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Unanticipated movements in aggregate demand and the business cycle
Economics Letters 2 (1979) 125-129 0 North-Holland Publishing Company
UNANTICIPATED MOVEMENTS IN AGGREGATE BUSINESS CYCLE Results from Variance Decompositions
Leonardo LEIDERMAN
*
Tel-Aviv University, Tel-Aviv, Israel Boston University, Boston, MA 02215, Received
DEMAND AND THE
USA
April 1979
What fraction of the variance in real economic activity over the business cycle is attributable to the variance of unanticipated movements in aggregate demand? This note reports calculations, based on a previous contribution by Lucas, aimed at answering this controversial question.
1. Introduction
The idea that aggregate economic fluctuations mostly arise as agents react to unanticipated changes in aggregate demand is at the center of the modern debate on the sources of business cycles. ’ In fact, this notion emerged with the recent advent of equilibrium models, that embody the assumptions of imperfect information and rational expectations; see for example Lucas (1973) Sargent (1973), and Barro (1976). The key testable proposition implied by these models is that unanticipated movements in aggregate demand account for most of the variance in real economic activity over the business cycle. Although this proposition has been commonly accepted as logically correct, many researchers have questioned its empirical plausibility. For example, Hall (1975) performed calculations that led him to conclude that only a trivial fraction of the variation in the unemployment rate of the postwar United States is attributable to price surprises. 2 If generally true, this finding would suggest support to the view that errors in price, or aggregate demand, forecasting play a small role in the generation of business cycles. Clearly, in order to evaluate these different views of the business cycle it is necessary to answer the question: what fraction of the variance in real economic activity * I would like to thank Robert E. Lucas, Jr. for his comments on an earlier version of table 1. Any errors are mine. ’ For recent views on the debate, see Lucas and Sargent (1978), and Modigliani (1977). 2 See also Sims’ comments on Hall’s paper, which appear immediately after the end of the latter. 125
126
L. Leiderman
/ Unanticipated movements in aggregate demand
is accounted for by unanticipaied fluctuations in aggregate demand? The purpose of this note is to report calculations aimed at answering this question. The calculations reported below involve variance decompositions performed on the basis of Lucas’ (1973) model and sample information. In section 2, a brief review of Lucas’ (1973) model is presented. Section 3 reports the pertinent calculations.
2. A brief review of Lucas’ (1973) model The model utilized below in order to quantitatively assess the role of unanticipated aggregate demand movements in the generation of business cycles is the one presented by Lucas (1973). The basic model consists of three equations,
yc,t(z>=yLP,(z)-E(P,II,(z))l
+bc,t-l(Z)
>
Pt(z)=P,tz, yt+P,=x,_l
(1) (2)
(3)
+Ax,=x,.
Eq. (1) is viewed as an output supply function under agents’ imperfect information. The equation posits that the cyclical component of output in a given market z, y,,(z), varies with the relative price perceived by suppliers, and with its own lagged value,y,,_, (z). Pt(z) is the (log of) the actual price in market z, and E(P,lZ~(z)) is the mean current, general price level, conditioned on the available information contained in Z,(z). Eq. (2) posits that the actual price differs from the economy-wide average, P,, by an amount that is orthogonal to Pt. That is, z is assumed to be normally distributed, independent of P,, with mean zero and variance r2. Eq. (3) is an aggregate demand equation. yr is the economy-wide output, obtained by adding economy-wide yC, t to a secular output component, 3 and xt denotes the log of nominal GNP. Furthermore Axt is normally distributed, with mean 6 and variance 2 OX.
With these assumptions, Lucas (1973) proceeds as follows: tirst, he averages output over markets, and thus obtains an aggregate supply function. Secondly, he assumes that Z,(z) : tit(z), 1,-r), where Z,_r is an information set that contains the relevant history of the economy up to period t - 1. Finally, the market-clearing assumption is imposed. The resulting equilibrium value of (the cyclical component of) real output is
yst = -IE
+ IIAx, + b~c,r-1,
where
3 Concretely,
yt = y ~,r +Y&t,
whereh,,r = a + Ptand t is a time trend variable. Also, 1~~1< 1.
(4)
L. Leiderman
/ Unanticipated movements in aggregate demand
121
Eq. (4) gives the model’s solution for Y~,~. As emphasized by Lucas (1973), the model asserts that changes in the average rate of nominal income (aggregate demand) growth will have no effects on real output. In contrast, unanticipated fluctuations in aggregate demand (i.e., changes in AX,) do have output effects. Moreover, the model involves the presumption that unanticipated movements in aggregate demand are the major source of business cycles, measured by fluctuations in y,,. Lucas estimated eq. (4) on the basis of annual data for eighteen countries covering the 1953-1967 period. For each country he reports goodness-of-fit statistics (R*‘s) and these pass the formal tests of significance. Also, his findings indicate support to the model’s hypothesis that the output effects of unanticipated demand shifts decline as the sample variance of Ax, increases. Although the reported results are informative, the specific question: what fraction of the variation in yc,* is attributed to the variation in Ax,? is not explicitly answered. The reported R*'sprovide some indication in this respect, but such indication is only partial, due to the inclusion of a lagged-dependent variable in (4). In order to answer this question, further calculations are required. Details on these calculations are reported in the next section.
3. Results from variance decompositions Our purpose here is to utilize Lucas’ (1973) model and sample information in order to quantitatively assess the role of unanticipated aggregate-demand fluctuations in the generation of business cycles. To accomplish this task, we first rewrite eq. (4) as a regression equation, yc,t = -II6
+ IIAx, + Ayc,r-1 + % ,
(5)
where er is the regression residual with the classical properties. The variance of yC, r is given by vary,,=II*
varti,th2varyC,,_,
+2nhcov(Ax,,y,,-l)+varEf.
Restricting the analysis to the stationary tion is implied:
distribution
of yc,r, the following condi-
Eq. (6) gives a decomposition of the variance of cyclical real output. The equation can be used in order to assess the fraction of the variance of real output that is attributable to variance in unanticipated aggregate demand. Such assessment, however, is not unique: It depends on the treatment of the covariance term appearing in (6). On the one hand, it is possible to attribute this term to the unanticipated aggregate demand variable; alternatively, one can exclude it from unanticipated aggregatedemand factors. These two alternatives lead to the two different variance decompositions that are reported below.
128
L. Leiderman
Table 1 Decompositions
of variance
Country
d
Argentina Austria Belgium Canada Denmark W. Germany Guatemala Honduras Ireland Italy Netherlands Norway Paraguay Sweden United Kingdom United States
/ Unanticipated movements in aggregate demand
ofy,,t
(1953-1967).
% Variance
a
in yc,r explained
by:
(1)
(2)
var Axr
var Axr and cov(Axr,
0.19 23.98 53.64
3.34 2.54 72.31 86.05 65.13 69.13 c
b
61.96 86.65 75.99 25.79 55.96 70.46 76.82 b
0.76 17.07 29.05
b
yo,r-t)
12.41 42.03 74.64 57.15 63.47 3.88 27.93 37.46 74.24
a Calculations based on eq. (6), and on the information reported by Lucas (1973). Note: Column (1) = [fI’/(l - h2)]var Axr/var yc,r, Column (2) = [l/(1 - A2)]var l r/var yc,r. b Percentage greater than 100%; the covariance term is of ‘large’ negative magnitude. c Percentage is negative; the covariance term is of ‘large’ negative magnitude. d Results for Puerto Rico are not reported, because for this country Ihl > 1. Results for Venezuela are not reported: in the first column, category b obtains; and in the second column, category c.
Table 1 reports the variance decompositions of interest. As mentioned before, the calculations were performed by using eq. (6) and the relevant information reported by Lucas (1973). The results reported in the first column of table 1 correspond to the case in which the variance of unanticipated aggregate demand includes only the first term appearing in eq. (6). In the case of seven out of the thirteen entries reported in this column, more than 50 percent of the variance iny,, is attributable to variance in unanticipated aggregate demand. In the calculations reported in the second column, the variance of unanticipated aggregate demand includes also the second term appearing in (6). In this case, eight out of the fifteen reported entries obtain values above 50 percent. Thus, for fifteen out of the twenty-eight reported entries the variance in unanticipated aggregate demand accounts for more than 50 percent of the variance in cyclical output. Furthermore, there are seven countries (Belgium,
L. Leiderman / Unanticipated movements in aggregate demand
129
Canada, W. Germany, Guatemala, Italy, Netherlands, United States) for which in at least one case aggregate demand surprises account for more than 70 percent of the variance in their cyclical output. Overall, then, in most cases unanticipated fluctuations in aggregate demand account for a substantial part of the variance in cyclical real output. 4 In future work, it would be interesting to perform similar calculations on the basis of other equilibrium models present in the macro literature. Also of interest would be an examination of the sensitivity of the results above to changes in the particular definitions of unanticipated aggregate demand and of cyclical output utilized above. In the meantime, however, our findings reveal that unanticipated movements in aggregate demand have an important role in the generation of business cycles.
References Barro, R.J., 1976, Rational expectations and the role of monetary policy, Journal of Monetary Economics 2, Jan., l-32. Hall, R.E., 1975, The rigidity of wages and the persistence of unemployment, Brookings Papers 6. Lucas, Jr., R.E., 1973, Some international evidence on output-inflation tradeoffs, American Economic Review 63, June, 326-334. Lucas, R.E. and T.J. Sargent, 1978, After Keynesian macroeconomics, Manuscript, July. Modigliani, F., 1977, The monetarist controversy, or should we forsake stabilization policies?, American Economic Review 67, March, 1-19. Sargent, T.J., 1973, Rational expectations, the real rate of interest, and the natural rate of unemployment, Brookings Papers 4.
4 Note that in the cases of Argentina and Paraguay only a trivial fraction in the variance of yc r is due to variance in unanticipated aggregate demand. This result is in conformity with Lucas’ hypothesis regarding H: In volatile aggregate demand countries (such as Argentina and Paraguay) nominal income changes are mainly associated with price movements and with negligible effects on output.
UNANTICIPATED MOVEMENTS IN AGGREGATE BUSINESS CYCLE Results from Variance Decompositions
Leonardo LEIDERMAN
*
Tel-Aviv University, Tel-Aviv, Israel Boston University, Boston, MA 02215, Received
DEMAND AND THE
USA
April 1979
What fraction of the variance in real economic activity over the business cycle is attributable to the variance of unanticipated movements in aggregate demand? This note reports calculations, based on a previous contribution by Lucas, aimed at answering this controversial question.
1. Introduction
The idea that aggregate economic fluctuations mostly arise as agents react to unanticipated changes in aggregate demand is at the center of the modern debate on the sources of business cycles. ’ In fact, this notion emerged with the recent advent of equilibrium models, that embody the assumptions of imperfect information and rational expectations; see for example Lucas (1973) Sargent (1973), and Barro (1976). The key testable proposition implied by these models is that unanticipated movements in aggregate demand account for most of the variance in real economic activity over the business cycle. Although this proposition has been commonly accepted as logically correct, many researchers have questioned its empirical plausibility. For example, Hall (1975) performed calculations that led him to conclude that only a trivial fraction of the variation in the unemployment rate of the postwar United States is attributable to price surprises. 2 If generally true, this finding would suggest support to the view that errors in price, or aggregate demand, forecasting play a small role in the generation of business cycles. Clearly, in order to evaluate these different views of the business cycle it is necessary to answer the question: what fraction of the variance in real economic activity * I would like to thank Robert E. Lucas, Jr. for his comments on an earlier version of table 1. Any errors are mine. ’ For recent views on the debate, see Lucas and Sargent (1978), and Modigliani (1977). 2 See also Sims’ comments on Hall’s paper, which appear immediately after the end of the latter. 125
126
L. Leiderman
/ Unanticipated movements in aggregate demand
is accounted for by unanticipaied fluctuations in aggregate demand? The purpose of this note is to report calculations aimed at answering this question. The calculations reported below involve variance decompositions performed on the basis of Lucas’ (1973) model and sample information. In section 2, a brief review of Lucas’ (1973) model is presented. Section 3 reports the pertinent calculations.
2. A brief review of Lucas’ (1973) model The model utilized below in order to quantitatively assess the role of unanticipated aggregate demand movements in the generation of business cycles is the one presented by Lucas (1973). The basic model consists of three equations,
yc,t(z>=yLP,(z)-E(P,II,(z))l
+bc,t-l(Z)
>
Pt(z)=P,tz, yt+P,=x,_l
(1) (2)
(3)
+Ax,=x,.
Eq. (1) is viewed as an output supply function under agents’ imperfect information. The equation posits that the cyclical component of output in a given market z, y,,(z), varies with the relative price perceived by suppliers, and with its own lagged value,y,,_, (z). Pt(z) is the (log of) the actual price in market z, and E(P,lZ~(z)) is the mean current, general price level, conditioned on the available information contained in Z,(z). Eq. (2) posits that the actual price differs from the economy-wide average, P,, by an amount that is orthogonal to Pt. That is, z is assumed to be normally distributed, independent of P,, with mean zero and variance r2. Eq. (3) is an aggregate demand equation. yr is the economy-wide output, obtained by adding economy-wide yC, t to a secular output component, 3 and xt denotes the log of nominal GNP. Furthermore Axt is normally distributed, with mean 6 and variance 2 OX.
With these assumptions, Lucas (1973) proceeds as follows: tirst, he averages output over markets, and thus obtains an aggregate supply function. Secondly, he assumes that Z,(z) : tit(z), 1,-r), where Z,_r is an information set that contains the relevant history of the economy up to period t - 1. Finally, the market-clearing assumption is imposed. The resulting equilibrium value of (the cyclical component of) real output is
yst = -IE
+ IIAx, + b~c,r-1,
where
3 Concretely,
yt = y ~,r +Y&t,
whereh,,r = a + Ptand t is a time trend variable. Also, 1~~1< 1.
(4)
L. Leiderman
/ Unanticipated movements in aggregate demand
121
Eq. (4) gives the model’s solution for Y~,~. As emphasized by Lucas (1973), the model asserts that changes in the average rate of nominal income (aggregate demand) growth will have no effects on real output. In contrast, unanticipated fluctuations in aggregate demand (i.e., changes in AX,) do have output effects. Moreover, the model involves the presumption that unanticipated movements in aggregate demand are the major source of business cycles, measured by fluctuations in y,,. Lucas estimated eq. (4) on the basis of annual data for eighteen countries covering the 1953-1967 period. For each country he reports goodness-of-fit statistics (R*‘s) and these pass the formal tests of significance. Also, his findings indicate support to the model’s hypothesis that the output effects of unanticipated demand shifts decline as the sample variance of Ax, increases. Although the reported results are informative, the specific question: what fraction of the variation in yc,* is attributed to the variation in Ax,? is not explicitly answered. The reported R*'sprovide some indication in this respect, but such indication is only partial, due to the inclusion of a lagged-dependent variable in (4). In order to answer this question, further calculations are required. Details on these calculations are reported in the next section.
3. Results from variance decompositions Our purpose here is to utilize Lucas’ (1973) model and sample information in order to quantitatively assess the role of unanticipated aggregate-demand fluctuations in the generation of business cycles. To accomplish this task, we first rewrite eq. (4) as a regression equation, yc,t = -II6
+ IIAx, + Ayc,r-1 + % ,
(5)
where er is the regression residual with the classical properties. The variance of yC, r is given by vary,,=II*
varti,th2varyC,,_,
+2nhcov(Ax,,y,,-l)+varEf.
Restricting the analysis to the stationary tion is implied:
distribution
of yc,r, the following condi-
Eq. (6) gives a decomposition of the variance of cyclical real output. The equation can be used in order to assess the fraction of the variance of real output that is attributable to variance in unanticipated aggregate demand. Such assessment, however, is not unique: It depends on the treatment of the covariance term appearing in (6). On the one hand, it is possible to attribute this term to the unanticipated aggregate demand variable; alternatively, one can exclude it from unanticipated aggregatedemand factors. These two alternatives lead to the two different variance decompositions that are reported below.
128
L. Leiderman
Table 1 Decompositions
of variance
Country
d
Argentina Austria Belgium Canada Denmark W. Germany Guatemala Honduras Ireland Italy Netherlands Norway Paraguay Sweden United Kingdom United States
/ Unanticipated movements in aggregate demand
ofy,,t
(1953-1967).
% Variance
a
in yc,r explained
by:
(1)
(2)
var Axr
var Axr and cov(Axr,
0.19 23.98 53.64
3.34 2.54 72.31 86.05 65.13 69.13 c
b
61.96 86.65 75.99 25.79 55.96 70.46 76.82 b
0.76 17.07 29.05
b
yo,r-t)
12.41 42.03 74.64 57.15 63.47 3.88 27.93 37.46 74.24
a Calculations based on eq. (6), and on the information reported by Lucas (1973). Note: Column (1) = [fI’/(l - h2)]var Axr/var yc,r, Column (2) = [l/(1 - A2)]var l r/var yc,r. b Percentage greater than 100%; the covariance term is of ‘large’ negative magnitude. c Percentage is negative; the covariance term is of ‘large’ negative magnitude. d Results for Puerto Rico are not reported, because for this country Ihl > 1. Results for Venezuela are not reported: in the first column, category b obtains; and in the second column, category c.
Table 1 reports the variance decompositions of interest. As mentioned before, the calculations were performed by using eq. (6) and the relevant information reported by Lucas (1973). The results reported in the first column of table 1 correspond to the case in which the variance of unanticipated aggregate demand includes only the first term appearing in eq. (6). In the case of seven out of the thirteen entries reported in this column, more than 50 percent of the variance iny,, is attributable to variance in unanticipated aggregate demand. In the calculations reported in the second column, the variance of unanticipated aggregate demand includes also the second term appearing in (6). In this case, eight out of the fifteen reported entries obtain values above 50 percent. Thus, for fifteen out of the twenty-eight reported entries the variance in unanticipated aggregate demand accounts for more than 50 percent of the variance in cyclical output. Furthermore, there are seven countries (Belgium,
L. Leiderman / Unanticipated movements in aggregate demand
129
Canada, W. Germany, Guatemala, Italy, Netherlands, United States) for which in at least one case aggregate demand surprises account for more than 70 percent of the variance in their cyclical output. Overall, then, in most cases unanticipated fluctuations in aggregate demand account for a substantial part of the variance in cyclical real output. 4 In future work, it would be interesting to perform similar calculations on the basis of other equilibrium models present in the macro literature. Also of interest would be an examination of the sensitivity of the results above to changes in the particular definitions of unanticipated aggregate demand and of cyclical output utilized above. In the meantime, however, our findings reveal that unanticipated movements in aggregate demand have an important role in the generation of business cycles.
References Barro, R.J., 1976, Rational expectations and the role of monetary policy, Journal of Monetary Economics 2, Jan., l-32. Hall, R.E., 1975, The rigidity of wages and the persistence of unemployment, Brookings Papers 6. Lucas, Jr., R.E., 1973, Some international evidence on output-inflation tradeoffs, American Economic Review 63, June, 326-334. Lucas, R.E. and T.J. Sargent, 1978, After Keynesian macroeconomics, Manuscript, July. Modigliani, F., 1977, The monetarist controversy, or should we forsake stabilization policies?, American Economic Review 67, March, 1-19. Sargent, T.J., 1973, Rational expectations, the real rate of interest, and the natural rate of unemployment, Brookings Papers 4.
4 Note that in the cases of Argentina and Paraguay only a trivial fraction in the variance of yc r is due to variance in unanticipated aggregate demand. This result is in conformity with Lucas’ hypothesis regarding H: In volatile aggregate demand countries (such as Argentina and Paraguay) nominal income changes are mainly associated with price movements and with negligible effects on output.