Policy Question
Preliminary Draft Do not Circulate or Quote Trade in Services: Elasticity Estimation for the U.S. Services Sector Jaime Marquez1 Federal Reserve Board December 2003 Introduction My goal in this paper is to discuss practical issues that arise in estimating of trade elasticities and their role in predicting U.S. external imbalances. I will focus on the elasticities associated with trade in services because little is known about them; important exceptions are Reeve (2001) and Deardorf et al. (2001). One may be tempted to infer from a thin literature that one can model service trade and merchandise trade in terms of the same forces—income and relative prices—and that knowing the elasticities for merchandise trade is enough to understand the behavior of service trade. Unfortunately, if service and merchandise trade respond to the same forces, then their responses have to differ in magnitude. Otherwise one cannot explain the growing divergences in the balances of service and trade since 1976, balances that were virtually identical to each other for nearly fifty years (figure 1): knowledge of trade elasticities for merchandise trade need not extend to service trade. And one factor that might explain a response differential is the recent advances in information technology. These advances have facilitated the tasks of creating, exploiting, and managing information effectively giving the United States a comparative advantage in those activities. 2 How could one detect if such a comparative advantage is present in econometric estimates of elasticities for service trade? 1
This material is part of a project I have undertaken with Cathy Mann who has given me numerous comments. I have also benefited from comments by Trevor Reeve and Joe Gagnon and from participants at seminars at the Federal Reserve Board and Johns Hopkins University-SAIS. The views 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. 2 I am grateful to Cathy Mann for pointing out this possibility.
2
Modeling Strategy I focus on two types of formulations: those that are “consistent with the data,” in the sense of overall fit and residual’s properties, and those that are “consistent with the theory,” in the sense of being strictly derived from optimization behavior. I have found that, when modeling merchandise trade, the further one pursues one type of consistency the farther away one moves from the other type of consistency. One question I address here is whether the modeling of service trade faces a comparable tradeoff. Of the formulations consistent with data, I limit my attention to the logarithmic case that implies that the associated parameters are constant elasticities. The specification for exports is Px + lags;η x > 0, ε x < 0, ln X = η x ln GDP * + ε x ln * Pgdp
(1)
where X represents exports of services in real terms, GDP* denotes foreign real GDP, Px denotes the export price of services, and Pgdp* denotes the foreign GDP deflator expressed in U.S. dollars. The specification for imports is Pm GDPV + lags;η m > 0, ε m < 0, + ε m ln ln M = η m ln Pgdp Pgdp
(2)
where M represents imports of services in real terms, GDPV denotes U.S. GDP in nominal terms, Pm denotes the U.S. import price of services, and Pgdp denotes the U.S.GDP deflator. Of the formulations consistent with optimizing behavior, I consider two possibilities: the Constant Elasticity of Substitution (CES) and the Almost Ideal Model. The CES formulation is
M Pd ln = δ + σ ln , D Pm
(3)
3
where D is service GDP (GDP minus Goods GDP), in real terms, Pd is the deflator for domestic services, and σ is the (constant) elasticity of substitution between domestic and foreign services. The Almost Ideal Model is given by Pm • M Pd Pm • M + Pd • D = w = α + β ln + δ ln (4) w 1− w Pm • M + Pd • D Pm Pm • Pd A key feature of this formulation is that income elasticity is 1 +
β w
, which varies with the share
of expenditures on services devoted to foreign services.
Estimation Strategy
The estimation sample is based on quarterly data from 1987 to 2001 and I examine both aggregate services and their components: fares, travel, other transportation, and other private services. In terms of estimation methods, I use ordinary least squares and full information maximum likelihood.3 For the formulations consistent with the data, parameter estimation involves starting from a general formulation and then, through the elimination of insignificant variables, arriving at a specific formulation. The only sort of generality that is allowed here is in the dynamic adjustment. So, I start with general formulation that allows, a priori, flexibility in dynamic response of trade to changes in income and relative prices. Then I exclude from the formulation irrelevant lags. For formulations consistent with theory, I estimate the parameters of the equations as provided by theory; I allow for a 1-quarter lag to make sure that there are not significant quarterly effects.
3
For FIML, I implement a VAR model for the logarithms of trade, economic activity, and relative prices; parameters are estimated with Johansen’s cointegration method.
4
Evaluating how much of that flexibility holds a posteriori can be a time-consuming process embodying statistical pitfalls. To avoid these two drawbacks, I rely on a search algorithm developed by Hendry and Krolzig (2001); their algorithm is fast, exhaustive, and controls for the effect of multiple search paths on the significance levels of the tests. To judge the statistical reliability of the estimates, I examine whether the residuals exhibit normality, serial independence, and conditional homoskedasticity. I also test the hypothesis of parameter constancy with half of the sample excluded from estimation and with the last eight observations excluded from estimation; these results are in the appendix.
Estimation Results
Table 1 presents the least square estimates for the long-run income and price elasticities for several categories of service exports and service imports. For aggregate services, the OLS results indicate that the income elasticity for export of services is larger than the income elasticity for import of services.4 Such a finding is in sharp contrast to the literature on elasticities for U.S. merchandise trade where the income elasticity for imports is greater than the income elasticity for exports. The results also indicate this asymmetry is not robust: For the FIML estimates, the income elasticity for imports is a bit higher than the one for exports. In terms of the parameter estimates of the components of service exports, the FIML estimates suggest unitary income elasticities for all the components save Other Private Services that has an income elasticity of three. Two interesting results are the case of Fares, where the search algorithm indicates that these exports follow a stationary process independent of income and relative prices, and the case of Other Private Services, where the two estimation methods give positive price elasticities. In terms of the parameter estimates of the components of service
5
imports, the FIML estimates suggest income elasticities ranging from 0.8 for travel to 3.3 for Other Private Services; the range of elasticity estimates for OLS is narrower. To examine the sensitivity of the results for imports, I consider variations of equation (2) for aggregate imports of services only; the focus on imports stems from the availability of the data needed for these variations. Specifically, the first alternative scales imports by U.S. population: ln
M GDPV Pm = M (ln , ln , lags ) . Pop Pop • Pgdp Pgdp
(5)
A second variation uses domestic services, instead of real GDP, as the domestic substitute: ln M = M (ln GDPV , ln Pm, ln Pd , lags ) .
(6)
Using homogeneity of degree one in income and prices, I re-express (6) as ln M = M (ln
GDPV Pm , ln , lags ) . Pd Pd
(7)
Note that equation (7) deflates nominal GDP using the deflator for domestic services instead of the GDP deflator as in equation (2). I also express (7) in per-capita terms to obtain ln
M GDPV Pm = M (ln , ln , lags ) . Pop Pop • Pd Pd
(8)
Finally, I modify (5) and (8) to allow for heterogeneity in the composition of population. Specifically, I examine whether the distinction between domestic and foreign-born population matters for the estimated elasticities: ln
4
Ms GDPV Pms ForeignBorn = M (ln , ln , ln , lags ) Pop Pop • Pgdp Pgdp Pop
(9)
The residuals of these formulations are white noise and one cannot reject parameter constancy.
6
ln
M GDPV Pm ForeignBorn = M (ln , ln , ln , lags ) Pop Pop • Pd Pd Pop
(10)
Table 2 has the OLS estimation results. The main conclusion from these results is that the estimates are quite sensitive to seemingly minor modifications in specification. For example, replacing the GDP deflator with the Service deflator (lines 1 and 4) raises the income elasticity from 1.4 to 2.2 and cuts the price elasticity nearly in half. These differences in estimates are important but one cannot choose between these two formulations on the basis of the statistical properties: both formulations exhibit white-noise residuals and parameter constancy. In terms of the estimated price elasticity, the results reveal that seemingly simple changes in formulation translate into quite noticeable changes in the estimated price elasticities. For the case where one uses the service deflator, the scaling by population changes the price elasticity from –0.5 (unscaled, line 4) to –1.2 (scaled, line 5). Again, these differences in estimates are important but one cannot choose between these two formulations because they exhibit white-noise residuals and parameter constancy. Table 3 has the OLS estimation results for the parameters of equations (3) and (4). For the CES formulation, I find that the elasticity of substitution is 1.8—that is, U.S. services are a good substitute for foreign services; the income elasticity equals one by design. For the Almost Ideal model, the coefficients on income add up to zero implying that the income elasticity is one.5 The residuals from these two formulations are serially independent and homoskedastic but do not exhibit normality.
5
The income elasticity is given by 1 +
β w
and what the results show is that β is zero in the long run.
7
Practical Implications
Overall, the formulations consistent with theory give quite satisfactory estimation results and the question is why not to use these formulations in forecasting exercises. I start addressing this question by evaluating the in-sample prediction errors of three formulations: the logarithmic (table 2, line 1), the CES, and the Almost Ideal. I will then re-estimate the parameters with data through 1995 and use those estimates to generate ex-post forecasts. I compute in-sample prediction errors as the difference between actual and predicted imports relative to the value of actual imports.6 Figure 2 shows the evolution of these prediction errors and the root-mean square prediction errors (RMSPE) for two periods: 1989-2001 and 1995-2001. According to the results, the CES formulation has the largest in-sample prediction errors with an RMSPE of 3.6 percent, more than twice as large as the RMSPE of the log-linear formulation. This differential in explanatory power is influenced by two large errors—in 1990 and in 1993. Thus I compute the RMSPE for the last seven years of the estimation sample and find a substantial narrowing in the RMSPE of the three formulations: from 1.2% for the loglinear formulation to 2.3% for the CES formulation. That relying on the CES might entail a decline in explanatory power, relative to that of the log-linear formulation, is not a new finding. What is new is that the explanatory power of the Almost Ideal formulation is virtually the same as that of the log-linear formulation. I interpret this similarity in explanatory power as implying a preference for the Almost Ideal formulation: it is based on optimizing behavior and it does not carry a deterioration of explanatory power within the estimation sample. The next question is whether one can use the out-of-sample performance as an additional criterion to select the type of formulation.
8
To generate out-of-sample predictions, I estimate the parameters of all three formulations (logarithmic, CES, and Almost Ideal) with data ending in 1995Q3 and then generate one-step ahead predictions using observations from 1995Q4 to 2001Q2. Figure 3 shows the actual and the ex-post predictions for the three formulations. Inspecting the evidence reveals two features of interest. First, there is a tendency in the three models to under-predict the actual volume of imports but the degree of under-prediction is small numerically. Second, there is virtually no difference in the predicted values among the three models. I interpret such similarity as strong evidence in favor of an optimization model for explaining service imports.
References
Deardorff, Alan V., Saul H. Hymans, Robert M. Stern, and Chong Xiang, "Forecasting U.S. Trade in Services" March 30, 2000. In Robert M. Stern, ed., Services in the International Economy: Measurement and Modeling, Sector and Country Studies, and Issues in the WTO Services Negotiations, University of Michigan Press, 2001, pp. 53-81 Hendry, D. F. and H. Krolzig, 2001, Automatic Econometric Model Selection Using PcGets, London: Timberlake. Reeve, T., 2001, “Trade in Services,” Federal Reserve Board, mimeo.
6
Because the three formulations that I am considering have different dependent variables, comparing their explanatory power involves re-expressing the equations in terms of the level of imports of aggregate services in real terms.
9
Billions of U.S. Dollars
U.S. External Balances
100
0
-100
Goods and Services Goods Services
-200
-300
-400
1930
1940
1950
1960
1970
1980
1990
Figure 1: External Balances in Services and Goods-United States, 1929-2002
2000
10
Table 1 Income and Price Elasticities for U.S. Trade in Services Alternative Groupings and Estimation Methods* 1987-2001
Category Aggregate a
Method
Exports Income Price
Imports Income Price
FIML b OLS
1.33* 1.74*
-0.37* -0.29*
1.43* 1.35*
-1.48* -1.56*
FIML OLS
1.10* c 0.00
-1.43* c 0.00
1.99* 2.30*
-0.60* -1.07*
FIML OLS
0.63* 1.11*
-0.78* -0.95*
0.84* 1.06*
-2.18* -1.49*
FIML OLS
0.90* 0.75*
-0.14 -0.04
0.94* 0.86*
-0.22 +0.00
FIML OLS
3.25* 3.23*
+0.49* +0.42*
3.32 1.56*
-0.37 -2.22*
Fares
Travel Other Private Transportation Other Private Services
* Statistically significant at the 5 percent level. a
Excludes trade in defense and royalties.
b
VAR model for the logarithms of trade, economic activity, and relative prices; parameters are estimated with Johansen’s FIML method. c
ln Xfares(t ) = 0.70 + 0.58 ln Xfares (t − 1) + 0.18 ln Xfares(t − 2)
11
Table 2 Income and Price Elasticities Results for Aggregate Service Imports: Sensitivity to Specification Formulation
Income
Price
1. Baseline (GDP Deflator)
1.35*
-1.56*
2. GDP Deflator & Per-capita
2.16*
3. GDP Deflator, Per-capita, Foreign Born
Chw1/2
Chw8
JB
AR
ARCH
√
√
√
√
√
+0.02
√
√
√
√
X
2.98*
+0.90*
√
√
√
√
√
4. Service Deflator
2.10*
-0.49*
√
√
√
√
√
5. Service Deflator & Per-capita
2.14*
-1.21*
√
√
√
√
√
6. Service Deflator, Per-capita, Foreign Born
2.73*
-0.70*
√
X
X
√
√
* Statistical significance at the 5 percent level; √: cannot reject null hypothesis at 5 percent significance level; X: cannot accept null hypothesis at 5 percent significance level. Chw1/2: Chow test for null hypothesis of parameter constancy with half of the sample excluded from estimation. Chw8: Chow test for null hypothesis of parameter constancy with the last eight observations excluded from estimation. JB: Test for null hypothesis of normality in the distribution of the residuals. AR: Test for null hypothesis of serial independence in the distribution of the residuals. ARCH: Test for null hypothesis of conditional homoskedasticity in the distribution of the residuals.
12
Table 3 Estimation Results for Formulations Consistent with Theory Constant-Elasticity Formulation M ln = D t (SE)
M 0.77* ln - 0.77 D t −1 (0.07) (0.26)
Pd + 1.09* ln Pm t (0.28)
-
Pd 0.69* ln Pm t −1 (0.31)
Short run σ: 1.09 Long run σ: 1.77 (=(1.09-0.69)/(1-0.77)) Rsquare = 0.97 Normality: cannot accept at the 5 percent significance level Serial Independence: cannot reject at the 5 percent significance level Homoskedasticity: cannot reject at the 5 percent significance level
Almost-Ideal Formulation Pm • M Pm • M EXP = 0.7832* + 1.219 + 4.8* ln P t Pm • M + Pd • D t Pm • M + Pd • D t −1
(SE)
(0.0698)
(1.63) (2.75)
Pd Pd EXP + 0.4246* ln - 4.947* ln + 0.1864* ln P t −1 Pm t −1 Pm t (0.89) (2.62) (0.901)
Rsquare = 0.89 Normality: cannot accept at the 5 percent significance level Serial Independence: cannot reject at the 5 percent significance level Homoskedasticity: cannot reject at the 5 percent significance level
13
1989-2001 1995-2001 7.5
RMSPE-CES RMSPE - Log-log RMSPE - Almost Ideal
3.6% 2.3 1.5% 1.2 1.8% 1.5
5.0
2.5
0.0
-2.5
-5.0
Prediction Errors: Actual - Fit (percent) -7.5 1989
1990
1991
1992
1993
1994
1995
1996
1997
Figure 2: In-sample Prediction Errors
1998
1999
2000
2001
14
Billions of Dollars, 1996 Prices 190
U.S. Service Imports Total ex. Defense and Royalties 180
170
160
Estimation Sample 1989-1995
Ex-post Forecasts
CES Logarithmic Almost Ideal Actual
150
140
130
1995
1996
1997
1998
1999
Figure 3: Out-of-Sample Predictions
2000
2001
15
Appendix Sensitivity Income and Price Elasticities to Composition of Aggregate Service Exports OLS, 1989-2001 Elasticities Income Price Aggregate This Study Reeve Travel This Study Reeve Fares This Study Reeve Other Transp. This Study Reeve Other Private This Study Reeve
Chw1/2
Statistical Tests Chw8 JB AR
ARCH
1.74* 1.97*
-0.29* -0.28*
√
√
√
√
√
1.11* 3.36*
-0.95* +0.42
√
√
√
√
√
0.00a 2.15*
0.00a -1.67*
√
√
√
√
√
0.75* 0.16
-0.04 -0.80*
√
√
√
√
√
3.23* 2.15*
+0.42* -0.22
X
√
X
√
√
* Satistical significance at the 5 percent level; √: cannot reject null hypothesis at 5 percent significance level; X: cannot accept null hypothesis at 5 percent significance level. a
ln Xfares(t ) = 0.70 + 0.58 ln Xfares(t − 1) + 0.18 ln Xfares(t − 2)
Reeve, T., 2001, “Trade in Services,” Federal Reserve Board, mimeo. The sample is quarterly over 1982-1996. The data for aggregate trade includes defense and royalties. JB: Test for null hypothesis of normality in the distribution of the residuals. AR: Test for null hypothesis of serial independence in the distribution of the residuals. ARCH: Test for null hypothesis of conditional homoskedasticity in the distribution of the residuals. Chw1/2: Chow test for null hypothesis of parameter constancy with half of the sample excluded from estimation. Chw8: Chow test for null hypothesis of parameter constancy with the last eight observations excluded from estimation.
16
Sensitivity Income and Price Elasticities to Composition of Aggregate Service Imports OLS, 1989-2001
Aggregate This Study Reeve Travel This Study Reeve Fares This Study Reeve Transportation This Study Reeve Other Private This Study Reeve
Income
Price
Chw1/2
Chw8 JB
AR
ARC H
1.35* 1.72*
-1.56* -0.84*
√
√
√
√
√
1.06* 1.72*
-1.49* -0.35
√
√
√
√
√
2.30* 1.65*
-1.07* -2.06
√
√
√
√
√
0.86* 1.15*
0.00 -0.53
√
√
√
√
√
1.56* 3.10*
-2.22* -0.44
√
√
√
√
√
* Statistical significance at the 5 percent level; √: cannot reject null hypothesis at 5 percent significance level; X: cannot accept null hypothesis at 5 percent significance level. Reeve, T., 2001, “Trade in Services,” Federal Reserve Board, mimeo. The sample is quarterly over 1982-1996. The data for aggregate trade includes defense and royalties. JB: Test for null hypothesis of normality in the distribution of the residuals. AR: Test for null hypothesis of serial independence in the distribution of the residuals. ARCH: Test for null hypothesis of conditional homoskedasticity in the distribution of the residuals. Chw1/2: Chow test for null hypothesis of parameter constancy with half of the sample excluded from estimation. Chw8: Chow test for null hypothesis of parameter constancy with the last eight observations excluded from estimation.
This material is part of a project I have undertaken with Cathy Mann who has given me numerous comments. I have also benefited from comments by Trevor Reeve and Joe Gagnon and from participants at seminars at the Federal Reserve Board and Johns Hopkins University-SAIS. The views 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. 2 I am grateful to Cathy Mann for pointing out this possibility.
2
Modeling Strategy I focus on two types of formulations: those that are “consistent with the data,” in the sense of overall fit and residual’s properties, and those that are “consistent with the theory,” in the sense of being strictly derived from optimization behavior. I have found that, when modeling merchandise trade, the further one pursues one type of consistency the farther away one moves from the other type of consistency. One question I address here is whether the modeling of service trade faces a comparable tradeoff. Of the formulations consistent with data, I limit my attention to the logarithmic case that implies that the associated parameters are constant elasticities. The specification for exports is Px + lags;η x > 0, ε x < 0, ln X = η x ln GDP * + ε x ln * Pgdp
(1)
where X represents exports of services in real terms, GDP* denotes foreign real GDP, Px denotes the export price of services, and Pgdp* denotes the foreign GDP deflator expressed in U.S. dollars. The specification for imports is Pm GDPV + lags;η m > 0, ε m < 0, + ε m ln ln M = η m ln Pgdp Pgdp
(2)
where M represents imports of services in real terms, GDPV denotes U.S. GDP in nominal terms, Pm denotes the U.S. import price of services, and Pgdp denotes the U.S.GDP deflator. Of the formulations consistent with optimizing behavior, I consider two possibilities: the Constant Elasticity of Substitution (CES) and the Almost Ideal Model. The CES formulation is
M Pd ln = δ + σ ln , D Pm
(3)
3
where D is service GDP (GDP minus Goods GDP), in real terms, Pd is the deflator for domestic services, and σ is the (constant) elasticity of substitution between domestic and foreign services. The Almost Ideal Model is given by Pm • M Pd Pm • M + Pd • D = w = α + β ln + δ ln (4) w 1− w Pm • M + Pd • D Pm Pm • Pd A key feature of this formulation is that income elasticity is 1 +
β w
, which varies with the share
of expenditures on services devoted to foreign services.
Estimation Strategy
The estimation sample is based on quarterly data from 1987 to 2001 and I examine both aggregate services and their components: fares, travel, other transportation, and other private services. In terms of estimation methods, I use ordinary least squares and full information maximum likelihood.3 For the formulations consistent with the data, parameter estimation involves starting from a general formulation and then, through the elimination of insignificant variables, arriving at a specific formulation. The only sort of generality that is allowed here is in the dynamic adjustment. So, I start with general formulation that allows, a priori, flexibility in dynamic response of trade to changes in income and relative prices. Then I exclude from the formulation irrelevant lags. For formulations consistent with theory, I estimate the parameters of the equations as provided by theory; I allow for a 1-quarter lag to make sure that there are not significant quarterly effects.
3
For FIML, I implement a VAR model for the logarithms of trade, economic activity, and relative prices; parameters are estimated with Johansen’s cointegration method.
4
Evaluating how much of that flexibility holds a posteriori can be a time-consuming process embodying statistical pitfalls. To avoid these two drawbacks, I rely on a search algorithm developed by Hendry and Krolzig (2001); their algorithm is fast, exhaustive, and controls for the effect of multiple search paths on the significance levels of the tests. To judge the statistical reliability of the estimates, I examine whether the residuals exhibit normality, serial independence, and conditional homoskedasticity. I also test the hypothesis of parameter constancy with half of the sample excluded from estimation and with the last eight observations excluded from estimation; these results are in the appendix.
Estimation Results
Table 1 presents the least square estimates for the long-run income and price elasticities for several categories of service exports and service imports. For aggregate services, the OLS results indicate that the income elasticity for export of services is larger than the income elasticity for import of services.4 Such a finding is in sharp contrast to the literature on elasticities for U.S. merchandise trade where the income elasticity for imports is greater than the income elasticity for exports. The results also indicate this asymmetry is not robust: For the FIML estimates, the income elasticity for imports is a bit higher than the one for exports. In terms of the parameter estimates of the components of service exports, the FIML estimates suggest unitary income elasticities for all the components save Other Private Services that has an income elasticity of three. Two interesting results are the case of Fares, where the search algorithm indicates that these exports follow a stationary process independent of income and relative prices, and the case of Other Private Services, where the two estimation methods give positive price elasticities. In terms of the parameter estimates of the components of service
5
imports, the FIML estimates suggest income elasticities ranging from 0.8 for travel to 3.3 for Other Private Services; the range of elasticity estimates for OLS is narrower. To examine the sensitivity of the results for imports, I consider variations of equation (2) for aggregate imports of services only; the focus on imports stems from the availability of the data needed for these variations. Specifically, the first alternative scales imports by U.S. population: ln
M GDPV Pm = M (ln , ln , lags ) . Pop Pop • Pgdp Pgdp
(5)
A second variation uses domestic services, instead of real GDP, as the domestic substitute: ln M = M (ln GDPV , ln Pm, ln Pd , lags ) .
(6)
Using homogeneity of degree one in income and prices, I re-express (6) as ln M = M (ln
GDPV Pm , ln , lags ) . Pd Pd
(7)
Note that equation (7) deflates nominal GDP using the deflator for domestic services instead of the GDP deflator as in equation (2). I also express (7) in per-capita terms to obtain ln
M GDPV Pm = M (ln , ln , lags ) . Pop Pop • Pd Pd
(8)
Finally, I modify (5) and (8) to allow for heterogeneity in the composition of population. Specifically, I examine whether the distinction between domestic and foreign-born population matters for the estimated elasticities: ln
4
Ms GDPV Pms ForeignBorn = M (ln , ln , ln , lags ) Pop Pop • Pgdp Pgdp Pop
(9)
The residuals of these formulations are white noise and one cannot reject parameter constancy.
6
ln
M GDPV Pm ForeignBorn = M (ln , ln , ln , lags ) Pop Pop • Pd Pd Pop
(10)
Table 2 has the OLS estimation results. The main conclusion from these results is that the estimates are quite sensitive to seemingly minor modifications in specification. For example, replacing the GDP deflator with the Service deflator (lines 1 and 4) raises the income elasticity from 1.4 to 2.2 and cuts the price elasticity nearly in half. These differences in estimates are important but one cannot choose between these two formulations on the basis of the statistical properties: both formulations exhibit white-noise residuals and parameter constancy. In terms of the estimated price elasticity, the results reveal that seemingly simple changes in formulation translate into quite noticeable changes in the estimated price elasticities. For the case where one uses the service deflator, the scaling by population changes the price elasticity from –0.5 (unscaled, line 4) to –1.2 (scaled, line 5). Again, these differences in estimates are important but one cannot choose between these two formulations because they exhibit white-noise residuals and parameter constancy. Table 3 has the OLS estimation results for the parameters of equations (3) and (4). For the CES formulation, I find that the elasticity of substitution is 1.8—that is, U.S. services are a good substitute for foreign services; the income elasticity equals one by design. For the Almost Ideal model, the coefficients on income add up to zero implying that the income elasticity is one.5 The residuals from these two formulations are serially independent and homoskedastic but do not exhibit normality.
5
The income elasticity is given by 1 +
β w
and what the results show is that β is zero in the long run.
7
Practical Implications
Overall, the formulations consistent with theory give quite satisfactory estimation results and the question is why not to use these formulations in forecasting exercises. I start addressing this question by evaluating the in-sample prediction errors of three formulations: the logarithmic (table 2, line 1), the CES, and the Almost Ideal. I will then re-estimate the parameters with data through 1995 and use those estimates to generate ex-post forecasts. I compute in-sample prediction errors as the difference between actual and predicted imports relative to the value of actual imports.6 Figure 2 shows the evolution of these prediction errors and the root-mean square prediction errors (RMSPE) for two periods: 1989-2001 and 1995-2001. According to the results, the CES formulation has the largest in-sample prediction errors with an RMSPE of 3.6 percent, more than twice as large as the RMSPE of the log-linear formulation. This differential in explanatory power is influenced by two large errors—in 1990 and in 1993. Thus I compute the RMSPE for the last seven years of the estimation sample and find a substantial narrowing in the RMSPE of the three formulations: from 1.2% for the loglinear formulation to 2.3% for the CES formulation. That relying on the CES might entail a decline in explanatory power, relative to that of the log-linear formulation, is not a new finding. What is new is that the explanatory power of the Almost Ideal formulation is virtually the same as that of the log-linear formulation. I interpret this similarity in explanatory power as implying a preference for the Almost Ideal formulation: it is based on optimizing behavior and it does not carry a deterioration of explanatory power within the estimation sample. The next question is whether one can use the out-of-sample performance as an additional criterion to select the type of formulation.
8
To generate out-of-sample predictions, I estimate the parameters of all three formulations (logarithmic, CES, and Almost Ideal) with data ending in 1995Q3 and then generate one-step ahead predictions using observations from 1995Q4 to 2001Q2. Figure 3 shows the actual and the ex-post predictions for the three formulations. Inspecting the evidence reveals two features of interest. First, there is a tendency in the three models to under-predict the actual volume of imports but the degree of under-prediction is small numerically. Second, there is virtually no difference in the predicted values among the three models. I interpret such similarity as strong evidence in favor of an optimization model for explaining service imports.
References
Deardorff, Alan V., Saul H. Hymans, Robert M. Stern, and Chong Xiang, "Forecasting U.S. Trade in Services" March 30, 2000. In Robert M. Stern, ed., Services in the International Economy: Measurement and Modeling, Sector and Country Studies, and Issues in the WTO Services Negotiations, University of Michigan Press, 2001, pp. 53-81 Hendry, D. F. and H. Krolzig, 2001, Automatic Econometric Model Selection Using PcGets, London: Timberlake. Reeve, T., 2001, “Trade in Services,” Federal Reserve Board, mimeo.
6
Because the three formulations that I am considering have different dependent variables, comparing their explanatory power involves re-expressing the equations in terms of the level of imports of aggregate services in real terms.
9
Billions of U.S. Dollars
U.S. External Balances
100
0
-100
Goods and Services Goods Services
-200
-300
-400
1930
1940
1950
1960
1970
1980
1990
Figure 1: External Balances in Services and Goods-United States, 1929-2002
2000
10
Table 1 Income and Price Elasticities for U.S. Trade in Services Alternative Groupings and Estimation Methods* 1987-2001
Category Aggregate a
Method
Exports Income Price
Imports Income Price
FIML b OLS
1.33* 1.74*
-0.37* -0.29*
1.43* 1.35*
-1.48* -1.56*
FIML OLS
1.10* c 0.00
-1.43* c 0.00
1.99* 2.30*
-0.60* -1.07*
FIML OLS
0.63* 1.11*
-0.78* -0.95*
0.84* 1.06*
-2.18* -1.49*
FIML OLS
0.90* 0.75*
-0.14 -0.04
0.94* 0.86*
-0.22 +0.00
FIML OLS
3.25* 3.23*
+0.49* +0.42*
3.32 1.56*
-0.37 -2.22*
Fares
Travel Other Private Transportation Other Private Services
* Statistically significant at the 5 percent level. a
Excludes trade in defense and royalties.
b
VAR model for the logarithms of trade, economic activity, and relative prices; parameters are estimated with Johansen’s FIML method. c
ln Xfares(t ) = 0.70 + 0.58 ln Xfares (t − 1) + 0.18 ln Xfares(t − 2)
11
Table 2 Income and Price Elasticities Results for Aggregate Service Imports: Sensitivity to Specification Formulation
Income
Price
1. Baseline (GDP Deflator)
1.35*
-1.56*
2. GDP Deflator & Per-capita
2.16*
3. GDP Deflator, Per-capita, Foreign Born
Chw1/2
Chw8
JB
AR
ARCH
√
√
√
√
√
+0.02
√
√
√
√
X
2.98*
+0.90*
√
√
√
√
√
4. Service Deflator
2.10*
-0.49*
√
√
√
√
√
5. Service Deflator & Per-capita
2.14*
-1.21*
√
√
√
√
√
6. Service Deflator, Per-capita, Foreign Born
2.73*
-0.70*
√
X
X
√
√
* Statistical significance at the 5 percent level; √: cannot reject null hypothesis at 5 percent significance level; X: cannot accept null hypothesis at 5 percent significance level. Chw1/2: Chow test for null hypothesis of parameter constancy with half of the sample excluded from estimation. Chw8: Chow test for null hypothesis of parameter constancy with the last eight observations excluded from estimation. JB: Test for null hypothesis of normality in the distribution of the residuals. AR: Test for null hypothesis of serial independence in the distribution of the residuals. ARCH: Test for null hypothesis of conditional homoskedasticity in the distribution of the residuals.
12
Table 3 Estimation Results for Formulations Consistent with Theory Constant-Elasticity Formulation M ln = D t (SE)
M 0.77* ln - 0.77 D t −1 (0.07) (0.26)
Pd + 1.09* ln Pm t (0.28)
-
Pd 0.69* ln Pm t −1 (0.31)
Short run σ: 1.09 Long run σ: 1.77 (=(1.09-0.69)/(1-0.77)) Rsquare = 0.97 Normality: cannot accept at the 5 percent significance level Serial Independence: cannot reject at the 5 percent significance level Homoskedasticity: cannot reject at the 5 percent significance level
Almost-Ideal Formulation Pm • M Pm • M EXP = 0.7832* + 1.219 + 4.8* ln P t Pm • M + Pd • D t Pm • M + Pd • D t −1
(SE)
(0.0698)
(1.63) (2.75)
Pd Pd EXP + 0.4246* ln - 4.947* ln + 0.1864* ln P t −1 Pm t −1 Pm t (0.89) (2.62) (0.901)
Rsquare = 0.89 Normality: cannot accept at the 5 percent significance level Serial Independence: cannot reject at the 5 percent significance level Homoskedasticity: cannot reject at the 5 percent significance level
13
1989-2001 1995-2001 7.5
RMSPE-CES RMSPE - Log-log RMSPE - Almost Ideal
3.6% 2.3 1.5% 1.2 1.8% 1.5
5.0
2.5
0.0
-2.5
-5.0
Prediction Errors: Actual - Fit (percent) -7.5 1989
1990
1991
1992
1993
1994
1995
1996
1997
Figure 2: In-sample Prediction Errors
1998
1999
2000
2001
14
Billions of Dollars, 1996 Prices 190
U.S. Service Imports Total ex. Defense and Royalties 180
170
160
Estimation Sample 1989-1995
Ex-post Forecasts
CES Logarithmic Almost Ideal Actual
150
140
130
1995
1996
1997
1998
1999
Figure 3: Out-of-Sample Predictions
2000
2001
15
Appendix Sensitivity Income and Price Elasticities to Composition of Aggregate Service Exports OLS, 1989-2001 Elasticities Income Price Aggregate This Study Reeve Travel This Study Reeve Fares This Study Reeve Other Transp. This Study Reeve Other Private This Study Reeve
Chw1/2
Statistical Tests Chw8 JB AR
ARCH
1.74* 1.97*
-0.29* -0.28*
√
√
√
√
√
1.11* 3.36*
-0.95* +0.42
√
√
√
√
√
0.00a 2.15*
0.00a -1.67*
√
√
√
√
√
0.75* 0.16
-0.04 -0.80*
√
√
√
√
√
3.23* 2.15*
+0.42* -0.22
X
√
X
√
√
* Satistical significance at the 5 percent level; √: cannot reject null hypothesis at 5 percent significance level; X: cannot accept null hypothesis at 5 percent significance level. a
ln Xfares(t ) = 0.70 + 0.58 ln Xfares(t − 1) + 0.18 ln Xfares(t − 2)
Reeve, T., 2001, “Trade in Services,” Federal Reserve Board, mimeo. The sample is quarterly over 1982-1996. The data for aggregate trade includes defense and royalties. JB: Test for null hypothesis of normality in the distribution of the residuals. AR: Test for null hypothesis of serial independence in the distribution of the residuals. ARCH: Test for null hypothesis of conditional homoskedasticity in the distribution of the residuals. Chw1/2: Chow test for null hypothesis of parameter constancy with half of the sample excluded from estimation. Chw8: Chow test for null hypothesis of parameter constancy with the last eight observations excluded from estimation.
16
Sensitivity Income and Price Elasticities to Composition of Aggregate Service Imports OLS, 1989-2001
Aggregate This Study Reeve Travel This Study Reeve Fares This Study Reeve Transportation This Study Reeve Other Private This Study Reeve
Income
Price
Chw1/2
Chw8 JB
AR
ARC H
1.35* 1.72*
-1.56* -0.84*
√
√
√
√
√
1.06* 1.72*
-1.49* -0.35
√
√
√
√
√
2.30* 1.65*
-1.07* -2.06
√
√
√
√
√
0.86* 1.15*
0.00 -0.53
√
√
√
√
√
1.56* 3.10*
-2.22* -0.44
√
√
√
√
√
* Statistical significance at the 5 percent level; √: cannot reject null hypothesis at 5 percent significance level; X: cannot accept null hypothesis at 5 percent significance level. Reeve, T., 2001, “Trade in Services,” Federal Reserve Board, mimeo. The sample is quarterly over 1982-1996. The data for aggregate trade includes defense and royalties. JB: Test for null hypothesis of normality in the distribution of the residuals. AR: Test for null hypothesis of serial independence in the distribution of the residuals. ARCH: Test for null hypothesis of conditional homoskedasticity in the distribution of the residuals. Chw1/2: Chow test for null hypothesis of parameter constancy with half of the sample excluded from estimation. Chw8: Chow test for null hypothesis of parameter constancy with the last eight observations excluded from estimation.
