Robert F. Mulligan, Ph.D.
Institut für Weltwirtschaft Kiel Advanced Studies in International Economic Policy Research

Applied Economics Letters 3 (1996), 275-279.


The Thirlwall-Hussain model (in which output growth depends on export growth) and an ad hoc alternative (in which growth depends on imports) are estimated and compared for the U.S. The Durbin-Wu-Hausman test is used to determine the endogeneity or exogeneity of exports and imports with respect to output. A Monte Carlo study reveals the small-sample behavior of the test statistics, which partly overturns the asymptotic results. Four sets of Monte Carlo simulations are performed. The first three assume the Thirlwall-Hussain model is correct and add: 1. standard normal, 2. log-normal, and 3. chi-square error terms. The fourth simulation uses the bootstrap method relying on the empirical distribution of the original data, and makes no assumption about the underlying data generating process. U.S. exports and imports are both endogenous with respect to output, a major difficulty for the Thirlwall-Hussain model.

1. Introduction

This essay evaluates a Keynesian model of economic growth using U.S. data. The Keynesian model emphasizes the role of demand factors, rather than supply factors like technology, human capital, and production inputs, as in neoclassical growth theory. In the Keynesian view, increases in the size of the labor force, the stock of capital, the stock of human capital, and technical change are mainly endogenous, adjusting passively in the long run to changes in demand (see McCombie, 1985; Thirlwall, 1979, 1991). The model, due to Thirlwall and Hussain (1982), assumes the balance-of-payments position is the main constraint on economic growth, imposing a limit on demand growth to which supply adapts.

The Thirlwall-Hussain model, and Keynesian models generally, unambiguously regard imports as endogenous with respect to output, and exports as exogenous. The Durbin-Wu-Hausman test (Durbin 1954, Wu 1973, Hausman 1978, often called the Wu-Hausman or Hausman specification test, here afterwards called DWH) will be used to evaluate this categorization.

An ad hoc alternative specification is estimated which allows output growth to be explained by import growth, rather than export growth as in the Thirlwall-Hussain model. This alternative specification is not related to the Thirlwall-Hussain model and is not theoretically justified, but is used to test for import endogeneity. Monte Carlo experiments are used to find the small-sample distributions of the DWH statistics. When evaluated in light of the Monte Carlo evidence, DWH tests indicate both exports and imports are endogenous with respect to output.

The essay is organized as follows. Section 2 presents the models to be tested. Section 3 presents data sources and issues. Section 4 presents DWH tests. Section 5 presents Monte Carlo simulations. Section 6 presents conclusions.

2. Theory

This section presents the two specifications used to test export and import endogeneity. Thirlwall and Hussain (1982) derive a model of balance-of-payments constrained growth. The derivation will not be repeated here.

Allowing an intercept and a stochastic error term, the reduced form is written in the form in which it will be estimated:

where Q1 > 0, y is output growth, x is export growth, (c - p) is growth of net capital inflows, and (p - pf) is change in the relative price level.

Empirical evidence supporting the Thirlwall-Hussain model has been provided by Thirlwall (1979) and Bairam (1988), both using data from industrial countries, and by Atesoglu (1993a) and Mulligan (1994) using time series for the U.S., Germany (Atesoglu 1994c), and Canada (Atesoglu 1994). Atesoglu (1994b) also found support for a more general Kaldorian model of which the Thirlwall-Hussain model is a special case.

As an ad hoc alternative specification, growth in imports (m) is substituted for exports:

The alternative is not explicitly derived from economic theory; it is an arbitrary specification that is required for testing import endogeneity, which cannot be tested with equation (1).

DWH tests allow direct evaluation of two central assumptions of the Thirlwall-Hussain model, and of Keynesian models in general: exports are exogenous with respect to output, and imports are endogenous.

3. Data

This section presents the data source and discusses data issues. The data is from the Economic Report of the President, 1994, appendix B, tables B-2, B-3, and B-114. The empirical measures are:

y: one hundred times the first difference of the logarithm of the gross domestic product at 1987 prices;
x: one hundred times the first difference of the logarithm of real exports;
m: one hundred times the first difference of the logarithm of real imports;
p: one hundred times the first difference of the logarithm of the GDP implicit price deflator;
pf : one hundred times the first difference of the logarithm of the imports of goods and services implicit price index;
c-p: percent change in real imports less real exports.
Following Atesoglu, each empirical measure was smoothed by a moving-average before estimation. This procedure is necessary to overcome the common cyclical fluctuations present in annual time series, which plague tests of economic growth models. This problem was recognized by McCombie (1983) and McCombie and de Ridder (1983), among others. Atesoglu (1993a, 1993b, 1994a, 1994b, 1994c) successfully mitigated effects of cyclical variation with a smoothing process.

The moving-average period was selected to be fairly long, eleven years, to effectively filter out cyclical variations. Atesoglu found sixteen-year moving-averages sufficient to remove cyclical fluctuations. Here, the eleven year moving-average period produced very similar estimates. The shorter period was chosen because gains in degrees of freedom improve asymptotic power and efficiency of the DWH test. Middle-of-period, or centered, moving-averages were used, rather than end-of-period, or lagged, moving-averages as in Atesoglu. Preliminary estimates showed little difference regardless of the kind of moving-average used.

Long-term moving-average smoothing imposes serial correlation on the data, so regressions were estimated by Beach-MacKinnon (1978) iterative maximum likelihood.

4. The Durbin-Wu-Hausman Test for Endogeneity

This section presents the DWH tests. Atesoglu (1994b) found the general Kaldorian model outperforms the Thirlwall-Hussain model, which is a special case. Kugler (1991), using the Juselius-Johansen procedure, found U.S. growth was not export-led, with mixed results for other countries.

Furthemore, estimates of equations (1) and (2) presented in table 1 show import growth is significant in explaining output growth, but export growth is not. Equation (2) regressions have higher adjusted coefficients of determination than equation (1), the Thirlwall-Hussain reduced form. In light of these results, it is relevant to examine the possibility that the Thirlwall-Hussain model's poor performance is due to endogeneity of import growth with output growth.

The DWH test was performed on each whole-sample 1953-1988 regression by comparing the Beach-MacKinnon residuals with residuals from a two-stage least squares regression. Davidson and MacKinnon (1993, 237-242) describe the DWH procedure. The 2SLS instrument lists consisted of a constant and current and one-year-lagged values of: output growth (the dependent variable,) growth in net capital inflows, and growth in relative prices.

The DWH test tests the null hypothesis that endogeneity between the instruments and the right-hand-side variables has no significant impact on the estimates. The null hypothesis is accepted if the two estimates of the same equation differ only randomly, i.e., if the actual data and the 2SLS fitted values are sufficiently similar, indicating the original right-hand-side variables excluded from the instrument list are endogenous.

The DWH test is often thought of as an explicit test for exogeneity of those variables in the instrument list not contained in the vector of right-hand-side variables, (see Wu (1973), Hausman (1978) and Nakamura and Nakamura (1981).) Davidson and MacKinnon (1993, 239) point out that the DWH test really tests whether possible endogeneity of the right-hand-side variables not contained in the instruments makes any difference to the coefficient estimates.

The 2SLS instrument lists exclude exports and imports, but include all other right-hand-side variables. This choice of instruments provides a test exclusively for endogeneity of exorts in equation (1) and of imports in equation (2). First-stage regressions return the own-values of the other right-hand-side variables included in the instrument list (net capital inflow and relative price level), focusing the test exclusively on possible export and import endogeneity.

For the Thirlwall reduced form (equation 1) the DWH test statistic is 0.4746, with an asymptotic probability level (chi-square with 2 degrees of freedom) of .789, indicating acceptance of the null hypothesis that endogeneity of exports with the instruments has no effect on the estimates. This result is consistent with endogeneity of export growth with output.

For the alternative specification (equation 2) the DWH statistic is 20.918, with an approximate asymptotic probability of .0001, indicating endogeneity of imports with the instruments has a highly significant effect on the estimate. This result is consistent with exogeneity of import growth with output.

The two results, that exports are endogenous with respect to output and imports are exogenous, flatly contradict the general Keynesian view, which is that exports are exogenous and imports are endogenous with respect to output.

5. Monte Carlo Simulations

This section presents the Monte Carlo simulations performed to obtain the small-sample behavior used to evaluate the DWH statistics. Asymptotic distribution theory often gives very misleading indications, so a Monte Carlo experiment was performed to examine small-sample properties of the DWH statistics.

Four sets of 1000 Monte Carlo iterations were performed. Each of the four experiments was initially seeded with 9378214, so the four simulations are not independent, but make different distributional assumptions. In the first three simulations, following the protocol of Godfrey and Pesaran (1983), the Thirlwall-Hussain model (equation 1) was assumed to be correct, and was used to generate the left-hand-side variable (growth in output, y).

The left-hand-side variables in the first three experiments were formed by adding a pseudo-random variable, simulating the error, to the fitted value from the original estimate. The first experiment added normally distributed errors with the same mean and variance as the residuals from the actual estimate of the Thirlwall model. The second experiment used log-normal errors, the third chi-square.

The fourth experiment went beyond the Godfrey-Pesaran protocol, and employed the bootstrap procedure, using the empirical distribution of the actual data to simulate all variables, not just the errors. Thus, the fourth experiment is free of any assumptions about the data generating process.

All pseudorandom variables were generated by the multiplicative congruential method described by L'Ecuyer (1990). In each iteration of each simulation, both DWH test statistics were calculated for each hypothesis, but note that the first three simulations assume equation (1) to be the data generating process.

The DWH test statistic is asymptotically chi-square distributed, and therefore one-tailed. The percent of Monte Carlo test statistics greater than the actual test statistic approximates the small-sample probability level, reported in table 2.

Low probability levels of DWH statistics for equation (1) indicate rejection of the null hypothesis that export endogeneity with respect to output has no impact on coefficient estimates, (i.e., indicate export exogeneity.) Similarly, low probability levels for equation (2) indicate import exogeneity.

Each of the four simulations gives similar results: both exports and imports are endogenous with respect to output. The bootstrap simulation would normally be considered most reliable since it is based on the empirical distribution of the actual data, but the first three simulations used the Thirlwall model to generate the simulated left-hand-side variable; it is striking that three simulations that assume the Thirlwall model is the true model and that export growth leads output growth, also find exports endogenous with respect to output.

6. Conclusion

This section presents conclusions. DWH tests indicate exports and imports are both endogenous with respect to output, and that this endogeneity is unlikely to affect the regression estimates. This result dramatically contradicts the Keynesian cross model of the impact of trade on output, in which imports are endogenous, depending on domestic income, and exports are exogenous, depending on foreign income. Thus, the DWH tests shed doubt on the Thirlwall-Hussain model.

The asymptotic behavior of the DWH statistics indicates exports are endogenous and imports exogenous, the exact opposite of the Keynesian view. The prospect improves when the correct small-sample behavior of the DWH statistics is used to evaluate the test, showing both quantities are endogenous with respect to output.

These results should be interpreted in light of the following cautions: (1) The DWH test is not robust to different instrument lists. The instruments used here are fairly obvious and meaningful, but other instruments might give different results.

(2) The a priori distinction between endogeneity and exogeneity contained in an economic model is not clearly related to the empirical/statistical distinction tested in this paper.

(3) Results for the U.S. may not be indicative of results for other countries. These results accord well with Kugler's (1991) finding that U.S. output growth is not export led. A possible explanation for U.S. export endogeneity is that for the earlier part of the post-war era, U.S. output was a large fraction of world output, which, along with relative prices, determines U.S. exports in the Thirlwall-Hussain model.

At the same time, DWH tests indicate the Thirlwall-Hussain model is correctly specified and efficiently estimated by the Beach-MacKinnon procedure. Instrumental variable or simultaneous equation techniques are not necessary to obtain efficient estimates, and provide relatively little gain in efficiency. However, the same holds true for the ad hoc alternative in which output is explained by imports.

The author thanks H. Sönmez Atesoglu, Clint Cummins, Harmen Lehment, Erwin Nijsse, R. Rajagopalan, Timo Tähtinen, and seminar participants at the Norwegian School of Management, and the Advanced Studies in International Economic Policy Research of the Kiel Institute of World Economics, for many helpful comments. Thanks are also due Robert Busch for invaluable computing support. I am solely responsible for errors.

Table 1.-Reduced form regressions, various sample periods
Part 1. H0: Thirlwall-Hussain reduced form: equation (1)
smpl  #obs Q0 Q1 Q2 Q3 r adj R2 DW
53-88 36 2.862
.3571 1.531
53-70 18  3.398
 .4668 1.251
71-88 18 2.006 [.000]**  .1144
.7024 1.634
Two-stage least squares, whole sample
53-88 36 2.815
 ---  0.233
Part 2. H1: Alternative specification: equation (2)
smpl  #obs G0 G1 G2 G3 r adj R2 DW
53-88  36  1.671 
 -.00002 [.990]  -.0372
 .8881 [.000]** .5637 1.924
53-70  18 2.025
.5028 1.685
71-88  18 1.001
.7630 1.983
Two-stage least squares, whole sample
53-88 36  -1.547
---   .2549 0.525

Estimates are by Beach-MacKinnon iterative maximum likelihood or two-stage least squares. Two-stage least squares estimates use as instruments: a constant, and current and one-lagged values of: output growth, growth in net capital inflows, and relative price level growth. Numbers in square brackets are two-tailed asymptotic probability levels. Coefficient estimates significant at the 1% level are indicated by **.

Table 2.-Durbin-Wu-Hausman Test for Endogenity of Exports-Imports with Monte Carlo simulation results: 
Approximate small sample probability levels of computed test statistics 1953-1988
DWH test statistic degrees of  freedom  asymptotic probability normal  residual log-normal residual chi-square residual  empirical distributions
H0: [Thirlwall-Hussain model, equation (1)]
Test for endogeneity of exports with respect to output
0.4746  2 .789  .932  .922  .914  .921
H1: [Alternative specification, equation (2)]
Test for endogeneity of imports with respect to output
20.918  3 .0001 .246 .207 .249 .383

Asymptotic probability levels are based on the chi-square distribution with degrees of freedom equal to the rank of the covariance matrix of the vector of differences between the coefficient vectors computed by Beach-MacKinnon iterative maximum likelihood and two-stage least squares, (i.e., "vector of contrasts," see Davidson and MacKinnon (1993, 238)).


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