2777
1999.25	-0.0240	0.4939
1999.5	-1.3982	0.2705
1999.75	-1.3468	0.6392
2000	-0.2779	0.5775
2000.25	1.5568	-0.0017
2000.5	-0.2134	-0.1238
2000.75	-0.3365	-0.4679
2001	0.6231	-0.7365
2001.25	3.9925	-2.4355
2001.5	0.4587	-3.1608
2001.75	0.4751	-3.6826
2002	-0.5486	-4.0461
2002.25	-1.5601	-3.7026
2002.5	-1.7272	-3.1703
2002.75	-0.3690	-3.2598
2003	-3.1754	-2.4228
2003.25	-1.0376	-2.1257
2003.5	0.4889	-2.2644
2003.75	-1.0384	-1.9621
2004	-0.4198	-1.2941
2004.25	1.1612	-1.6468
2004.5	-3.0142	-0.9593
2004.75	0.0885	-0.6691
2005	0.3693	-0.1165
2005.25	-1.9514	0.3880
2005.5	-0.3220	0.2094
2005.75	-3.6163	1.0969
2006	1.6565	0.7607
2006.25	0.7587	0.5993
2006.5	-2.1775	1.5311
2006.75	-0.7866	2.0188

Figure 3:  Cross-Correlogram for Inflation, 1983-2006

Data for Figure 3

Leads	Cross-correlogram with Relative Import Prices	Cross-correlograms with Real Unit Labor Costs
0	0.3370	0.2325
1	0.3031	0.1918
2	0.2929	0.1665
3	0.3029	0.1459
4	0.2962	0.1684
5	0.3101	0.1895
6	0.3089	0.2246
7	0.3142	0.2291
8	0.3089	0.3125
9	0.3457	0.2772
10	0.3733	0.2697
11	0.3971	0.2338
12	0.4132	0.2251

Figure 4:  Properties of Estimated Demand Curve

Figure 5:  Actual and Predicted Inflation from Alternative Specifications (4-Quarter Moving Average)

Predicted inflation is defined in equation (27) in the text. The estimated parameters used in constructing the predicted series for the VES specification are reported in the second column of Table 2, labelled "VES without indexation." The parameters used for the CES case appear in the fourth column of Table 2, labelled "CES without indexation."

Data for Figure 5

Dates	Predicted Inflation (CES, no indexation)	Predicted Inflation (VES, no indexation)	Observed Goods Price Inflation
1984	1.1602	1.8822	2.8205
1984.25	1.0296	1.7314	2.0081
1984.5	1.0721	1.6632	2.0266
1984.75	1.1276	1.6001	1.9682
1985	1.1055	1.3242	2.3464
1985.25	1.1354	1.0966	2.6709
1985.5	1.0880	0.8859	2.6248
1985.75	1.1313	0.8786	2.3091
1986	1.1291	1.0739	1.0112
1986.25	1.1165	1.2357	0.5787
1986.5	1.1894	1.5331	0.6104
1986.75	1.2873	1.7655	0.4906
1987	1.4488	1.9962	1.2676
1987.25	1.5916	2.2868	1.1203
1987.5	1.7241	2.4928	0.9537
1987.75	1.7139	2.6838	1.0664
1988	1.7501	2.9674	0.2261
1988.25	1.8165	3.2363	0.6520
1988.5	1.8471	3.3777	1.6431
1988.75	1.8017	3.4319	2.1673
1989	1.6615	3.3012	2.8328
1989.25	1.4036	2.9599	3.5090
1989.5	1.1241	2.6424	3.1062
1989.75	1.0028	2.3844	2.8247
1990	0.8982	2.1127	3.6406
1990.25	0.9499	2.0001	3.8264
1990.5	1.0702	2.0352	3.5811
1990.75	1.1998	2.1228	3.2703
1991	1.2840	2.1616	3.3642
1991.25	1.3179	2.1182	3.0540
1991.5	1.2780	1.9411	3.0264
1991.75	1.1705	1.7012	3.0305
1992	1.0986	1.5601	1.8598
1992.25	1.0117	1.4419	1.1065
1992.5	0.9811	1.4793	0.4519
1992.75	0.8237	1.3409	0.6169
1993	0.7170	1.1473	1.2063
1993.25	0.6971	1.1297	0.9605
1993.5	0.6249	1.0112	1.1328
1993.75	0.5769	0.9181	1.3914
1994	0.5084	0.8832	0.8425
1994.25	0.3246	0.7234	0.8620
1994.5	0.1602	0.6119	0.8271
1994.75	-0.0058	0.5500	0.1700
1995	-0.1093	0.5639	-0.3554
1995.25	-0.1454	0.6506	-0.5626
1995.5	-0.1401	0.7140	-0.4902
1995.75	-0.0605	0.7500	-0.0685
1996	-0.0884	0.6102	0.7325
1996.25	-0.1718	0.3228	1.2936
1996.5	-0.2654	0.0078	1.3680
1996.75	-0.3688	-0.3016	1.3276
1997	-0.3928	-0.5039	1.1013
1997.25	-0.4594	-0.7148	0.9636
1997.5	-0.5516	-0.8836	0.4543
1997.75	-0.4989	-0.9140	0.1324
1998	-0.3752	-0.9017	-0.2721
1998.25	-0.0453	-0.6980	-0.5012
1998.5	0.3269	-0.5182	0.0505
1998.75	0.5685	-0.4053	-0.0370
1999	0.7716	-0.2913	0.1662
1999.25	0.8328	-0.3073	0.2630
1999.5	0.8474	-0.2925	-0.4057
1999.75	0.8715	-0.2665	-0.5437
2000	1.1732	0.0248	-0.7617
2000.25	1.2792	0.1557	-0.3665
2000.5	1.6116	0.4826	-0.0703
2000.75	1.8643	0.7199	0.1823
2001	1.8955	0.7661	0.4075
2001.25	1.9828	0.7547	1.0164
2001.5	1.8556	0.4996	1.1844
2001.75	1.6860	0.1765	1.3873
2002	1.3689	-0.2873	1.0944
2002.25	1.2616	-0.4055	-0.2937
2002.5	1.0746	-0.5431	-0.8402
2002.75	0.9745	-0.5792	-1.0512
2003	0.8964	-0.5143	-1.7079
2003.25	0.7886	-0.5428	-1.5773
2003.5	0.6022	-0.6572	-1.0233
2003.75	0.5269	-0.6481	-1.1906
2004	0.3337	-0.7621	-0.5017
2004.25	0.0042	-0.9778	0.0480
2004.5	-0.1092	-0.9541	-0.8278
2004.75	-0.2567	-0.9723	-0.5461
2005	-0.3613	-0.9884	-0.3488
2005.25	-0.4333	-0.9427	-1.1269
2005.5	-0.5666	-1.0036	-0.4539
2005.75	-0.7476	-1.0101	-1.3801
2006	-0.6898	-0.9091	-1.0583
2006.25	-0.7110	-0.8919	-0.3808
2006.5	-0.7134	-0.7882	-0.8446
2006.75	-0.5431	-0.6416	-0.1372

Figure 6:  Sampling Distribution of Estimates from Alternative Specifications

The VES specification described in equation (26) in the text is used as the data-generating process.

Data for Figure 6 - Panel A:  Indexation, δ_D

Indexation, δD	True Value of ψ = 0.67: Probability Density Function of Estimator from CES Specification	True Value of ψ = 0.67: Probability Density Function of Estimator from VES specification	True Value of ψ = 0.9: Probability Density Function of Estimator from CES Specification	True Value of ψ = 0.9: Probability Density Function of Estimator from VES specification
0.000	0.0019293	0.0431689	0.0001354	0.0414185
0.005	0.0026540	0.0557006	0.0000155	0.0556164
0.010	0.0037207	0.0671857	-0.0000699	0.0688779
0.015	0.0051167	0.0776919	-0.0001216	0.0812418
0.020	0.0068292	0.0872867	-0.0001401	0.0927468
0.025	0.0088453	0.0960378	-0.0001263	0.1034317
0.030	0.0111522	0.1040130	-0.0000806	0.1133354
0.035	0.0137371	0.1112798	-0.0000039	0.1224965
0.040	0.0165871	0.1179060	0.0001032	0.1309538
0.045	0.0196895	0.1239591	0.0002401	0.1387461
0.050	0.0230315	0.1295068	0.0004061	0.1459123
0.055	0.0266001	0.1346168	0.0006005	0.1524910
0.060	0.0303826	0.1393568	0.0008226	0.1585210
0.065	0.0343662	0.1437943	0.0010719	0.1640412
0.070	0.0385379	0.1479971	0.0013476	0.1690902
0.075	0.0428851	0.1520328	0.0016490	0.1737069
0.080	0.0473949	0.1559691	0.0019755	0.1779300
0.085	0.0520544	0.1598736	0.0023265	0.1817983
0.090	0.0568509	0.1638139	0.0027012	0.1853506
0.095	0.0617715	0.1678578	0.0030990	0.1886256
0.100	0.0668033	0.1720728	0.0035192	0.1916622
0.105	0.0719306	0.1764914	0.0039634	0.1944858
0.110	0.0771249	0.1810044	0.0044415	0.1970694
0.115	0.0823549	0.1854675	0.0049660	0.1993725
0.120	0.0875892	0.1897363	0.0055490	0.2013548
0.125	0.0927965	0.1936664	0.0062028	0.2029759
0.130	0.0979452	0.1971134	0.0069398	0.2041954
0.135	0.1030042	0.1999330	0.0077720	0.2049729
0.140	0.1079418	0.2019808	0.0087120	0.2052682
0.145	0.1127269	0.2031124	0.0097718	0.2050407
0.150	0.1173279	0.2031834	0.0109637	0.2042501
0.155	0.1217237	0.2021023	0.0122960	0.2028732
0.160	0.1259339	0.1999889	0.0137600	0.2009546
0.165	0.1299885	0.1970160	0.0153430	0.1985561
0.170	0.1339171	0.1933563	0.0170325	0.1957397
0.175	0.1377498	0.1891825	0.0188158	0.1925671
0.180	0.1415163	0.1846673	0.0206802	0.1891001
0.185	0.1452465	0.1799833	0.0226130	0.1854005
0.190	0.1489702	0.1753034	0.0246015	0.1815302
0.195	0.1527172	0.1708003	0.0266331	0.1775510
0.200	0.1565175	0.1666466	0.0286952	0.1735246
0.205	0.1603816	0.1629686	0.0307781	0.1694974
0.210	0.1642430	0.1597069	0.0328847	0.1654535
0.215	0.1680157	0.1567557	0.0350208	0.1613613
0.220	0.1716140	0.1540091	0.0371925	0.1571894
0.225	0.1749521	0.1513612	0.0394056	0.1529063
0.230	0.1779440	0.1487062	0.0416660	0.1484806
0.235	0.1805041	0.1459382	0.0439797	0.1438809
0.240	0.1825463	0.1429513	0.0463525	0.1390755
0.245	0.1839849	0.1396398	0.0487905	0.1340331
0.250	0.1847341	0.1358977	0.0512994	0.1287223
0.255	0.1847366	0.1316511	0.0538813	0.1231275
0.260	0.1840494	0.1269531	0.0565227	0.1172976
0.265	0.1827581	0.1218887	0.0592060	0.1112975
0.270	0.1809483	0.1165430	0.0619137	0.1051920
0.275	0.1787055	0.1110011	0.0646284	0.0990460
0.280	0.1761155	0.1053479	0.0673325	0.0929243
0.285	0.1732637	0.0996685	0.0700086	0.0868918
0.290	0.1702357	0.0940479	0.0726391	0.0810135
0.295	0.1671172	0.0885711	0.0752066	0.0753540
0.300	0.1639937	0.0833233	0.0776936	0.0699783
0.305	0.1609314	0.0783704	0.0800897	0.0649368
0.310	0.1579181	0.0737026	0.0824138	0.0602214
0.315	0.1549224	0.0692911	0.0846916	0.0558096
0.320	0.1519128	0.0651072	0.0869490	0.0516789
0.325	0.1488577	0.0611220	0.0892120	0.0478068
0.330	0.1457257	0.0573067	0.0915064	0.0441708
0.335	0.1424853	0.0536325	0.0938581	0.0407484
0.340	0.1391048	0.0500707	0.0962930	0.0375171
0.345	0.1355529	0.0465925	0.0988370	0.0344544
0.350	0.1317979	0.0431689	0.1015160	0.0315376
0.355	0.1278182	0.0397806	0.1043402	0.0287490
0.360	0.1236315	0.0364447	0.1072580	0.0260888
0.365	0.1192651	0.0331879	0.1102018	0.0235619
0.370	0.1147465	0.0300366	0.1131044	0.0211732
0.375	0.1101032	0.0270174	0.1158983	0.0189275
0.380	0.1053626	0.0241569	0.1185162	0.0168298
0.385	0.1005521	0.0214815	0.1208907	0.0148849
0.390	0.0956993	0.0190179	0.1229543	0.0130978
0.395	0.0908314	0.0167924	0.1246398	0.0114733
0.400	0.0859761	0.0148318	0.1258798	0.0100162
0.405	0.0811589	0.0131529	0.1266297	0.0087284
0.410	0.0763982	0.0117342	0.1269365	0.0075989
0.415	0.0717104	0.0105448	0.1268701	0.0066136
0.420	0.0671120	0.0095535	0.1265002	0.0057584
0.425	0.0626196	0.0087294	0.1258967	0.0050193
0.430	0.0582497	0.0080414	0.1251294	0.0043822
0.435	0.0540188	0.0074584	0.1242682	0.0038329
0.440	0.0499433	0.0069494	0.1233829	0.0033575
0.445	0.0460398	0.0064833	0.1225432	0.0029418
0.450	0.0423249	0.0060292	0.1218192	0.0025717
0.455	0.0388121	0.0055622	0.1212630	0.0022354
0.460	0.0355042	0.0050834	0.1208573	0.0019292
0.465	0.0324010	0.0045998	0.1205672	0.0016519
0.470	0.0295024	0.0041187	0.1203578	0.0014022
0.475	0.0268083	0.0036473	0.1201941	0.0011786
0.480	0.0243184	0.0031928	0.1200414	0.0009798
0.485	0.0220327	0.0027624	0.1198646	0.0008045
0.490	0.0199511	0.0023634	0.1196290	0.0006514
0.495	0.0180734	0.0020029	0.1192995	0.0005190
0.500	0.0163994	0.0016882	0.1188414	0.0004061
0.505	0.0149253	0.0014243	0.1182273	0.0003111
0.510	0.0136323	0.0012076	0.1174610	0.0002327
0.515	0.0124981	0.0010323	0.1165536	0.0001690
0.520	0.0115001	0.0008928	0.1155165	0.0001185
0.525	0.0106158	0.0007832	0.1143609	0.0000795
0.530	0.0098228	0.0006977	0.1130981	0.0000504
0.535	0.0090986	0.0006307	0.1117395	0.0000295
0.540	0.0084208	0.0005763	0.1102962	0.0000153
0.545	0.0077669	0.0005287	0.1087796	0.0000060
0.550	0.0071144	0.0004823	0.1072009	0.0000000
0.555	0.0064465	0.0004325	0.1055704	-0.0000041
0.560	0.0057687	0.0003793	0.1038943	-0.0000066
0.565	0.0050921	0.0003243	0.1021779	-0.0000080
0.570	0.0044279	0.0002687	0.1004263	-0.0000082
0.575	0.0037872	0.0002139	0.0986447	-0.0000077
0.580	0.0031812	0.0001613	0.0968384	-0.0000065
0.585	0.0026210	0.0001123	0.0950125	-0.0000050
0.590	0.0021177	0.0000681	0.0931722	-0.0000032
0.595	0.0016824	0.0000302	0.0913228	-0.0000015
0.600	0.0013264	0.0000000	0.0894694	0.0000000
0.605	0.0010569	-0.0000217	0.0876161	0.0000011
0.610	0.0008658	-0.0000355	0.0857619	0.0000018
0.615	0.0007411	-0.0000426	0.0839049	0.0000021
0.620	0.0006709	-0.0000441	0.0820431	0.0000022
0.625	0.0006432	-0.0000412	0.0801743	0.0000021
0.630	0.0006460	-0.0000350	0.0782966	0.0000018
0.635	0.0006674	-0.0000266	0.0764078	0.0000013
0.640	0.0006954	-0.0000172	0.0745060	0.0000009
0.645	0.0007181	-0.0000080	0.0725891	0.0000004
0.650	0.0007235	0.0000000	0.0706551	0.0000000
0.655	0.0007024	0.0000058	0.0687018	-0.0000003
0.660	0.0006570	0.0000095	0.0667266	-0.0000005
0.665	0.0005920	0.0000114	0.0647268	-0.0000006
0.670	0.0005124	0.0000118	0.0626997	-0.0000006
0.675	0.0004230	0.0000110	0.0606425	-0.0000006
0.680	0.0003287	0.0000094	0.0585525	-0.0000005
0.685	0.0002344	0.0000071	0.0564270	-0.0000004
0.690	0.0001449	0.0000046	0.0542634	-0.0000002
0.695	0.0000652	0.0000021	0.0520588	-0.0000001
0.700	0.0000000	0.0000000	0.0498105	0.0000000
0.705	-0.0000469	-0.0000016	0.0475185	0.0000001
0.710	-0.0000768	-0.0000025	0.0451930	0.0000001
0.715	-0.0000922	-0.0000031	0.0428470	0.0000002
0.720	-0.0000955	-0.0000032	0.0404934	0.0000002
0.725	-0.0000892	-0.0000030	0.0381451	0.0000001
0.730	-0.0000758	-0.0000025	0.0358150	0.0000001
0.735	-0.0000577	-0.0000019	0.0335161	0.0000001
0.740	-0.0000374	-0.0000012	0.0312613	0.0000001
0.745	-0.0000173	-0.0000006	0.0290635	0.0000000
0.750	0.0000000	0.0000000	0.0269356	0.0000000
0.755	0.0000127	0.0000004	0.0248888	0.0000000
0.760	0.0000208	0.0000007	0.0229280	0.0000000
0.765	0.0000250	0.0000008	0.0210560	0.0000000
0.770	0.0000260	0.0000009	0.0192758	0.0000000
0.775	0.0000243	0.0000008	0.0175906	0.0000000
0.780	0.0000208	0.0000007	0.0160032	0.0000000
0.785	0.0000159	0.0000005	0.0145168	0.0000000
0.790	0.0000104	0.0000003	0.0131343	0.0000000
0.795	0.0000049	0.0000002	0.0118587	0.0000000
0.800	0.0000000	0.0000000	0.0106930	0.0000000
0.805	-0.0000037	-0.0000001	0.0096385	0.0000000
0.810	-0.0000062	-0.0000002	0.0086890	0.0000000
0.815	-0.0000077	-0.0000003	0.0078367	0.0000000
0.820	-0.0000083	-0.0000003	0.0070735	0.0000000
0.825	-0.0000081	-0.0000003	0.0063917	0.0000000
0.830	-0.0000073	-0.0000002	0.0057833	0.0000000
0.835	-0.0000059	-0.0000002	0.0052403	0.0000000
0.840	-0.0000042	-0.0000001	0.0047550	0.0000000
0.845	-0.0000021	-0.0000001	0.0043192	0.0000000
0.850	0.0000000	0.0000000	0.0039253	0.0000000
0.855	0.0000021	0.0000001	0.0035652	0.0000000
0.860	0.0000042	0.0000001	0.0032310	0.0000000
0.865	0.0000059	0.0000002	0.0029148	0.0000000
0.870	0.0000073	0.0000002	0.0026087	0.0000000
0.875	0.0000081	0.0000003	0.0023048	0.0000000
0.880	0.0000083	0.0000003	0.0019952	0.0000000
0.885	0.0000077	0.0000003	0.0016720	0.0000000
0.890	0.0000062	0.0000002	0.0013272	0.0000000
0.895	0.0000037	0.0000001	0.0009530	0.0000000
0.900	0.0000000	0.0000000	0.0005414	0.0000000

Data for Figure 6 - Panel B:  Calvo Probability, θ

Calvo Probability, θ	True Value of ψ = 0.67: Probability Density Function of Estimator from CES Specification	True Value of ψ = 0.67: Probability Density Function of Estimator from VES specification	True Value of ψ = 0.9: Probability Density Function of Estimator from CES Specification	True Value of ψ = 0.9: Probability Density Function of Estimator from VES specification
0.300	0.0000000	0.0000000	0.0009475	0.0002707
0.305	0.0000000	0.0000002	0.0006967	0.0006234
0.310	0.0000000	0.0000003	0.0004958	0.0008889
0.315	0.0000000	0.0000002	0.0003391	0.0010864
0.320	0.0000000	0.0000001	0.0002208	0.0012349
0.325	0.0000000	0.0000000	0.0001354	0.0013535
0.330	0.0000000	-0.0000001	0.0000769	0.0014613
0.335	0.0000000	-0.0000002	0.0000399	0.0015773
0.340	0.0000000	-0.0000003	0.0000185	0.0017207
0.345	0.0000000	-0.0000002	0.0000071	0.0019104
0.350	0.0000000	0.0000000	0.0000000	0.0021657
0.355	0.0000000	0.0000003	-0.0000072	0.0025015
0.360	0.0000000	0.0000006	-0.0000133	0.0029175
0.365	0.0000000	0.0000008	-0.0000158	0.0034092
0.370	0.0000000	0.0000006	-0.0000122	0.0039722
0.375	0.0000000	0.0000000	0.0000000	0.0046021
0.380	0.0000000	-0.0000011	0.0000221	0.0052959
0.385	0.0000000	-0.0000023	0.0000511	0.0060573
0.390	0.0000000	-0.0000029	0.0000825	0.0068914
0.395	0.0000000	-0.0000023	0.0001120	0.0078032
0.400	0.0000000	0.0000000	0.0001354	0.0087981
0.405	0.0000000	0.0000043	0.0001492	0.0098839
0.410	0.0000000	0.0000086	0.0001544	0.0110807
0.415	0.0000000	0.0000109	0.0001526	0.0124113
0.420	0.0000000	0.0000088	0.0001457	0.0138987
0.425	0.0000000	0.0000000	0.0001354	0.0155658
0.430	0.0000000	-0.0000159	0.0001238	0.0174361
0.435	0.0000000	-0.0000322	0.0001143	0.0195356
0.440	0.0000000	-0.0000406	0.0001106	0.0218912
0.445	0.0000000	-0.0000327	0.0001164	0.0245294
0.450	0.0000000	0.0000000	0.0001354	0.0274770
0.455	0.0000000	0.0000641	0.0001690	0.0307379
0.460	0.0000000	0.0001588	0.0002094	0.0342246
0.465	0.0000000	0.0002819	0.0002465	0.0378270
0.470	0.0000000	0.0004307	0.0002703	0.0414347
0.475	0.0000000	0.0006029	0.0002707	0.0449377
0.480	0.0000000	0.0007938	0.0002410	0.0482666
0.485	0.0000000	0.0009896	0.0001884	0.0515152
0.490	0.0000000	0.0011743	0.0001236	0.0548184
0.495	0.0000000	0.0013321	0.0000573	0.0583110
0.500	0.0000000	0.0014470	0.0000000	0.0621278
0.505	0.0000001	0.0015119	-0.0000393	0.0663530
0.510	0.0000001	0.0015552	-0.0000590	0.0708680
0.515	0.0000002	0.0016143	-0.0000589	0.0755039
0.520	0.0000001	0.0017266	-0.0000393	0.0800913
0.525	0.0000000	0.0019293	0.0000000	0.0844613
0.530	-0.0000003	0.0022575	0.0000571	0.0885035
0.535	-0.0000005	0.0027364	0.0001232	0.0923436
0.540	-0.0000007	0.0033888	0.0001878	0.0961657
0.545	-0.0000005	0.0042376	0.0002405	0.1001543
0.550	0.0000000	0.0053057	0.0002707	0.1044938
0.555	0.0000010	0.0066128	0.0002724	0.1092571
0.560	0.0000020	0.0081663	0.0002571	0.1140718
0.565	0.0000025	0.0099707	0.0002410	0.1184539
0.570	0.0000020	0.0120303	0.0002402	0.1219195
0.575	0.0000000	0.0143495	0.0002707	0.1239848
0.580	-0.0000037	0.0169405	0.0003424	0.1243521
0.585	-0.0000075	0.0198478	0.0004402	0.1234689
0.590	-0.0000094	0.0231236	0.0005427	0.1219689
0.595	-0.0000076	0.0268202	0.0006287	0.1204859
0.600	0.0000000	0.0309900	0.0006768	0.1196535
0.605	0.0000138	0.0356756	0.0006732	0.1199039
0.610	0.0000279	0.0408814	0.0006341	0.1208625
0.615	0.0000352	0.0466023	0.0005833	0.1219530
0.620	0.0000284	0.0528330	0.0005445	0.1225994
0.625	0.0000000	0.0595683	0.0005414	0.1222252
0.630	-0.0000514	0.0667958	0.0005924	0.1203973
0.635	-0.0001043	0.0744738	0.0006941	0.1172534
0.640	-0.0001315	0.0825535	0.0008381	0.1130744
0.645	-0.0001058	0.0909861	0.0010157	0.1081411
0.650	0.0000000	0.0997227	0.0012182	0.1027342
0.655	0.0001917	0.1086750	0.0014435	0.0971026
0.660	0.0003891	0.1175968	0.0017155	0.0913681
0.665	0.0004907	0.1262027	0.0020642	0.0856204
0.670	0.0003949	0.1342070	0.0025201	0.0799495
0.675	0.0000000	0.1413240	0.0031132	0.0744450
0.680	-0.0007134	0.1473591	0.0038652	0.0691750
0.685	-0.0014368	0.1524807	0.0047634	0.0641189
0.690	-0.0017793	0.1569482	0.0057863	0.0592344
0.695	-0.0013502	0.1610210	0.0069128	0.0544790
0.700	0.0002412	0.1649584	0.0081213	0.0498105
0.705	0.0030740	0.1688898	0.0093946	0.0452002
0.710	0.0059811	0.1724248	0.0107319	0.0406751
0.715	0.0074839	0.1750433	0.0121364	0.0362759
0.720	0.0061037	0.1762248	0.0136112	0.0320434
0.725	0.0003618	0.1754492	0.0151597	0.0280184
0.730	-0.0099618	0.1723481	0.0169862	0.0242418
0.735	-0.0200519	0.1671618	0.0201000	0.0207552
0.740	-0.0238346	0.1602823	0.0257115	0.0176003
0.745	-0.0152362	0.1521017	0.0350310	0.0148188
0.750	0.0118172	0.1430122	0.0492691	0.0124526
0.755	0.0611784	0.1333655	0.0692906	0.0105245
0.760	0.1278157	0.1233520	0.0945789	0.0089821
0.765	0.2044764	0.1131213	0.1242717	0.0077540
0.770	0.2839077	0.1028235	0.1575067	0.0067689
0.775	0.3588569	0.0926082	0.1934218	0.0059556
0.780	0.4226355	0.0826146	0.2307651	0.0052508
0.785	0.4708121	0.0729386	0.2667267	0.0046236
0.790	0.4995195	0.0636650	0.2981071	0.0040509
0.795	0.5048905	0.0548791	0.3217067	0.0035100
0.800	0.4830580	0.0466659	0.3343259	0.0029778
0.805	0.4325537	0.0391042	0.3337561	0.0024397
0.810	0.3615047	0.0322484	0.3217518	0.0019140
0.815	0.2804372	0.0261469	0.3010584	0.0014273
0.820	0.1998773	0.0208478	0.2744213	0.0010060
0.825	0.1303509	0.0163994	0.2445858	0.0006768
0.830	0.0799476	0.0128187	0.2139852	0.0004575
0.835	0.0470104	0.0099978	0.1838036	0.0003320
0.840	0.0274455	0.0077976	0.1549132	0.0002756
0.845	0.0171594	0.0060789	0.1281859	0.0002634
0.850	0.0120584	0.0047028	0.1044938	0.0002707
0.855	0.0087035	0.0035510	0.0844905	0.0002770
0.860	0.0062749	0.0025899	0.0679563	0.0002791
0.865	0.0046073	0.0018067	0.0544530	0.0002781
0.870	0.0035354	0.0011888	0.0435423	0.0002749
0.875	0.0028940	0.0007235	0.0347861	0.0002707
0.880	0.0025179	0.0003981	0.0277462	0.0002665
0.885	0.0022419	0.0001998	0.0219845	0.0002633
0.890	0.0019007	0.0001160	0.0170626	0.0002623
0.895	0.0013290	0.0001340	0.0125425	0.0002644
0.900	0.0003618	0.0002412	0.0079859	0.0002707

Appendix

This appendix is divided into three sections. In Appendix A, we derive the demand curves of the final goods producer as well as the log-linearized expression for inflation, i.e., equation (19). Appendix B compares the small sample properties of our approach with estimating the moment condition implied by equation (23). In Appendix C, we show the results from the system estimation of the forecasting VAR.

A  Theoretical Derivations

A.1  Deriving the Demand of a Domestically-Produced Good

To derive the demand curves for domestically-produced goods, recall that the representative final goods producer maximizes equation (1) subject to the demand aggregator implied by equations (2)-(4). The first order conditions associated with this problem are:

$$P_{Dt}(i) = \frac{\Lambda_{t}}{A_{t}}\left[ \frac{1-\nu}{1-\omega} \frac{A_{Dt}(i)}{A_{t}}+\nu\right] ^{\gamma_{t}-1} \left[ V_{Dt}^{1/\rho }+V_{Mt}^{1/\rho}\right] ^{\rho-1} V_{Dt}^{\frac{1}{\rho}-1} (1-\omega )^{\rho-1},$$ (A.1)

$$P_{Mt}(i) = \frac{\Lambda_{t}}{A_{t}}\left[ \frac{1-\nu}{\omega} \frac {A_{Mt}(i)}{A_{t}}+\nu\right] ^{\gamma_{t}-1} \left[ V_{Dt}^{1/\rho} +V_{Mt}^{1/\rho}\right] ^{\rho-1} V_{Mt}^{\frac{1}{\rho}-1} \omega^{\rho -1},$$ (A.2)

where $\Lambda_{t}$ is the Lagrange multiplier associated with equation (2). Before deriving the demand curves, we need to define $P_{Ft} = \frac{\Lambda_{t}}{A_{t}}$ and show that $P_{Ft}$ satisfies equation (8).

To do so, rewrite equations (A.1)-(A.2) as:

$$\left[ \frac{1-\nu}{\omega} \frac{A_{Dt}(i)}{A_{t}}+\nu\right] = \left( \frac{P_{Dt}(i)}{P_{Ft}}\right) ^{\frac{1}{\gamma_{t}-1}} \left[ V_{Dt}^{1/\rho}+V_{Mt}^{1/\rho}\right] ^{\frac{1-\rho}{\gamma_{t}-1}} V_{Dt}^{\frac{\rho-1}{\rho(\gamma_{t}-1)}} (1-\omega)^{\frac{1-\rho} {\gamma_{t}-1}},$$

$$\left[ \frac{1-\nu}{\omega} \frac{A_{Mt}(i)}{A_{t}}+\nu\right] = \left( \frac{P_{Mt}(i)}{P_{Ft}}\right) ^{\frac{1}{\gamma_{t}-1}} \left[ V_{Dt}^{1/\rho}+V_{Mt}^{1/\rho}\right] ^{\frac{1-\rho}{\gamma_{t}-1}} V_{Mt}^{\frac{\rho-1}{\rho(\gamma_{t}-1)}} \omega^{\frac{1-\rho}{\gamma_{t} -1}}.$$

Substituting these expressions into equations (3)-(4), we can express $V_{Dt}$ and $V_{Mt}$ as:

$$V_{Dt} = \frac{1}{(1-\nu)\gamma_{t}} \left( \frac{P_{Dt}}{P_{Ft}}\right) ^{\frac{\gamma_{t}}{\gamma_{t}-1}} \left[ V_{Dt}^{1/\rho}+V_{Mt}^{1/\rho }\right] ^{\frac{\gamma_{t}(1-\rho)}{\gamma_{t}-1}} V_{Dt}^{\frac{\gamma _{t}(\rho-1)}{(\gamma_{t}-1)\rho}} (1-\omega)^{\frac{\gamma_{t}-\rho} {\gamma_{t}-1}}$$ (A.3)

$$V_{Mt} = \frac{1}{(1-\nu)\gamma_{t}} \left( \frac{P_{Mt}}{P_{Ft}}\right) ^{\frac{\gamma_{t}}{\gamma_{t}-1}} \left[ V_{Dt}^{1/\rho}+V_{Mt}^{1/\rho }\right] ^{\frac{\gamma_{t}(1-\rho)}{\gamma_{t}-1}} V_{Mt}^{\frac{\gamma _{t}(\rho-1)}{(\gamma_{t}-1)\rho}} \omega^{\frac{\gamma_{t}-\rho}{\gamma _{t}-1}},$$ (A.4)

where the price indices, $P_{Dt}$ and $P_{Mt}$, are defined in equation (7). Using equations (A.3) and (A.4), the ratio of $V_{Dt}$ to $V_{Mt}$ is given by:

$$\left( \frac{V_{Dt}}{V_{Mt}}\right) ^{\frac{1}{\rho}} = \left( \frac{P_{Dt} }{P_{Mt}}\right) ^{\frac{\gamma_{t}}{\gamma_{t}-\rho}} \frac{(1-\omega )}{\omega}.$$ (A.5)

Since optimal behavior by a final goods producer implies that equation (2) holds with equality, we can rewrite it as:

$$\left[ \left( \frac{V_{Dt}}{V_{Mt}}\right) ^{\frac{1}{\rho}} + 1 \right] ^{\rho} V_{Mt} = \frac{1}{(1-\nu)\gamma_{t}}.$$ (A.6)

It is useful to express equation (A.4) as:

$$V_{Mt} = \frac{1}{(1-\nu)\gamma_{t}} \left( \frac{P_{Mt}}{P_{Ft}}\right) ^{\frac{\gamma_{t}}{\gamma_{t}-1}} \left[ \left( \frac{V_{Dt}}{V_{Mt}}\right) ^{\frac{1}{\rho}} + 1 \right] ^{\frac{\gamma_{t}(1-\rho)}{\gamma_{t}-1}} \omega^{\frac{\gamma_{t}-\rho}{\gamma_{t}-1}}.$$

Substituting this expression and equation (A.5) into equation (A.6), we have:

$$\left[ \left( \frac{P_{Dt}}{P_{Mt}}\right) ^{\frac{\gamma_{t}}{\gamma_{t} -\rho}} \frac{(1-\omega)}{\omega} + 1 \right] ^{\frac{\gamma_{t}-\rho} {\gamma_{t}-1}} \omega^{\frac{\gamma_{t}-\rho}{\gamma_{t}-1}} P_{Mt} ^{\frac{\gamma_{t}}{\gamma_{t}-1}}=P_{Ft}^{\frac{\gamma_{t}}{\gamma_{t}-1}}.$$

This expression, with some manipulation, can be written as:

$$P_{Ft}=\left[ (1-\omega)P_{Dt}^{\frac{\gamma_{t}}{\gamma_{t}-\rho}}+\omega P_{Mt}^{\frac{\gamma_{t}}{\gamma_{t}-\rho}}\right] ^{\frac{\gamma_{t}-\rho }{\gamma_{t}}},$$

which is equation (8).

With $P_{Ft}$ defined in this way, we can now turn to deriving the demand curve for a domestically-produced good, i.e., equation (6). We begin by re-expressing equation (A.1) as:

$$\left[ \frac{1-\nu}{1-\omega} \frac{A_{Dt}(i)}{A_{t}}+\nu\right] = \left( \frac{P_{Dt}(i)}{P_{Ft}}\right) ^{\frac{1}{\gamma_{t}-1}} \left[ 1 + \left( \frac{V_{Mt}}{V_{Dt}}\right) ^{\frac{1}{\rho}} \right] ^{\frac{1-\rho} {\gamma_{t}-1}} (1-\omega)^{\frac{1-\rho}{\gamma_{t}-1}}.$$ (A.7)

Note that equation (A.5) implies:

$$1 + \left( \frac{V_{Mt}}{V_{Dt}}\right) ^{\frac{1}{\rho}} = \frac {P_{Dt}^{\frac{\gamma_{t}}{\rho-\gamma_{t}}}}{1-\omega} \left[ (1-\omega )P_{Dt}^{\frac{\gamma_{t}}{\gamma_{t}-\rho}}+\omega P_{Mt}^{\frac{\gamma_{t} }{\gamma_{t}-\rho}}\right] ,$$

or

$$1 + \left( \frac{V_{Mt}}{V_{Dt}}\right) ^{\frac{1}{\rho}} = \frac{1}{1-\omega} \left( \frac{P_{Ft}}{P_{Dt}}\right) ^{\frac{\gamma_{t}}{\gamma_{t}-\rho} }.$$ (A.8)

Substituting equation (A.8) into equation (A.7) yields:

$$\left[ \frac{1-\nu}{1-\omega} \frac{A_{Dt}(i)}{A_{t}}+\nu\right] = \left( \frac{P_{Dt}(i)}{P_{Ft}}\right) ^{\frac{1}{\gamma_{t}-1}} \left( \frac{P_{Dt} }{P_{Ft}}\right) ^{\frac{\gamma_{t}}{\gamma_{t}-\rho}\frac{1-\rho}{\gamma _{t}-1}}.$$

Rearranging this expression, we get equation (6):

$$A_{Dt}(i)=(1-\omega)\left[ \frac{1}{1-\nu}\left( \frac{P_{Dt}(i)}{P_{Dt} }\right) ^{\frac{1}{\gamma_{t}-1}}\left( \frac{P_{Dt}}{P_{Ft}}\right) ^{\frac{\rho}{\gamma_{t}-\rho}}-\frac{\nu}{1-\nu}\right] A_{t}.$$

A.2  Deriving the Log-Linearized Pricing Equation

To derive equation (19), we begin by defining the contract price, $P^{c}_{Dt}(i) = \frac{P_{Dt}(i)}{P_{Dt}}$, for a firm that optimally chooses its price at date $t$. Using this definition in equation (10) and log-linearizing, we get:

$$\hat{P}^{c}_{Dt}(i) = \sum^{\infty}_{j=1} (\beta\theta)^{j} (\hat{\pi} _{Dt+j}-\delta_{D} \hat{\pi}_{Dt+j-1})+ (1-\beta\theta) \sum^{\infty}_{j=0} (\beta\theta)^{j} \left[ \hat{s}_{t+j} - \frac{1}{\epsilon-1} \hat{\epsilon }_{t+j}(i) \right] .$$ (A.9)

In the above equation, $\hat{\epsilon}_{t}(i)$ is the log-linearized version of the elasticity of demand for good $i$ given by:

$$\hat{\epsilon}_{t+j}(i) = \nu\epsilon\left( \hat{P}^{c}_{Dt}(i) - \sum ^{j}_{k=1} (\hat{\pi}_{Dt+k}-\delta_{D} \hat{\pi}_{Dt+k-1})\right) -\nu\epsilon_{A} \hat{p}_{Ft+j} + \frac{\gamma}{1-\gamma} \hat{\gamma}_{t+j},$$ (A.10)

where $\hat{p}_{Ft}$ is the log-linearized price index consisting of all of the prices of a firm's competitors relative to the domestic price index, (i.e., $p_{Ft}= \frac{P_{Ft}}{P_{Dt}}$). Substituting this expression for the elasticity of demand into equation (A.9), we have:

$$\hat{P}^{c}_{Dt}(i) = \sum^{\infty}_{j=1} (\beta\theta)^{j} (\hat{\pi} _{Dt+j}-\delta_{D} \hat{\pi}_{Dt+j-1})+ \frac{1-\beta\theta}{1+\frac {\nu\epsilon}{\epsilon-1}} \left[ \hat{s}_{t+j} + \frac{\nu\epsilon} {\epsilon-1} \frac{\epsilon_{A}}{\epsilon} \hat{p}_{Ft+j} - \left( \frac {1}{\epsilon-1}\right) \left( \frac{\gamma}{1-\gamma}\right) \hat{\gamma }_{t+j}\right] .$$

Using the definition of the steady state markup (i.e., $\mu= \frac{\epsilon }{\epsilon-1}$) and the definition of $\Psi$ (i.e., $\Psi= \frac{\nu\mu} {1+\nu\mu}$), this expression, after quasi-differencing, can be rewritten as:

$$\hat{P}^{c}_{Dt}(i) - \beta\theta\hat{P}^{c}_{Dt+1}(i) = \beta\theta(\hat{\pi }_{Dt+1}-\delta_{D} \hat{\pi}_{Dt}) + (1-\beta\theta) \left[ (1-\Psi) \hat {s}_{t} + \Psi\frac{\epsilon_{A}}{\epsilon} \hat{p}_{Ft} + (2 \Psi-1) \hat{\gamma}_{t} \right] .$$ (A.11)

From the log-linearized version of the first expression in equation (7), the contract price at date $t$ can be related to traded goods inflation via:

$$\hat{P}^{c}_{Dt}(i) = \frac{\theta}{1-\theta} (\hat{\pi}_{Dt}-\delta_{D} \hat{\pi}_{Dt-1}).$$ (A.12)

Substituting this expression into equation (A.11), we get an expression relating domestic price inflation to real marginal cost and $p_{Ft}$:

$$\hat{\pi}_{Dt}-\delta_{D} \hat{\pi}_{Dt-1} = \beta(\hat{\pi}_{Dt+1}-\delta_{D} \hat{\pi}_{Dt}) + \kappa\left[ (1-\Psi) \hat{s}_{t} + \Psi\frac{\epsilon_{A} }{\epsilon} \hat{p}_{Ft} + (2 \Psi-1) \hat{\gamma}_{t} \right] .$$ (A.13)

The log-linearized version of equation (8) implies that

$$\hat{p}_{Ft} = \omega\hat{p}_{Mt}.$$

Using this expression in equation (A.13) yields equation (<19).

B  Small Sample Properties of our GMM Estimator

This appendix investigates the small-sample properties of our estimator through a Monte Carlo exercise. We bootstrapped 10,000 Monte Carlo samples for traded goods inflation, relative import prices, and the labor share using equation (26), and its auxiliary forecasting processes, which we have reproduced below:

$$\hat{\pi}_{t}=\delta_{D}\hat{\pi}_{t-1}+\kappa_{D} \left[ \frac{1-\Psi} {1-\beta\rho_{s}} \hat{s}_{t}+ \omega\frac{\epsilon_{A}}{\epsilon} \frac {\Psi(1+\beta\rho_{M2}L)}{1-\beta\rho_{M1}-\beta^{2} \rho_{M2}} \hat{p} _{Mt}\right] + \epsilon_{\pi t}$$ (B.1)

$$\hat{s}_{t} = \rho_{s} \hat{s}_{t-1} + u_{st}$$

$$\hat{p}_{Mt} = \rho_{M1} \hat{p}_{Mt-1} + \rho_{M2} \hat{p}_{Mt-2} + u_{p_{M}t}.$$

As pseudo-true parameter values for our Monte Carlo exercise, we used the GMM estimates of $\theta$, $\Psi$, $\delta_{D}$, $\rho_{s}$, $\rho_{M1}$, and $\rho_{M2}$ from equations (B.1) with the following instrument set: one lag of the labor share, two lags of goods inflation, and two lags of relative import prices as instruments (see the second column of Table 5). In bootstrapping, we jointly sampled with replacement from this model's estimates of $\epsilon_{\pi t}$ and the residuals from the two autoregressive processes, which maintains the correlation structure across residuals. Each bootstrapped sample had a length of 96 observations, which is the same length as the 1983Q1-2006Q4 sample.

For each sample of artificial data, we used the generalized method of moments to re-estimate $\theta$, $\Psi$, and $\delta_{D}$ based, again, on equations (B.1) using the same instrument set as above. The solid lines in Figure 7 show the parameters' sampling distributions. As discussed earlier in the text, the estimates of $\theta$ and $\delta_{D}$ based on the closed form solution do not display much small-sample bias. The estimate of $\Psi$ does display some modest upward bias, and the small-sample confidence intervals appear a little wider than intervals based on asymptotic derivations. Overall, however, our GMM estimator performs well in small samples.

Figure 7 also compares the performance of our baseline estimator with an alternative GMM estimator that uses equation (23) as the moment condition (reproduced below):

$$\hat{\pi}_{t}-\delta_{D}\hat{\pi}_{t-1}=\beta E_{t}[\hat{\pi}_{t+1} -\delta_{D}\hat{\pi}_{t}]+ \kappa_{D} \left[ (1-\Psi) \hat{s}_{t}+ \Psi \omega\frac{\epsilon_{A}}{\epsilon}\hat{p}_{Mt}+\varphi\widehat{\gamma} _{t}\right]$$ (B.2)

$$\kappa_{D} = \frac{(1-\beta\theta)(1-\theta)}{\theta[1+\epsilon\frac {\alpha}{1-\alpha} (1-\Psi)]}.$$

For convenience, we shall refer to our baseline estimator as "closed-form", while we dub the alternative, based on equation (B.2), "single-equation."

In contrast to our closed-form estimator, we find that single-equation GMM estimator performs poorly in small samples. In particular, we found that estimates of $\Psi$ were severely upward-biased with the mass of the distribution near unity - the upper bound of feasible values for $\Psi$. Accordingly, we do not show sampling distributions when we jointly estimate $\Psi$ with $\theta$ and $\delta_{D}$. Instead, the dashed lines labelled "Single-Equation Estimates" in the middle and lower panels of Figure 7 show the sampling distributions for $\theta$ and $\delta_{D}$, conditional on $\psi$ being at its pseudo-true value of $0.67$. We find it remarkable that, despite using the pseudo-true value of $\Psi$, the GMM estimator based on equation (B.2) still performs worse in small samples than the GMM estimator based on the closed form solution. The former yields estimates of $\delta_{d}$ that are more upward-biased and the distribution of estimates for $\theta$ has fatter tails.

Figure 7:  Comparison of Small Sample Properties of Alternative Estimators

Data for Figure 7 - Panel A:  Degree of Real Rigidity, ψ

Degree of Real Rigidity (ψ)	Probability Density Function of Closed-Form Estimates
0.200	0.0072431
0.205	0.0068022
0.210	0.0064418
0.215	0.0061571
0.220	0.0059434
0.225	0.0057958
0.230	0.0057097
0.235	0.0056804
0.240	0.0057029
0.245	0.0057727
0.250	0.0058850
0.255	0.0060350
0.260	0.0062180
0.265	0.0064292
0.270	0.0066639
0.275	0.0069173
0.280	0.0071847
0.285	0.0074614
0.290	0.0077426
0.295	0.0080235
0.300	0.0082994
0.305	0.0085670
0.310	0.0088286
0.315	0.0090882
0.320	0.0093495
0.325	0.0096162
0.330	0.0098924
0.335	0.0101816
0.340	0.0104878
0.345	0.0108148
0.350	0.0111664
0.355	0.0115450
0.360	0.0119468
0.365	0.0123669
0.370	0.0128002
0.375	0.0132415
0.380	0.0136859
0.385	0.0141281
0.390	0.0145632
0.395	0.0149861
0.400	0.0153916
0.405	0.0157777
0.410	0.0161543
0.415	0.0165342
0.420	0.0169304
0.425	0.0173558
0.430	0.0178232
0.435	0.0183456
0.440	0.0189358
0.445	0.0196067
0.450	0.0203712
0.455	0.0212378
0.460	0.0221970
0.465	0.0232349
0.470	0.0243376
0.475	0.0254913
0.480	0.0266821
0.485	0.0278959
0.490	0.0291191
0.495	0.0303376
0.500	0.0315376
0.505	0.0327079
0.510	0.0338477
0.515	0.0349589
0.520	0.0360435
0.525	0.0371034
0.530	0.0381406
0.535	0.0391570
0.540	0.0401544
0.545	0.0411350
0.550	0.0421005
0.555	0.0430547
0.560	0.0440083
0.565	0.0449738
0.570	0.0459638
0.575	0.0469906
0.580	0.0480669
0.585	0.0492051
0.590	0.0504177
0.595	0.0517171
0.600	0.0531160
0.605	0.0546209
0.610	0.0562148
0.615	0.0578746
0.620	0.0595775
0.625	0.0613004
0.630	0.0630205
0.635	0.0647148
0.640	0.0663602
0.645	0.0679340
0.650	0.0694130
0.655	0.0707830
0.660	0.0720639
0.665	0.0732845
0.670	0.0744732
0.675	0.0756588
0.680	0.0768699
0.685	0.0781350
0.690	0.0794829
0.695	0.0809420
0.700	0.0825411
0.705	0.0843017
0.710	0.0862173
0.715	0.0882740
0.720	0.0904583
0.725	0.0927565
0.730	0.0951550
0.735	0.0976399
0.740	0.1001978
0.745	0.1028149
0.750	0.1054776
0.755	0.1081706
0.760	0.1108727
0.765	0.1135609
0.770	0.1162126
0.775	0.1188047
0.780	0.1213146
0.785	0.1237193
0.790	0.1259959
0.795	0.1281218
0.800	0.1300739
0.805	0.1318384
0.810	0.1334362
0.815	0.1348972
0.820	0.1362514
0.825	0.1375285
0.830	0.1387586
0.835	0.1399715
0.840	0.1411970
0.845	0.1424651
0.850	0.1438056
0.855	0.1452345
0.860	0.1467115
0.865	0.1481826
0.870	0.1495935
0.875	0.1508902
0.880	0.1520185
0.885	0.1529243
0.890	0.1535534
0.895	0.1538516
0.900	0.1537649
0.905	0.1532391
0.910	0.1522200
0.915	0.1506535
0.920	0.1484855
0.925	0.1456618
0.930	0.1421282
0.935	0.1378307
0.940	0.1327151
0.945	0.1267272
0.950	0.1198129
0.955	0.1119180
0.960	0.1029885
0.965	0.0929701
0.970	0.0818088
0.975	0.0694503
0.980	0.0558406
0.985	0.0409254
0.990	0.0246507

Data for Figure 7 -Panel B:  Indexation, δ_D

Indexation (δD)	Probability Density Function of Single Equation Estimates	Probability Density Function of Closed-Form Estimates
0.000	0.0341029	0.0179568
0.005	0.0418360	0.0305635
0.010	0.0493784	0.0423497
0.015	0.0567244	0.0533660
0.020	0.0638681	0.0636630
0.025	0.0708038	0.0732914
0.030	0.0775256	0.0823017
0.035	0.0840278	0.0907444
0.040	0.0903044	0.0986702
0.045	0.0963497	0.1061297
0.050	0.1021578	0.1131734
0.055	0.1077231	0.1198519
0.060	0.1130395	0.1262159
0.065	0.1181014	0.1323158
0.070	0.1229029	0.1382023
0.075	0.1274382	0.1439260
0.080	0.1317015	0.1495375
0.085	0.1356870	0.1550873
0.090	0.1393888	0.1606260
0.095	0.1428011	0.1662043
0.100	0.1459182	0.1718726
0.105	0.1487417	0.1776501
0.110	0.1513031	0.1834294
0.115	0.1536415	0.1890717
0.120	0.1557959	0.1944381
0.125	0.1578053	0.1993898
0.130	0.1597088	0.2037880
0.135	0.1615454	0.2074937
0.140	0.1633541	0.2103681
0.145	0.1651739	0.2122724
0.150	0.1670439	0.2130678
0.155	0.1689824	0.2126664
0.160	0.1709251	0.2111852
0.165	0.1727867	0.2087923
0.170	0.1744821	0.2056555
0.175	0.1759264	0.2019430
0.180	0.1770342	0.1978228
0.185	0.1777206	0.1934628
0.190	0.1779003	0.1890311
0.195	0.1774884	0.1846958
0.200	0.1763996	0.1806247
0.205	0.1745805	0.1769411
0.210	0.1721045	0.1735885
0.215	0.1690764	0.1704655
0.220	0.1656013	0.1674709
0.225	0.1617841	0.1645032
0.230	0.1577298	0.1614611
0.235	0.1535433	0.1582432
0.240	0.1493296	0.1547483
0.245	0.1451936	0.1508749
0.250	0.1412404	0.1465218
0.255	0.1375473	0.1416206
0.260	0.1340814	0.1362353
0.265	0.1307825	0.1304630
0.270	0.1275900	0.1244006
0.275	0.1244436	0.1181453
0.280	0.1212828	0.1117940
0.285	0.1180473	0.1054437
0.290	0.1146767	0.0991916
0.295	0.1111105	0.0931346
0.300	0.1072884	0.0873699
0.305	0.1031708	0.0819719
0.310	0.0988020	0.0769264
0.315	0.0942471	0.0721962
0.320	0.0895712	0.0677445
0.325	0.0848395	0.0635344
0.330	0.0801171	0.0595291
0.335	0.0754690	0.0556915
0.340	0.0709605	0.0519849
0.345	0.0666567	0.0483722
0.350	0.0626226	0.0448167
0.355	0.0589076	0.0412924
0.360	0.0554976	0.0378181
0.365	0.0523628	0.0344235
0.370	0.0494732	0.0311384
0.375	0.0467989	0.0279924
0.380	0.0443101	0.0250154
0.385	0.0419769	0.0222370
0.390	0.0397694	0.0196871
0.395	0.0376577	0.0173954
0.400	0.0356119	0.0153916
0.405	0.0336077	0.0136945
0.410	0.0316433	0.0122792
0.415	0.0297223	0.0111097
0.420	0.0278486	0.0101502
0.425	0.0260257	0.0093646
0.430	0.0242576	0.0087172
0.435	0.0225478	0.0081720
0.440	0.0209001	0.0076931
0.445	0.0193182	0.0072445
0.450	0.0178059	0.0067904
0.455	0.0163663	0.0063026
0.460	0.0149994	0.0057840
0.465	0.0137051	0.0052454
0.470	0.0124828	0.0046974
0.475	0.0113322	0.0041507
0.480	0.0102528	0.0036159
0.485	0.0092442	0.0031039
0.490	0.0083061	0.0026252
0.495	0.0074380	0.0021906
0.500	0.0066395	0.0018108
0.505	0.0059098	0.0014936
0.510	0.0052462	0.0012359
0.515	0.0046459	0.0010318
0.520	0.0041058	0.0008753
0.525	0.0036229	0.0007604
0.530	0.0031942	0.0006814
0.535	0.0028167	0.0006321
0.540	0.0024875	0.0006066
0.545	0.0022035	0.0005991
0.550	0.0019617	0.0006036
0.555	0.0017591	0.0006141
0.560	0.0015928	0.0006247
0.565	0.0014597	0.0006294
0.570	0.0013569	0.0006224
0.575	0.0012813	0.0005976
0.580	0.0012300	0.0005492
0.585	0.0011999	0.0004712
0.590	0.0011880	0.0003576
0.595	0.0011915	0.0002025
0.600	0.0012072	0.0000000

Data for Figure 7 -Panel C:  Calvo Probability, θ

Calvo Probability (θ)	Probability Density Function of Single Equation Estimates	Probability Density Function of Closed-Form Estimates
0.500	0.0098084	0.0025653
0.505	0.0100779	0.0021378
0.510	0.0107764	0.0018524
0.515	0.0118381	0.0017205
0.520	0.0131972	0.0017531
0.525	0.0147880	0.0019617
0.530	0.0165599	0.0023573
0.535	0.0185226	0.0029513
0.540	0.0207008	0.0037549
0.545	0.0231196	0.0047794
0.550	0.0258035	0.0060359
0.555	0.0287533	0.0075318
0.560	0.0318728	0.0092582
0.565	0.0350414	0.0112025
0.570	0.0381387	0.0133518
0.575	0.0410442	0.0156934
0.580	0.0436867	0.0182365
0.585	0.0461916	0.0210786
0.590	0.0487333	0.0243392
0.595	0.0514862	0.0281378
0.600	0.0546250	0.0325939
0.605	0.0582772	0.0377815
0.610	0.0623830	0.0435922
0.615	0.0668356	0.0498724
0.620	0.0715283	0.0564683
0.625	0.0763543	0.0632262
0.630	0.0811919	0.0700522
0.635	0.0858595	0.0770921
0.640	0.0901605	0.0845519
0.645	0.0938983	0.0926374
0.650	0.0968764	0.1015542
0.655	0.0989711	0.1113649
0.660	0.1003507	0.1215579
0.665	0.1012563	0.1314780
0.670	0.1019292	0.1404704
0.675	0.1026105	0.1478799
0.680	0.1034747	0.1532625
0.685	0.1044287	0.1570180
0.690	0.1053128	0.1597571
0.695	0.1059672	0.1620907
0.700	0.1062321	0.1646295
0.705	0.1060044	0.1677632
0.710	0.1054081	0.1709967
0.715	0.1046239	0.1736141
0.720	0.1038323	0.1748993
0.725	0.1032141	0.1741361
0.730	0.1028707	0.1708065
0.735	0.1025864	0.1651843
0.740	0.1020664	0.1577411
0.745	0.1010158	0.1489487
0.750	0.0991399	0.1392787
0.755	0.0962446	0.1291501
0.760	0.0925398	0.1187703
0.765	0.0883360	0.1082942
0.770	0.0839439	0.0978764
0.775	0.0796741	0.0876716
0.780	0.0757615	0.0778191
0.785	0.0721382	0.0683954
0.790	0.0686603	0.0594616
0.795	0.0651843	0.0510786
0.800	0.0615663	0.0433077
0.805	0.0577150	0.0362031
0.810	0.0537481	0.0297927
0.815	0.0498357	0.0240977
0.820	0.0461480	0.0191394
0.825	0.0428550	0.0149389
0.830	0.0400754	0.0114980
0.835	0.0377224	0.0087410
0.840	0.0356576	0.0065727
0.845	0.0337427	0.0048979
0.850	0.0318394	0.0036215
0.855	0.0298348	0.0026565
0.860	0.0277173	0.0019486
0.865	0.0255004	0.0014515
0.870	0.0231981	0.0011191
0.875	0.0208239	0.0009054
0.880	0.0184262	0.0007641
0.885	0.0161914	0.0006490
0.890	0.0143407	0.0005141
0.895	0.0130950	0.0003131
0.900	0.0126754	0.0000000

C  Empirical Estimates of the Forecasting VAR

Before we show the estimates of the forecasting VAR, recall that the VAR that we use can be rewritten in companion form as:

$$X_{t} = A X_{t-1} + u_{t},$$

where $X_{t}^{\prime}=[\hat{s}_{t}$ $\hat{p}_{Mt}$ $\hat{s}_{t-1}$ $\hat {p}_{Mt-1}]$, and $u_{t}^{\prime}=[u_{st}$ $u_{p_{M}t}$ 0 $0]^{\prime}$. Accordingly, the first row of the matrix $A$ corresponds to the estimated process for the labor share ( $\hat{s}_{t}$) and the second row corresponds to the estimated process for relative import prices ( $\hat{p}_{Mt}$).

Table 6 shows the estimates of $A$ for three different specifications: the benchmark VES, CES with lagged indexation, and the VES specification using an unrestricted VAR(2). (We show only the results in which capital is firm-specific, since the estimates of $A$ are unchanged under the alternative assumption of full capital mobility within a country.) The table confirms that both the labor share and relative import prices are well-approximated by simple, univariate processes. In particular, for the benchmark VES specification, there is no evidence of feedback between the two variables (i.e., $A_{12} = A_{14} = A_{21} = A_{23}=0$).

Table 6:  System Estimates of VAR

Parameters	Benchmark VES	CES with indexation	VAR(2) Forecasting Model
Labor Share Equation: A11	0.89 (0.03)	0.91 (0.03)	0.71 (0.14)
Labor Share Equation: A12	0 (-)	0 (-)	0.00 (0.05)
Labor Share Equation: A13	0 (-)	0 (-)	0.21 (0.16)
Labor Share Equation: A14	0 (-)	0 (-)	0.00 (0.05)
Labor Share Equation: R2	0.83	0.83	0.83
Labor Share Equation: Durbin-Watson Statistic	2.26	2.32	1.78
Relative Import Price Equation: A21	0 (-)	NA	-0.01 (0.11)
Relative Import Price Equation: A22	1.37 (0.06)	NA	1.39 (0.06)
Relative Import Price Equation: A23	0 (-)	NA	0.00 (0.1)
Relative Import Price Equation: A24	-0.39 (0.06)	NA	-0.41 (0.06)
Relative Import Price Equation: R2	0.98	NA	0.98
Relative Import Price Equation: Durbin-Watson Statistic	1.91	NA	1.92

^a  Standard errors are reported in parentheses.

Footnotes

^*  The authors thank Chris Erceg, Alejandro Justiniano, Mike Kiley, Sylvain Leduc, Andy Levin, Andrea Pescatori, Argia Sbordone, Rob Vigfusson, and Jonathan Wright as well as seminar participants at the Federal Reserve Board and Federal Reserve Bank of Philadelphia for useful comments and suggestions. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. Return to text

^†  Corresponding Author: Christopher Gust, Telephone 202-452-2383, Fax 202-452-4926. Email addresses: [email protected], [email protected], [email protected] Return to text

1.   For arguments in favor of view that global factors have changed the behavior of inflation, see Borio and Filardo (2006) and Rogoff (2003). For evidence that the effect has been limited, see Ihrig, Kamin, Lindner, and Marquez (2007). Ball (2006) takes an even more extreme position, arguing that there is no effect of foreign variables on U.S. inflation. Return to text

2.   Our paper is related to a longstanding literature that includes import prices in the estimation of reduced-form Phillips curves such as Gordon (1973) and Dornbusch and Fischer (1984). However, our paper differs from these earlier works by providing estimates from a structural model. Return to text

3.   Important work estimating the standard NKPC includes Galí and Gertler (1999), Galí, Gertler, and López-Salido (2001) and Sbordone (2002). Return to text

4.   This estimate is broadly consistent with the micro evidence of Nakamura and Steinsson (2007), who find a median duration of non-sale prices of 8-11 months using prices for both consumers and producer's finished goods. Return to text

5.   For simplicity, we assume that the underlying dynamic general equilibrium model implies zero growth in both the tradable and non-tradable sectors. As a result, $\beta$ has the interpretation as the representative household's discount factor. If we did allow for positive and differential growth rates in these two sectors, the appropriate discount rate in equation (19) would be $\tilde{\beta} = \frac{\beta\pi}{ v_{A} \pi_{A}}$, where $\pi$ is the steady state inflation rate for tradable goods, $\pi_{A}$ is the steady state inflation rate of consumer prices, and $v_{A}$ is the growth rate of consumption. For U.S. data, consumption growth and the inflation differential between traded goods prices and consumer prices are small enough that taking into account these extra terms in equation (19) has a negligible effect on our results. Return to text

6.   Our specification has some similarities with Vega and Winkelried (2005), who derive a NKPC in which world prices affect domestic inflation in a small open economy with Rotemberg (1982) style price contracts and a translog demand curve. Also, Sbordone (2007) analyzes how the entry of new competitors affects the slope of the NKPC in a closed economy context using the preferences of Dotsey and King (2005). Return to text

7.   We construct a Laspeyres index for domestic goods prices by netting out the index for export prices from the overall index for goods prices. Return to text

8.   A measure that corresponded more closely to costs in the tradable sector is the labor share for the manufacturing sector, but it is only available on an annual basis beginning in 1986. Return to text

9.   Ideally, we would like to have data on a basket of imported goods that matches the basket of domestically-produced goods. However, no such series are available, and instead our measure of imported prices excludes oil prices, reflecting that oil's share of imports is much larger than its share of domestic goods production. Later, as sensitivity analysis, we use an import price series that excludes other commodity prices. Return to text

10.   For a summary of this literature, see chapter 7 of Campbell, Lo, and MacKinlay (1997). For an early application of this approach to inflation dynamics, see Sbordone (2002). Return to text

11.   If the markup shock evolved according to an AR(1) process, then the lagged endogenous variables that we use would be invalid instruments. In our empirical work, we test for the presence of serially correlated errors to evaluate this possibility. Return to text

12.   For our commodity price measure, we use the raw industrials spot commodity price series from the Commodity Research Bureau. Return to text

13.   For a discussion of the macro estimates and estimates after trade liberalizations, see Ruhl (2005). Return to text

14.   For the CES demand curves, we exclude relative import prices from the instrument set, since the estimated system of equations no longer involves import prices. Return to text

15.   More specifically, we use the critical values from Tables 1 and 2 of Stock and Yogo (2004). For the VES specification with $\delta_{D} = 0$, we have 2 endogenous regressors ($n = 2$), and 9 instruments ($K_{2} = 9$) excluding exogenous variables such as the constant. The critical value for the test based on a desired maximal bias of 5 percent relative to OLS is 18.76, and the critical value for a 10 percent desired maximal size of a 5 percent Wald test is 29.32. Return to text

16.   These values for $\Psi$, $\delta_{D}$, and $\theta$ correspond to the GMM estimates of the VES specification using a smaller set of instruments. In particular, we eliminated the lags of commodity price inflation and the interest rate spread from the instrument set to facilitate the Monte Carlo exercise. The results of this smaller instrument set are shown in Table 5. For additional details regarding the Monte Carlo experiment, see Appendix B.  Return to text

17.   See Podivinsky (1999) for a review of the literature using Monte Carlo simulations to evaluate the small sample properties of GMM. Return to text

18.   This alternative parameterization holds fixed the value of $\kappa_{D} (1-\Psi)$, the reduced-form slope coefficient of real unit labor cost in equation (23).  Return to text

19.   For example, for a very high markup, such as the one estimated by Dossche, Heylen, and den Poel (2007), the variation in the desired markup can be substantial without much variation in the demand elasticity. Return to text

20.   The findings of Nakamura and Steinsson (2007) are also in line with earlier micro studies surveyed in Taylor (1999). In contrast, Bils and Klenow (2004) find a much higher frequency of price adjustment using micro data on consumer prices. The lower frequency of price changes in Nakamura and Steinsson (2007) largely reflects that they exclude temporary sales in measuring price changes, while Bils and Klenow (2004) include sales. Return to text

21.   If we included the 1991 dummy into our analysis, the estimates of $\theta$ and $\delta_{D}$ would fall and the overall fit of the model would improve. However, we take a more conservative approach and exclude the dummy from our analysis. Return to text

22.   In this case, we use the benchmark set of instruments. Return to text

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