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Board of Governors of the Federal Reserve System

International Finance Discussion Papers

Number 918, January 2008---Screen Reader
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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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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

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 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.

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.

At time , an aggregate final good, , 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, , , imported goods, , , and to maximize profits:

(1) |

subject to . In maximizing profits, the firm takes the prices of the domestic, , imported goods, , and the final good, , as given.

For , 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:

(2) |

In turn, is an aggregator of domestically-produced goods given by

(3) |

and is an aggregator of imported goods given by

(4) |

In the above, influences the substitutability between domestic and foreign goods. The share parameter 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 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 evolves according to:

(5) |

where
is an identically and
independently distributed (*iid*) process with zero-mean and
standard deviation,
. 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
and
, 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 and
, 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:

where and .

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

(6) |

In these demand curves, , and are price indices of domestic and imported goods given by:

and | (7) |

while is a price index consisting of all the prices of a firm's competitors:

(8) |

As in Dotsey and King (2005), when , 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, .

Intermediate good 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, .

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 faces a constant probability of being able to re-optimize its price. This probability is assumed to be independent across time, firms, and countries. If firm can not re-optimize its price at time , the firms resets its price based on lagged inflation as in Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2003). In particular, , where , and the parameter captures the degree of indexation to past inflation. In this specification corresponds to indexation to steady state inflation (), and implies full indexation to past inflation. When firm can re-optimize in period , it maximizes:

(9) |

taking , its demand schedule, and the indexing scheme, as given. In the above, is the stochastic discount factor with steady state value, , and denotes the conditional expectations operator at date . The first-order condition from this problem is:

(10) |

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

(11) |

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

(12) |

where the lower case variables denote relative prices (i.e., and ).

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

(13) |

where is the log-linearized desired markup and . The desired markup is given by:

(14) |

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

(15) |

and 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 , which is the elasticity of the elasticity with respect to a firm's relative price. For , this elasticity measures how much 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 , then the demand curves are CES absent the markup shock, and .

The second source of variation in a firm's desired markup arises from foreign competition. This effect depends on , where

(16) |

is the elasticity of substitution between home and foreign goods. The elasticity of the elasticity with respect to the relative import price, , measures how much rises when relative import prices fall. In that case, a firm faces stiffer competition from abroad and will lower its desired markup. For , the CES case, there is no effect of foreign competitiveness on domestic markups and . The importance of foreign competitiveness on the desired markups of domestic firms depends on the degree of trade openness () 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 and , the desired markup can be expressed as:

(17) |

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

(18) |

In our empirical work, we focus on estimating while calibrating the values of and . These three parameters uniquely determine the demand curve parameters discussed earlier - , , and - via equations (15), (16), and (18).

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:

(19) | ||

where |

and where
is domestic price inflation
expressed as a log deviation from steady state, represents real marginal cost (defined using
), and the composite parameter,
, influences the sensitivity of
inflation to exogenous variations in the markup and is given by
.^{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, , affects the responsiveness of inflation to real
marginal cost through its effect on .
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 .

The importance of this foreign competitiveness effect on domestic inflation depends on the degree of trade openness () and the import price elasticity ( ) as well as . We use to gauge the extent of the real rigidity associated with pricing complementarities between firms. A higher value of 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 , 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 . 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 and in the closed economy () using equation (19). As a result, many researchers opt to calibrate the value of with little empirical guidance. However, when , 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 and .

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.

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

(20) |

where is a firm's demand for labor and is a common technological factor. Finally, 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 can be interpreted as those factors which are variable in the short run.

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

(21) |

where and 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, , 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., ), the first order condition for a firm that re-optimizes its price at date is:

(22) |

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

(23) |

where , and and 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 with . 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 or .

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, , 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.

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:

(24) |

where 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 as a vector of variables that includes and , our VAR in companion form can be written as:

(25) |

where is a matrix of VAR coefficients, and
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 using
the relationship:
.

For our benchmark specification of the VAR, we include only measures of real unit labor costs and relative import prices in . 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:

(26) |

where is the lag operator, and is the autoregressive coefficient for unit labor costs, and , and 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
. 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 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 , ,
, as well as , the
coefficients from the VAR used to forecast unit labor costs and
import prices (for our benchmark specification, the relevant
elements of are ,
, and ). We
calibrate , , and
. Given considerable
uncertainty about the values of these parameters, we report results
for alternative calibrations in our sensitivity analysis.
Throughout our analysis, we set
.

For our benchmark calibration, we choose ,
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 implies
. We choose
, 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 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 , which in our model corresponds to the steady state import share. For our benchmark calibration, we choose , 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 .

Table 1
reports our estimates of , , and
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 ,
the estimates are very similar. With firm-specific capital, the
second column of Table 2 shows that our
estimate of 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,
, implying an average contract
duration of over 5 quarters.^{14} Since our estimate of 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 implies a demand elasticity that is far from constant, as the value of 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 for different values of and compares it to the CES demand curve (i.e., ). 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 implies that demand for good 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 ) and the lower elasticity between home and foreign goods ( ) than between home goods (). 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 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, , defined as:

(27) |

using our estimates for and . The dashed red line in the figure shows a four-quarter moving-average of , 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 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 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 "", predicted inflation accounts for nearly 75 percent of the volatility of observed inflation, with movements in relative import prices accounting for about a 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
, 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
,
,
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 .

The top panels of Figure 6 plot the
sampling distributions of our estimates for
and for the
first Monte Carlo experiment in which the pseudo-true value of
is . The estimate
of
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
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
and above
their pseudo-true values. As shown in the bottom panels, the bias
for
and becomes
more severe, when we increase the value of from
0.67 to 0.9 and lower from 0.75 to
0.67.^{18} In particular, the mean estimate of
is 0.47 compared to its
pseudo-true value of when . 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
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, 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 depends on both the steady state demand
elasticity or markup, and the elasticity of the demand elasticity
with respect to a firm's price,
. 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 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
, 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 is
0.67, quite close to our estimate. In our view, is the relevant metric for comparing results, since
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
, 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}

Table 4 considers the sensitivity of our estimates to the calibrated values of , , and . 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 determines the share of imports in goods production. For our benchmark calibration, we choose , the sample average of the ratio of nonfuel goods imports to goods production from 1983-2006. As an alternative, we set , its value in 2004. With , our estimate of 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, , from its benchmark value of 1.5 to 0.5, a value consistent with short-run estimates. In this case, the estimate of rises to 0.84, still within the 90% confidence interval of the benchmark model. Alternatively, an increase in to 2 lowers our estimate of to 0.56. This fall in , 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 , 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 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.

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 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):
with
and
0
.^{22} For the VAR, it is
useful to define the vectors
0 0
and
0
to pick out the the first element,
, and the second element,
, from the vector . Using the previous definitions, we can solve equation
(23), the second
order difference equation governing inflation dynamics and rewrite
it as:

(28) |

The third column of Table 5 reports the estimation results under this alternative forecasting model, and Appendix C reports our estimate of . As shown in Table 5, the estimate of 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.

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 of overall GDP. A rough estimate would suggest that foreign competition has lowered overall GDP inflation about 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 (-) |

δ_{D} | 0.10 (0.08) | 0 (-) | 0.34 (0.08) | 0 (-) |

0.73 | 0.76 | 0.34 | 0.28 | |

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] |

g_{min} | 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
at lags 1 and 4.
Probability values of Q-statistics are reported in brackets.
refers
to the ratio of the volatility of predicted inflation to the
volatility of actual inflation, and
refers to the contribution of the relative import price to
inflation volatility.

^{b} The estimated
inflation equation from Section 4 is:

where .

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 (-) |

δ_{D} | 0.10 (0.08) | 0 (-) | 0.34 (0.08) | 0 (-) |

0.73 | 0.76 | 0.34 | 0.28 | |

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] |

g_{min} | 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
at lags 1 and 4.
Probability values of Q-statistics are reported in brackets.
refers
to the ratio of the volatility of predicted inflation to the
volatility of actual inflation, and
refers to the contribution of the relative import price to
inflation volatility.

^{b} The estimated
inflation equation is:

where .

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

Study | ||||
---|---|---|---|---|

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 | ε = 0.5_{A} | ε = 2_{A} | 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) |

δ_{D} | 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:

where .

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

Parameter | Benchmark VES ^{b} | Smaller Instrument Set ^{c} | VAR(2) Forecasting Model ^{d} | Alternative Import Price Series ^{e} | 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) |

δ_{D} | 0.10 (0.08) | 0.16 (0.09) | 0.11 (0.08) | 0.18 (0.09) | 0.11 (0.08) |

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] |

g_{min} | 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
at lags 1 and 4,
respectively. Probability values of the Q-statistics are reported
in brackets.
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:

where .

^{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, δ _{D} | 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 |

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.

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:

(A.1) | |

(A.2) |

where is the Lagrange multiplier associated with equation (2). Before deriving the demand curves, we need to define and show that satisfies equation (8).

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

Substituting these expressions into equations (3)-(4), we can express and as:

(A.3) |

(A.4) |

where the price indices, and , are defined in equation (7). Using equations (A.3) and (A.4), the ratio of to is given by:

(A.5) |

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

(A.6) |

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

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

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

which is equation (8).

With 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:

(A.7) |

Note that equation (A.5) implies:

or

(A.8) |

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

Rearranging this expression, we get equation (6):

To derive equation (19), we begin by defining the contract price, , for a firm that optimally chooses its price at date . Using this definition in equation (10) and log-linearizing, we get:

(A.9) |

In the above equation, is the log-linearized version of the elasticity of demand for good given by:

(A.10) |

where 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., ). Substituting this expression for the elasticity of demand into equation (A.9), we have:

Using the definition of the steady state markup (i.e., ) and the definition of (i.e., ), this expression, after quasi-differencing, can be rewritten as:

(A.11) |

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

(A.12) |

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

(A.13) |

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

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

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:

(B.1) | ||

As pseudo-true parameter values for our Monte Carlo exercise, we used the GMM estimates of , , , , , and 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 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 , , and 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 and based on the closed form solution do not display much small-sample bias. The estimate of 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):

(B.2) | ||

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 were severely upward-biased with the mass of the distribution near unity - the upper bound of feasible values for . Accordingly, we do not show sampling distributions when we jointly estimate with and . Instead, the dashed lines labelled "Single-Equation Estimates" in the middle and lower panels of Figure 7 show the sampling distributions for and , conditional on being at its pseudo-true value of . We find it remarkable that, despite using the pseudo-true value of , 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 that are more upward-biased and the distribution of estimates for 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 (δ _{D}) | 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 |

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

where , and 0 . Accordingly, the first row of the matrix corresponds to the estimated process for the labor share ( ) and the second row corresponds to the estimated process for relative import prices ( ).

Table 6 shows the estimates of 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 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., ).

Table 6: System Estimates of VAR

Parameters | Benchmark VES | CES with indexation | VAR(2) Forecasting Model |
---|---|---|---|

Labor Share Equation: A_{11} | 0.89 (0.03) | 0.91 (0.03) | 0.71 (0.14) |

Labor Share Equation: A_{12} | 0 (-) | 0 (-) | 0.00 (0.05) |

Labor Share Equation: A_{13} | 0 (-) | 0 (-) | 0.21 (0.16) |

Labor Share Equation: A_{14} | 0 (-) | 0 (-) | 0.00 (0.05) |

Labor Share Equation: R^{2} | 0.83 | 0.83 | 0.83 |

Labor Share Equation: Durbin-Watson Statistic | 2.26 | 2.32 | 1.78 |

Relative Import Price Equation: A_{21} | 0 (-) | NA | -0.01 (0.11) |

Relative Import Price Equation: A_{22} | 1.37 (0.06) | NA | 1.39 (0.06) |

Relative Import Price Equation: A_{23} | 0 (-) | NA | 0.00 (0.1) |

Relative Import Price Equation: A_{24} | -0.39 (0.06) | NA | -0.41 (0.06) |

Relative Import Price Equation: R^{2} | 0.98 | NA | 0.98 |

Relative Import Price Equation: Durbin-Watson Statistic | 1.91 | NA | 1.92 |

^{a} Standard errors are reported in
parentheses.

^{*} 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:
luca.guerrieri@frb.gov, christopher.gust@frb.gov,
david.lopez-salido@frb.gov 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, 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 , where is the steady state inflation rate for tradable goods, is the steady state inflation rate of consumer prices, and 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 , we have 2 endogenous regressors (), and 9 instruments () 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 , , and 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 , 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 and 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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