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Measuring U.S. International Relative Prices: A WARP View of the World

Charles P. Thomas, Jaime Marquez, and Sean Fahle*

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to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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

In this paper we construct a new measure of U.S. prices relative to those of its trading partners and use it to reexamine the behavior of U.S. net exports. Our measure differs from existing measures of the dollar's real effective exchange rate (REER) in that it explicitly incorporates both the difference in price levels between the United States and developing economies and the growing importance of these developing economies in world trade. Unlike existing REERs, our measure shows that relative U.S. prices have increased significantly over the past 15 years. In terms of simple correlations, the relationship between our measure of relative prices and U.S. net exports is much more coherent than that between existing REERs and net exports. To explore this relationship further, we use our measure to construct an index of foreign prices relevant for U.S. export volumes and reexamine several export equations. We find that export equations with the new index dominate those with previous measures in terms of in-sample fit, out-of-sample fit, and parameter constancy. In addition, we find that with the new index of foreign prices the estimated elasticity of U.S. exports with respect to foreign income is a good bit higher than the unitary elasticity found in previous studies using other price measures. This has implications for U.S. current account adjustment.

Keywords: Automated model specification, China, competitiveness, IMF, FRB, geometric aggregation, Penn World Tables, real effective exchange rates, trade elasticities

JEL classification: C82, F41, C22


1  Introduction

In this paper we assemble a new measure of international relative prices to gauge the average amount by which U.S. prices differ from foreign prices. Interest in developing such measures in international economics is not new.1 What is new in this paper is the focus on the interactions between the dispersion of prices across countries and the increased trade with emerging economies. Recognition of these interactions yields a picture of U.S. international relative prices that is fundamentally different from the one given by existing measures of the real effective exchange rate. Indeed, unlike existing measures of relative prices, we find a significant increase in U.S. prices relative to its trading partners over the past 15 years. Further, most of this increase owes to greater trade with developing economies rather than increases in U.S. prices relative to individual countries.

Our measure differs from those currently available for two reasons. First, we measure bilateral relative price levels, as opposed to bilateral relative price indexes. Second, we use an aggregation method that retains the information embodied in those levels. In contrast, existing measures of relative prices are constructed by either chaining or averaging indexes - that is, they begin with price and exchange rate indexes constructed to have a value of 100 in a base year so that the value of the index in a given period indicates how much prices have changed since the base year. Thus, multi-country aggregates of these indexes measure the average change relative to the base year. Such methods are ideal if the purpose is to measure average changes in bilateral real exchange rates but not for measuring the level of U.S. prices relative to prices elsewhere.

Of course, others have recognized the importance of differentiating price indexes from price levels.2 But the implications of combining that distinction with the increased role of developing countries in world trade has not received attention. In particular, the fact that prices in some developing economies are systematically below those in developed economies, combined with the fact that emerging economies' share of world trade has been increasing, has led to a decline in the average world price of traded goods even though prices in individual countries have not fallen. Aggregates based on price indexes cannot capture this interaction between price levels and trade shares. Our weighted average relative price (WARP) is designed specifically to capture this interaction and does so by using a geometric aggregate where the weights capture the change in the structure of U.S. external trade.

Section 2 reviews the evolution of several well known real effective exchange rate indexes. Although these indexes differ in source data and aggregation scheme, they generally paint a similar picture: U.S. prices relative to foreign prices have risen and fallen since 1975 but, on balance, they show no trend. Section 3 presents the WARP, discusses a few of its properties, and compares it to other measures. According to WARP, U.S. prices have risen significantly relative to its trading partners' prices since 1975 with most of the increase occurring since 1990.

This upward trend in U.S. international relative prices constitutes the main result of this paper. Section 4 examines several factors responsible for this upward trend: choice of price data, aggregation method, and currency basket. We find that the upward trend owes to the aggregation of relative price levels as such and to the shift in U.S. trade patterns away from the relatively high-price industrial countries toward the lower-price developing economies. Section 5 examines the sensitivity of this upward trend to both parametric structures and measurement errors; we find that the upward trend of U.S. international relative prices is robust.

Section 6 addresses whether WARP can be thought of as a measure of competitiveness. A point that comes clearly from the analysis is that any reasonable measure of competitiveness will necessarily incorporate the prices of non-traded goods and services as well as the prices of traded goods and services. Indeed, with analytical examples we show why the suitability of a measure of competitiveness to a particular application is largely an empirical question. With this in mind, Section 6 also looks at the relationship between WARP and the U.S. trade balance. We find that in terms of simple correlations, the relationship between relative prices and the U.S. trade balance (as a share of GDP) is much tighter when one uses WARP than when using conventional measures of real effective exchange rates. To explore why this might be the case, we examine several econometric specifications for the volume of U.S. exports. The focus is on assessing the implications for parameter estimates of using WARP-based and other measures of foreign prices to construct a relative price of exports. Our goal is not to offer detailed specifications for exports but, rather, to see if the WARP passes the "proof of concept" test. The evidence suggests that it does.

2  Existing Measures of Relative Prices

Existing measures of the dollar's real effective exchange rate (REER) are designed to reflect how much, on average, U.S. prices have changed relative to the prices of its trading partners.3 The top panel of figure 1 shows the measures constructed by the Federal Reserve, the OECD, and the IMF, all of which are based on relative CPIs.4 Though they differ from one another in many important methodological respects, they all show two common features. First, over the past thirty years, U.S. relative prices have changed little on average, a property that is at odds with the growing U.S. current-account deficit. Second, over shorter periods, U.S. international relative prices deviate substantially from their long-term mean and indeed these prices reached a historical peak in 1985.5

These three measures are constructed by aggregating bilateral real exchange rate indexes. That is, they begin with bilateral nominal exchange rate indexes and adjust them by relative movements in U.S. and foreign consumer price indexes. These bilateral real exchange rates (indexed to 1973=100) are shown in the middle panels of figure 1. The left panel plots the indexes vis-a-vis selected industrial countries; the right panel plots the indexes vis-a-vis selected emerging economies.6 There is clearly a good deal of dispersion among these bilateral indexes, indicating that the CPI-adjusted value of the dollar has risen relative to some countries' currencies and fallen relative to others. On a bilateral basis, these real exchange rates can be interpreted as changes in relative prices.

Given the dispersion of bilateral real exchange rates across countries, it is hard to tell if there is a general pattern to the movements. This is the point of a REER: to distill these various movements into a single measure. To do so requires a weighting scheme. The aggregates shown in the top panels use weights based on trade shares. The weights used by the Federal Reserve Board in its Broad Real Index are representative; a selection of these is given in the bottom panels of figure 1.7 We note the increasing weight given to developing economies, especially China and Mexico, since 1990.

Figure 1:  Real Exchange Rate Indexes:  Effective and Bilateral - Selected Institutions

Figure 1 has five panels arranged in three rows. For the top panel, the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of three measures of the real effective exchange rate for the U.S. dollar; these values range from 70 to 160. These measures are from the Federal Reserve Board, the IMF, and the OECD, and all are rescaled such that their value in 2000 equals 100. Each of these measures is represented by a line and the three lines move in tandem. The lines decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak of about 130 in 1984; they decrease from 1985 to 1987 and remain relatively unchanged until 1994. The lines start increasing in 1995 and reach a peak of 110 towards the end of 2001. The lines decrease from 2002 to 2006. Middle-left panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of the CPI-adjusted bilateral exchange rates for five countries: Canada, Japan, Portugal, Sweden, and Switzerland. The range of the vertical axis is from zero to five hundred. Each of these bilateral exchange rates is represented by a line and the five lines tend to move together varying between 50 and 180, showing no trend from 1970 to 2006. Middle-right panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of the CPI-adjusted bilateral exchange rates for five countries: Colombia, Indonesia, Mexico, China, and Singapore. The range of the vertical axis is from zero to five hundred. Each of these bilateral exchange rates is represented by a line and the five lines move quite differently. The lines for Colombia, Mexico, and Singapore vary between 90 and 170, showing no trend from 1970 to 2006. The line for China shows an upward trend that reaches 500 in 1994, declining to a little above 400 by 2006. The line for Indonesia is relative constant through 1996, increases to 500 in 1997, and declines subsequently to about 260. Bottom-left panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the weights of the Federal Reserve Boards broad measure for five countries: Canada, Germany, Japan, Mexico, and China. The range of the vertical axis is from zero to 0.35. Each of these weights is represented by a line and the five lines move quite differently. The line for Canada declines gradually from 0.21 in 1970 to 0.16 in 2006; the line for Mexico rises from 0.04 to 0.10; the line for China rises from 0 to 0.15; the line for Japan declines from 0.18 to 0.09; the line for Germany is relatively constant at 0.06. Bottom-right panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the weights of the Federal Reserve Boards broad measure for two country blocs: industrial economies and emerging economies. The range of the vertical axis is from zero to 0.75. Each of these weights is represented by a line and the two lines move quite differently. The line for industrial countries declines from 0.75 in 1970 to 0.5 in 2006. The line for emerging economies increases gradually from 0.25 in 1970 to 0.50 in 2006.


How has the increased weight of the developing economies affected the REERs? If one looks at both the increase (depreciation) in China's bilateral real exchange rate since 1973 (middle right panel) and the increase in China's weight in U.S. trade since 1990 (bottom left) one might conclude that China's real exchange rate has had a significant impact on the dollar's REER. However, in fact, China's real exchange rate has had a relatively small effect on the dollar's REER. The mechanics for this result vary with the particular REER used but, in general, the reason is that most of the increase in the dollar real bilateral exchange rate vis-a-vis China occurred prior to 1990, a time when China's weight in U.S. trade was relatively small. REERs designed to show average changes do not get much of a boost from the Yuan's real depreciation prior to 1990 because China's weight in the index was small during that period. Conversely, despite the increase in the weight of China after 1990, there has not been much real depreciation of the Yuan during the period when the weight was large, so, again, the REERs do not get much of a boost. In general, what matters for existing measures of real effective exchange rates is whether the bilateral exchange rates are changing and, if they are not changing much, then increasing the weights on these countries does not cause the REER to change.

If the sole objective is to measure changes in the real effective exchange rates, then one can hardly improve upon existing measures. What we argue is that an exclusive focus on such changes carries a loss of information, that this loss is more than a theoretical possibility, and that the increased participation of low-cost producers in the world economy gives it economic significance.

3  The WARP

3.1  Intuition

The basic idea behind our aggregate is simple. Suppose, for expository ease only, that we have the foreign-currency price of a basket of goods in a foreign country $ i$ (call it $ P_{i}$), and that we also have the dollar price of the same basket in the United States (call it $ P_{us}$). As shown in equation (1), by multiplying the ratio of these prices by the market exchange rate we can define a bilateral relative price $ q_{i}$ as

$\displaystyle q_{i}=\frac{P_{us}}{P_{i}}\cdot E_{i/\$}.$ (1)

This relative price is unitless and easy to interpret: A value of 2 means that the basket is twice as expensive in the United States as it is in country $ i$.

To combine these bilateral relative prices into an aggregate measure for the United States, we use a weighted geometric mean where the weights vary over time and reflect each country's importance in U.S. trade. Specifically,

$\displaystyle Q_{t}^{g}=\left( q_{1t}\right) ^{w_{1t}}\cdot\left( q_{2t}\right) ^{w_{2t}}\cdot\cdots\cdot\left( q_{nt}\right) ^{w_{nt}},$ (2)

where $ w_{it}$ is the time-varying weight associated with the $ ith$ country.8 Two features of $ Q^{g}$ are worth noting. First, the level of the aggregate has meaning: a value of 1.5 means that U.S. prices are on average fifty percent above foreign prices and this value is not arbitrarily determined by the choice of base year. Second, the aggregate can change even if all bilateral relative prices are fixed.

An obvious alternative to $ Q^{g}$ is the commonly used chained aggregate, which is a weighted average of the growth rates of bilateral relative prices:

$\displaystyle \frac{Q_{t}^{c}}{Q_{t-1}^{c}}= {\textstyle\prod\limits_{i=1}^{N}} \left( \frac{q_{it}}{q_{i,t-1}}\right) ^{w_{it}}.$ (3)

By convention, $ Q_{t=base}^{c}$ is set equal to 100 in a given base period and the level of the index for all other periods is defined recursively. Chained aggregation has two important features to recommend it. First, the index is independent of the levels of its constituent $ q$'s, implying that we do not have to choose a meaningful base period for them. Second, changes in the aggregate index only reflect changes in the underlying relative prices. That is, if these rates do not change over a given period, then the aggregate index will not change, even if the weights do. Thus, $ Q_{t}^{c}$ may be ideal for measuring the average change in the dollar's bilateral relative prices.9

Given these aggregation formulas, how can, in the aggregate, U.S. prices rise relative to foreign prices? Holding all else equal, there are four channels:

  1. The nominal dollar exchange rate can appreciate.
  2. U.S. prices can rise.
  3. Foreign prices can fall.
  4. The weight of relatively low-price foreign economies can increase.

The first three channels operate through their impact on the bilateral relative prices - the $ q$'s - and they are fully captured in both the geometric and chained aggregates. However, the chained index does not attempt to capture the fourth channel whereas the geometric aggregate does so explicitly. Specifically, logarithmic differentiation of equations (2) and (3) with respect to time yields

$\displaystyle d\ln Q_{t}^{g}$$\displaystyle = {\textstyle\sum\limits_{i}} w_{it}\cdot d\ln\left( q_{it}\right) + {\textstyle\sum\limits_{i}} dw_{it}\cdot\ln\left( q_{it}\right)$
$\displaystyle d\ln Q_{t}^{c}$$\displaystyle = {\textstyle\sum\limits_{i}} w_{it}\cdot d\ln\left( q_{it}\right) ,$

which implies that

$\displaystyle d\ln Q_{t}^{g}-d\ln Q_{t}^{c}= {\textstyle\sum\limits_{i}} d\omega_{it}\cdot\ln\left( q_{it}\right) . $

Thus the difference in growth rates between the geometric and the chained aggregate is $ {\textstyle\sum\limits_{i}} d\omega_{it}\cdot\ln\left( q_{it}\right) .$ This term captures the interaction between each period's distribution of the level of bilateral relative prices and the evolution of the weights; if the weights are constant, then the two growth rates are identical.

3.2  Implementation

The previous discussion assumed, for expository convenience, the availability of data for the price levels of the foreign and domestic baskets. Thus the first step in implementing our measure is to obtain the bilateral relative prices-the $ q$'s. Data for bilateral relative prices are particularly difficult to obtain because they require comparability of products across countries.10 To this end we use the Penn World Tables, which offer data on purchasing power parties.11

Greatly simplified, Penn collects data on spending and prices for products that are comparable across countries to estimate bilateral purchasing power parities. To avoid the calculations being sensitive to the choice of base country, Penn introduces the concept of "international dollars." This strategy generates a system of simultaneous equations-the Geary-Khamis system-in which the PPP estimates depend on the international dollar and vice versa.12 Specifically, given the international-dollar price of the $ jth$ product, $ \pi_{j}$, the purchasing power parity for the $ ith$ country is

$\displaystyle PPP_{\frac{i}{\$}}=\frac{ {\textstyle\sum\limits_{j=1}^{m}} P_{j}^{i}\cdot Y_{j}^{i}}{ {\textstyle\sum\limits_{j=1}^{m}} \pi_{j}\cdot Y_{j}^{i}},$  $\displaystyle i=1,...n,$ (4)

where $ P_{j}^{i}$ is the price of the $ jth$ product in the $ ith$ country, $ Y_{j}^{i}$ is the amount produced of the $ jth$ product in the $ ith$ country, and the $ j$ index runs over the list of goods and services included in GDP.13 The numerator equals the nominal GDP of the $ ith$ country expressed in local-currency terms whereas the denominator is the value of $ ith$ country's GDP expressed in international dollars. Given $ PPP_{\frac{i}{\$}},$ the international dollar price for the $ jth$ product is computed as

$\displaystyle \pi_{j}= {\textstyle\sum\limits_{i=1}^{n}} \frac{P_{j}^{i}}{PPP_{\frac{i}{\$}}}\cdot\left( \frac{Y_{j}^{i}}{ {\textstyle\sum\limits_{i=1}^{n}} Y_{j}^{i}}\right) ,$  $\displaystyle j=1,...m,$ (5)

where the first term is the price of the $ jth$ product in the $ ith$ country expressed in international dollars and the second term is the $ ith$'s country share in world output of the $ jth$ product.

The system given by (4) and (5) consists of $ m+n$ equations, of which only $ m+n-1$ are linearly independent. To address the over-determined character of the system, Penn uses the United States as the numeraire country meaning that the international dollar has the same purchasing power over total U.S. GDP as the U.S. dollar. Thus the average U.S. price relative to the average price of the $ ith$ country can be estimated as the market exchange rate divided by Penn's PPP:

$\displaystyle q_{i}=\frac{E_{\frac{i}{\$}}}{PPP_{\frac{i}{\$}}}.$ (6)

There are several drawbacks to the Penn data for studying the open-economy implications of movements of U.S. international relative prices. First, the data are released with long delays: the most recent release (release 6.2 in 2006) has data ending in 2004.14 Second, the data are annual. To address these two limitations, the paper develops a method to extend Penn's annual parities and to estimate the associated quarterly observations.15 Finally, the data are subject to errors and section 5 examines the implications of these errors for our measure of U.S. international relative prices.

With these considerations in mind, the top panels of figure 2 show the evolution of the levels of bilateral relative prices for selected countries.16 Among the industrial countries (left panel), U.S. prices are highest relative to Portugal and lowest relative to Switzerland with most measures near or a little below one. As shown to the right, among emerging economies, there is a good deal more dispersion with relative prices ranging between 1.5 and 6.

Figure 2:  U.S. International Relative Prices: Bilateral and Aggregate

Figure 2 has four panels arranged as two columns and two rows. For the top-left panel, the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of the U.S. bilateral relative prices for four countries: Canada, Japan, Portugal, and Switzerland. The range of the vertical axis is from zero to seven. Each of these bilateral relative prices is represented by a line. The lines for Canada, Japan, and Switzerland tend to move together varying between 0.8 and 1.5; the line for Portugal declines from 2.5 in 1970 to 1.0 in 2006. Top-right panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of the U.S. bilateral relative prices for five countries: China, India, Hong Kong, Mexico, and Thailand. The range of the vertical axis is from zero to seven. Each of these bilateral relative prices is represented by a line and the five lines move quite differently. The lines for Mexico and Hong Kong vary between one and 2.5 with no trend from 1970 to 2006. The line for China shows an upward trend that reaches 5.2 in 1994, declining to 3.6 by 2006. The line for India rises from 2.8 in 1970 to 5.2 in 2006; the line for Thailand starts and ends at 3 but it shows large fluctuations in between. Bottom-left panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the weights of the Federal Reserve Boards broad measure for six countries and two blocs: Canada, Germany, Japan, Mexico, China, the United Kingdom, the bloc of industrial economies, and the bloc of emerging economies. The range of the vertical axis is from zero to 0.75. Each of these weights is represented by a line. The line for Canada declines gradually from 0.21 in 1970 to 0.16 in 2006; the line for Mexico rises from 0.04 to 0.10; the line for China rises from 0 to 0.15; the line for Japan declines from 0.18 to 0.09; the line for Germany is relatively constant at 0.06. The line for the bloc of industrial countries declines from 0.75 in 1970 to 0.5 in 2006. The line for the bloc of emerging economies increases gradually from 0.25 in 1970 to 0.50 in 2006. Bottom-right panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the value of the weighted average of U.S. international relative prices (WARP). The range of the vertical axis is from 0.65 to 1.65. The line for the WARP declines gradually from 1.4 in 1970 to 1.1 in 1980; the line rises from 1.1 in 1981 to 1.5 in 1984. Subsequently, the line declines to 1.1 in 1987 and remain relatively unchanged until 1994. Since then, the line rises reaching 1.5 by the end of 2001; the line declines to 1.35 by 2006.


For aggregation we use the same trade weights as those in the Federal Reserve's Broad Real Dollar index (shown in lower left panel). Note that between 1980 and 1990 the total weight of emerging economies held steady near 25 percent, but since 1990 it has doubled to near 50 percent, reflecting rising weights for China and Mexico. The weight for industrial countries has declined, with Japan's weight declining the most. The weighted average of the 34 bilateral relative prices is constructed using equation (2) and shown in the lower right panel. The aggregate of U.S. international relative prices shows an upward trend since the end of the Bretton-Woods period. Indeed, by this measure, U.S. prices are roughly 40 percent above those of its trading partners.

Figure 3 compares the evolution of this measure to the real effective exchange rates from the Federal Reserve and the IMF, rescaled by their own 1971-1991 sample means. The three measures move in near lockstep between 1971 and 1986. As such, neither the choice of aggregation method nor the measure of bilateral price has a noticeable effect on the aggregate measure of U.S. international relative prices through 1986. Since then, however, the aggregates tend to diverge. Specifically, the WARP shows a sustained increase and by 2002 it reaches the same value it had in 1985. In contrast, the other measures remain well below their 1985 peaks. This more recent divergence of U.S. international relative prices might be of interest in assessing the likelihood of a dollar depreciation large enough to address the U.S. external imbalance. Specifically, if one were to apply the 1985-1987 dollar depreciation to the 2006 values of the aggregates based on bilateral price indexes, then these aggregates would fall to levels not recorded in history. In contrast, applying the same depreciation to WARP would bring it to its 1986 value and, by this historical standard, such a depreciation would be consistent with previous experience.

Figure 3:  Alternative Measures of International Relative Prices

In Figure 3 the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of three measures of U.S. international relative prices; these values range from 80 to 135. These measures are the weighted average of U.S. international relative prices, and the real effective exchange rates for the U.S. dollar from the Federal Reserve Board and from the IMF. All measures are rescaled by their own mean, taken over the period from 1971 through 1991. Each of these measures is represented by a line and the three lines move in tandem from 1970 to 1992. Specifically, these lines decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak of about 130 in 1984; they decrease from 1985 to 1987 and remain relatively unchanged until 1992. Subsequently, the line for the WARP increases and reaches a peak of 128 towards the end of 2001, the line declines to 114 by 2006. The lines for the two real effective exchange rates increase also and reach a peak of 112 by 2001; these two lines decrease to 95 by 2006.


4  Explaining the Rise in U.S. International Relative Prices

We now look at why our WARP has risen much more than the other measures since the late 1980s. To ease the exposition, we abstract from differences involving country coverage and weighting scheme to focus on the measurement of bilateral prices and aggregation methods.

4.1  Aggregation Methods and Price Measures

WARP differs from existing measures in both the choice of aggregation formula and the measure of bilateral prices, raising the question of which of these two factors explains the different trends in the aggregates. To address this question, we construct similar aggregates to those reported by other institutions where their bilateral relative prices are replaced with ours$ .$ This strategy ensures that any difference can be interpreted as due to the choice of aggregation method.

The Federal Reserve reports chained aggregates of bilateral CPI-adjusted exchange-rate indexes:

$\displaystyle \frac{ \mathbb{Q} _{t}^{c}}{ \mathbb{Q} _{t-1}^{c}}= {\textstyle\prod\limits_{i=1}^{N}} \left( \frac{r_{\frac{i}{\$},t}}{r_{\frac{i}{\$},t-1}}\right) ^{w_{it} },$ (7)

where

$\displaystyle r_{it}=\left( \frac{CPI_{us,t}}{CPI_{i,t}}\right) \cdot\left( \frac{E_{\frac{i}{\$},t}}{E_{_{\frac{i}{\$},t_{0}}}}\right) , $

$ t_{0}$ represents the base period, $ CPI_{us}$ is the U.S. consumer price index, and $ CPI_{i}$ is the consumer price index for the $ ith$ country. The IMF reports a fixed-weight geometric aggregate of bilateral CPI-adjusted exchange rates:

$\displaystyle \mathbb{Q} _{t}^{gI}= {\textstyle\prod\limits_{i=1}^{N}} \left( r_{\frac{i}{\$},t}\right) ^{w_{i}}.$ (8)

There are several differences between $ q_{it}$ and $ r_{it}$ that are potentially relevant for explaining differences between WARP and existing measures of real effective exchange rates. First, $ q_{it}$ measures the level of bilateral relative prices whereas $ r_{it}$ measures the percent change in bilateral relative prices. Second, the basket used for $ q_{it}$ refers to GDP items and thus includes consumption, investment, government purchases, and exports. The basket used for $ r_{it}$ is limited to consumption items both from domestic and foreign sources. Finally, the baskets embodied in $ q_{it}$ are the same for U.S. and foreign prices whereas the baskets embodied in $ r_{it}$ are not the same for U.S. and foreign prices.

To examine whether the upward trend in our WARP is due to differences between $ q_{it}$ and $ r_{it},$ we construct the same aggregates reported by other institutions while using $ q_{it}$ instead of $ r_{it}.$ Following the FRB's methodology, we construct a chained aggregate of bilateral relative prices$ ,$ $ Q_{t}^{c},$ substituting the Penn parities (the $ q_{i^{\prime}s}) $ into equation (7). We also report the geometric average of indexes of bilateral relative prices, similar to the IMF's methodology:

$\displaystyle Q_{t}^{gI}= {\textstyle\prod\limits_{i=1}^{N}} \left( \frac{q_{it}}{q_{it_{0}}}\right) ^{\omega_{i}}, $

where, following the IMF, we set $ q_{it_{0}}$ as the mean of the values of $ q_{it}$ in 2000 and $ \omega_{i}$ as the sample mean from 1989 to 1991.

Figure 4 shows the evolution of the real effective exchange rates from the Federal Reserve and the IMF along with $ Q_{t}^{g},$ $ Q_{t} ^{c},$ and $ Q_{t}^{gI};$ for comparison purposes, we rescale these measures by their own 1971-1991 sample means. The results indicate that aggregates based on Penn's bilateral relative price indexes, $ Q_{t}^{c}$ and $ Q_{t}^{gI}$, show a downward trend meaning that the upward trend in WARP is not due to differences between $ q_{it}$ and $ r_{it}$ but, rather, to the choice of aggregation method.

Figure 4:  U.S. International Relative Price - Sensitivity to Price Data and Aggregation Method

In Figure 4 the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of five measures of U.S. international relative prices; these values range from 60 to 135. These measures are the weighted average of U.S. international relative prices, the real effective exchange rate for the U.S. dollar from the Federal Reserve Board and the IMF, the chained aggregate of Penn parities, and the geometric aggregate of indexes of Penn parities. All measures are rescaled by their own mean, taken over the period from 1971 through 1991. Each of these measures is represented by a line and the five lines move in tandem from 1970 to 1989. Specifically, these lines decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak of about 130 in 1984; they decrease from 1985 to 1989. Subsequently, the line for the WARP increases and reaches a peak of 128 towards the end of 2001, the line declines to 114 by 2006. The line for the chained aggregate of Penn parities falls from 1989 to 1994 reaching a trough of 73. Subsequently, this line rises reaching a peak of 100 towards the end of 2001, and then declines to 83 by 2006. The lines for the remaining three measures increase starting in 1990 and reach a peak of 112 by 2001; these three lines decrease to 95 by 2006.


4.2  Currency Baskets and Weighting Schemes

We now ask what factors in our aggregation method are responsible for the upward trend in $ Q^{g}.$ Is this trend due to the composition of the currency basket or to our weighting scheme? To address these questions, figure 5 reports separate geometric sub-aggregates for industrial countries and for emerging economies; the $ w_{it}^{\prime}s$ for each group are renormalized to add up to one. The thick blue line is the aggregate of U.S. prices relative to other industrial countries alone. It has been trending down slowly, and it indicates that in 2006 U.S. prices were on average 10 percent below those in other industrial countries. The thick black line plots U.S. prices relative to the prices of emerging economies alone. It has been trending up sharply. These calculations suggest that a key factor accounting for the upward trend in our WARP, the thick red line, is the shift in U.S. trade patterns. Specifically, within the overall aggregate, trade has shifted away from the relatively high-price industrial countries toward the lower-price emerging economies, which tends to raise the overall measure of U.S. prices relative to our trading partners. Within the emerging economies sub-aggregate (the black line), trade has shifted toward the lowest-price economies, such as China; this shift tends to raise U.S. prices relative to the group.

Figure 5:  U.S. International Relative Price - Sensitivity to Currency Basket and Weighting Scheme.

In Figure 5 the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of six measures of U.S. international relative prices; these values range from 0.50 to 2.5. These measures are the geometric weighted average of U.S. international relative prices, with and without allowing for changing weights; the geometric weighted average of U.S. international relative prices for the group of industrial countries, with and without allowing for changing weights; the geometric weighted average of U.S. international relative prices for the group of emerging economies, with and without allowing for changing weights. Each of these measures is represented by a line. The lines for the aggregate of industrial economies decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak of about 1.25 in 1984; they decrease from 1985 to 1989. Subsequently, the lines rise reaching a peak of 1.10 towards the end of 2001, and then decline to 0.8 by 2006. For the group of emerging economies, the line for the aggregate allowing weights to vary decreases from 1970 to 1980; the line increases from 1981 and reaches a peak of about 2.2 in 1984; the line decreases from 1985 to 1989 and subsequently, the line rises reaching a peak of 2.4 towards the end of 2001, and then declines to 2.18 by 2006. The line for the corresponding aggregate with the weights fixed at their 1991 values follows the same contour but the 2001 peak is 2.1 instead of 2.4. For the aggregate of all countries allowing varying weights, the line decreases gradually from 1970 to 1980; increases from 1981 and reaches a peak of about 1.5 in 1984; the line decreases from 1985 to 1989. Subsequently, the line increases and reaches a peak of 1.5 towards the end of 2001; the line declines to 1.30 by 2006. The line for the corresponding aggregate with the weights fixed at their 1991 values follows the same contour but the 2001 peak is 1.25 instead of 1.5.

*  Weights after 1991 are fixed at their 1991 values for the Emerging and Industrial Economies and the Broad with fixed weights series.


To illustrate the importance of the increased weight of the low-price economies, we construct a counterfactual where we ask what would have happened if the weights after 1991 were fixed at their 1991 values. As shown by the dashed red line, in this world, our measure would have U.S. prices only about 10 percent above those of our trading partners--roughly unchanged since 1975 and near the 30-year average. Further, this fixed-weight aggregate has a downward trend with a historical peak in 1985, quite similar to the pattern of the standard aggregates shown earlier. The key question, however, is whose weights exert the strongest influence. To address that question, the figure reports the fixed-weight aggregate for the industrial countries. Appearing as the dashed blue line, this aggregate exhibits a downward trend with a historical peak in 1985, similar to the case of variable weights and to the associated aggregates reported by other institutions. This result implies that fixing industrial-country weights does not change the evolution of that sub-aggregate. In contrast, fixing the weights of each emerging economy induces a downward trend in the associated sub-aggregate (dashed black line), which leaves the historical peak back in 1985, unlike its variable-weight version which peaks in 2003. Therefore, the upward trend in our measure of U.S. international relative prices is due to the increased weight of the low-price economies in the U.S. basket.

5  Sensitivity Analysis

5.1  Sensitivity to Parameterization

A well known result is that our $ Q^{g}$ is a particular case of the CES function

$\displaystyle Q_{t}^{ces}=\left[ {\textstyle\sum\limits_{i=1}^{n}} w_{it}\cdot\left( q_{it}\right) ^{\sigma-1}\right] ^{\frac{1}{\sigma-1}}, $

where $ \sigma$ is the elasticity of substitution among purchases of foreign products and $ Q^{g}=\lim_{\sigma\rightarrow1}Q_{t}^{ces}$. In the absence of econometric evidence supporting $ \sigma=1,$ a relevant question to pose is how sensitive is the upward trend in U.S. international relative prices to alternative values of σ.17

Figure 6 shows the sensitivity of $ Q_{t}^{ces}$ to values of $ \sigma$ ranging from high substitutability ( $ \sigma=2.5$) to near complementarity ( $ \sigma=0.05$). The calculations reveal three findings. First, there is a direct association between the value of $ \sigma$ and the slope of the trend of U.S. relative prices. Second, the 2006 level of WARP is sensitive to extreme values of $ \sigma$ (2.5 and 0.05); using using less extreme values of $ \sigma$ (1.1 and 0.9) yields values of $ Q_{t}^{ces}$ quite close to the values taken by $ Q^{g}$. Finally, if one interprets the large swings in U.S. bilateral trade shares as suggesting high substitutability among foreign products ($ \sigma>2$), then the upward trend in $ Q^{g}$ understates the extent to which U.S. international relative prices have been increasing.

Figure 6:  U.S. International Relative Prices - CES Aggregates with Alternative Values of σ

In Figure 6 the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of seven measures of U.S. international relative prices; these values range from 0.8 to 2.2. These measures are the geometric weighted average of U.S. international relative prices for alternative values of the degree of substitution among foreign products (sigma). There is a line for each of seven different values of sigma. For values of sigma less than one, the lines decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak of about 1.5 in 1984; they decrease from 1985 to 1989 and subsequently, the lines rise reaching a peak towards the end of 2001. The value of the 2001 peak ranges from 1.5 for sigma equal one to 1.33 for sigma equal 0.05. Subsequently, these lines decline from 2002 through 2006. The lines for values of sigma above one follow a similar contour, but their 2001 peaks are much higher than their peaks in 1984.


Overall, we interpret these results as suggesting that the upward trend in U.S. international relative prices associated with WARP is not due to our reliance on a unitary elasticity of substitution among foreign products. This conclusion, however, abstracts from the quality of the data of relative prices, an issue that we examine next.

5.2  Sensitivity to Measurement Errors

We now ask how sensitive is the upward trend of WARP to measurement errors. Other things equal, a lower estimate of $ PPP_{i/\$}$ raises $ q_{i},$ our measure of U.S. prices relative to country $ i.$ Thus a relevant question is whether we are over-estimating $ q_{i},$ and hence WARP, because Penn is underestimating $ PPP_{i/\$}$ for emerging economies. We consider three approaches to address this question: examining alternative measures of purchasing power parity for China; adjusting our measures of relative prices by imputing correction factors larger than those of Kravis and Lipsey (1990); and comparing the WARP to calculations based on the price data for the Big Mac.

5.2.1  Alternative Estimates of PPP for China

A focus on China's purchasing power parity can be motivated in two ways. First, the weight for China has experienced the largest increase and it now has the second largest value in our weighting scheme. Second, the price data for China are of questionable reliability. Figure 7 compares the estimates for China's $ PPP_{i/\$}$ from Penn (solid dark-blue line) to the IMF's estimates from seven recent vintages.18 Prior to 1994, Penn's estimate is never more than eight percent below the IMF estimates. For all the post-1994 period, Penn's estimate is at least as large as any of the estimates from the IMF. Thus there does not seem to be a systematic undervaluation of Penn's parities relative to those of the IMF. The one estimate we could find that is above that of Penn is that of the OECD.19 For 2004, the OECD estimate for China is 2.3, compared to Penn's estimate of 2.1. Thus the OECD estimate is roughly 10 percent above the Penn estimate.

Figure 7:  IMF PPP $ Rates for China - Sensitivity to Data Vintage

In Figure 7 the horizontal axis records time-units measured as years from 1986 to 2005. The vertical axis contains the values for eight measures of the purchasing power parity for China; these values range from 0.7 to 2.3. These measures correspond to seven different releases from the IMFs World Economic Outlook and the 6.2 release of the Penn World Tables. Each of these measures is represented by a line and the eight lines move in tandem. Over the period, the lines increase from values near one in 1986 to values near two in 2005. Specifically, they decline from 1980 to 1983, rise from 1984 to 1997, decline from 1998 to 2002, and rise from 2003 to 2005.


We do not interpret this scant evidence as suggesting that Penn's estimates are relatively error free but that, perhaps, comparing inaccurate measures across institutions is not informative. Thus we examine below the implications of imputing large measurement errors to the relative prices of emerging economies. Indeed, we find that even if the relative prices for all emerging economies were 20 percent below what Penn estimates, this is still not enough to overturn the basic upward trend of WARP.

5.2.2  Imputation of Correction Factors

We now impute measurement errors to the relative prices of emerging economies to examine the sensitivity of WARP to such mismeasurement. Specifically, we denote $ \widetilde{q}_{it}$ as the error-free but unobserved bilateral relative price and postulate that $ \widetilde{q}_{it}=\left( q_{it}\right) ^{\theta}$ where $ \theta$ is the imputed correction factor. We could impute the value of $ \theta$ using either the 13 percent estimated by Kravis and Lipsey (1990) or the 10 percent wedge implied by the OECD estimate for China. To encompass these sources and to allow for even larger errors, we apply $ \theta=3/4$ to the relative prices of China, of Latin American countries, of all emerging economies excluding both China and Latin America countries, and of all emerging economies. Note that the magnitude of the error is directly related to the value of $ q.$ If $ q=2$ and $ \theta=3/4,$ then $ \widetilde {q}=1.68$ implying an error of 19 percent.

The top panel of figure 8 shows how U.S. prices relative to those of emerging economies respond to the imputed correction factor. We find that if $ \theta=3/4$ is applied to all developing countries, then reliance on Penn data overstates U.S. relative prices vis-a-vis this countries by about 20 percent. The bottom panel of figure 8 shows how these measurement errors affect WARP.20 When the correction is applied only to China, the upward trend in U.S. international relative prices remains in place. Applying the correction factor to either Latin America or other emerging economies (except China and Latin American countries), leaves the trend rate of WARP largely unchanged. Finally, applying the correction factor to all of the emerging economies dampens the upward trend of WARP but by no means eliminates it.

Figure 8:  U.S. International Relative Prices - Sensitivity to Measurement Error

Figure 8 has two panels arranged in a single column with two rows. The horizontal axis in the top panel records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of five measures of U.S. international relative prices for the group of emerging economies; these values range from 1.25 to 2.45. Each of these measures is represented by a line and the five lines move in tandem. Specifically, these lines decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak in 1984; they decrease from 1985 to 1989. Subsequently, the lines reach a higher peak towards the end of 2001 and decline through 2006. The ordering of the lines since 1998, from highest to lowest, is: no correction to price data, correction applied only to Latin America, correction applied to Other Emerging economies (that is, excluding China and Latin America), correction applied to China, and correction applied to all emerging economies. The horizontal axis in the bottom panel records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of five measures of U.S. international relative prices for the aggregate of all countries; these values range from 0.95 to 1.60. Each of these measures is represented by a line and the five lines move in tandem. Specifically, these lines decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak in 1984; they decrease from 1985 to 1989. Subsequently, the lines reach a higher peak towards the end of 2001 and decline through 2006. The ordering of the lines since 1998, from highest to lowest, is: no correction to price data, correction applied only to Latin America, correction applied to Other Emerging economies (that is, excluding China and Latin America), correction applied to China, and correction applied to all emerging economies.


5.2.3  Consistency with Big Mac Prices

We now evaluate whether the results from using Penn's parities are unique by comparing them to the prices of McDonald's Big Mac reported by The Economist. This alternative is of interest because The Economist reports the absolute dollar-price levels for the Big Mac. Figure 9 shows the cross-country dispersion of dollar prices for the Big Mac from 1986 to 2007 for 31 countries.21 The data reveal that the number of countries with prices below the U.S. price has increased markedly over the years. As for the range of prices, Switzerland tends to have the highest price whereas China generally has the lowest price.

Figure 9:  Cross-country Dispersion of Big Mac Prices - 1986-2007

In Figure 9 the horizontal axis records time-units measured as years from 1986 to 2007. The vertical axis contains the values for the Economists Big Mac; the axis ranges from zero to six. For each year, the figure shows the value of a Big Mac (in dollars per burger) for all the countries included in the Economists survey; the number of countries varies from year to year. For 1986, the price of a Big Mac ranges from one dollar to 2.5 dollars. For 2007, the price of a Big Mac ranges from 1.1 dollars to five dollars. For most years, China has the lowest price and Switzerland has the highest price.


Given these prices, we construct the U.S. bilateral relative price of a Big Mac, $ q_{B,i},$ as

$\displaystyle q_{B,i}=\frac{P_{B,us}}{P_{B,i}}, $

where $ P_{B,us}$ is the dollar price of a Big Mac in the United States and $ P_{B,i}$ is dollar price of a Big Mac in the $ ith$ country. Given $ q_{B,i}$, the associated geometric aggregate is

$\displaystyle Q_{Bt}^{g}= {\textstyle\prod\limits_{i=1}^{N_{B}}} \left( q_{B,it}\right) ^{\omega_{Bit}},$ (9)

where $ N_{B}$ is the number of countries included in the aggregate and $ \omega_{Bit}$ is the trade weight for the $ ith$ country; we construct this aggregate for countries that are included in the Federal Reserve's Broad Real Dollar index.

Because the list of countries reported by the Economist varies across time, we construct $ Q_{Bt}^{g}$ for two groups that differ in the span of continuous data: group A with data since 1994 and group B with data since 1999; the list of countries in group B includes the countries of group A along with the euro area and other emerging economies. For comparison purposes, we also compute the chained aggregate of relative CPIs and the geometric aggregate of Penn's parities for each country group. Figure 10 shows that the aggregate of Big Mac relative prices and the WARP for group B increase from 1999 to 2006. In contrast, the chained aggregate of relative CPIs for group B declines between these two dates; comparisons based on group A give the same result. Thus we interpret these features as corroborating the evidence embodied in the WARP: U.S. international relative prices have increased.

Figure 10:  U.S. International Relative Prices: WARP and Big Mac

In Figure 10 the horizontal axis records time-units measured as years from 1986 to 2006. The vertical axis contains the values of seven measures of U.S. international relative prices; these values range from 0.95 to 1.5. These measures show data for two country groups. Group A has countries with data since 1994: Argentina, Australia, Brazil, Britain, Canada Chile, Hong Kong, Japan, Malaysia, Mexico, South Korea, Switzerland, and Thailand. Group B has countries with data since 1999: Group A, the Euro Area, Indonesia, Russia, Singapore, Sweden, and Taiwan. The lines correspond to the weighted average of Big Mac relative prices for country groups A and B; the real effective exchange rate for the U.S. dollar from the Federal Reserve Board for country groups A and B; and the geometric aggregate of Penn parities for all countries and for country groups A and B. These lines decline from 1986 to 1989 and increase afterwards. The three geometric aggregate measures and the two Federal Reserve Board measures reach their peaks by the end of 2001, declining subsequently. The 2001 peaks of the geometric aggregate measures vary between 1.43 and 1.48 and are somewhat higher than their Federal Reserve Board counterparts, which peak between 1.23 and 1.25. The Big Mac measures for Groups A and B peak slightly later, in 2004, around 1.41 and 1.30, respectively, also declining subsequently.

*  The FRB Group A and B series are re-scaled by the 1971 value of the Geometric aggregate of levels from Penn's parities.


6  Applications

6.1  WARP and Competitiveness

One question of interest is whether our WARP is informative for issues involving international competitiveness. A priori, one could argue that the WARP is not informative because it depends importantly on the prices of non-tradeables. There are, however, developments related to relative prices across countries that are not well reflected in standard measures of real exchange rates and yet have important influences on trade and other macro variables. In particular, our aim is to assess whether the WARP captures some aspects of what people have in mind when they use the term 'competitiveness' in a macro context.

It is common practice to consider competitiveness in terms of the prices of tradeable products at home and abroad:

A change in the relative price of a manufactured product (tradable good) between any two suppliers is defined as a change in price competitiveness. (McGuirk, 1986, page 3)

However, this view is somewhat sharpened by recognizing that prices of competing goods influence each other and that differences in competitiveness are determined by differences in costs that manifest themselves as differences in margins in the tradeable sector:

One might say that an industry is internationally competitive if it produces tradables and is profitable. A reduction in competitiveness is then a reduction in profitability in some or all tradables industries. (Corden 1994, page 267)

This latter view is a return to Keynes' view in 1925 when he wrote

My own guess is that, compared with 1913, sheltered [non-tradeable] prices here are, at the present rates of sterling exchange, perhaps as much as ten per cent. too high in comparison with the unsheltered [tradeable] prices, and that the injury thus caused to the competitive position of our exports in the international market is aggravated by the fact that in Germany, France, Belgium and Italy the sheltered prices are fully ten per cent. too low. (Keynes, 1925, page 301). Emphasis and bracketed entries added.

Keynes recognized that the prices of "unsheltered" (traded) products would be nearly equalized in the world market and that competitiveness would be determined by, and reflected in, the relative prices of sheltered (non-traded) goods.

One way to illustrate Keynes' point is to show that conventional measures of competitiveness are not invariant to developments in the non-tradeable sector. Specifically, following Corden, we express competitiveness for the $ jth$ product as the ratio of producers' markups

$\displaystyle \rho_{j}=\left( \frac{\overline{p}_{j}}{c_{j}^{1}}\right) /\left( \frac{\overline{p}_{j}}{c_{j}^{u}}\right) =\frac{c_{j}^{u}}{c_{j}^{1}}, $

where we assume that the law of one price holds with $ \overline{p}_{j}$ being the associated price; $ c_{j}^{1}$ is the marginal cost of the $ jth$ product in country 1 and $ c_{j}^{u}$ is the U.S. counterpart. If $ \rho_{j}>1,$ then "Country 1 is said to be more competitive than the United States in the $ jth$ industry."

To determine marginal costs, appendix A.4 develops a simple, three country model in which production takes place with a Leontieff technology using labor and both tradeable and non-tradeable intermediate inputs. With these assumptions, marginal costs are linear functions of factor prices and input requirements. Thus, as detailed in equation (31) in appendix A.4, the effect on $ \rho_{j}$ of a one percent decrease in U.S. productivity of the non-tradeable input is

$\displaystyle \frac{d\rho_{j}}{\rho_{j}}=\nu_{jn}^{u}>0, $

where $ \nu_{jn}^{u}$ is the share of the non-tradeable input in $ c_{j}^{u}$. This dependency of $ \rho_{j}$ on non-tradeables embodies Keynes' point: questions of competitiveness cannot be usefully examined by abstracting from the prices of non-tradeables.

Overall, this example shows that if the essence of a measure of competitiveness is invariance to developments in the non-tradeable sector, then a popular measure offered in the literature is as deficient as our WARP. This difficulty, which arises independently of any compounding issues related to data availability, is simply an example of where it makes sense to tailor one's tool to address the question at hand. Thus the suitability of a given measure to a particular application is largely an empirical question which we now examine using WARP.

6.2  WARP and U.S. External Imbalances

A central tenet of macroeconomic theory for open economies is that, other things equal, an increase in a country's prices relative to prices abroad will result in a deterioration in net exports. There is less agreement, however, on how to measure relative prices, and the empirical validity of the tenet clearly depends on how prices are measured. To examine this idea, figure 11 shows scatter diagrams between U.S. non-oil net exports, as a share of GDP, and the contemporaneous value of U.S. international relative prices using three measures: chained aggregate of Penn parities ($ Q_{t}^{c}$), chained aggregate of relative CPIs ( $ \mathbb{Q} ^{c}$), and the geometric aggregate of levels of Penn parities ($ Q^{g}$). For 1971 to 2006 (left panels), the data indicate that whether net exports are inversely related to the U.S. international relative price depends on how one measures that price. Indeed, the association is absent if one uses $ Q_{t}^{c}$ where it is present if one uses $ \mathbb{Q} ^{c}$ and strongest if one uses $ Q^{g}$.

Figure 11:  U.S. Non-oil Net Exports and U.S. International Relative Prices

Figure 11 has six panels arranged as two columns and three rows. The top two panels show two scatter diagrams between U.S. non-oil exports as a share of GDP (vertical axis) and the chained aggregate of Penn parities. The scatter on the top-left panel shows no correlation between these two variables for the period 1971-2006. The scatter on the top-right panel splits the sample into two periods: 1971-1986 and 1987-2006; for each sub-sample there is an inverse association between the two variables. The two middle panels show two scatter diagrams between U.S. non-oil exports as a share of GDP (vertical axis) and the chained aggregate of relative CPIs. The scatter on the middle-left panel shows an inverse correlation between these two variables for the period 1971-2006. The scatter on the middle-right panel splits the sample into two periods: 1971-1986 and 1987-2006; for each sub-sample there is an inverse association between the two variables. The two bottom panels show two scatter diagrams between U.S. non-oil exports as a share of GDP (vertical axis) and the geometric aggregate of Penn parities. The scatter on the bottom-left panel shows an inverse correlation between these two variables for the period 1971-2006. The scatter on the bottom-right panel splits the sample into two periods: 1971-1986 and 1987-2006; for each sub-sample there is an inverse association between the two variables.


A full understating of these disparate correlations involves recognizing that the character of bivariate associations is influenced by the level of aggregation and by omitted factors, such as foreign income and dynamic adjustments, the role of which could depend on the measure of relative prices. We will examine the role of these factors below but, in the meantime, closer inspection of the scatter plots reveals clusters of observations in which net exports and relative prices are inversely related. These clusters are most distinct for the plots using the chained aggregates of either CPIs or Penn's parities. Thus the panels on the right re-examine the relationship using the clusters formed with data from 1971-1986 and from 1987-2006; this dating is motivated by the evidence of figure 3 showing a break in the trends of international relative prices in 1987. For each of these subsamples, there is an inverse relationship between external balances and the U.S. international relative price. For the first sample, the strength of the relationship is comparable across the three measures of international relative prices. For the second sample, the association strengthens only for the geometric aggregate.

Overall, reliance on $ Q^{g}$ offers the greatest empirical support for the textbook proposition that net exports and relative prices are inversely related. Nevertheless, the evidence raises several questions that need to be addressed before declaring that $ Q^{g}$ passes the "proof of concept" test. Specifically, as indicated earlier, how can one be sure that these correlations are not unduly influenced by the absence of other key factors such as foreign economic activity?

6.3  WARP and Trade Modeling: The Case of Exports

To address these questions, we examine whether the alternative measures of international relative prices have implications for characterizing the behavior of aggregate U.S. exports. We focus on exports because they are directly related to foreign prices, the objective of WARP. Indeed, if one postulates that exports respond to foreign economic activity and to the price of exports ($ P_{x})$ relative to the foreign price ($ P_{f})$, $ \frac{P_{x} }{P_{f}},$ then one needs to construct a measure for $ \frac{P_{x}}{P_{f}}.$ The current practice is to use official statistics for $ P_{x}$ and to measure $ P_{ft}$ as a chained aggregate of foreign CPIs, expressed in U.S. dollars. In contrast, our approach is to measure $ P_{f}$ so as to ensure consistency with the evolution of U.S. international relative prices. For example, if one adopts $ Q^{g}$ as the relevant measure of international relative prices, then $ P_{f}^{g}=\frac{P_{us}}{Q^{g}}$ where $ P_{us}$ is the U.S. GDP deflator.22 Alternatively, if one adopts $ Q_{t}^{c}$, then the aggregate measure of foreign prices is $ P_{f}^{c}=\frac{P_{us}} {Q^{c}},$ where $ P_{us}$ is either the U.S. GDP deflator or the U.S. CPI, depending on how $ Q^{c}$ is constructed.

We do not focus on modeling imports because the advantage of our measure of international relative prices is less obvious given the availability of official statistics for the components of the relative price of imports$ .$ 23

Measuring the Relative Price of Exports   To measure the relative price of exports we first tailor the weighting scheme to exports and re-compute our three measures of U.S. international relative prices using weights that exclude the contribution of imports and include the role of bilateral exports and third-country markets; figure 12 shows that the choice of weights has a relatively minor effect on our three measures of international relative prices. Second, we solve for the implied $ P_{f}:$

$\displaystyle P_{fx}^{g}$ $\displaystyle =\frac{P_{us}}{Q_{x}^{g}}:$geometric of levels of Penn parities (10)
       
$\displaystyle P_{fx}^{c}$ $\displaystyle =\frac{P_{us}}{Q_{x}^{c}}:$chained of levels of Penn parities    
       
$\displaystyle CPI_{fx}^{c}$ $\displaystyle =\frac{CPI_{us}}{ \mathbb{Q} _{x}^{c}}:$chained of CPIs,    

where a sub-script 'x' denotes the use of export weights and $ CPI_{fx}^{c}$ is the chained aggregate of foreign CPIs, expressed in U.S. dollars. Third, given $ Px,$ we obtain the three measures of relative export prices as

$\displaystyle rpx^{geo}$ $\displaystyle =\frac{P_{x}}{P_{fx}^{g}}$ (11)
       
$\displaystyle rpx^{c}$ $\displaystyle =\frac{P_{x}}{P_{fx}^{c}}$    
       
$\displaystyle rpx^{cpi}$ $\displaystyle =\frac{P_{x}}{CPI_{fx}^{c}}.$    

Figure 12:  U.S. International Relative Prices - Sensitivity to Weighting Scheme

Figure 12 has three panels arranged as one column with three rows. For the top panel, the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of the geometric aggregate of Penn parities using two types of weights: the ones from the Federal Reserve Boards broad index and the bilateral export weights. The range of the vertical axis is from 0.9 to 1.6. Each of these measures is represented by a line and the lines move in tandem. Both lines decline gradually from 1.5 in 1970 to 1.1 in 1980; the lines rise from 1.1 in 1981 to 1.5 in 1984. Subsequently, the lines decline to 1.1 in 1987 and remain relatively unchanged until 1994. Since then, the lines rise reaching 1.5 by the end of 2001; the lines decline to 1.4 by 2006. Middle panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of the chained aggregate of Penn parities using two types of weights: the ones from the Federal Reserve Boards broad index and the bilateral export weights. The range of the vertical axis is from 0.9 to 1.6. Each of these measures is represented by a line and the lines move in tandem. Both lines decline gradually from 1.5 in 1970 to 1.1 in 1980; the lines rise from 1.1 in 1981 to 1.5 in 1984. Subsequently, the lines decline to 1.1 in 1987 and remain relatively unchanged until 1994. Since then, the lines rise reaching 1.2 by the end of 2001; the lines decline to 1.0 by 2006. Bottom panel: The horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of the chained aggregate of relative CPIs using two types of weights: the ones from the Federal Reserve Boards broad index and the bilateral export weights. The range of the vertical axis is from 0.9 to 1.6. Each of these measures is represented by a line and the lines move in tandem. Both lines decline gradually from 1.5 in 1970 to 1.1 in 1980; the lines rise from 1.1 in 1981 to 1.5 in 1984. Subsequently, the lines decline to 1.1 in 1987 and remain relatively unchanged until 1994. Since then, the lines rise reaching 1.4 by the end of 2001; the lines decline to 1.2 by 2006.

*  The Chained of Penn Parities and Chained of Relative CPIs series are re-scaled by the 1971q2 value of Geometric aggregate of levels from Penn's parities.


How important are differences in international relative prices for the profile of the relative price of exports? Figure 13 documents the data and the steps taken to arrive at the three measures of $ rpx$. The top-left panel shows the (export-weighted) measures of international relative prices; the right panel shows the three U.S. price indexes: $ P_{us}, $ $ CPI_{us},$ and $ P_{x}.$ The bottom-left panel shows the implied measures of aggregate foreign prices (equation 10); the series have upward trends and move together through 1987 but diverge afterwards with $ P_{fx}^{g} $ flattening while the other two series continue their upward trends, albeit at a lower rate. The flattening of $ P_{fx}^{g}$ reflects the increasing importance of low-price economies, a phenomenon captured only by $ Q_{x}^{g}$. The bottom-right panel shows the three measures for $ rpx$ (equation 11); $ rpx_{t}^{c}$ and $ rpx_{t}^{cpi}$ move together and have downward trends reflecting the upward trend in their measures of foreign prices. In contrast, $ rpx_{t}^{geo}$ trends down through 1990 and flattens afterwards, reflecting the flattening of $ P_{fx}^{g}$. Overall, aggregation schemes that recognize interactions between price levels and the increased trade with emerging economies, as captured by $ Q_{x}^{g},$ yield a picture of U.S. relative export prices that is fundamentally different from the one given by existing aggregation methods. We now examine whether this difference matters for characterizing the response of U.S. exports to income and relative prices.

Figure 13:  Derivation of U.S. Relative Export Price

Figure 13 has four panels arranged in two columns and two rows. For the top-left panel, the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of three measures of U.S. international relative prices; these values range from 0.9 to 1.6. These measures are the geometric weighted average of U.S. international relative prices, the chained aggregate of Penn parities, and the chained aggregate of relative CPIs. Each of these measures is represented by a line and the three lines move in tandem from 1970 to 1989. Specifically, these lines decrease gradually from 1970 to 1980; they increase from 1981 and reach a peak of about 1.6 in 1984; they decrease from 1985 to 1989. Subsequently, the line for the geometric aggregate of Penn parities increases and reaches a peak of 1.55 towards the end of 2001, the line declines to 1.14 by 2006. The line for the chained aggregate of Penn parities falls from 1989 to 1994 reaching a trough of 0.95. Subsequently, this line rises reaching a peak of 1.2 towards the end of 2001, and then declines to 1 by 2006. The line for the chained of relative CPIs increases starting in 1990 and reaches a peak of 1.4 by 2001; this line decreases to 1.2 by 2006. For the top-right panel, the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of three measures of U.S. price indexes; these values range from 0.9 to 5.7. These measures are the U.S. CPI, the U.S. export deflator (goods and services), and the GDP deflator. Each of these measures is represented by a line and the three lines increase gradually from 1970 to 1982. Subsequently, the line for the export price index experiences a substantial decrease in its upward trend. The lines for the CPI and GDP deflator continue their gradual increase through 2006. For the bottom-left panel, the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of three measures of the aggregate foreign prices; these values range from 0.5 to 4.5. These measures are the chained aggregate of foreign CPIs, the chained aggregate of foreign GDP deflators, and the geometric aggregate of foreign GDP deflators. Each of these measures is represented by a line and the three lines increase gradually from 1970 to 1986. Subsequently, the line for the geometric aggregate of foreign GDP deflators experiences a substantial decrease in its upward trend. The lines for the chained aggregates of foreign CPI and foreign GDP deflators continue their gradual increase through 2006. For the bottom-right panel, the horizontal axis records time-units measured as quarters from 1970 to 2006. The vertical axis contains the values of three measures of the relative price of exports; these values range from 0.6 to 1.6. The measures are the export price relative to the chained aggregate of foreign CPIs, relative to the chained aggregate of foreign GDP deflators, and relative to the geometric aggregate of foreign GDP deflators. Each of these measures is represented by a line and the three lines decrease gradually from 1970 to 1986. Subsequently, the line for the price of exports relative to the geometric aggregate of foreign GDP deflators experiences a substantial reduction in its downward trend. The lines for the relative prices of exports using the chained aggregates of CPI and GDP deflator continue their gradual decrease through 2006.

*  International Relative Prices with Export Weights are re-scaled by the 1971q2 value of Geometric aggregate of levels from Penn's parities. The U.S. Domestic Price indexes are re-scaled to 1971q2 = 1.


Econometric Formulation   To model U.S. exports, we assume that foreign and domestic products are imperfect substitutes for each other (see Goldstein and Khan, 1985) and postulate an error-correction formulation:

$\displaystyle \Delta\ln X_{t}$ $\displaystyle =\underset{short-run}{\underbrace{\alpha^{j}+ {\textstyle\sum\limits_{i=1}^{4}} \beta_{i}^{j}\cdot\Delta\ln X_{t-i}+ {\textstyle\sum\limits_{i=0}^{4}} \phi_{i}^{j}\cdot\Delta\ln Y_{t-i}^{\ast}+ {\textstyle\sum\limits_{i=0}^{4}} \mu_{i}^{j}\cdot\Delta\ln rpx_{t-i}^{j}}}+$ (12)
       
  $\displaystyle +$ $\displaystyle \underset{long-run}{\underbrace{\theta_{x}^{j}\cdot(\ln X_{t-1}-\eta_{x}^{j}\cdot\ln Y_{t-1}^{\ast}-\varepsilon_{x}^{j}\cdot\ln rpx_{t-1}^{j})}}+$  $\displaystyle u_{t}^{j},$ $\displaystyle u_{t}^{j}\symbol{126} IN(0,\sigma_{j}^{2})$    
$\displaystyle j$ $\displaystyle =$geo, c, cpi,    

where $ X$ is the volume of exports of goods and services; $ Y^{\ast}$ is the foreign real GDP; $ \eta_{x}^{j}>0$ is the long-run income elasticity; and $ \varepsilon_{x}^{j}<0$ is the long-run price elasticity.24 Equation (12) assumes that the growth rate of exports responds to short- and long-run factors. Specifically, movements in income and relative prices induce cyclical swings in exports. But, even if income and relative prices were fixed, exports could be changing as they adjust to their long run level given by $ \eta _{x}^{j}\cdot\ln Y^{\ast}+\varepsilon_{x}^{j}\cdot\ln rpx^{j}.$ This gradual adjustment is captured by the term in parentheses where $ \theta_{x}^{j}<0$ represents the speed of adjustment. Finally, finding that $ \alpha^{j}\neq0$ means that exports would automatically change over time regardless of the evolution of income and relative prices and thus we interpret a significant $ \alpha^{j}$ as evidence of misspecification.

As formulated, equation (12) is non-linear in the parameters. To avoid the associated estimation difficulties, we re-express this equation as linear in the parameters:

$\displaystyle \Delta\ln X_{t}$ $\displaystyle =\alpha^{j}+ {\textstyle\sum\limits_{i=1}^{4}} \beta_{i}^{j}\cdot\Delta\ln X_{t-i}+ {\textstyle\sum\limits_{i=0}^{4}} \phi_{i}^{j}\cdot\Delta\ln Y_{t-i}^{\ast}+ {\textstyle\sum\limits_{i=0}^{4}} \mu_{i}^{j}\cdot\Delta\ln rpx_{t-i}^{j}+$ (13)
       
  $\displaystyle +\theta_{x}^{j}\cdot\ln X_{t-1}+\theta_{y}^{j}\cdot\ln Y_{t-1}^{\ast} +\theta_{p}^{j}\cdot\ln rpx_{t-1}^{j}+u_{t}^{j},$  $\displaystyle u_{t}^{j} \symbol{126}IN(0,\sigma_{j}^{2})$    
       
   $\displaystyle j=$geo, c, cpi; $\displaystyle \theta_{y}^{j}>0$ and $\displaystyle \theta _{p}^{j}<0.$    

Using a '^' to denote an estimated value, we use the least squares values of $ \widehat{\theta}_{x}^{j}$ , $ \widehat{\theta}_{y}^{j}$ , and $ \widehat{\theta}_{p}^{j}$ to compute the implied elasticities as $ \widehat{\eta}_{x}^{j}=-\frac{\widehat{\theta}_{y}^{j}}{\widehat{\theta} _{x}^{j}}$ and $ \widehat{\varepsilon}_{x}^{j}=-\frac{\widehat{\theta}_{p}^{j} }{\widehat{\theta}_{x}^{j}}.$ Note that these elasticity estimates are ratios of normal variables and thus the associated distributions are not known in advance.25 Thus the associated confidence intervals are constructed using Monte Carlo simulations$ ;$ appendix A.5 has the details.

Estimation Strategy   For parameter estimation we apply least squares to equation (13) using observations from 1972Q3 to 2004Q4 with data from 1971Q2 to 1972Q2 reserved for lags and data from 2005Q1 to 2006Q4 reserved for evaluating out-of-sample predictive accuracy. One may argue that there are gains in precision of the estimates if one were to exclude insignificant variables from the model. To avoid the statistical pitfalls associated with the joint nature of model specification and parameter estimation, we rely on a computer-automated algorithm, developed by Hendry and Krolzig (2001).26 Their algorithm combines least squares with a selection criteria that excludes insignificant coefficients and tests for both parameter constancy and white-noise residuals; the critical values for rejection are not fixed in advance but, rather, are calculated sequentially. We report results for equation (13), labeled the General formulation, and for the simplified formulation, labeled the Specific formulation.

To examine the potential for simultaneity bias, we postulate a vector-autoregressive model explaining exports, income, and relative prices and then apply Johansen's cointegration method to estimate the cointegration vector; this approach treats income and prices as endogenous.27

Econometric Results   Table 1 shows estimation and test results for all three measures of relative export prices. The signs for $ \theta_{x}^{j},$ $ \theta_{y}^{j},$ and $ \theta_{p}^{j}$ are consistent with expectations and their magnitudes are roughly comparable across measures of relative export prices. In terms of in-sample fit, the standard error of the regression has a narrow range of variation: from 1.89% for $ rpx_{t}^{geo}$ to 1.93% for $ rpx_{t}^{cpi}.$ Furthermore, the Chow tests cannot reject the hypothesis of parameter stability, and the residuals exhibit normality, serial independence, and homoskedasticity.

Table 1:  Parameter Estimates for Model of U.S. Exports of Goods and Services: Sensitivity to Measure of Foreign Prices - OLS, 1971Q2-2004Q4

Measures of Sensitivity
Geometric Foreign PGDP: General
(1)
Geometric Foreign PGDP: Specific
(2)
Chained Foreign PGDP: General
(3)
Chained Foreign PGDP: Specific
(4)
Chained Foreign CPIs: General
(5)
Chained Foreign CPIs: Specific
(6)
Chained Foreign
CPIs: ex. Intercept
(7)
Parameter Estimate: Intercept α
0.0392
0e
0.1618
0e
0.1966
0.1574
set to zero
Parameter Estimate: Intercept α - se
0.0474
-
0.0664
-
0.0725
0.0657
-
Parameter Estimate: Sum of coefficients for ∆lnX
-0.3157
0e
-0.1478
0e
-0.1448
0e
0e
Parameter Estimate: Sum of coefficients for ∆lnX - se
0.192
-
0.186
-
0.1829
-
-
Parameter Estimate: Sum of coefficients for ∆lnY
4.3777
2.899
3.8885
3.1748
4.0375
2.8246
3.2633
Parameter Estimate: Sum of coefficients for ∆lnY - se
0.79
0.4267
0.7885
0.4247
0.786
0.4571
0.4267
Parameter Estimate: Sum of coefficients for ∆lnRP
0.088
0e
0.1178
0e
0.1661
0e
0e
Parameter Estimate: Sum of coefficients for ∆lnRP - se
0.1885
-
0.2109
-
0.1954
-
-
Parameter Estimate: Lagged Exports: θx
-0.1244
-0.1157
-0.1547
-0.1249
-0.1534
-0.1295
-0.1138
Parameter Estimate: Lagged Exports: θx - se
0.0368
0.0260
0.0448
0.0299
0.0404
0.0292
0.0289
Parameter Estimate: Lagged Foreign Income:  θy
0.1735
0.1709
0.1860
0.1813
0.1748
0.1508
0.1644
Parameter Estimate: Lagged Foreign Income:  θy - se
0.0579
0.0391
0.0626
0.0443
0.0552
0.0423
0.0427
Parameter Estimate: Lagged Relative Price:  θp
-0.1314
-0.0982
-0.1439
-0.0755
-0.1503
-0.1195
-0.0675
Parameter Estimate: Lagged Relative Price:  θp - se
0.0315
0.0203
0.0368
0.0171
0.0358
0.0269
0.0162
Measures of Fit: SER
1.807%
1.890%
1.812%
1.917%
1.809%
1.895%
1.930%
Measures of Fit: Adj Rqrd
0.459
0.408
0.455
0.391
0.457
0.405
0.382
Measures of Fit: No. of Parameters
18
5
18
5
18
6
5
Hypotheses (p-values)(a): Parameter Stability: Half Sample
0.999
0.998
0.996
0.996
0.998
0.995
0.998
Hypotheses (p-values)(a): Parameter Stability: Last 8 quarters
0.973
0.858
0.835
0.589
0.908
0.788
0.700
Hypotheses (p-values)(a): Properties of residuals: Normality
0.020
0.108
0.039
0.175
0.026
0.153
0.170
Hypotheses (p-values)(a): Properties of residuals: Serial Independence
0.841
0.093
0.578
0.417
0.174
0.174
0.299
Hypotheses (p-values)(a): Properties of residuals: Homoskedasticity
0.009
0.137
0.016
0.188
0.150
0.150
0.140
Implied Elasticities: Income = -θy / θx
1.39
1.48
1.20
1.45
1.14
1.16
1.44
Implied Elasticities: Price= -θp / θx
-1.06
-0.85
-0.93
-0.60
-0.98
-0.92
-0.59

0e:  Algorithm finds the variable to be statistically irrelevant and sets the coefficient to zero.
(a)  An entry less that 0.01 means that the associated hypothesis can be rejected at the 1% significance level.


The sole dissonant note in these results is the presence of a positive and statistically significant intercept in the specific formulation using the chained of relative CPIs (column 6). Finding that $ \alpha^{cpi}>0$ means that exports would expand even if income and relative prices were fixed. We do not see an economic justification for such a result and treat this finding as an instance in which an algorithm delivering an otherwise statistically reliable model is not delivering an economically meaningful model. Thus we also re-estimate the parameters of the model constraining the intercept to zero and find (column 7) that the constrained model exhibits a slight deterioration of fit, which is not surprising, and that the values for the remaining parameter estimates based on $ rpx_{t}^{cpi}$ are close to the estimates based on $ rpx_{t}^{c}.$ (This finding is reassuring given the similarity in the data for these two relative prices.) In terms of the elasticities, the implied income elasticity is positive and ranges from 1.4 for $ rpx_{t}^{cpi}$ to 1.5 for $ rpx_{t}^{geo}$; the implied price elasticity is negative and ranges from -0.6 for $ rpx_{t}^{cpi}$ to -0.9 for $ rpx_{t}^{geo}$.

To assess the statistical properties of these estimates, figure 14 shows the 95% (Monte Carlo) confidence intervals for the estimated income elasticity along with the confidence bands for estimates from the Johansen method; figure 18 in appendix A.5 shows the densities for all income elasticities. The results indicate that the median income elasticity is positive; greater than one; significantly greater than zero; and quite similar to the implied income elasticity of table 1. Furthermore, the median elasticity based on $ rpx_{t}^{geo}$ exceeds the median elasticity for the other measures of relative prices, a result robust to estimation method. Note that the proximity of the medi