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Intracountry evidence on the Lucas variance hypothesis.

INTRACOUNTRY EVIDENCE ON THE LUCAS VARIANCE HYPOTHESIS

I. INTRODUCTION

Seventeen years ago, Lucas [1973] developed an aggregate supply model that challenged the efficacy of neo-Keynesian macroeconomic stabilization policy and significantly altered the development of macroeconomic models. Employing a multimarket stochastic equilibrium model, Lucas assumed that individual suppliers decide how much to produce based on perceived relative prices. Their expectations of future prices and demand are formed rationally using all available information about specific and economy-wide prices and quantities. Changes in market specific prices, however, reflect both changes in the general price level and changes in relative prices. Individual suppliers cannot distinguish between the two, because they have instantaneous information about their own specific prices but only lagged information about the general price level. Thus, a current unanticipated increase in the general price level and nominal aggregate demand will be partly perceived by suppliers as an increase in the relative price and relative demand for their goods, and they will temporarily increase output. A highly volatile general price level and nominal aggregate demand result in less general-relative price confusion and a fading of the short-run output-inflation tradeoff. That is, the Lucas model suggests an inverse relationship between the output-inflation tradeoff and the variability of the rate of inflation and the rate of change of nominal aggregate demand.

Lucas used a sample of eighteen countries to " ... see whether the terms of the output-inflation `tradeoff' vary across countries in the way predicted by the natural rate theory" [1973, 330]. He estimated the following reduced form supply function over the 1952-67 period:

[Y.sub.ct] = [-pi Delta] + [pi Delta chi.sub.t] + [lambda y.sub.c,t-1'] (1) where [y.sub.ct] is detrended real GNP, [Delta chi.sub.t] is the change of the natural logarithm of nominal GNP, [delta] is the average rate of demand expansion, [lambda] is the speed of adjustment with [~lambda~] [less than to] 1, and [pi] is the response of real GNP to changes in nominal GNP, with 0 [is not less than or equal to] [pi] [is not less than or equal to] 1. Since the estimated tradeoff coefficients were found to be greater for a group of sixteen "stable" countries than for two highly volatile countries, Lucas argued that " ... the behavior of the estimated [pi] values across countries is in striking conformity with the natural rate hypothesis" [1973, 332].

Empirical studies by Albero [1981], Attfield and Duck [1983], Fernadez [1977], Hanson [1980], Hercowitz [1983] and Koskela and Viren [1980] have generally provided cross-country evidence supporting Luca's variance hypothesis. Froyen and Waud [1980] however, presented, in addition to cross-country evidence, limited intracountry evidence for (1) the relationship between the output-inflation tradeoff and the variances of the rate of change of nominal income and the price level and (2) the relationship between the variance of the inflation rate and the variance of the rate of change of nominal income. They examined these relationships on a cross-country basis for each of ten industrialized countries for the 1956-76 period. The 1957-66 and 1967-76 subperiods were used to make intracountry comparisons in individual countries. Their cross-country and intracountry results support the hypothesis that the output-inflation tradeoff and the variance of the inflation rate are inversely related. Their evidence, however, does not suggest that the variance of the rate of change of nominal income is negatively correlated with the tradeoff and positively correlated with the variance of the inflation rate.(1)

It is assumed in Luca's paper and all other cross-country studies that the slope of the supply curve, [gamma] and the variance of the specific market price from its economy-wide average, [tau.sup.2] , are stable across countries and within countries over time. The cross-country stability assumption is probably unrealistic, however, for the samples of countries employed by Lucas [1973], Albero [1981], Hercowitz [1983] and Koskela and Viren [1980]. Besides the " ... large measurement errors on the supply elasticities for each country individually" (Lucas [1977, 731]), there may exist significant structural and social differences among countries that violate Lucas's stability assumption.[2] Thus, cross-country comparisons may be an improper test of Lucas's variance hypothesis and generalizations based on such evidence may be questionable.

A country-by-country analysis, on the other hand, provides a more rigorous test because it allows the relaxation of the cross-country stability assumption. Froyen and Waud's [1980] intracountry analysis is the first attempt at intertemporal examination of the variance hypothesis. Their evidence, however, is derived " ... by comparing the behavior of a given country in what are apparently two regimes (i.e., the 1957-66 and 1967-76 subperiods)" [1980, 409]. Since the 1970s were characterized by significant supply disturbances and erratic changes in aggregate nominal income, the over-time within-country stability assumption may be violated and intracountry comparisons may be biased.[3] In a more recent paper, Froyen and Waud [1984] developed an extended Lucas-type model that incorporates explicitly the effects of the variability of aggregate demand, of supply-side shocks, and of inflation-rate variability on the output-inflation tradeoff. They tested their model for the United Kingdom from 1957 to 1980. Their evidence provides support for the significance of the last two effects but not for the first.[4]

Jung [1985] argued that, since there is no reason to believe that [gamma] and [tau.sup.2] are constant across countries, partial rather than simple correlations are the correct measures of these relationships. Employing a sample of fifty-six countries - nineteen developed and thirty-seven less-developed countries, he reexamined these three relationships by calculating both simple and partial correlation coefficients. The partial correlations were calculated after controlling for cross-country changes in [gamma]. Jung found that both simple and partial correlations tests provided support for a negative relationship between the output-inflation tradeoff and the variances of the rate of change of nominal income and of the price level for the developed countries, but the evidence was weak for the less-developed countries. On the other hand, the positive association between the two variances was significant for both groups. He concluded that his evidence " ... is in conflict with what Froyen and Waud [1980] found for the group of ten industrial countries, but strengthens Lucas's [1973] and Albero's [1981] findings in the sense that the result is based on a measure of nonspurious, partial correlation" [1985, 111]. Katsimbris [1990], however, argues that the partial correlation coefficients Jung claims are nonspurious measures may be subject to aggregation bias because of possible within-country over-time changes of the parameters [gamma] and [tau.sup.2].

All cross-country studies are subject to aggregation bias associated with the use of measures of variability of nominal income and of the general price level that are calculated over the entire sample period. These statistics are point-estimates and are consistent with a number of distributions. Thus, it is possible, for instance, that a number of countries have the same variability of nominal aggregate demand but different output-inflation tradeoff coefficients, and vice-versa, if the distributions underlying their variability measures are different (i.e., they have different skewness). The purpose of this paper is to provide additional intertemporal evidence on the relationships examined by Froyen and Waud [1980; 1984] and Jung [1985]. Section II describes the methodology used in this study, section III discusses the empirical results, and section IV examines the cross-country structural stability. Section V summarizes the results and their implications.

II. METHODOLOGY

This study employs annual data of real and nominal GNP[5] on thirtynine countries(6) for the period 1953-85. The intertemporal examination of the Lucas variance hypothesis requires that aggregate supply is stable over time. There is no reason, however, to believe that the aggregate supply schedules of the individual countries were stable over the sample period, especially during the 1970s. In order to capture these supply shocks, equation (1) was extended to include as an explanatory variable the price of crude oil. Crude oil price shocks shift the aggregate supply curve, resulting in significant changes in real output.[7] Thus, equation (1) becomes

[y.sub.ct] = [-pi Delta] +[pi DELTA X.sub.t] + [lambda y.sub.c,t-1] + [alpha mu.sub.t] +

[u.sub.t] where [alpha] [less than to] 0, and [mu] is the aggregate supply shock proxied by the change in the natural logarithm of the relative price of crude oil; [mu] is measured by the difference of the first difference of the natural logarithms of the nominal price of Saudi crude oil and the implicit price deflator. Given the level of nominal aggregate demand, a positive (negative) aggregate supply shock causes real output to decline (increase) and the price level to increase (decrease). In this augmented Lucas supply function, suggested by Froyen and Waud [1980],[8] the output-inflation tradeoff coefficient [pi] is given by

[MATHEMATICAL EXPRESSION OMITTED]

where [MATHEMATICAL EXPRESSION OMITTED] are the variances of the market specific demand shocks, of the nominal aggregate demand shocks, and of the aggregate supply shocks, respectively. In addition, the inflation variance is given by

[MATHEMATICAL EXPRESSION OMITTED]

where

[MATHEMATICAL EXPRESSION OMITTED]

From equation (2), it follows that [pi] is a negative function of both [MATHEMATICAL EXPRESSION OMITTED]. Given that [theta] is a decreasing function of both [MATHEMATICAL EXPRESSION OMITTED] equation (3) implies a positive relationship between [MATHEMATICAL EXPRESSION OMITTED] and both these variances.(9) Finally, equations (2) and (3) suggest a negative association between [pi] and [MATHEMATICAL EXPRESSION OMITTED].

In this paper, equation (1') and the following three equations were estimated:

[pi] = [a.sub.o] + [a.sub.1] [sigma.sub.x] + [a.sub.2] [sigma.mu] + [u.sub.1'],

[pi] = [b.sub.o] + [b.sub.1] [sigma.sub.p] + [u.sub.2'],

and

[sigma.sub.p] = [c.sub.o] + [c.sub.1] [sigma.sub.x] + [c.sub.2] [sigma.sub.mu] + [u.sub.3]

where [sigma.sub.x] is the standard deviation of the first difference of the natural logarithm of nominal GNP, [sigma.sub.mu] is the standard deviation of the rate of change of the relative crude oil price, [sigma.sub.p] is the standard deviation of the difference of the first differences of the natural logarithms of nominal and real GNP, [a.sub.i], [b.sub.i], and [c.sub.1] are parameters to be estimated, and the [U.sub.i] are random errors. The model suggests that [a.sub.1], [a.sub.2], [b.sub.1] [is less than] 0 and [C.sub.1], [C.sub.2] [is greater than] 0. The method of estimation is ordinary least squares. When autocorrelation was detected for equation (1'), the instrumental variable approach suggested by Wallis [1967] was employed, while for equations (4), (5), and (6), the regressions were adjusted using the Cochrane-Orcutt transformation technique. Descriptive statistics for the thirty-nine countries in the sample are given in Table I.

Table : [TABULAR DATA DOMITTED]

III. EMPIRICAL RESULTS

First, equation (1') was estimated for each individual country over the 1954-85 period. 10 The estimated output-inflation tradeoff coefficients, [pi], and the corresponding adjusted coefficients of determination, [R.sup.2], are reported in the first two columns of Table I.(11) The estimated coefficients satisfy the model's restrictions of 0 [is not less than or equal to] [Pi] [is not less than or equal to] 1 in thirty-three, [~lambda~] [is less than to] 1 in thirty-one, and [alpha] [is greater than to] 0 in twenty countries.(12) The [pi] estimates were positive and significantly different from zero for seventeen countries at the 1 percent level and eight countries at the 5 percent level. These results are, in general, consistent with those in Lucas [1973], Albero [1981], and Jung [1985]. The estimated coefficient [alpha], on the other hand, was negative and significant in one country at the 1 percent level and four countries at the 5 percent level. It was positive and significant for only two countries at the 1 and 5 percent levels, respectively. These findings are consistent with those in Froyen and Waud [1984] where the aggregate supply shocks, proxied by the energy prices, were not found to be significant.(13)

In order to generate time-series data to test the empirical validity of the relationships described by equations (4), (5), and (6) over time on a country-by-country basis, equation (1') was estimated as a rolling regression,(14) and [sigma.sub.x1] [sigma.sub.p1] and [sigma.sub.mu] were calculated over a fifteen-year moving period. That is, equation (1') was estimated and three standard deviations were calculated, first over 1955-69, next over 1956-70,..., and over 1971-85.(15) In this way, a moving series of estimates and calculated [sigma.sub.x], [sigma.sub.p1] and [sigma.sub.mu] values for each country weregenerated. The cross-country and within-country over-time stability assumption of the slope of the supply curve is relaxed. The estimated slope coefficients and the corresponding [R.sup.2s] of equations (4), (5), and (6) are reported in

Table : [TABULAR DATA OMITTED]

First, the relationship between the output-inflation tradeoff coefficient, [pi], and the variability of nominal aggregate demand, [sigma.sub.x1] was found to be negative for twenty-five of the thirty-nine countries; but it was significant for only five countries at the 1 percent level and five countries at the 5 percent level. Moreover, for the fourteen countries with a positive coefficient, it was significant for one country at the 1 percent level and two at the 5 percent level. On the other hand, the coefficient of the variability of the aggregate supply shock, [a.sub.2], was found to be negative in twenty-two cases, but it was significant in only six cases at the 1 percent level and three cases at the 5 percent level; whereas the coefficient was positive and significant in four cases at the 1 percent level and two cases at the 5 percent level.

Second, the relationship between the output-inflation tradeoff, [pi], and the variability measure of the inflation rate, [sigma.sub.p1] was found to be negative for thirty-two of the thirty-nine countries and significant for nine countries at the 1 percent and six countries at the 5 percent levels. [Tabular Data Omitted]

Finally, the relationship between the variability of the inflation rate and the variability of the rate of change of nominal aggregate demand was found to be positive in thirty cases and significant for twenty-three and two countries at the 1 percent and 5 percent levels, respectively. For the nine countries with a negative coefficient, only one was significant at the 1 percent level. The relationship between the variability of the inflation rate and the variability of the aggregate supply shock was positive for twenty-seven and significant for nineteen countries at the 1 percent level, the coefficient was negative and significant for five countries at the 1 percent level.(16)

The above results suggest that the Lucas hypothesis of a negative association between the output-inflation tradeoff and the variability of nominal aggregate demand is supported for about 26 percent of the countries, and the output-inflation tradeoff is negatively related to the variability of the inflation rate in 39 percent of the countries. The variability of nominal aggregate demand and the variability of the inflation rate appear to be positively related in about 64 percent of the countries. On the other hand, the hypothesis of a negative association between the output-inflation tradeoff and the variability of aggregate supply shocks was supported in only 23 percent of the countries, and that of a positive association between the variability of the inflation rate and aggregate supply shocks received support in 49 percent of the countries.

In general, the intracountry evidence presented here provides weak support for Luca's variance hypotheses and is consistent with the findings of Froyen and Waud [1980; 1984] who concluded that their" ... results are not consistent with the sequence hypothesized by Lucas wherein differences in inflation variance and, consequently, differing output-inflation tradeoffs are the results of differences in aggregate demand variance" [1980, 409-10]. Our estimates also provide weak evidence for a supply-side effect. Froyen and Waud [1984], when they use the energy price measure to proxy aggregate supply shocks, found similar results. On the other hand, the results are in sharp contrast with Jung [1985] who found strong support for all hypotheses for developed countries. The three relationships are significant for only 28, 32 and 42 percent of the group of nineteen developed countries contained in our sample, respectively.

Froyen and Waud [1980] suggest that the lack of strong empirical support for Luca's variance hypotheses may be due to cross-country or over-time aggregate supply disturbances and/or to possible interdependencies between the distributions of relative and aggregate demand disturbances. In our case, however, such an explanation is not plausible. First, the evidence is based on a country-by-country analysis. And second, equation (1') contains an explanatory variable to capture over-time supply disturbances.

IV. CROSS COUNTRY STRUCTURAL STABILITY

Having estimated individual regressions for each country, one can test to see if there are significant structural differences between countries. We estimated pooled regressions for equations (4), (5), and (6). The pooled coefficient estimates and the corresponding [R.sup.2s] are reported at the end of Table II. For equation (4), the relationship between [pi] and [sigma.sub.chi] was found to be significantly negative and that between [pi] and [sigma.sub.mu] was found to be significantly positive at the 5 and 1 percent levels, respectively. For equation (5), the correlation between [pi] and [sigma.sub.p] was found to be negative and significant at the 5 percent level; and for equation (6), all the estimated coefficients were positive and significant at the 1 percent level. The Chow test for the three equations gave the F-statistics reported at the end of Table II. The null hypothesis of no structural differences between countries is rejected for all equations at the 1 percent level. This suggests that there are significant structural differences among countries that are inherent and/or due to "large measurement errors" (Lucas [1977, 731]). Thus, cross-country comparisons are subject to aggregate bias.

IV. SUMMARY AND CONCLUSION

Three basic implications that follow from Lucas's aggregate supply model are (1) there is a negative association between the variability of nominal aggregate demand and the output-inflation tradeoff, (2) there is a negative association between the variability of the inflation rate and the output-inflation tradeoff, and (3) a positive relationship exists between inflation-rate variability and aggregate demand variability. The time-series evidence produced in this study fails to provide strong support for these three propositions. The three relationships had the correct sign and were significant for only 23, 36 and 64 percent of the thirty-nine countries in our sample. These findings differ from those of most of the previous empirical studies based on cross-country comparisons, which found support for Lucas's variance hypotheses, but are consistent with Froyen and Waud's [1980; 1984] results. The evidence also suggests that cross-country comparisons may be subject to aggregation bias due to existing significant structural differences between countries. Though the results are only tentative, they suggest that cross-sectional generalizations about the empirical relevance of Lucas's propositions must be received with great caution. (1.) Arak [1977] criticized Lucas's assumption that nominal aggregate demand is exogenously determined. She developed tests of the Lucas model that did not support Lucas's underlying theory for the United States. Lucas [1977], however, replied that Arak's results occurred because her tests assume an exact fit of the solutions of his model.

(2.) Froyen and Waud [1980] chose to include in their sample only similarly developed countries in order to satisfy Lucas's cross-country stability assumption.

(3.) See Alberto [1981, 241] and Froyen and Waud [1984, 53].

(4.) See also Froyen and Waud [1985] for a U.S. model.

(5.) When GNP was not available, GDP was used. The data were taken from the International Financial Statistics Yearbook, 1983 for years 1953-58, Yearbook 1987 for years 1959-80, and February 1988 for 1981-85.

(6.) In order to maintain an acceptable number of degrees of freedom when the functional relationships are tested, only the countries with a minimum of thirty-one annual observations were included in the sample.

(7.) The inclusion of the price of Saudi crude oil was suggested by an anonymous referee. Froyen and Waud [1984] have also used the crude oil price as a proxy for aggregate supply shocks. The relationship between the crude oil price and the macroeconomy is suggested by a number of studies. Berndt and Wood [1975; 1979] and Wilcox [1983] found strong support for the hypothesis that energy and capital are complementary in the U.S. before and after 1973. Hamilton [1983] presented evidence supporting a strong correlation between the crude oil price and U.S. recessions well before 1972. And Gisser and Goodwin [1986] showed that the functional relationship between the crude oil price and the U.S. macroeconomy has been stable over the postwar period.

(8.) See also Froyen and Waud [1984; 1988].

(9.) Froyen and Waud [1984; 1988] also showed that [alpha], the coefficient of the aggregate supply shocks, is an increasing function of both [sigma.sup.2.sub.chi] and [sigma.sup.2.sub.mu], and a positive function of [sigma.sup.2.sub.p].

(10.) The 1953 observation was lost when the real and nominal income growth rates were calculated. Due to lack of data, for Ecuador and Sri Lanka the equations were estimated over 1954-84, for New Zealand and Spain over 1955-85, and for Turkey over 1954-83.

(11.) In order to economize on space, only the estimated coefficients and the corresponding adjusted [R.sup.2] s are reported. The complete regression results are available from the author on request.

(12.) [lambda] was significantly greater than one in only one case.

(13.) Froyen and Waud [1984], however, found significant supply-side effects when [mu] is measured by the import prices.

(14.) An anonymous referee raised the question of whether the supply shock variable becomes significant for rolling regressions that cover the later portion of the sample period. The findings show that for one-third of the countries in the sample the significance of the supply shocks increased over the post 1959-73 moving periods, whereas for the remaining two-thirds there was no noticeable change.

(15.) Similar procedures have been adopted elsewhere. See Klein [1975], Katsimbris [1985], and Katsimbris and Miller [1982].

(16.) We also tested two additional implications of this extended version of Lucas's model, namely, (1) that there is a positive relationship between [alpha] which measures the output effect of aggregate supply shocks in equation (1'), and the variabilities of nominal aggregate demand and supply shocks, [sigma.sub.chi] and [sigma.sub.mu], and (2) that there is a positive relationship between [alpha] and the variability of the inflation rate, [sigma.sub.p]. We found a positive correlation between [alpha] and [sigma.sub.p] in twenty-six cases, but it was significant in only two countries at the 1 and 5 percent levels, respectively. The correlation between [alpha] and [sigma.sub.mu] was positive in fourteen cases and significant for four at the 1 percent level and one at the 5 percent level, whereas the correlation was negative and significant for four at the 1 percent and four at the 5 percent level. On the other hand, the correlation between [alpha] and [sigma.sub.p] was found to be positive in twenty cases but significant in only three at the 1 percent and one at the 5 percent level, and it was negative and significant in one at the 1 percent and four cases at the 5 percent level. The complete results are available from the author on request.

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(*) Professor of Economics. The author is indebted for useful comments and suggestions to an anonymous referee and Professors S. M. Miller, S. Ray, and F. W. Ahking of the University of Connecticut. Additional thanks also go to E. Kaparakis and S. D' Souza for valuable computer assistance. Nevertheless, the author is solely responsible for the contents of this paper.
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Date:Oct 1, 1990
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