BALANCE OF PAYMENTS CONSTRAINED GROWTH: EVIDENCE FOR THE U.S. FROM TWO ECONOMETRIC METHODOLOGIES Robert F. Mulligan, Ph.D. CLARKSON UNIVERSITY SCHOOL OF MANAGEMENT Department of Economics & Finance (1994)

mulligan@wcu.edu

SUMMARY

Recursive Chow tests are used to examine parameter stability and search for optimal break points. The Thirlwall growth model (in which GDP growth depends on export growth) and an ad hoc alternative (in which growth depends on imports) are estimated and compared for explanatory power using a battery of Cox-type non-nested tests (F, J, J_A, Ñ, and W tests.) A Monte Carlo study is performed to obtain the small sample behavior of the test statistics, which is found to be quite different from the asymptotic distributions commonly used to evaluate these tests. Four sets of Monte Carlo simulations are performed. The first three assume the Thirlwall model is the correct model and add 1. standard normal, 2. log-normal, and 3. chi-square error terms. The fourth simulation uses the bootstrap method and relies on the empirical distribution of the original data. The Thirlwall model performs poorly, and the ad hoc alternative performs well. Finally, Durbin-Wu-Hausman tests are performed which show that imports are endogenous and exports are exogenous, accounting for the Thirlwall model's poor performance. Thus, the Thirlwall model provides a good explanation of economic growth.

1. Introduction

This essay uses Cox-type tests to evaluate 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 to changes in demand (see McCombie, 1985; Thirlwall, 1979, 1991). The model is due to Thirlwall and Hussain (1982). This model assumes that the balance-of-payments position is the main constraint on economic growth, because the balance-of-payments imposes a limit on the growth of demand to which supply can adapt.

An alternative specification is presented which allows output growth to be explained by import growth, rather than export growth, as in the Thirlwall-Hussain model. The two alternatives are tested against each other employing a battery of Cox-type non-nested hypothesis tests. Monte Carlo experiments are used to find the small-sample distributions of Cox-type test statistics. Finally, Durbin-Wu-Hausman (DWH) tests are used to determine whether exports or imports are endogenous. The Cox-type tests support the alternative specification over the Thirlwall model, but the DWH tests indicate that this result is due to the endogeneity of imports with output.

The essay is organized as follows. Section 2 presents the theoretical models to be tested. Section 3 presents data sources and issues. Section 4 presents estimates of the models. Section 5 presents the Cox-type non-nested hypothesis tests. Section 6 presents the Monte Carlo simulation showing the small-sample properties of the Cox-type test statistics. Section 7 presents the Durbin-Wu-Hausman tests for endogeneity. Section 8 presents the conclusions.

2. Theory

This section presents a description of the Thirlwall-Hussain (1982) model of balance-of-payments constrained growth. The Thirlwall-Hussain model can be represented by three equations. The first represents growth in exports (x) as an increasing function of real world income (w), and a negative function of the difference between growth in domestic prices (p) and growth in foreign prices (p_f) (measured in domestic currency):

The second equation represents growth in real imports (m) as an increasing function of real domestic income (y) and the difference in the growth rates of domestic and foreign price levels (p - p_f):

The third equation is the equilibrium condition of the model stated in rate of change form: where c is net capital inflows. Usually, equations (2) and (3) are solved for a reduced form for y in terms of x, c, p, and p_f (e.g., Atesoglu, 1993a):

If there are no capital inflows, relative prices remain constant, and nominal exports equal nominal imports, then a = 1, and equation 4 reduces to the dynamic form of the Harrod foreign trade multiplier, where growth in output is determined by the inverse of the income elasticity of demand (1/p ) times growth in exports. This result is also know as Thirlwall's law; empirical evidence supporting this prediction has been provided by Thirlwall (1979) and Bairam (1988), both using data from industrial countries, and by Atesoglu using time series for the U.S. (Atesoglu 1993a), and Canada (Atesoglu 1994).

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

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

The alternative is not derived from economic theory; it is an arbitrary specification that happens to be convenient for testing the Thirlwall model. The Cox-type non-nested tests require two alternative specifications.

3. Data

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

Following Atesoglu, each empirical measure was smoothed by a moving-average before being used in 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, 1994) 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 power and efficiency of Cox-type tests. 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.

4. Estimates

Table 1 gives estimates of the two alternatives, equations 5 and 6, for various post World War II samples. First-order serial correlation was corrected with Beach and MacKinnon's (1978) iterative maximum likelihood procedure. The Beach-MacKinnon procedure is preferable to Cochrane-Orcutt in this case because it allows retention of the first observation in the sample, which gives a larger gain in efficiency the smaller the sample size, (Davidson and MacKinnon, 1993, 348-349.)

In part 1, the slope on real export growth in equation 6 has the correct sign and significance level for the 1971-1979 sample only. Growth in capital inflows is significant and positive for the 1971-1979 and 1980-1988 periods, while growth in relative prices is significant and negative for the 1980-1988 period only.

In part 2, the slope on real import growth is always significant and positive. This could be due to causality, reverse causality, high correlation, or high endogeneity, of real import growth with real output growth, (or any combination.) Capital inflows are never significant. Growth in relative prices is significant and negative for the 1971-1988, 1971-1979, and 1980-1988 periods.

Tables 2 and 3 give iterative Chow test statistics for the Thirlwall reduced form (equation 5) for different break points. Although asymptotic probability levels suggest structural breaks nearly everywhere, the samples are very small, so it is not clear that the coefficients are not, in fact, structurally stable, or even where the optimal break point might lie. Because iterative Chow tests do not indicate an optimal break point, the postwar sample was divided in half, at 70-71, and the second half was then divided in half, at 79-80. These break points give the sample ranges for the estimates shown in table 1.

5. Hypothesis Tests

Table 4 gives a battery of Cox-type non-nested test statistics based on the full period 1953-1988 regressions using procedures based on the seminal papers by Cox (1961, 1962) and Atkinson (1970). These statistics were all calculated using formulae and procedures given by Godfrey and Pesaran (1983). (The necessary formulae for unbiased estimators of variances of residuals are given by Pesaran (1974, 158-159).) The Cox-type tests are the Davidson-MacKinnon (1981) J test, the Fisher-McAleer (1981) J_A test (which employs the Atkinson adjustment,) and the Godfrey-Pesaran (1983) Ñ and W tests. Fisher's conventional F test is shown for comparison. All five tests are asymptotically equivalent. The full 1953-1988 postwar sample was used, to take as much advantage as possible of the asymptotic properties of the tests.

The F test is performed with an artificially nested regression including all regressors contained in either sub-model. Because each alternative contains only one regressor which is unique to that model, the F test statistic is merely the square of the t statistic for the coefficient on growth in imports (testing equation 5) or exports (testing equation 6,) and has the same asymptotic probability value as the t statistic. Godfrey and Pesaran (1983, 143) give a brief discussion of the testing procedure.

The probability values given in table 4 are all asymptotic: the F statistics are asymptotically F distributed with 1 and 32 degrees of freedom; the J and J_A test statistics are both asymptotically standard normal and asymptotically t distributed with degrees of freedom equal to the number of observations minus the number of regressors, (probability levels given for these tests assume the t distribution with 32 degrees of freedom;) the Ñ and W test statistics are asymptotically standard normal. Asymptotic probability levels are often misleading in small samples, as will be seen below.

Based solely on asymptotic distribution theory, the F test supports the alternative specification over the Thirlwall model, the J and J_A tests support neither, while the Ñ and W tests support both models (suggesting that the true model is a combination of the two.) The Ñ and W are more powerful and would generally be taken to govern.

6. Monte Carlo Simulations

Asymptotic distribution theory often gives very misleading indications, so a Monte Carlo experiment was performed to examine small-sample properties of the tests. Four sets of 1000 Monte Carlo iterations were performed. Each of the four experiments was initially seeded with 9378214. In the first three simulations, the Thirlwall model 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 of the left-hand-side variable. The first experiment added a normally distributed set of 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 used chi-squared errors. Godfrey and Pesaran (1983, 140) describe transformations to construct the log-normal and chi-squared errors. The fourth and last experiment employed the bootstrap procedure, using the empirical distribution of the actual data to simulate all variables, not just the errors.

All pseudorandom variables were generated by the multiplicative congruential method described by L'Ecuyer (1990). In each iteration of each simulation, all five Cox-type test statistics were calculated for each hypothesis (equations 5 and 6). Following Hine and Bischoff (1994), the percentage of Monte Carlo test statistics greater in absolute value than the actual test statistic approximates the small-sample probability level.

Low probability levels of the test statistics for H₀ indicate rejection of the null hypothesis that the additional regressors in H₁do not add any explanatory value not already contained in H₀. In other words, low probability levels for the H₀ test statistics indicate support for H₁ (the alternative specification,) and low probability levels for the H₁ test statistics indicate support for H₀ (the Thirlwall model.)

The F test, which is not size adjusted, supports the alternative specification, as do both the size adjusted Ñ and W tests. The J and J_A tests give no decision. The results of Ñ and W test govern, because they are more powerful in small samples (Godfrey and Pesaran, 1983); thus, according to the Cox-type tests, the alternative specification is better supported by postwar U.S. data.

Each of the four simulations gives similar results. 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 left-hand-side variable; it is striking that the three simulations that assume the Thirlwall model is the true model support the alternative.

7. Tests for Endogeneity

In light of these results, it is relevant to examine the possibility that the Thirlwall model's poor performance is due to endogeneity of import growth with output growth. The Durbin-Wu-Hausman (DWH) test (see Davidson and MacKinnon, 1993, 237-242) was performed on each whole-sample 1953-1988 regression by comparing the Beach-MacKinnon residuals with residuals from a two-stage least squares regression. The 2SLS instrument lists consisted of the current and one year lagged values of the dependent variable, net capital inflows, relative prices, and either import growth (in the case of the Thirlwall model,) or export growth, (in the case of the alternative,) i.e., the instruments included the explanatory variable unique to the other model, and excluded the explanatory variable unique to the model being tested.

The instruments allow testing the hypothesis that endogeneity between the instruments and the right-hand-side variables has significant impact on the estimates. Under fairly general conditions given by Davidson and MacKinnon (1993, 239) the distribution of the DWH statistics is exact.

The DWH test is often thought of as a 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 for endogeneity of the right-hand-side variables not contained in the instruments.

For the Thirlwall reduced form (equation 5) the DWH test statistic is 0.8949, with a probability level (chi-square with 3 degrees of freedom) of .827, which indicates acceptance of the null hypothesis that endogeneity of the regressors with the instruments (including import growth) has little effect on the estimates. This result is consistent with exogeneity of export growth with output.

For the alternative specification (equation 6) the DWH statistic is 31.141, with an approximate probability of 0.000, indicating endogeneity of the regressors with the instruments (here including export growth) has a highly significant effect on the estimate. This result is consistent with endogeneity of import growth with output.

8. Conclusions

The DWH tests support the conclusion that import endogeneity is likely to affect the regression estimates, and export endogeneity is not. This result accords well with 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 support to Thirlwall model.

At the same time, the DWH results suggest that equation 6 is misspecified and cannot be efficiently estimated by least squares, but requires instrumental variable or simultaneous equation techniques. In contrast, the Thirlwall model appears to be correctly specified and efficiently estimated by Beach-MacKinnon.

Furthermore, if exports are indeed endogenous with domestic output, while imports are exogenous, this explains the Cox-type test results: the alternative specification outperforms the Thirlwall model merely because it contains a variable (import growth) that is endogenous with the left-hand-side variable (output growth). Thus, the Thirlwall model stands as an explanatory model of output growth.

The author thanks H. Sönmez Atesoglu, Clint Cummins, Harmen Lehment, Erwin Nijsser, R. Rajagapolan, Timo Tätinien, 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 alone am responsible for errors.
 
 

Table 2.-Chow tests for different break points within the 1953-1988 period
 
 

break test stat probability

59-60 14.39345 .00005

60-61 13.48746 .00007

61-62 13.66947 .00007

62-63 13.60446 .00007

63-64 13.80567 .00006

64-65 17.45659 .00001

65-66 22.09041 .00000

66-67 21.90824 .00000

67-68 22.09574 .00000

68-69 21.15040 .00000

69-70 9.107343 .00085

70-71 3.473791 .04442

71-72 3.392967 .04742

72-73 3.005283 .06515

73-74 3.487498 .04393

74-75 3.769181 .03507

75-76 3.142994 .05815

76-77 4.935696 .01429

77-78 0.988018 .38450

78-79 0.9209397 .40948

79-80 1.760959 .18977

80-81 2.524967 .09752

81-82 2.248939 .12360

82-83 2.006137 .15275
 
 

Statistics are asymptotically F distributed with 2 and 29 degrees of freedom. Probability values are asymptotic upper tail areas.
 
 
 
 
 
 

Table 3.-Chow tests for different break points within the 1971-1988 period
 
 

break test stat probability

76-77 5.788097 .01917

77-78 7.092261 .01051

78-79 7.670290 .00821

79-80 12.82188 .00134

80-81 17.90840 .00035

81-82 18.89503 .00028

82-83 14.77277 .00077
 
 

Statistics are asymptotically F distributed with 2 and 11 degrees of freedom. Probability levels are asymptotic upper tail areas.
 
 
 
 
 
 

Table 4.-Summary of Cox-type test statistics
 
 

H₀ test Asymptotic H₁ test Asymptotic

test statistic probability statistic probability
 
 

F 18.431 1.527x10^-04** 0.511 0.480

(R.A. Fisher)
 
 

J 0.308 0.760 0.361 0.721

(Davidson-MacKinnon, 1981)
 
 

J_A 1.160 0.255 0.222 0.826

(Fisher-McAleer, 1981)
 
 

Ñ -5.907 3.494x10^-09** -2.804 0.00505**

(Godfrey-Pesaran, 1983)
 
 

W -3.716 2.024x10^-04** -2.253 0.0243*

(Godfrey-Pesaran, 1983)
 
 

Statistics significant at the 5% level indicated by *; those significant at the 1% level indicated by **.
 
 
 
 
 
 

Table 5.-Monte Carlo simulation results: Approximate small sample probability levels of computed test statistics
 
 

normal log-normal chi-square empirical (bootstrap)

test residual residual residual distributions
 
 

H₀: [Thirlwall model, equation (5)]
 
 

F .587 .587 .588 .583

J .832 .846 .840 .831

J_A .149 .151 .151 .244

Ñ .001 .001 0 .008

W .008 .008 .006 .010
 
 
 
 

H₁: [Alternative specification, equation (6)]
 
 

F .018 .018 .021 .017

J .836 .834 .821 .810

J_A .771 .781 .792 .817

Ñ .431 .321 .407 .596

W .431 .324 .412 .585
 
 

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