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Using cross-country variances to evaluate growth theories
ELSEVIER
Journal of Economic Dynamics and Control 20 (1996) 102771049
Using cross-country variances to evaluate growth theories Paul Evans Department
ofEconomics,Ohio
State University,
Columbus, OH 43210-1172,
USA
(Received January 1995; final version received August 1995)
Abstract Much of the empirical growth literature has attempted to evaluate growth theories by estimating regressions that relate the growth rate of per capita output for a sample of countries to initial per capita output and country characteristics. The resulting inferences are shown to be invalid except under strong conditions. An alternative method that uses cross-country variances is formulated and shown to produce valid inferences under weak conditions. Applying this method to data from thirteen countries over the period 1870-1989 provides no evidence that their per capita outputs have different trend growth rates and much evidence that they revert toward a common trend. This evidence does not support those endogenous growth theories that predict appreciably different trend growth rates across countries. words: Growth theory; Endogenous/exogenous deterministic trends; Monte Carlo simulation JEL classification: C22; C23; C15; 040 Key
growth;
Unit
roots;
Common/
1. Introduction
In recent years, a large empirical literature has attempted to evaluate growth theories. Many of the contributors to this literature have fitted regressions relating the average growth rate of per capita output over some time period for a sample of countries to initial per capita output and country characteristics. They have then applied standard methods of inference to the estimated
I am grateful for helpful comments from In Choi, James Hamilton, Georgios Karras, Manfred Keil, and two anonymous
referees.
0165-1889/96/%15.00 0 1996 Elsevier Science B.V. All rights SSDI 016S-1889(95)00888-3
reserved
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coefficients.’ Milton Friedman (1992) has argued that inferences based on such growth regressions are invalid and has suggested an alternative approach for evaluating growth theories. This approach exploits the time-series properties of the cross-country variance of the logarithm of per capita output.’ In this paper, I show that the inferences from growth regressions are indeed invalid except under highly restrictive conditions and that Friedman’s alternative approach is valid under weak conditions. I also develop a formal econometric apparatus for analyzing the time-series properties of cross-country variances. Employing this apparatus, I find no evidence of different trend growth rates for Australia, Austria, Belgium, Canada, Denmark, Finland, France, Italy, Norway, Sweden, the United Kingdom, the United States, and West Germany over the period 1870-1989 and strong evidence of reversion toward a common trend.3 Furthermore, the trend growth rates of 51 countries do not appear to differ over the period 1950-1992. Therefore, either endogenous growth models that predict different trend growth rates for countries are fundamentally flawed, or else the effects that they predict are unimportant for these countries. The basic idea of the paper is straightforward. If countries have different trend growth rates as predicted by endogenous growth theories, the logarithms of their per capita outputs should wonder away from each other at positive rates and hence their cross-country variance should be integrated of order one around an upward quadratic trend. By contrast, if the countries follow parallel balanced growth paths as predicted by exogenous growth theories, the logarithms of their per capita outputs should not wonder away from each other and hence their cross-country variance should be stationary around a constant positive mean. To illustrate this idea, consider Fig. 1, which plots the cross-country variance of the logarithm of per capita output for the thirteen industrial countries mentioned above. The figure indicates that identical growth rates cannot be rejected since their cross-country variance did not trend upward quadratically. Moreover, the rapid elimination of the effects of World Wars I and II provides
’ A far from exhaustive list of such contributors is Kormendi and Meguire (1985), Baumol(1986) De Long (1988), Barro (1991) De Long and Summers (1991), Dollar (1991), Barro and Sala-i-Martin (1992), Mankiw, Romer, and Weil (1992), Levine and Renelt (1992), King and Levine (1993), De Gregorio (1993), Easterly (1993, 1994), Easterly and Rebel0 (1993), Easterly, Kremer, Pritchett, and Summers (1993), Alesina and Rodrik (1994), and Persson and Tabellini (1994). * Quah (1993b) has also evaluated by examining time. In his 1993a and output in different years evolved over time.
criticized growth regressions and has argued that growth theories should be the evolution of the cross-country distribution of per capita output over 1994 papers, he discretized the cross-sectional distributions of per capita and fitted Markov models in order to characterize how these distributions
3 Baumol, Blackman, and Woolf (1989) Ben-David (1993), Parente and Prescott (1993), and Keil and Vohtra (1993) have also employed cross-country variances to evaluate growth theories. Their analyses are much more informal and qualitative than that of this paper.
P. Evans J Journal of Economic Dynamics and Control 20 (1996) 1027-1049
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0.05 -
0.00 m,,,, I875 Fig. 1. Cross-country
I999 variances
I923
of log per capita
I947 output
1971 for 13 industrial
countries
some evidence for parallel balanced growth paths. These visual impressions are confirmed later in the paper. The rest of the paper is organized as follows. Section 2 discusses the conditions under which inferences from growth regressions are valid and develops a formal econometric apparatus for using cross-country variances to evaluate growth theories. Section 3 applies the apparatus. Section 4 summarizes the paper and draws some conclusions. 2. Econometric discussion Let y,, denote the logarithm of per capita output for country n during period t, which has been observed for countries 1,2, . . . , N and periods 0, 1, . . . , T. Suppose that each y,, is difference-stationary and invertible and satisfies standard regularity conditions. Endogenous growth theories predict that countries have different trend growth rates since they differ in technologies, preferences, market structures, and government policies inter alia. This prediction suggests that the following hypothesis be considered: HI: The stochastic processes yl,,yzt, . . . , yNt are difkrence-stationary
with no cointegration among themselves, and Ayl,, Ay2,, . . , AyN, have different unconditional means.
The differences in growth rates arise from differences in the incentives to accumulate a set of reproducible factors that can be produced domestically with
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P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-l 049
constant social returns in terms of themselves. Exogenous growth theories, by contrast, predict that countries grow at the same trend rate and share a common trend. They further predict that the unconditional mean of the logarithmic difference between any two countries’ per capita outputs should differ from zero since countries have different technologies, preferences, market structures, and government policies. These predictions suggest that a second hypothesis be considered: Hz: A unique difirence-stationary
series a, exists such that yl, - a,, yZt - a,, . . . , yNt - a, are stationary with nonzero means and the unconditional mean of I,“= 1 (y., - a,) is zero.
The common trend a, arises because of the worldwide acquisition of useful technical knowledge, and the dynamics of each y,, - a, reflect the diffusion of technical knowledge across countries as well as the domestic accumulation of reproducible factors of production. The latter is subject to diminishing own social returns. The worldwide stock of technical knowledge may grow at an endogenously determined rate, however. I assume that Hr and H2 are mutually exclusive; that is, either endogenous growth theories or else exogenous growth theories characterize growth in all of the countries. Dispensing with this assumption is difficult and contrary to the spirit of these models, which intend the phenomena that they describe to be universal. Virtually all of the empirical growth literature has evaluated growth models by applying ordinary least squares to growth regressions of the form
gn= @z + By,, + y’x. + w,,
n=12 2 ,..., IV,
(2.1)
where gn is the average growth rate of per capita output between periods 0 and T, x, is a vector of variables that control for cross-country variation in either the level or the growth rate of y,,, c1and /I are parameters and y is a vector of parameters, and w, is an error term with a zero mean and finite variance. A significantly negative estimate of /I is interpreted to imply that ceteris paribus, a high (low) initial value of per capita output is followed by low (high) growth in per capita output and hence that per capita output reverts toward a common trend. For this reason, Hr (endogenous growth) is rejected in favor of H2 (exogenous growth). By contrast, an estimate of /I insignificantly different from zero is interpreted to imply that countries do not revert toward a common trend and, indeed, have different trend growth rates if y is estimated to be significantly different from the zero vector. In this case, Hi is not rejected in favor of HZ. The estimate of the parameter vector y is then used to infer how a country’s characteristics affect either its per capita output level relative to that of the other
P. Evans J Journal of Economic Dynamics and Control 20 (1996)
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countries under H2 or its growth rate relative to that of the other countries under H 1. Unfortunately, growth regressions can be interpreted in this fashion only under the highly implausible conditions stated in: Theorem 1. As N approaches infinity, the estimator p obtained by applying ordinary least squares to Eq. (2. I) converges in probability to zero under H 1 and to negative value under Hz if (i) x, can controlfor
all variation in either mean levels
or growth rates and (ii) ylr - a,, yz, - a,, . . , yNt - a, are,first-orde_r autoregressions. If either condition (i) or condition (ii) is violated, plim,,
m b need not be
zero under H, and need not be negative under Hz. Even ifconditions satisfied, accounting
only for heteroskedasticity
need not produce
(i) and (ii) are a consistent
estimate of the variance of fl unless (iii) the innovations to y,, - a,, yzl - a,, . y,, - a, are contemporaneously uncorrelated with each other.
,
The validity of Theorem 1 is fairly apparent merely from examining Eq. (2.1). For a proof, however, see the Appendix. If conditions (ii) and (iii) are satisfied, H, can be tested against H2 using a procedure formulated in my 1995a paper. The estimates reported there indicate that condition (i) is violated. Condition (ii) also appears to be violated. My 1995b paper shows that this condition can be rejected for a sample of 48 countries over the period 1950 -1990 at a significance level less than lOma, given that condition (iii) is satisfied. If condition (ii) is indeed violated, growth regressions cannot be readily interpreted and no simple fix is available. Finally, international trade in goods and assets probably makes condition (iii) invalid. Without this condition, valid inference is difficult if not impossible. Consider the series defined by
(2.2) with (2.3)
The following theorems provide formal econometric justification time-series properties of V, to evaluate growth theories. Theorem 2.
for using the
Suppose that either H, or else Hz holds. Then V, has the representa-
tion
d’,=i+9t+pl/,-l+
~ i=l
Cpid1/,..i+V,,
(2.4)
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where [,n,p, and the cp’s are parameters, p is a nonnegative integer, and v, is a serially uncorrelated error term with a zero mean and jnite variance 02. In Eq. (2.4), HI imposes the restrictions
and H2 imposes the restrictions p <
0,
? = 0,
[>O.
Theorem 3. Let b be the estimator of p obtained by applying ordinary least squares to Eq. (2.4) and let z(p) be its t-ratio. Under HI, T 5’2t? converges in distribution to N{O, 720 [a(1 - Cy= 1 Cpi)/q12} and z(B) converges in distribution to N(O,l) as T approaches injinity. Under HZ, both diverge as T approaches infinity. Theorem 4. Let rj be the estimator of n obtained by applying ordinary least squares to Eq. (2.4) and let z(q) be its t-ratio. Under Hz, T 3121jconverges in distribution to N(0, 12a’) and z(q) converges in distribution to N(0, 1) as T approaches infinity. Under HI, both diverge as T approaches injinity.
The proofs are sketched in the Appendix. Theorem 2 states that V, is integrated of order one around an upward quadratic trend if the hypothesis HI holds and that V, is stationary around a constant positive mean if the hypothesis H2 holds.4 Theorems 3 and 4 then state that standard methods of inference are valid asymptotically. Theorem 3 may be surprising since unit-root tests typically produce statistics that become functions of Brownian motion asymptotically. Normality and convergence at rate T 5’2 hold here because V, is eventually dominated by a deterministic quadratic trend. According to Theorems 2 and 3, HI can be treated as the null hypothesis in testing H, against H2, and HI can be rejected if r(b) exceeds an appropriately chosen critical value. By contrast, Theorems 2 and 4 imply that H2 can also be treated as the null hypothesis in testing H2 against HI and that H2 can be rejected if r(9) exceeds its critical value. The ability to test both ways adds credibility to the inferences obtained if exactly one of the hypotheses can be 4A widespread fallacy is that the failure of V, to decline monotonically to zero is evidence against what the literature infelicitously calls convergence and what this paper calls reversion toward a common trend. According to Theorem 2, reversion is supported if V, fluctuates around a positive mean and is not supported only if V, trends upward quadratically. Collapse toward zero is not possible for the data-generating processes considered here.
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rejected. In that case, one might reasonably be rejected.
accept whichever
hypothesis
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cannot
3. Empirical analysis This section applies the methods developed in the previous section to four series of cross-country variances of the logarithm of per capita output. The first is calculated for a group of thirteen industrial countries over the period 1870-1989. This is the principal series considered here because it is relatively long, it is calculated from fairly high-quality data, and it includes four countries that have long had large economies (France, the United Kingdom, the United States, and West Germany). The second is calculated for a group of 5 1 countries over the shorter period 1950-1992.5 Finally, the third and fourth are calculated for two mutually exclusive subsets of these 51 countries. See the Appendix for a description of the data and their sources. 3. I. Finite-sample
critical values of z(b) and the power of tests based on it
In finite samples, the critical values of r(b) may be very different from the fractiles of the standard normal distribution to which they converge as T approaches infinity. Fortunately, these critical values can be estimated using Monte Carlo simulations. To estimate them, I first obtained 10,000 samples of y’s generated according to’ Y”1 =
6” + Yn.,-
1 +
Unt,
n = 1,2,...,
13,
t = 1,2 ,...,
119,
(3.1)
with u,,~ w N11D(0,0.0511*),
(3.2)
y,, - NIID(O,0.3519*),
(3.3)
6” - NIID(O, 1*),
(3.4)
’ The group is Australia, Austria, Belgium, Bolivia, Brazil, Canada, Chile, Colombia, Costa Rica, Cyprus, Denmark, the Dominican Republic, Ecuador, Egypt, El Salvador, Finland, France, Guatemala, Honduras, Iceland, Ireland, India, Italy, Japan, Kenya, Luxembourg, Mauritius, Mexico, Morocco, Netherlands, New Zealand, Nigeria, Norway, Panama, Pakistan, Paraguay, Peru, Philippines, Portugal, South Africa, Spain, Sweden, Switzerland, Thailand, Turkey, the United Kingdom, the United States, Uganda, Uruguay, Venezuela, and West Germany. 6 By convention, p = 0 is said to hold for the case in which lags of dy., or AV, do not appear in the right-hand members of Eqs. (3.3) and (2.4). Performing the simulations for this case is natural because the asymptotic distribution of r(P) does not depend on the nuisance parameters (&} and because its finite-sample distribution typically depends on them only slightly.
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P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-1049
for c = 0.000,0.005,0.010,0.015,0.020,0.030, and 0.040. The figure 0.0511 is the minimal standard error obtained by fitting y,, to fixed country and time effects and up to six lags of itself, 0.3519 is the cross-country standard deviation of y,, in 1870, and c is the assumed cross-country standard deviation of trend growth rates. Next, I applied Eqs. (2.3) and (2.2) to the generated samples of data, obtaining 10,000 samples of V,, each with 120 observations.7 Finally, I fitted the regression (2.4) with p = 0 to each of these samples of V,, obtaining samples with 10,000 values of z(s) from which I tabulated the empirical distributions. The c&h fractiles of these empirical distributions are then the desired estimates of the critical values of size CL. The second column of Table 1 reports the estimated critical values of size 0.05. Clearly, they fall as i: becomes large but nonetheless are appreciably less than - 1.645, their asymptotic value. These results confirm what one would expect intuitively: the larger the cross-country variance in trend growth rates predicted by endogenous growth models, the less negative r(b) must be for any given sample size in order to provide evidence of any given strength against such models. I have also estimated the power of tests of the null hypothesis H1 against specific alternative hypotheses using Monte Carlo simulations. First, I obtained 10,000 samples of data generated according to Y,,-Y,*=I(y,,,-i
-y,*)++
n= 1,2 ,..., 13, t= 1,2 ,..., 119,
(3.5)
with Yno = Y:,
(3.6)
u,,*N NIID(0,0.05112) ,
(3.7)
y,* N NIID(O,O.3519*),
(3.8)
for Iz = 0.90,0.94, and 0.98. Using growth regressions, Barro and Sala-i-Martin and Mankiw, Romer, and Weil obtained the figure 0.98. My 1995a paper explained why this estimate is probably biased upward appreciably and obtained an unbiased estimate of 0.90 on the assumption that condition (ii) of Theorem 1 is satisfied. The intermediate value 0.94 is the average of the median unbiased estimates of Athat I obtained for 48 countries in my 1995b paper. Next, I applied Eqs. (2.3) and (2.2) to the generated samples of data, obtaining 10,000 samples of V,. I then fitted the regression (2.4) with p = 0 to each of these
’ Note that subtracting j, removes any common time-specific effects in ynt. For this reason, including such effects in the data-generating processes (3.1)-(3.4) and (3.5)-(3.8) below is unnecessary.
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Table Critical
of Economic Dynamics and Control 20 (1996) 1027-1049
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1 values and power of tests of HI vs. Hz based on t(P) with sizes of 0.05 Power
i
Critical
0.000 0.005 0.010 0.015 0.020 0.030 0.040
- 3.327 -3.176 - 2.740 -2.411 -2.243 -2.107 -2.038
values
of the tests with 1, =
0.90
0.94
0.98
0.3653 0.4603 0.7111 0.8832 0.9340 0.9604 0.9709
0.1936 0.2528 0.4920 0.6892 0.7762 0.8406 0.8693
0.0833 0.1165 0.2673 0.4370 0.5336 0.6080 0.6488
samples of V,, obtaining samples with 10,000 values of z(b). Finally, I determined the fraction of the 10,000 estimates of z(b) that fell below the critical values reported in the first column of Table 1. The third, fourth, and fifth columns of Table 1 report the resulting estimates of power for i, = 0.90,0.94, and 0.98. According to the table, power is large if 2 < 0.90 and [ 2 0.015, small if I 2 0.98 and c I 0.005, and moderate otherwise. These results are qualitatively plausible since the greater the cross-country variation in trend growth rates predicted by endogenous growth models and the more rapidly countries revert toward a common trend, the more easily should Hi be rejected in favor of H2. 3.2. Finite-sample
critical values of z( 4) and the power qf tests based on it
Using Monte Carlo simulations similar to the ones described above, I also estimated the 0.05 critical values of ~(0 under H2 and its power under Hi. Table 2 reports the results, which are interesting in at least two ways. First, the critical values lie well above 1.645, their asymptotic value. Indeed, basing inference on the asymptotic distribution of r(B) would frequently lead one to conclude that V, trends downward when it is actually stationary around a constant mean. Second, the test has moderate power even for L near one and [ near zero but does not have great power unless < is rather large. 3.3. Empirical
analysis for thirteen industrial countries
Tables 3 and 4 report the values of fi, z(B), 6, and z(Q) for lag lengths of 1, 2, and 3 years in the second and third columns and estimates of their marginal significance levels in the remaining columns. I estimated the marginal significance levels from the empirical distributions produced by the Monte Carlo simulations described in Sections 3.1 and 3.2. The t-ratio and marginal significance levels for p = 0 are not reported because p = 0 can be rejected in favor of
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Table 2 Critical values and power of tests of H2 vs. Hi based on r(rj) with sizes of 0.05 Power I
Critical
0.90 0.94 0.98
values
-2.181 - 1.709 -1.163
Table 3 r(P) and its marginal 1870-1989
significance
of the tests with [ =
0.005
0.010
0.015
0.020
0.030
0.040
0.4582 0.3705 0.1924
0.4934 0.404 1 0.2174
0.4953 0.4159 0.2446
0.5942 0.5241 0.3628
0.8212 0.7792 0.6603
0.9364 0.9182 0.8636
levels estimated
for 13 industrial
Marginal P
b
1 2 3
-0.1296 -0.1488 -0.1631
G) -3.380 - 3.709 -3.818
The marginal significance a Monte Carlo simulation
Table 4 r(rj) and its marginal 1870-1989
levels were estimated with 10,000 replications.
significance
levels estimated
from
ri
1 2 3
-0.000149 -0.ooo168 -0.000183
The marginal significance a Monte Carlo simulation
- 2.568 -2.798 -2.931 from
levels for [ =
O.C!4M
0.010
0.0440 0.0189 0.0129
0.0295 0.0142 0.0089
0.0082 0.0023 0.0021
an empirical
for 13 industrial
+l)
levels were estimated with 10,000 replications.
significance
over the period
0.000
Marginal P
countries
distribution
countries
generated
over the period
significance
levels for I =
0.90
0.94
0.98
0.9678 0.9814 0.9856
0.9689 0.9789 0.9838
0.9791 0.9865 0.9904
an empirical
distribution
by
generated
by
p = 1 at very low significance levels. (The t-ratio for $1 is 3.804 when p = 1.) Although the Akaike and Schwarz criteria select p = 1, Tables 3 and 4 also report the results for p = 2 and p = 3 in order to verify that the results for p = 1 are robust. According to Table 3, HI can be rejected in favor of Hz at the 0.05
P. Evans 1 Journal of Economic Dynamics and Control 20 (1996)
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1037
significance level, however small the assumed cross-country variance of trend growth rates is. In turn, Table 4 implies that H2 cannot be rejected in favor of H 1 at any reasonable significance level. Indeed, the test reveals a significant downward trend in V,. I conclude that no evidence exists for even very small differences in trend growth rates of the per capita outputs for the countries considered here’ and that strong evidence exists for their reversion toward a common trend.’ I can offer two plausible explanations for the significant downward trend in V,. First, it may reflect an initial value I’, greatly exceeding the unconditional mean of V’,and transitional dynamics lasting an appreciable fraction of the sample period. The availability of untapped natural resources in Australia, Canada, and the United States during the late nineteenth century may have produced the large initial value. Second, it may reflect selection bias for the countries considered here. As argued by De Long, selecting a group of countries that has become rich by the end of the sample tends to make the per capita outputs of the countries closer together at the end of the sample than at the beginning. ’ O 3.4. Empirical analysis for three other groups of countries Because of the potential selection bias, I also consider a series calculated for 51 countries over the period 1950-1992. Fig. 2, which plots the cross-country variances for this group of countries, provides little evidence of an upward quadratic trend. Furthermore, the jaggedness of the series provides some evidence that it may fluctuate around a constant mean. On net, H2 appears somewhat more strongly supported than Hi.
‘Jones (1995) reaches a similar conclusion from the fact that trend growth rates of individual countries appear not to have changed appreciably over time. Note that if growth is endogenous, a strong parallel exists between variation of trend growth rates at a given time across countries and variation for a given country over time. ’ Using a sample similar to the one used here, Bernard and Durlauf (1991) have tested whether jai, - yj, has a unit-root for all possible pairs (i,j) of countries. They find fairly strong evidence against a unit root for the pairs within a few subsets of the countries but weak evidence for all other pairs. Their conclusion probably differs from mine for two reasons. First, their null hypothesis rules out the possibility of different trend growth rates for countries and hence most endogenous growth models a priori. Second, the standard univariate tests that they employ have much lower power than the tests employed here. ” Selection should bias the test of Hz against HI much more than the test of H, against Hz since including the intercept and time trend in Eq. (2.4) should remove much of the induced decline in V,. For this reason, the rejection of HI in favor of Hz in this section provides fairly strong evidence in favor of exogenous growth theories.
1038
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of Economic Dynamics and Control 20 (1996) 1027-1049
0.80 -
0.75 -
0.70 -
0.65 -
0.60 ’
3 0 ~8 I952
Fig. 2. Cross-country
3 1961 variances
I970
1373
of log per capita
output
I366 for 51 countries.
Unfortunately, formal statistical analysis does not provide strong evidence for either hypothesis. Pretesting revealed that V, is well characterized by the process &=0.125+0.000542t-0.13561/,-r (2.48) (2.00) (2.36)
+0.3474&r, (2.30)
(3.9)
where the figures in parentheses are absolute values of c-ratios. Monte Carlo simulations showed that the estimated coefficient on t is not significantly positive at the 0.05 level and that the coefficient on V,_ 1 is not significantly negative unless 5, the cross-country standard deviation of trend growth rates, is at least 1.7 percent a year.’ 1 Consequently, Hz and HI cannot be readily rejected. Figs. 3 and 4 show that this conclusion can be strengthened somewhat by dividing the 51 countries into two mutually exclusive groups consisting of 22 OECD countries plus Cyprus and the other 28 countries. The two series of cross-country variances do not trend upward quadratically and are quite jagged, implying that H, cannot be rejected and suggesting that HI can be rejected for
I1 A description of these simulations results are available upon request.
and those mentioned
below and two tables summarizing
their
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Dynamics and Control 20 (1996) 1027-1049
1039
0.275 0.250
0.125 0.100 ~7-7
0.075 1952
Fig. 3. Cross-country
0.38
variances
,
1961
1979
1970
of log per capita
output
-7TV_1988
for 22 OECD
countries
plus Cyprus
..____
n
0.28 -
1952
Fig. 4. Cross-country
variances
1961
1970
of log per capita
1979
output
1988
for 28 nonOECD
fairly small values of [. Pretesting revealed that the cross-country the 22 OECD countries plus Cyprus is well characterized by AC = 0.057 - 0.00808t - 0.26071/,_ 1 + O.l67AV’_, , (1.42) (2.8 1) (2.58) (2.46)
countries.
variance for
(3.10)
1040
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and the cross-country variance for the 28 nonOECD countries is well characterized by AC = 0.146 - 0.00300t - 0.40521:_ 1 + 0.265Al’_ 1, (1.62) (2.99) (1.23) (3.02)
(3.11)
Monte Carlo simulations showed that the estimated coefficients on t are not significantly positive at the 0.05 level and that the coefficients on V,_ 1 are significantly negative if [ is at least 1.1 percent a year. Consequently, H2 cannot be rejected at all and Hr can be rejected more readily for each subset of countries than for the entire group.”
4. Summary and conclusions If countries have different trend growth rates as predicted by endogenous growth theories, the cross-country variance of the logarithms of their per capita outputs should be integrated of order one around an upward quadratic trend. By contrast, if they follow parallel balanced growth paths as predicted by exogenous growth theories, the cross-country variance should be stationary around a constant positive mean. The paper considered four series of crosscountry variances: one for 13 industrial countries over the period 1870-1989, a second for 51 countries over the period 1950-1992, a third for 22 OECD countries plus Cyprus over the period 1950-1992, and a fourth for 28 nonOECD countries. None appears to trend upward quadratically, and the first provides strong evidence of stationarity around a constant positive mean. Thus, no evidence is found that growth rates differ endogenously, and considerable evidence is found for parallel balanced growth paths. Therefore, either endogenous growth models that predict different trend growth rates for countries are fundamentally flawed, or else the effects that they predict must be relatively unimportant for the countries considered here.
t2 Durlauf and Johnson (1992) and Bernard and Durlauf (1995) have argued that the logarithms of per capita output for groups of countries like the first one considered in this section do not revert toward a single common trend. Rather, several mutually exclusive subsets of the countries revert toward distinct common trends, which are not cointegrated with each other and presumably drift at different rates. Cross-country variances can be used to evaluate this hypothesis. It is strongly supported if the cross-country variances for large groups of countries trend upward quadratically while the cross-country variances for appropriately selected smaller groups of countries are stationary around a constant positive mean. The results presented here provide at most a modicum of support for this hypothesis.
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These empirical results suggest that individual countries should be modeled as facing diminishing social returns to the accumulation of domestically produced reproducible factors of production. The assumption of constant returns to scale should be reserved for large collections of countries, if not the world as the whole.
Appendix The data
The principal series investigated in the paper is the cross-country variance of the logarithm of per capita output for Australia, Austria, Belgium, Canada, Denmark, Finland, France, Italy, Norway, Sweden, the United Kingdom, the United States, and West Germany over the period 1870-1989. In order to calculate that series, I first obtained data on real per capita gross domestic products by dividing the indices of real gross domestic product in Tables A.6-A.8 of Angus Maddison (1991) by the corresponding population figures in his Tables B.2-B.4 and normalizing the resulting ratios to equal the per capita gross domestic products of the countries in 1989 converted to U.S. dollars at the exchange rates prevailing then. Maddison adjusted his indices of real gross domestic product to account for frontier changes, and 1 made similar adjustments to his population figures. I then applied Eqs. (2.3) and (2.2) to the logarithms of these data. I also consider the cross-country variance of the logarithm of per capita output for a group of 51 countries and for two mutually exclusive subsets of 23 and 28 countries over the period 1950-1992. The 51 countries comprise all those for which the data are available over the entire sample period. I applied Eqs. (2.3) and (2.2) to logarithms of per capita gross domestic product in constant international prices (RGDPCH) from the Penn World Tables 5.6, which are described in Robert Summers and Alan Heston (1991). Proof of Theorem 1
Without essential loss of generality, I consider only the case in which the vector x, is null. In that case, if conditions (i), (ii), and (iii) are satisfied, the data-generating process for y,, - a, can be written as Y”, -
at =
&(Yn.*-1
-4-J
+ %t,
(A.11
where I, is a parameter and u,* is a serially and mutually uncorrelated error term with a zero mean and finite variance. Solving Eq. (A.l) backward from period
1042
P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-1049
T to period 1 yields
gn-(UT -uo)/T = Bn(Yno -UO) + f
‘$’ X%T-i, 1-O
64.2)
where B,, = (1,’ - 1)/T. The estimator obtained by applying ordinary least squares to Eq. (2.1) is asymptotically equivalent to the estimator obtained by regressing gn - (ar - ao)/T on ynO- ao. I therefore consider the estimator
ij = f
(Y,O - ao)Cs,
-
(UT- ao)lTl
PI=1
i (Y~O- ~0)‘. i
Substituting Eq. (A.2) into Eq. (A.3) and taking probability members of the resulting equation gives
plim pI = N+CC
ctt [
k f
(Y,o
-
a012P.
n-1
(A-3)
n=l
I/[pi;$
limits of both
(Y,o-uo)2
n-1
1
=b,
(A.4) since y,, - a, is uncorrelated with future realizations of its own innovations. UnderH,,L, = A2 = ... =&= landhence/?, =/?2= ... =&=O.Eq.(A.4) therefore implies that plimN+, p = 0. I assume that under HZ, (Y,,~- uo)2 and /ln are jointly distributed across countries with finite first and second moments and with only positive support for (yno - a# and only negative support for bn. Therefore, the quantity b does indeed exist and is negative under Hz. Moreover, the error term is uncorrelated across countries because u,~is contemporaneously uncorrelated across countries by supposition. Accounting for heteroskedasticity then suffices to produce a consistent standard error. A general form for (A.l) is
(A.9
Ynt-ut=r”+n,cY..,-,-u,-l)+Unt.
If(i) is violated, the t’s differ from zero; if (ii) is violated, U,, is serially correlated; and if (iii) is violated, U,, is contemporaneously correlated across countries.
Solving Eq. (A.5) backward results in
gn- (UT- UO)/T = Bn(Yno - UO)+ f
72’ %(rn+ un,T-i). 1-O
64.6)
P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
1043
Clearly, y,, - a0 is cross-sectionally correlated with 5, if(i) is violated and with at least V,, if (ii) is violated, and w, is correlated across countries if (iii) is violated. Hence, fi need not converge in probability to b as N approaches infinity. In particular, the probability limit need not be zero under H1 and need not be negative under HZ. Even if conditions (i) and (ii) are satisfied, accounting for heteroskedasticity does not suffice to produce a consistent standard error unless (iii) is also satisfied. Proof of Theorem 2
H1 and H2 are nested by the data-generating
Ynt -
4
=
Pn
+
6nt
+
f
$ni%,t-i7
n=
process
1,2 ,..., N,
(A.7)
i=O
where the p’s, 6’s, and 1,4’sare parameters and [ulr, u2,, . . . , u,,] is a serially uncorrelated error vector that has a zero mean vector and a positive definite covariance matrix and satisfies
for n1,n2,n3,n4 = 1,2, . . . . N and tl, tz, t3, t4 = 0,1,. . . , T. Without loss of generality, a, can be defined so that the sample averages of pL,,a,, p,, C,‘To I//,iu”,t_i, and 6, cl: o tjmiu,,f _ i are zero. Under H 2, the 6’s are zero, the p’s vary across countries, the $‘s are square-summable, and all roots of I,:0 @niLilie outside the unit circle. Under H1, the 6’s are nonzero and vary across countries, and all roots of (1 - L) C,c, Il/,iL’ lie outside the unit circle.13 Hence, lim $“/ni= 6,
(A.9)
i+cc
where I,$ is a finite nonzero parameter. Because the sample averages of the p’s and 6’s are normalized to zero, subtracting the average of each member of Eq. (A.6) from itself results in
(A. 10) j=l
i=O
I3 This formulation has the problem that the data-generating process (A.7) cannot have held forever since the realizations of every y., between periods 0 and T would then be infinite. I assume that if H1 holds, the process began at some finite date in the past and the u’s before that date were zero.
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P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
where 0nji is (N - l)@ni/N ifj = n and -~ji/N ifj # n. Note that under Hz the B’s are square-summable and under H, they satisfy lim
9nji
=
(A.1 1)
e,j,
i-rcc
where 8”j is (N - 1)&/N is j = n and -&,,lN ifj # n. Furthermore, the sample averages pL,IF= 1 C,p”_,e,jiu,,-i and 6, I:= 1 C,zo fl”jiU,,_i are zero. Squaring both members of Eq. (A.lO) and averaging across countries then results in
+ f
“i’ f “f’(k$
i=(J j=l
i,“,cy,1
I=1
snjienlk)
Uj,t-iUl,,-k.
(A.12)
n-l
Suppose that hypothesis Hz holds so that 6, = & = ... = 6, = 0 and the B’s are square-summable. In that case, (A.12) and (A.@ imply that the mean and autocovariances of V, exist. Hence, it is stationary with a constant positive mean. Suppose instead that H, holds so that 6r # & # ... # a,?,,# 0 and the O’sare not square-summable. Eq. (A.12) then implies that the mean of V, approaches infinity as t approaches infinity because the second and third terms in the first line and the expectation of the fourth term do so. Imposing Hr in Eq. (A.12), differencing, and taking expectations produces
(A.13) where w.’ is the variance of u,,~.Without essential loss of generality, I assume here that Eu,,,~,+, = 0 if n, # n,. l4 Eq. (A.1 1) implies that C,p”_ 1 (O:i - Ot,j,i- 1)
I4 Eq. (A.lO) can always be reparameterized terms with zero means and finite variances.
in terms of N mutually
and serially uncorrelated
error
P. Evans / Journal of Economic Dynamics and Control 20 (1996)
1027-1049
1045
exists. Eq. (A.13) therefore implies that the mean of AV, - 2((1/N)Cr=, 6.‘)~ exists under Hr. Straightforward, but tedious, calculation reveals that its autocovariances also exist. Hence, V, is integrated of order one around an upward deterministic quadratic trend. Suppose that V, can be adequately approximated as a (p + l)th-order autoregression. The discussion above then implies that Eq. (2.4) holds. Under HI, p = 0 and 9 > 0 hold; under Hz, p < 0, q = 0, and i > 0 hold. Proqf
qf Theorem 3
Without essential loss of generality, I assume that p = 0 in Eq. (2.4).” Let 9’ G [Q, 4, (1 and 8 be the ordinary least squares estimator of 9. It then follows that
(A.14)
Premultiplying produces
both
members
of Eq. (A.14) by B = diag [T 5’2, T3’*, T ‘I*]
E(B - 9)
c v,-
lV,
c tv, _ Cvt T-51
v:_,
T-4Ctt&
T-31
V,_,
T-3x
t*
T-*Et
T-*x
t
T-lx
1
-l
I[
1
T-5/*1
K-IV,
T-3’2x
tv,
T-‘/*x
v,
. 1 (A.15)
I5 If p > 0, the analogue
to q below is q/(1 - Cf’=,
cp,)
1046
P. Evans / Journal of Economic Dynamics and Control 20 (1996)
1027-1049
Under H,, Eq. (2.4) can be solved backward from period t to period 1 to obtain V, = VI)+ ([ + q/2) t + (q/2) t2 + i
(A.16)
Vi.
i=l
Eq. (A.16) implies that V, is eventually dominated 9 > 0. For this reason,
-“I
T-'x I’_,
tK_1
T-4x
tVml
T-3x
t2
T-‘xt
T-3x
v,_,
T-*x
t
T-lx
1
s
1
1
0
(q,2)j-o1 s3ds
s,‘s’ds
1 s 1
s3ds (v/2)
s4ds (v/2)
by the term (u/2) t* since
s*ds
-
0
IO1sds 1
(q/2)
’ s* ds
s0 -~*I20 ~18
_ rlf6 and
ds
’ sds
s0
v/8
~16
l/3
l/2
l/2
1
1
s0
(A.17)
= Q
(rll+W- 2 Jo1 sW(d}ds
s 1
W(l) -
W(s) ds
0
W(l)
(A.18) I J
P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
1041
as T approaches infinity. In Eqs. (A.17) and (A.18), s is a variable of integration and W(s) is the Wiener process. The term in the last line of Eq. (A.18) is distributed as N(0, .‘Q).“j Hence,
Z(8- 9)
sN[O,Q-'(a2Q)Q-'1
=N(O,a'Q-')
(A.19)
as T approaches infinity. Under H Ir p = 0, and the (1,l)th entry of Q ’ is ~~120
~18
~116
~18
l/3
l/2
~116
l/2
1
= 7201~~.
(A.20)
Therefore. T “‘d 5 N [0,720(0/~#]
(A.21)
.
I6 W(s) is normally distributed with a mean of zero and variance of s for all s on [0,11.Moreover. EW(s)W(r) = min(r,s) for all s and r on [O,l]. The covariance matrix is a*Q because
+‘(I,-ZjO’sW(s)ds]
=1
-4j~s’di+4j~‘[j~sr’dr+j~‘s’rdr]ds=:.
E[W(I)-j;
W(r)ds]
=,-2
jO’sds+
EW(l)*
=
+‘(I)
=,
jS’sdr]ds=:.
I,
- 2 jo’ sW(s)ds][W(l/
- j;sdr--2
E[W(I)-2
E[H’(l)-
jO’[j;rdr+
j0’s’ds+2
j+V(s)ds]W(,)=l-2
j~‘W(r)ds]W(I1=l-j~‘sds=:
- jO’ Wids]
jO’[j;srdr+
jzls’dr]ds=l,
j0’s2ds=_:,
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P. Evans /Journal
of Economic
Dynamics and Control 20 (1996) 1027-1049
The t-ratio t(b) is defined to be j? divided by the square root of (A.22) times the (l,l)th
entry of
As T approaches infinity, the former converges in probability to g2 because 3 is consistent and T 5 times the latter converges in probability to the (l,l)th entry of Q-r. Consequently, (A.21) implies that z( 6) is N(0,1). Finally, T “‘6 and r( 6) must diverge under H2 because 3 is also consistent in that case. Proof of Theorem 4
See Chapter 16 of James Hamilton (1993).
References Alesina, Albert0 and Dani Rodrik, 1994, Distributive politics and economic growth, Quarterly Journal of Economics 109,465-490. Barro, Robert J., 1991, Economic growth in a cross section of countries, Quarterly Journal of Economics 106,407-443. Barro, Robert J. and Xavier Sala-i-Martin, 1992, Convergence, Journal of Political Economy 100, 223-251. Baumol, William J., 1986, Productivity growth, convergence, and welfare: What the long-run data show, American Economic Review 76, 1072-1085. Baumol, William J., Sue Anne Batey Blackman, and Edward N. Wolff, 1989, Productivity and American leadership: The long view (MIT Press, Cambridge, MA). Bernard, Andrew B. and Steven N. Durlauf, 1991, Convergence of international output movements, NBER working paper no. 3996, May. Bernard, Andrew B. and Steven N. Durlauf, 1995, Convergence in international output, Journal of Applied Econometrics 10, 977108. Ben-David, Dan, 1993, Equalizing exchange: Trade liberalization and income convergence, Quarterly Journal of Economics 108,653-680. Cass, David M., 1965, Optimal growth in an aggregate model of capital accumulation, Review of Economic Studies 32, 2333240. De Long, Bradford, 1965, Productivity growth, convergence, and welfare: Comment, American Economic Review 78,233-240. De Long, Bradford and Lawrence H. Summers, 1991, Equipment investment and economic growth, Quarterly Journal of Economics 106, 4077444.
P. Evans 1 Journal of Economic Dynamics and Control 20 (1996) 1027-1049
De Gregorio, Jose, 1993, Inflation, 31, 271-298.
taxation,
and long-run
growth,
Journal
of Monetary
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Economics
Dollar, David, 1991, Outward-oriented developing economies really do grow more rapidly: Evidence from 95 LDCs, 197661985, Economic Development and Cultural Change 40, 523-544. Durlauf, Steven N. and Paul A. Johnson, 1992, Local versus global convergence across national economies, NBER working paper no. 3996, February. Easterly, William, 1993, How much do distortions affect growth?, Journal of Monetary Economics 32, 1877212. Easterly, William, 1994, Economic stagnation, fixed factors, and policy thresholds. Journal of Monetary Economics 33, 5255558. Easterly, William and Sergio Rebelo, 1993, Fiscal policy and growth: An empirical investigation, Journal of Monetary Economics 32,417 458. Easterly, William, Michael Kremer, Lant Pritchett, and Lawrence H. Summers, 1993, Good policy or good luck? Country growth performance and temporary shocks. Journal of Monetary Economics 32,459 483. Evans, Paul, 1995a, How to estimate growth regressions consistently, Unpublished manuscript. January. Evans, Paul, 1995b, How fast do economies converge?, Review of Economics and Statistics, forthcoming. Friedman, Milton, 1992, Do old fallacies ever die?, Journal of Economic Literature 30, 2129 2032. Hamilton, James D., 1994, Time series analysis (Princeton University Press, Princeton, NJ). Jones, Charles I., 1995, Time series tests of endogenous growth models, Quarterly Journal of Economics 1 IO, 495 526. Keil, Manfred W. and R.A. Vohra, 1993, What’s Wyoming got that we don’t’?, Unpublished manuscript (Northeastern University, Evanston, IL). King, Robert G. and Ross Levine, 1993, Finance and growth: Schumpeter might be right, Quarterly Journal of Economics 108, 717-738. Kormendi, Roger and Philip Meguire, 1985, Macroeconomic determinants of growth: Crosscountry evidence, Journal of Monetary Economics 16, 141~~163. Levine, Ross and David Renelt, 1992, A sensitivity analysis of cross-country growth regressions, American Economic Review 82,942-963. Maddison, Angus, 1991, Dynamic forces of capitalist development (Oxford University Press, New York, NY). Mankiw, N. Gregory, David Romer, and David N. Weil, 1992, A contribution to the empirics of economic growth, Quarterly Journal of Economics 107, 4077438. Parente, Stephen L. and Edward C. Prescott, 1993, Changes in the wealth of nations, Federal Reserve Bank of Minneapolis Quarterly Review 17, 3 16. Persson, Torsten and Guido Tabellini, 1994, Is inequality harmful?, American Economic Review 84, 60@ 621. Quah, Danny, 1993a, Empirical cross-section dynamics in economic growth, European Economic Review 37, 426- 434. Quah. Danny, 1993b, Gabon’s fallacy and tests of the convergence hypothesis, Scandinavian Journal of Economics 95,427-443. Quah, Danny, 1994, Empirics of economic growth and convergence. Unpublished manuscript (London School of Economics, London). Summers, Robert and Alan Heston, 1991, The Penn world tables (Mark 5): An expanded set of international comparisons, 1950-1988, Quarterly Journal of Economics 106, 3277368.
Journal of Economic Dynamics and Control 20 (1996) 102771049
Using cross-country variances to evaluate growth theories Paul Evans Department
ofEconomics,Ohio
State University,
Columbus, OH 43210-1172,
USA
(Received January 1995; final version received August 1995)
Abstract Much of the empirical growth literature has attempted to evaluate growth theories by estimating regressions that relate the growth rate of per capita output for a sample of countries to initial per capita output and country characteristics. The resulting inferences are shown to be invalid except under strong conditions. An alternative method that uses cross-country variances is formulated and shown to produce valid inferences under weak conditions. Applying this method to data from thirteen countries over the period 1870-1989 provides no evidence that their per capita outputs have different trend growth rates and much evidence that they revert toward a common trend. This evidence does not support those endogenous growth theories that predict appreciably different trend growth rates across countries. words: Growth theory; Endogenous/exogenous deterministic trends; Monte Carlo simulation JEL classification: C22; C23; C15; 040 Key
growth;
Unit
roots;
Common/
1. Introduction
In recent years, a large empirical literature has attempted to evaluate growth theories. Many of the contributors to this literature have fitted regressions relating the average growth rate of per capita output over some time period for a sample of countries to initial per capita output and country characteristics. They have then applied standard methods of inference to the estimated
I am grateful for helpful comments from In Choi, James Hamilton, Georgios Karras, Manfred Keil, and two anonymous
referees.
0165-1889/96/%15.00 0 1996 Elsevier Science B.V. All rights SSDI 016S-1889(95)00888-3
reserved
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P. Evans 1 Journal of Economic Dynamics and Conirol20
(1996) 1027-1049
coefficients.’ Milton Friedman (1992) has argued that inferences based on such growth regressions are invalid and has suggested an alternative approach for evaluating growth theories. This approach exploits the time-series properties of the cross-country variance of the logarithm of per capita output.’ In this paper, I show that the inferences from growth regressions are indeed invalid except under highly restrictive conditions and that Friedman’s alternative approach is valid under weak conditions. I also develop a formal econometric apparatus for analyzing the time-series properties of cross-country variances. Employing this apparatus, I find no evidence of different trend growth rates for Australia, Austria, Belgium, Canada, Denmark, Finland, France, Italy, Norway, Sweden, the United Kingdom, the United States, and West Germany over the period 1870-1989 and strong evidence of reversion toward a common trend.3 Furthermore, the trend growth rates of 51 countries do not appear to differ over the period 1950-1992. Therefore, either endogenous growth models that predict different trend growth rates for countries are fundamentally flawed, or else the effects that they predict are unimportant for these countries. The basic idea of the paper is straightforward. If countries have different trend growth rates as predicted by endogenous growth theories, the logarithms of their per capita outputs should wonder away from each other at positive rates and hence their cross-country variance should be integrated of order one around an upward quadratic trend. By contrast, if the countries follow parallel balanced growth paths as predicted by exogenous growth theories, the logarithms of their per capita outputs should not wonder away from each other and hence their cross-country variance should be stationary around a constant positive mean. To illustrate this idea, consider Fig. 1, which plots the cross-country variance of the logarithm of per capita output for the thirteen industrial countries mentioned above. The figure indicates that identical growth rates cannot be rejected since their cross-country variance did not trend upward quadratically. Moreover, the rapid elimination of the effects of World Wars I and II provides
’ A far from exhaustive list of such contributors is Kormendi and Meguire (1985), Baumol(1986) De Long (1988), Barro (1991) De Long and Summers (1991), Dollar (1991), Barro and Sala-i-Martin (1992), Mankiw, Romer, and Weil (1992), Levine and Renelt (1992), King and Levine (1993), De Gregorio (1993), Easterly (1993, 1994), Easterly and Rebel0 (1993), Easterly, Kremer, Pritchett, and Summers (1993), Alesina and Rodrik (1994), and Persson and Tabellini (1994). * Quah (1993b) has also evaluated by examining time. In his 1993a and output in different years evolved over time.
criticized growth regressions and has argued that growth theories should be the evolution of the cross-country distribution of per capita output over 1994 papers, he discretized the cross-sectional distributions of per capita and fitted Markov models in order to characterize how these distributions
3 Baumol, Blackman, and Woolf (1989) Ben-David (1993), Parente and Prescott (1993), and Keil and Vohtra (1993) have also employed cross-country variances to evaluate growth theories. Their analyses are much more informal and qualitative than that of this paper.
P. Evans J Journal of Economic Dynamics and Control 20 (1996) 1027-1049
1029
0.05 -
0.00 m,,,, I875 Fig. 1. Cross-country
I999 variances
I923
of log per capita
I947 output
1971 for 13 industrial
countries
some evidence for parallel balanced growth paths. These visual impressions are confirmed later in the paper. The rest of the paper is organized as follows. Section 2 discusses the conditions under which inferences from growth regressions are valid and develops a formal econometric apparatus for using cross-country variances to evaluate growth theories. Section 3 applies the apparatus. Section 4 summarizes the paper and draws some conclusions. 2. Econometric discussion Let y,, denote the logarithm of per capita output for country n during period t, which has been observed for countries 1,2, . . . , N and periods 0, 1, . . . , T. Suppose that each y,, is difference-stationary and invertible and satisfies standard regularity conditions. Endogenous growth theories predict that countries have different trend growth rates since they differ in technologies, preferences, market structures, and government policies inter alia. This prediction suggests that the following hypothesis be considered: HI: The stochastic processes yl,,yzt, . . . , yNt are difkrence-stationary
with no cointegration among themselves, and Ayl,, Ay2,, . . , AyN, have different unconditional means.
The differences in growth rates arise from differences in the incentives to accumulate a set of reproducible factors that can be produced domestically with
1030
P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-l 049
constant social returns in terms of themselves. Exogenous growth theories, by contrast, predict that countries grow at the same trend rate and share a common trend. They further predict that the unconditional mean of the logarithmic difference between any two countries’ per capita outputs should differ from zero since countries have different technologies, preferences, market structures, and government policies. These predictions suggest that a second hypothesis be considered: Hz: A unique difirence-stationary
series a, exists such that yl, - a,, yZt - a,, . . . , yNt - a, are stationary with nonzero means and the unconditional mean of I,“= 1 (y., - a,) is zero.
The common trend a, arises because of the worldwide acquisition of useful technical knowledge, and the dynamics of each y,, - a, reflect the diffusion of technical knowledge across countries as well as the domestic accumulation of reproducible factors of production. The latter is subject to diminishing own social returns. The worldwide stock of technical knowledge may grow at an endogenously determined rate, however. I assume that Hr and H2 are mutually exclusive; that is, either endogenous growth theories or else exogenous growth theories characterize growth in all of the countries. Dispensing with this assumption is difficult and contrary to the spirit of these models, which intend the phenomena that they describe to be universal. Virtually all of the empirical growth literature has evaluated growth models by applying ordinary least squares to growth regressions of the form
gn= @z + By,, + y’x. + w,,
n=12 2 ,..., IV,
(2.1)
where gn is the average growth rate of per capita output between periods 0 and T, x, is a vector of variables that control for cross-country variation in either the level or the growth rate of y,,, c1and /I are parameters and y is a vector of parameters, and w, is an error term with a zero mean and finite variance. A significantly negative estimate of /I is interpreted to imply that ceteris paribus, a high (low) initial value of per capita output is followed by low (high) growth in per capita output and hence that per capita output reverts toward a common trend. For this reason, Hr (endogenous growth) is rejected in favor of H2 (exogenous growth). By contrast, an estimate of /I insignificantly different from zero is interpreted to imply that countries do not revert toward a common trend and, indeed, have different trend growth rates if y is estimated to be significantly different from the zero vector. In this case, Hi is not rejected in favor of HZ. The estimate of the parameter vector y is then used to infer how a country’s characteristics affect either its per capita output level relative to that of the other
P. Evans J Journal of Economic Dynamics and Control 20 (1996)
1027-1049
1031
countries under H2 or its growth rate relative to that of the other countries under H 1. Unfortunately, growth regressions can be interpreted in this fashion only under the highly implausible conditions stated in: Theorem 1. As N approaches infinity, the estimator p obtained by applying ordinary least squares to Eq. (2. I) converges in probability to zero under H 1 and to negative value under Hz if (i) x, can controlfor
all variation in either mean levels
or growth rates and (ii) ylr - a,, yz, - a,, . . , yNt - a, are,first-orde_r autoregressions. If either condition (i) or condition (ii) is violated, plim,,
m b need not be
zero under H, and need not be negative under Hz. Even ifconditions satisfied, accounting
only for heteroskedasticity
need not produce
(i) and (ii) are a consistent
estimate of the variance of fl unless (iii) the innovations to y,, - a,, yzl - a,, . y,, - a, are contemporaneously uncorrelated with each other.
,
The validity of Theorem 1 is fairly apparent merely from examining Eq. (2.1). For a proof, however, see the Appendix. If conditions (ii) and (iii) are satisfied, H, can be tested against H2 using a procedure formulated in my 1995a paper. The estimates reported there indicate that condition (i) is violated. Condition (ii) also appears to be violated. My 1995b paper shows that this condition can be rejected for a sample of 48 countries over the period 1950 -1990 at a significance level less than lOma, given that condition (iii) is satisfied. If condition (ii) is indeed violated, growth regressions cannot be readily interpreted and no simple fix is available. Finally, international trade in goods and assets probably makes condition (iii) invalid. Without this condition, valid inference is difficult if not impossible. Consider the series defined by
(2.2) with (2.3)
The following theorems provide formal econometric justification time-series properties of V, to evaluate growth theories. Theorem 2.
for using the
Suppose that either H, or else Hz holds. Then V, has the representa-
tion
d’,=i+9t+pl/,-l+
~ i=l
Cpid1/,..i+V,,
(2.4)
P. Evans /Journal
1032
of Economic Dynamics and Control 20 (1996) 1027-1049
where [,n,p, and the cp’s are parameters, p is a nonnegative integer, and v, is a serially uncorrelated error term with a zero mean and jnite variance 02. In Eq. (2.4), HI imposes the restrictions
and H2 imposes the restrictions p <
0,
? = 0,
[>O.
Theorem 3. Let b be the estimator of p obtained by applying ordinary least squares to Eq. (2.4) and let z(p) be its t-ratio. Under HI, T 5’2t? converges in distribution to N{O, 720 [a(1 - Cy= 1 Cpi)/q12} and z(B) converges in distribution to N(O,l) as T approaches injinity. Under HZ, both diverge as T approaches infinity. Theorem 4. Let rj be the estimator of n obtained by applying ordinary least squares to Eq. (2.4) and let z(q) be its t-ratio. Under Hz, T 3121jconverges in distribution to N(0, 12a’) and z(q) converges in distribution to N(0, 1) as T approaches infinity. Under HI, both diverge as T approaches injinity.
The proofs are sketched in the Appendix. Theorem 2 states that V, is integrated of order one around an upward quadratic trend if the hypothesis HI holds and that V, is stationary around a constant positive mean if the hypothesis H2 holds.4 Theorems 3 and 4 then state that standard methods of inference are valid asymptotically. Theorem 3 may be surprising since unit-root tests typically produce statistics that become functions of Brownian motion asymptotically. Normality and convergence at rate T 5’2 hold here because V, is eventually dominated by a deterministic quadratic trend. According to Theorems 2 and 3, HI can be treated as the null hypothesis in testing H, against H2, and HI can be rejected if r(b) exceeds an appropriately chosen critical value. By contrast, Theorems 2 and 4 imply that H2 can also be treated as the null hypothesis in testing H2 against HI and that H2 can be rejected if r(9) exceeds its critical value. The ability to test both ways adds credibility to the inferences obtained if exactly one of the hypotheses can be 4A widespread fallacy is that the failure of V, to decline monotonically to zero is evidence against what the literature infelicitously calls convergence and what this paper calls reversion toward a common trend. According to Theorem 2, reversion is supported if V, fluctuates around a positive mean and is not supported only if V, trends upward quadratically. Collapse toward zero is not possible for the data-generating processes considered here.
P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
rejected. In that case, one might reasonably be rejected.
accept whichever
hypothesis
1033
cannot
3. Empirical analysis This section applies the methods developed in the previous section to four series of cross-country variances of the logarithm of per capita output. The first is calculated for a group of thirteen industrial countries over the period 1870-1989. This is the principal series considered here because it is relatively long, it is calculated from fairly high-quality data, and it includes four countries that have long had large economies (France, the United Kingdom, the United States, and West Germany). The second is calculated for a group of 5 1 countries over the shorter period 1950-1992.5 Finally, the third and fourth are calculated for two mutually exclusive subsets of these 51 countries. See the Appendix for a description of the data and their sources. 3. I. Finite-sample
critical values of z(b) and the power of tests based on it
In finite samples, the critical values of r(b) may be very different from the fractiles of the standard normal distribution to which they converge as T approaches infinity. Fortunately, these critical values can be estimated using Monte Carlo simulations. To estimate them, I first obtained 10,000 samples of y’s generated according to’ Y”1 =
6” + Yn.,-
1 +
Unt,
n = 1,2,...,
13,
t = 1,2 ,...,
119,
(3.1)
with u,,~ w N11D(0,0.0511*),
(3.2)
y,, - NIID(O,0.3519*),
(3.3)
6” - NIID(O, 1*),
(3.4)
’ The group is Australia, Austria, Belgium, Bolivia, Brazil, Canada, Chile, Colombia, Costa Rica, Cyprus, Denmark, the Dominican Republic, Ecuador, Egypt, El Salvador, Finland, France, Guatemala, Honduras, Iceland, Ireland, India, Italy, Japan, Kenya, Luxembourg, Mauritius, Mexico, Morocco, Netherlands, New Zealand, Nigeria, Norway, Panama, Pakistan, Paraguay, Peru, Philippines, Portugal, South Africa, Spain, Sweden, Switzerland, Thailand, Turkey, the United Kingdom, the United States, Uganda, Uruguay, Venezuela, and West Germany. 6 By convention, p = 0 is said to hold for the case in which lags of dy., or AV, do not appear in the right-hand members of Eqs. (3.3) and (2.4). Performing the simulations for this case is natural because the asymptotic distribution of r(P) does not depend on the nuisance parameters (&} and because its finite-sample distribution typically depends on them only slightly.
1034
P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-1049
for c = 0.000,0.005,0.010,0.015,0.020,0.030, and 0.040. The figure 0.0511 is the minimal standard error obtained by fitting y,, to fixed country and time effects and up to six lags of itself, 0.3519 is the cross-country standard deviation of y,, in 1870, and c is the assumed cross-country standard deviation of trend growth rates. Next, I applied Eqs. (2.3) and (2.2) to the generated samples of data, obtaining 10,000 samples of V,, each with 120 observations.7 Finally, I fitted the regression (2.4) with p = 0 to each of these samples of V,, obtaining samples with 10,000 values of z(s) from which I tabulated the empirical distributions. The c&h fractiles of these empirical distributions are then the desired estimates of the critical values of size CL. The second column of Table 1 reports the estimated critical values of size 0.05. Clearly, they fall as i: becomes large but nonetheless are appreciably less than - 1.645, their asymptotic value. These results confirm what one would expect intuitively: the larger the cross-country variance in trend growth rates predicted by endogenous growth models, the less negative r(b) must be for any given sample size in order to provide evidence of any given strength against such models. I have also estimated the power of tests of the null hypothesis H1 against specific alternative hypotheses using Monte Carlo simulations. First, I obtained 10,000 samples of data generated according to Y,,-Y,*=I(y,,,-i
-y,*)++
n= 1,2 ,..., 13, t= 1,2 ,..., 119,
(3.5)
with Yno = Y:,
(3.6)
u,,*N NIID(0,0.05112) ,
(3.7)
y,* N NIID(O,O.3519*),
(3.8)
for Iz = 0.90,0.94, and 0.98. Using growth regressions, Barro and Sala-i-Martin and Mankiw, Romer, and Weil obtained the figure 0.98. My 1995a paper explained why this estimate is probably biased upward appreciably and obtained an unbiased estimate of 0.90 on the assumption that condition (ii) of Theorem 1 is satisfied. The intermediate value 0.94 is the average of the median unbiased estimates of Athat I obtained for 48 countries in my 1995b paper. Next, I applied Eqs. (2.3) and (2.2) to the generated samples of data, obtaining 10,000 samples of V,. I then fitted the regression (2.4) with p = 0 to each of these
’ Note that subtracting j, removes any common time-specific effects in ynt. For this reason, including such effects in the data-generating processes (3.1)-(3.4) and (3.5)-(3.8) below is unnecessary.
P. Evans /Journal
Table Critical
of Economic Dynamics and Control 20 (1996) 1027-1049
1035
1 values and power of tests of HI vs. Hz based on t(P) with sizes of 0.05 Power
i
Critical
0.000 0.005 0.010 0.015 0.020 0.030 0.040
- 3.327 -3.176 - 2.740 -2.411 -2.243 -2.107 -2.038
values
of the tests with 1, =
0.90
0.94
0.98
0.3653 0.4603 0.7111 0.8832 0.9340 0.9604 0.9709
0.1936 0.2528 0.4920 0.6892 0.7762 0.8406 0.8693
0.0833 0.1165 0.2673 0.4370 0.5336 0.6080 0.6488
samples of V,, obtaining samples with 10,000 values of z(b). Finally, I determined the fraction of the 10,000 estimates of z(b) that fell below the critical values reported in the first column of Table 1. The third, fourth, and fifth columns of Table 1 report the resulting estimates of power for i, = 0.90,0.94, and 0.98. According to the table, power is large if 2 < 0.90 and [ 2 0.015, small if I 2 0.98 and c I 0.005, and moderate otherwise. These results are qualitatively plausible since the greater the cross-country variation in trend growth rates predicted by endogenous growth models and the more rapidly countries revert toward a common trend, the more easily should Hi be rejected in favor of H2. 3.2. Finite-sample
critical values of z( 4) and the power qf tests based on it
Using Monte Carlo simulations similar to the ones described above, I also estimated the 0.05 critical values of ~(0 under H2 and its power under Hi. Table 2 reports the results, which are interesting in at least two ways. First, the critical values lie well above 1.645, their asymptotic value. Indeed, basing inference on the asymptotic distribution of r(B) would frequently lead one to conclude that V, trends downward when it is actually stationary around a constant mean. Second, the test has moderate power even for L near one and [ near zero but does not have great power unless < is rather large. 3.3. Empirical
analysis for thirteen industrial countries
Tables 3 and 4 report the values of fi, z(B), 6, and z(Q) for lag lengths of 1, 2, and 3 years in the second and third columns and estimates of their marginal significance levels in the remaining columns. I estimated the marginal significance levels from the empirical distributions produced by the Monte Carlo simulations described in Sections 3.1 and 3.2. The t-ratio and marginal significance levels for p = 0 are not reported because p = 0 can be rejected in favor of
P. Evans / Journal of Economic L@amics
1036
and Control 20 (I 996) 1027-I 049
Table 2 Critical values and power of tests of H2 vs. Hi based on r(rj) with sizes of 0.05 Power I
Critical
0.90 0.94 0.98
values
-2.181 - 1.709 -1.163
Table 3 r(P) and its marginal 1870-1989
significance
of the tests with [ =
0.005
0.010
0.015
0.020
0.030
0.040
0.4582 0.3705 0.1924
0.4934 0.404 1 0.2174
0.4953 0.4159 0.2446
0.5942 0.5241 0.3628
0.8212 0.7792 0.6603
0.9364 0.9182 0.8636
levels estimated
for 13 industrial
Marginal P
b
1 2 3
-0.1296 -0.1488 -0.1631
G) -3.380 - 3.709 -3.818
The marginal significance a Monte Carlo simulation
Table 4 r(rj) and its marginal 1870-1989
levels were estimated with 10,000 replications.
significance
levels estimated
from
ri
1 2 3
-0.000149 -0.ooo168 -0.000183
The marginal significance a Monte Carlo simulation
- 2.568 -2.798 -2.931 from
levels for [ =
O.C!4M
0.010
0.0440 0.0189 0.0129
0.0295 0.0142 0.0089
0.0082 0.0023 0.0021
an empirical
for 13 industrial
+l)
levels were estimated with 10,000 replications.
significance
over the period
0.000
Marginal P
countries
distribution
countries
generated
over the period
significance
levels for I =
0.90
0.94
0.98
0.9678 0.9814 0.9856
0.9689 0.9789 0.9838
0.9791 0.9865 0.9904
an empirical
distribution
by
generated
by
p = 1 at very low significance levels. (The t-ratio for $1 is 3.804 when p = 1.) Although the Akaike and Schwarz criteria select p = 1, Tables 3 and 4 also report the results for p = 2 and p = 3 in order to verify that the results for p = 1 are robust. According to Table 3, HI can be rejected in favor of Hz at the 0.05
P. Evans 1 Journal of Economic Dynamics and Control 20 (1996)
1027-1049
1037
significance level, however small the assumed cross-country variance of trend growth rates is. In turn, Table 4 implies that H2 cannot be rejected in favor of H 1 at any reasonable significance level. Indeed, the test reveals a significant downward trend in V,. I conclude that no evidence exists for even very small differences in trend growth rates of the per capita outputs for the countries considered here’ and that strong evidence exists for their reversion toward a common trend.’ I can offer two plausible explanations for the significant downward trend in V,. First, it may reflect an initial value I’, greatly exceeding the unconditional mean of V’,and transitional dynamics lasting an appreciable fraction of the sample period. The availability of untapped natural resources in Australia, Canada, and the United States during the late nineteenth century may have produced the large initial value. Second, it may reflect selection bias for the countries considered here. As argued by De Long, selecting a group of countries that has become rich by the end of the sample tends to make the per capita outputs of the countries closer together at the end of the sample than at the beginning. ’ O 3.4. Empirical analysis for three other groups of countries Because of the potential selection bias, I also consider a series calculated for 51 countries over the period 1950-1992. Fig. 2, which plots the cross-country variances for this group of countries, provides little evidence of an upward quadratic trend. Furthermore, the jaggedness of the series provides some evidence that it may fluctuate around a constant mean. On net, H2 appears somewhat more strongly supported than Hi.
‘Jones (1995) reaches a similar conclusion from the fact that trend growth rates of individual countries appear not to have changed appreciably over time. Note that if growth is endogenous, a strong parallel exists between variation of trend growth rates at a given time across countries and variation for a given country over time. ’ Using a sample similar to the one used here, Bernard and Durlauf (1991) have tested whether jai, - yj, has a unit-root for all possible pairs (i,j) of countries. They find fairly strong evidence against a unit root for the pairs within a few subsets of the countries but weak evidence for all other pairs. Their conclusion probably differs from mine for two reasons. First, their null hypothesis rules out the possibility of different trend growth rates for countries and hence most endogenous growth models a priori. Second, the standard univariate tests that they employ have much lower power than the tests employed here. ” Selection should bias the test of Hz against HI much more than the test of H, against Hz since including the intercept and time trend in Eq. (2.4) should remove much of the induced decline in V,. For this reason, the rejection of HI in favor of Hz in this section provides fairly strong evidence in favor of exogenous growth theories.
1038
P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
0.80 -
0.75 -
0.70 -
0.65 -
0.60 ’
3 0 ~8 I952
Fig. 2. Cross-country
3 1961 variances
I970
1373
of log per capita
output
I366 for 51 countries.
Unfortunately, formal statistical analysis does not provide strong evidence for either hypothesis. Pretesting revealed that V, is well characterized by the process &=0.125+0.000542t-0.13561/,-r (2.48) (2.00) (2.36)
+0.3474&r, (2.30)
(3.9)
where the figures in parentheses are absolute values of c-ratios. Monte Carlo simulations showed that the estimated coefficient on t is not significantly positive at the 0.05 level and that the coefficient on V,_ 1 is not significantly negative unless 5, the cross-country standard deviation of trend growth rates, is at least 1.7 percent a year.’ 1 Consequently, Hz and HI cannot be readily rejected. Figs. 3 and 4 show that this conclusion can be strengthened somewhat by dividing the 51 countries into two mutually exclusive groups consisting of 22 OECD countries plus Cyprus and the other 28 countries. The two series of cross-country variances do not trend upward quadratically and are quite jagged, implying that H, cannot be rejected and suggesting that HI can be rejected for
I1 A description of these simulations results are available upon request.
and those mentioned
below and two tables summarizing
their
P. Evans /Journal
of Economic
Dynamics and Control 20 (1996) 1027-1049
1039
0.275 0.250
0.125 0.100 ~7-7
0.075 1952
Fig. 3. Cross-country
0.38
variances
,
1961
1979
1970
of log per capita
output
-7TV_1988
for 22 OECD
countries
plus Cyprus
..____
n
0.28 -
1952
Fig. 4. Cross-country
variances
1961
1970
of log per capita
1979
output
1988
for 28 nonOECD
fairly small values of [. Pretesting revealed that the cross-country the 22 OECD countries plus Cyprus is well characterized by AC = 0.057 - 0.00808t - 0.26071/,_ 1 + O.l67AV’_, , (1.42) (2.8 1) (2.58) (2.46)
countries.
variance for
(3.10)
1040
P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-1049
and the cross-country variance for the 28 nonOECD countries is well characterized by AC = 0.146 - 0.00300t - 0.40521:_ 1 + 0.265Al’_ 1, (1.62) (2.99) (1.23) (3.02)
(3.11)
Monte Carlo simulations showed that the estimated coefficients on t are not significantly positive at the 0.05 level and that the coefficients on V,_ 1 are significantly negative if [ is at least 1.1 percent a year. Consequently, H2 cannot be rejected at all and Hr can be rejected more readily for each subset of countries than for the entire group.”
4. Summary and conclusions If countries have different trend growth rates as predicted by endogenous growth theories, the cross-country variance of the logarithms of their per capita outputs should be integrated of order one around an upward quadratic trend. By contrast, if they follow parallel balanced growth paths as predicted by exogenous growth theories, the cross-country variance should be stationary around a constant positive mean. The paper considered four series of crosscountry variances: one for 13 industrial countries over the period 1870-1989, a second for 51 countries over the period 1950-1992, a third for 22 OECD countries plus Cyprus over the period 1950-1992, and a fourth for 28 nonOECD countries. None appears to trend upward quadratically, and the first provides strong evidence of stationarity around a constant positive mean. Thus, no evidence is found that growth rates differ endogenously, and considerable evidence is found for parallel balanced growth paths. Therefore, either endogenous growth models that predict different trend growth rates for countries are fundamentally flawed, or else the effects that they predict must be relatively unimportant for the countries considered here.
t2 Durlauf and Johnson (1992) and Bernard and Durlauf (1995) have argued that the logarithms of per capita output for groups of countries like the first one considered in this section do not revert toward a single common trend. Rather, several mutually exclusive subsets of the countries revert toward distinct common trends, which are not cointegrated with each other and presumably drift at different rates. Cross-country variances can be used to evaluate this hypothesis. It is strongly supported if the cross-country variances for large groups of countries trend upward quadratically while the cross-country variances for appropriately selected smaller groups of countries are stationary around a constant positive mean. The results presented here provide at most a modicum of support for this hypothesis.
P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-1049
1041
These empirical results suggest that individual countries should be modeled as facing diminishing social returns to the accumulation of domestically produced reproducible factors of production. The assumption of constant returns to scale should be reserved for large collections of countries, if not the world as the whole.
Appendix The data
The principal series investigated in the paper is the cross-country variance of the logarithm of per capita output for Australia, Austria, Belgium, Canada, Denmark, Finland, France, Italy, Norway, Sweden, the United Kingdom, the United States, and West Germany over the period 1870-1989. In order to calculate that series, I first obtained data on real per capita gross domestic products by dividing the indices of real gross domestic product in Tables A.6-A.8 of Angus Maddison (1991) by the corresponding population figures in his Tables B.2-B.4 and normalizing the resulting ratios to equal the per capita gross domestic products of the countries in 1989 converted to U.S. dollars at the exchange rates prevailing then. Maddison adjusted his indices of real gross domestic product to account for frontier changes, and 1 made similar adjustments to his population figures. I then applied Eqs. (2.3) and (2.2) to the logarithms of these data. I also consider the cross-country variance of the logarithm of per capita output for a group of 51 countries and for two mutually exclusive subsets of 23 and 28 countries over the period 1950-1992. The 51 countries comprise all those for which the data are available over the entire sample period. I applied Eqs. (2.3) and (2.2) to logarithms of per capita gross domestic product in constant international prices (RGDPCH) from the Penn World Tables 5.6, which are described in Robert Summers and Alan Heston (1991). Proof of Theorem 1
Without essential loss of generality, I consider only the case in which the vector x, is null. In that case, if conditions (i), (ii), and (iii) are satisfied, the data-generating process for y,, - a, can be written as Y”, -
at =
&(Yn.*-1
-4-J
+ %t,
(A.11
where I, is a parameter and u,* is a serially and mutually uncorrelated error term with a zero mean and finite variance. Solving Eq. (A.l) backward from period
1042
P. Evans / Journal of Economic Dynamics and Control 20 (1996) 1027-1049
T to period 1 yields
gn-(UT -uo)/T = Bn(Yno -UO) + f
‘$’ X%T-i, 1-O
64.2)
where B,, = (1,’ - 1)/T. The estimator obtained by applying ordinary least squares to Eq. (2.1) is asymptotically equivalent to the estimator obtained by regressing gn - (ar - ao)/T on ynO- ao. I therefore consider the estimator
ij = f
(Y,O - ao)Cs,
-
(UT- ao)lTl
PI=1
i (Y~O- ~0)‘. i
Substituting Eq. (A.2) into Eq. (A.3) and taking probability members of the resulting equation gives
plim pI = N+CC
ctt [
k f
(Y,o
-
a012P.
n-1
(A-3)
n=l
I/[pi;$
limits of both
(Y,o-uo)2
n-1
1
=b,
(A.4) since y,, - a, is uncorrelated with future realizations of its own innovations. UnderH,,L, = A2 = ... =&= landhence/?, =/?2= ... =&=O.Eq.(A.4) therefore implies that plimN+, p = 0. I assume that under HZ, (Y,,~- uo)2 and /ln are jointly distributed across countries with finite first and second moments and with only positive support for (yno - a# and only negative support for bn. Therefore, the quantity b does indeed exist and is negative under Hz. Moreover, the error term is uncorrelated across countries because u,~is contemporaneously uncorrelated across countries by supposition. Accounting for heteroskedasticity then suffices to produce a consistent standard error. A general form for (A.l) is
(A.9
Ynt-ut=r”+n,cY..,-,-u,-l)+Unt.
If(i) is violated, the t’s differ from zero; if (ii) is violated, U,, is serially correlated; and if (iii) is violated, U,, is contemporaneously correlated across countries.
Solving Eq. (A.5) backward results in
gn- (UT- UO)/T = Bn(Yno - UO)+ f
72’ %(rn+ un,T-i). 1-O
64.6)
P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
1043
Clearly, y,, - a0 is cross-sectionally correlated with 5, if(i) is violated and with at least V,, if (ii) is violated, and w, is correlated across countries if (iii) is violated. Hence, fi need not converge in probability to b as N approaches infinity. In particular, the probability limit need not be zero under H1 and need not be negative under HZ. Even if conditions (i) and (ii) are satisfied, accounting for heteroskedasticity does not suffice to produce a consistent standard error unless (iii) is also satisfied. Proof of Theorem 2
H1 and H2 are nested by the data-generating
Ynt -
4
=
Pn
+
6nt
+
f
$ni%,t-i7
n=
process
1,2 ,..., N,
(A.7)
i=O
where the p’s, 6’s, and 1,4’sare parameters and [ulr, u2,, . . . , u,,] is a serially uncorrelated error vector that has a zero mean vector and a positive definite covariance matrix and satisfies
for n1,n2,n3,n4 = 1,2, . . . . N and tl, tz, t3, t4 = 0,1,. . . , T. Without loss of generality, a, can be defined so that the sample averages of pL,,a,, p,, C,‘To I//,iu”,t_i, and 6, cl: o tjmiu,,f _ i are zero. Under H 2, the 6’s are zero, the p’s vary across countries, the $‘s are square-summable, and all roots of I,:0 @niLilie outside the unit circle. Under H1, the 6’s are nonzero and vary across countries, and all roots of (1 - L) C,c, Il/,iL’ lie outside the unit circle.13 Hence, lim $“/ni= 6,
(A.9)
i+cc
where I,$ is a finite nonzero parameter. Because the sample averages of the p’s and 6’s are normalized to zero, subtracting the average of each member of Eq. (A.6) from itself results in
(A. 10) j=l
i=O
I3 This formulation has the problem that the data-generating process (A.7) cannot have held forever since the realizations of every y., between periods 0 and T would then be infinite. I assume that if H1 holds, the process began at some finite date in the past and the u’s before that date were zero.
1044
P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
where 0nji is (N - l)@ni/N ifj = n and -~ji/N ifj # n. Note that under Hz the B’s are square-summable and under H, they satisfy lim
9nji
=
(A.1 1)
e,j,
i-rcc
where 8”j is (N - 1)&/N is j = n and -&,,lN ifj # n. Furthermore, the sample averages pL,IF= 1 C,p”_,e,jiu,,-i and 6, I:= 1 C,zo fl”jiU,,_i are zero. Squaring both members of Eq. (A.lO) and averaging across countries then results in
+ f
“i’ f “f’(k$
i=(J j=l
i,“,cy,1
I=1
snjienlk)
Uj,t-iUl,,-k.
(A.12)
n-l
Suppose that hypothesis Hz holds so that 6, = & = ... = 6, = 0 and the B’s are square-summable. In that case, (A.12) and (A.@ imply that the mean and autocovariances of V, exist. Hence, it is stationary with a constant positive mean. Suppose instead that H, holds so that 6r # & # ... # a,?,,# 0 and the O’sare not square-summable. Eq. (A.12) then implies that the mean of V, approaches infinity as t approaches infinity because the second and third terms in the first line and the expectation of the fourth term do so. Imposing Hr in Eq. (A.12), differencing, and taking expectations produces
(A.13) where w.’ is the variance of u,,~.Without essential loss of generality, I assume here that Eu,,,~,+, = 0 if n, # n,. l4 Eq. (A.1 1) implies that C,p”_ 1 (O:i - Ot,j,i- 1)
I4 Eq. (A.lO) can always be reparameterized terms with zero means and finite variances.
in terms of N mutually
and serially uncorrelated
error
P. Evans / Journal of Economic Dynamics and Control 20 (1996)
1027-1049
1045
exists. Eq. (A.13) therefore implies that the mean of AV, - 2((1/N)Cr=, 6.‘)~ exists under Hr. Straightforward, but tedious, calculation reveals that its autocovariances also exist. Hence, V, is integrated of order one around an upward deterministic quadratic trend. Suppose that V, can be adequately approximated as a (p + l)th-order autoregression. The discussion above then implies that Eq. (2.4) holds. Under HI, p = 0 and 9 > 0 hold; under Hz, p < 0, q = 0, and i > 0 hold. Proqf
qf Theorem 3
Without essential loss of generality, I assume that p = 0 in Eq. (2.4).” Let 9’ G [Q, 4, (1 and 8 be the ordinary least squares estimator of 9. It then follows that
(A.14)
Premultiplying produces
both
members
of Eq. (A.14) by B = diag [T 5’2, T3’*, T ‘I*]
E(B - 9)
c v,-
lV,
c tv, _ Cvt T-51
v:_,
T-4Ctt&
T-31
V,_,
T-3x
t*
T-*Et
T-*x
t
T-lx
1
-l
I[
1
T-5/*1
K-IV,
T-3’2x
tv,
T-‘/*x
v,
. 1 (A.15)
I5 If p > 0, the analogue
to q below is q/(1 - Cf’=,
cp,)
1046
P. Evans / Journal of Economic Dynamics and Control 20 (1996)
1027-1049
Under H,, Eq. (2.4) can be solved backward from period t to period 1 to obtain V, = VI)+ ([ + q/2) t + (q/2) t2 + i
(A.16)
Vi.
i=l
Eq. (A.16) implies that V, is eventually dominated 9 > 0. For this reason,
-“I
T-'x I’_,
tK_1
T-4x
tVml
T-3x
t2
T-‘xt
T-3x
v,_,
T-*x
t
T-lx
1
s
1
1
0
(q,2)j-o1 s3ds
s,‘s’ds
1 s 1
s3ds (v/2)
s4ds (v/2)
by the term (u/2) t* since
s*ds
-
0
IO1sds 1
(q/2)
’ s* ds
s0 -~*I20 ~18
_ rlf6 and
ds
’ sds
s0
v/8
~16
l/3
l/2
l/2
1
1
s0
(A.17)
= Q
(rll+W- 2 Jo1 sW(d}ds
s 1
W(l) -
W(s) ds
0
W(l)
(A.18) I J
P. Evans /Journal
of Economic Dynamics and Control 20 (1996) 1027-1049
1041
as T approaches infinity. In Eqs. (A.17) and (A.18), s is a variable of integration and W(s) is the Wiener process. The term in the last line of Eq. (A.18) is distributed as N(0, .‘Q).“j Hence,
Z(8- 9)
sN[O,Q-'(a2Q)Q-'1
=N(O,a'Q-')
(A.19)
as T approaches infinity. Under H Ir p = 0, and the (1,l)th entry of Q ’ is ~~120
~18
~116
~18
l/3
l/2
~116
l/2
1
= 7201~~.
(A.20)
Therefore. T “‘d 5 N [0,720(0/~#]
(A.21)
.
I6 W(s) is normally distributed with a mean of zero and variance of s for all s on [0,11.Moreover. EW(s)W(r) = min(r,s) for all s and r on [O,l]. The covariance matrix is a*Q because
+‘(I,-ZjO’sW(s)ds]
=1
-4j~s’di+4j~‘[j~sr’dr+j~‘s’rdr]ds=:.
E[W(I)-j;
W(r)ds]
=,-2
jO’sds+
EW(l)*
=
+‘(I)
=,
jS’sdr]ds=:.
I,
- 2 jo’ sW(s)ds][W(l/
- j;sdr--2
E[W(I)-2
E[H’(l)-
jO’[j;rdr+
j0’s’ds+2
j+V(s)ds]W(,)=l-2
j~‘W(r)ds]W(I1=l-j~‘sds=:
- jO’ Wids]
jO’[j;srdr+
jzls’dr]ds=l,
j0’s2ds=_:,
1048
P. Evans /Journal
of Economic
Dynamics and Control 20 (1996) 1027-1049
The t-ratio t(b) is defined to be j? divided by the square root of (A.22) times the (l,l)th
entry of
As T approaches infinity, the former converges in probability to g2 because 3 is consistent and T 5 times the latter converges in probability to the (l,l)th entry of Q-r. Consequently, (A.21) implies that z( 6) is N(0,1). Finally, T “‘6 and r( 6) must diverge under H2 because 3 is also consistent in that case. Proof of Theorem 4
See Chapter 16 of James Hamilton (1993).
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