The effect of recessions on the relationship between output variability and growth.

Olekalns, Nilss

Olan T. Henry (*)

Nilss Olekalns (+)

This paper investigates the relationship between output volatility and growth using postwar real GDP data for the United States. We expand on recent research by Beaudry and Koop (1993), documenting the asymmetric effect of recessions on output growth. The results presented in this paper suggest that output volatility is highest when the economy is contracting. While we find that the economy expands most rapidly following a recession, this expansion is offset by the negative impact of output uncertainty.

1. Introduction

Traditional approaches to understanding the dynamics of economic growth emphasized the role played by factor inputs. In contrast, the new empirical growth literature makes use of a wide variety of auxiliary explanatory factors. (1) In this paper, we focus on the role played by one of these auxiliary factors, namely, the part that recessions play in affecting the dynamics of real output growth. The paper reflects a recent trend in macroeconomics of bringing together analyses of the business cycle and of economic growth, areas that until comparatively recently have been treated by macroeconomists as separate. (2)

Our paper extends the idea, first found in Beaudry and Koop (1993), that the "current depth of recession" (hereafter CDR) produces an asymmetry in output growth. This asymmetry is reflected in what is sometimes known as a "bounce-back" effect, namely, that output growth recovers strongly following a recent recession. The CDR approach treats the historical maximum level of output as an attractor that influences the dynamics of output growth when output falls below its previous peak. Beaudry and Koop (1993) hypothesize that there is a nonlinearity in this "peak reversion"; the further output falls from its peak, the greater is the pressure that builds up for output to return to its historical maximum. As a result, the speed at which output recovers varies according to the severity of the recession.

The question of whether CDR asymmetries can be detected in the data has been examined by Beaudry and Koop (1993), Bradley and Jansen (1997), Bodman and Crosby (1998), and Jansen and Oh (1999). The empirical evidence, in general, provides support for the inclusion of CDR asymmetries in empirical models of economic growth dynamics. (3)

Our research differs from these previous analyses of CDR asymmetries in that we use an empirical specification that also allows for the possible existence of a relationship between output volatility and growth. Whether such a relationship exists has been the subject of debate at both the theoretical and the empirical level. Business cycle models based on the natural rate hypothesis deny the existence of any relation between output variability and growth. According to these models, deviations of output around its long-run trend are the result of signal extraction problems relating to workers' and firms' difficulties in distinguishing relative from absolute price level changes. Long-run output growth, being dependent on human and physical capital accumulation, is independent from these information difficulties (Friedman 1968). An alternate view emphasizes the increased riskiness associated with investment when output is volatile. This acts to retard real output growth (Woodford 1990). A third view argues that o utput volatility will be positively associated with economic growth since sufficient compensation to make risky investments worthwhile exists only when growth rates are high (Black 1987).

Attempts to distinguish between these views have tended to find support for a positive association between output volatility and average growth rates. This is true for studies based on international comparisons (Kormendi and Meguire 1985; Grier and Tullock 1989) as well as for analyses using high-frequency time-series data and GARCH-in-mean (GARCH-M) techniques (Caporale and McKiernan 1996, 1998). (4) A notable exception is the cross-country study by Ramey and Ramey (1995), who identify a negative volatility effect on growth.

Our paper brings these previously disparate strands of the literature together. We consider a CDR-GARCH-M time-series representation for U.S. output growth. This specification enables an investigation of CDR asymmetries in the data while allowing for the possibility of a volatility effect on output. Our specification also allows for the possibility that output volatility itself may be subject to an asymmetry as well.

The results in the paper strongly support the CDR-GARCH-M specification. We find that the economy tends to expand fastest in the period immediately after a recession. However, this expansion is offset by a negative effect of output volatility on growth; we find that the estimated conditional variance of output is highest in the periods following a negative innovation to growth and that this acts to dampen growth. (5)

The empirical model is subjected to a battery of diagnostic tests, including tests designed to detect bias due to the size and the sign of innovations and bias due to the omission of depth of recession effects. We find that the CDR-GARCH-M specification is not rejected by the data. In contrast, the more usual homoskedastic CDR model of output growth is revealed as being misspecified.

The paper is organized as follows. In section 2, we describe the CDR-GARCH-M methodology. The results are presented in section 3, and section 4 concludes the paper.

2. Methodology

Our starting point is the standard ARMA representation of output growth,

[THETA](L)[DELTA][y.sub.t] = [micro] + [PHI](L)[[member of].sub.t], (1)

where [DELTA] is the first-difference operator, [y.sub.t] is a measure of the natural logarithm of real output, [micro] is a term capturing any drift in growth, [[member of].sub.t] is an i.i.d. error term, and [THETA](L) and [PHI](L) are polynomials in the lag operator L. Equation 1 can be used to forecast the effect of an innovation on output out to some time horizon j according to [[SIGMA].aup.j.sub.i=1][[psi].sub.i][[member of].sub.i], where

[psi](L) = [PHI](L)/[THETA](L) = [summation over ([infinity]/i=0)] [[psi].sub.i][L.sup.i]. (2)

These forecasts will be conditional on there being a symmetric output response to positive and negative innovations. However, measures of shock persistence derived from Equation 1 and forecasts derived from Equation 2 will be biased if the data are not fully consistent with the symmetry assumption.

One way of relaxing the symmetry constraint is to follow Beaudry and Koop (1993) and augment Equation I with a measure of the CDR. This is defined as the gap between the current level of output and its historical maximum level, that is, [CDR.sub.t] = max[{[y.sub.t-s]}.sup.t.sub.s=0] - [y.sub.t] CDR will take nonzero values either when output dips below its trend value because of a negative shock or in the aftermath of a positive shock as output returns to trend. The implication of adding the CDR term to Equation 1 is that the conditional expectation of future output is influenced by whether the current level of output is above, below, or at its historical maximum.

With the introduction of the CDR term, Equation 1 is modified according to

[THETA](L)[DELTA][y.sub.t] = [micro] + {[PSI](L) - 1}[CDR.sub.t] + [PHI](L)[[member of].sub.t] (3)

where the lag polynomial [THETA](L) is of order p, [PHI](L) is of order q, and [PSI](L) is of order r with (0) = 1. This parameterization for [DELTA][y.sub.t] is very simple and nests the ARMA model (Eqn. 1) while allowing for the possibility of asymmetries associated with different stages of the business cycle.

Our concern in this paper is not just to identify an asymmetry in output growth but to determine whether output volatility affects growth once asymmetries are accounted for. Therefore, we model the conditional variance of output and the conditional mean using a modified GARCH-M specification. Conventional GARCH models allow for both "volatility clustering" (i.e., the tendency for periods of high [low] volatility to follow periods of high [low] volatility) and for the conditional variance to affect the conditional mean (Engle 1982; Bollerslev 1986; Engle, Lilien, and Robins 1987). However, unlike conventional GARCH models, we generalize the conditional mean and conditional variance to allow for the possibility that they are affected by the state of the business cycle. This leads to the following CDR-GARCH-M model:

[THETA](L)[DELTA][y.sub.t] = [micro] + {[PSI](L)- 1}[CDR.sub.t] + [lambda][square root of ([h.sub.t])] + [PHI](L)[[member of].sub.1] [[member of].sub.t] ~ N(0, [h.sub.t])

[h.sub.t] = [omega] + [alpha](L)[h.sub.t-1] + [beta](L)[[member of].sup.2.sub.t-1] + [gamma](L)[I.sub.t-1][[member of].sup.2.sub.t-1] (4)

where

[I.sub.t] = {1 if {max[{[y.sub.t-us]}.sup.t.sub.s=0] - [y.sub.t]} > 0

0 otherwise,

where the persistence of shocks to the conditional mean and to the conditional variance can vary according to the phase of the business cycle. In essence, the model proposes a threshold in the variance. Periods where the CDR term is nonzero will lead to higher variance of output compared with periods where the CDR term is zero, if the coefficient [gamma] is statistically significant. This model is an extension of the Threshold GARCH of Glosten, Jagannathan, and Runkle (1993).

3. Results

The data used are U.S. real GNP measured in billions of chained (1992) dollars, originally sourced from the Bureau of Economic Analysis and reformatted by Economic Information Systems. The data are quarterly and span the period 1947:1 to 1998:4. The upper panel of Figure 1 plots the quarterly growth rate in the data while the lower panel plots the CDR variable.

Table 1 reports maximum likelihood estimates of Equation 3 using the scheme proposed by Campbell and Mankiw (1987) to determine the order of the lag polynomials, p, q, and r. This involves setting the maximum values of p, q, and r to 3 and p + q + r = 6. The optimal model for each number of parameters was chosen using the Akaike (1974) and Schwarz (1978) information criteria.

The results in Table 1 are broadly consistent with those reported by Beaudry and Koop (1993), namely, that growth is higher in periods following recessions. Moreover, it is not possible to exclude the CDR term from Equation 3 on statistical grounds.

Given that the aim of this study is to examine the volatility of output over the business cycle, we move to the CDR-GARCH-M model discussed previously. Table 2 reports the maximum likelihood estimates of the model obtained using the quasi-maximum-likelihood approach of Bollerslev and Wooldridge (1992). The lag order in the conditional mean equation was determined initially by assuming that p = q = r = 3 and testing down. The preferred model was an ARMA(2, 2) p = q = 2, with one lag of the CDR term, r = 1, and a GARCH in mean coefficient [lambda].

The results of a Ljung-Box test on the standardized residuals, Q(4), suggest that the preferred model is free from serial correlation in the mean equation. Furthermore, the choice of a Threshold GARCH(1, 1) parameterization for the conditional variance equation appears appropriate. We also evaluate the adequacy of the GARCH specification by using Pagan and Sabau's (1992) moment-based test. This test has its basis in the implication derived from a GARCH model that E([[member of].sup.2.sub.t] = [h.sub.t]. The test is based on the satisfaction of the restriction [H.sub.0]:[[delta].sub.0] = 0, [delta] = 1, in the auxiliary regression [[member of].sup.2.sub.t] = [[delta].sub.0] + [[delta].sub.1][h.sub.t] + [[upsilon].sub.t], where [[upsilon].sub.t] represents a white-noise innovation. As reported in Table 2, the estimated CDR-GARCH model satisfies the moment condition at all usual levels of significance. Moreover, on the basis of a Ljung-Box test for serial correlation in the squared, standardized residuals, [Q.sup.2] (4), we are unable to detect evidence of misspecification in the conditional variance.

The CDR-GARCH-M model collapses to a CDR model under the restrictions [lambda] = [alpha] = [beta] = [gamma] = 0. However, these restrictions are overwhelmingly rejected using a Wald test. The test statistic is 43.1249, which under a [chi square](4) distribution indicates that the marginal significance of the test is 0.0000. The CDR-GARCH-M model appears to offer a superior conditional data characterization to the standard CDR model of Beaudry and Koop (1993).

The positive and significant coefficient associated with [CDR.sub.t-1] in the conditional mean equation indicates the asymmetry in economic growth, that is, growth is fastest in the recovery phase of a recession. The model also displays evidence of an asymmetric volatility response to innovations in the growth rate. That is, periods where the CDR term is nonzero lead to higher levels of output volatility than periods where the CDR term is zero. This suggests that periods where output has fallen below its historical maximum are inherently more volatile than booms.

This asymmetry in volatility is clearly depicted in Figures 2 and 3. These figures show the news impact curves, depicting the relationship between innovations to growth and the volatility of output, holding information constant at time t - 1 and before. (6) The equation for the GARCH(1, 1) news impact curve is [h.sub.t] = A + [beta][[member of].sup.2.sub.t-1], where A = [omega] + [alpha][h.sub.t-1]. In our model, the news impact curve during nonrecessionary periods, that is, when [CDR.sub.t] = 0 is [h.sub.t] = A + [beta][[member of].sup.2.sub.t-1]. In recessionary periods, that is, when [CDR.sub.t] > 0, the news impact curve is [h.sub.t] = A + ([beta] + [gamma])[[member of].sup.2.sub.t-1]. (7)

The simulated news impact curve in Figure 2 is consistent with the absence of a relationship between news about growth and the volatility of output during nonrecessionary periods. Conversely, in recessionary periods, the significance of the [gamma] parameter results in large innovations in growth being associated with large values of [h.sub.t]. The significant and negative coefficient on the GARCH in mean term points toward a negative volatility feedback on growth. The news impact curve confirms that this feedback will be strongest after a period of recession. In effect, recessions result in increased output uncertainty, which in turn serves to retard recovery following unfavorable news about output.

All these conclusions are dependent on the model we have selected being an adequate representation of the data-generating process. The Ljung-Box and Pagan-Sabau tests reported previously suggest that our model specification is appropriate. However, there is the potential for bias due to asymmetric response to the sign or magnitude of the innovation, [[member of].sub.t], or to bias associated with periods where the CDR term is nonzero.

To identify potential misspecification of the conditional variance due to asymmetric response to innovations in growth, [[member of].sub.t], we calculated Engle and Ng's (1993) test for size and sign bias in conditionally heteroskedastic models. Define [S.sup.-.sub.t-1] as an indicator dummy that takes the value of 1 if [[member of].sub.t-1] < 0 and the value zero otherwise. The test for sign bias is based on the significance of [[phi].sub.1] in

[[member of].sup.2.sub.1] - [[phi].sub.0] + [[phi].sub.1][S.sup.-.sub.t-1] + [v.sub.t], (5)

where [v.sub.t] is a white-noise error term. If positive and negative innovations in [[member of].sub.t] impact on the conditional variance of growth differently to the prediction of the model, then [[phi].sub.1] will be statistically significant. It may also be the case that the source of the bias is caused not only by the sign but also by the magnitude of the shock. The negative size bias test is based on the significance of the slope coefficient [[phi].sub.1] in

[[member of].sup.2.sub.t] = 0 [[phi].sub.1][S.sup.-.sub.t-1][[member of].sub.t-1] + [v.sub.t]. (6)

Likewise, defining [[S.sup.+.sub.t-1]] = 1 - [S.sup.-.sub.t-1], then the Engle and Ng (1993) joint test for asymmetry in variance is based on the regression

[[member of].sup.2.sub.t] = 0 + [[phi].sub.1][S.sup.-.sub.t-1] + [[phi].sub.2] [S.sup.-.sub.t-1] [[member of].sub.t-1] + [[phi].sub.3][S.sup.+.sub.t-1][[member of].sub.t-1] + [v.sub.t], (7)

where [v.sub.t] is a white-noise disturbance term. Significance of the parameter [[phi].sub.1] indicates the presence of sign bias. That is, positive and negative realizations of [[member of].sub.t] affect future volatility differently to the prediction of the model. Similarly, significance of [[phi].sub.2] or [[phi].sub.3] would suggest size bias, where not only the sign but also the magnitude of innovation in growth is important. A joint test for sign and size bias, based on the Lagrange multiplier principle, may be performed as T.[R.sup.2] from the estimation of Equation 7. The results in Table 2 suggest that the model is free from size and sign bias.

Finally, we test for bias in the conditional variance arising from the failure to adequately capture the effects of recessions. Again, this test may be performed using a Lagrange multiplier approach based on the auxiliary regression

[[member of].sup.2.sub.t] = [[phi].sub.0] + [[phi].sub.1][I.sub.t-1] + [[phi].sub.2][CDR.sub.t-1] + [v.sub.1], (8)

where [I.sub.t] = 1 if {max[{[y.sub.t-s]}.sup.t.sub.s=0] - [y.sub.t]} > 0 and is zero otherwise. In other words, [I.sub.t] captures periods when the economy is "in recession." The CDR term is used to capture biases due to the depth of the recession. The test is based on the restrictions [[phi].sub.1] = 0, [[phi].sub.2] = 0, and is distributed as a [chi square](2). The test statistic of 1.9836 indicates a marginal significance level of 0.3709. There is no evidence of bias due to recessions in the model.

4. Conclusion

In common with the results of Beaudry and Koop (1993), Bradley and Jansen (1997), and Jansen and Oh (1999), the evidence in this paper suggests that U.S. economic growth differs according to whether the economy's level of output has fallen relative to its historical maximum. However, we demonstrate that the CDR-GARCH-M model provides a superior conditional characterization for U.S. postwar GNP to the standard CDR model. This indicates that the volatility of U.S. economic growth is also affected by the level of output relative to the historical maximum; contractionary periods tend to be more volatile than expansions of similar magnitude. The CDR-GARCH-M model passes a battery of standard specification tests as well as a series of LM tests designed to detect bias due to size and sign of innovation and bias due to recessions.

Our results provide further support for the view that there exists a significant asymmetry in the U.S. growth rate, with growth accelerating as the economy recovers from recessions. However, we also document an increase in output volatility that accompanies recessions and that acts to dampen subsequent growth. A theoretical explanation for why an increase in output volatility might accompany a recession is a topic worthy of future research.

(*.) Department of Economics, University of Melbourne, Victoria, 3010, Australia; E-mail oth@unimelb.edu.au.

(+.) Department of Economics, University of Melbourne, Victoria, 3010, Australia; E-mail nilss@unimelb.edu.au; corresponding author.

The authors thank Michelle Barnes, participants at the Fifth Annual Australian Macroeconomics Workshop, and an anonymous referee for their comments on earlier drafts on this paper. Any errors in the paper are our responsibility.

Received June 2000; accepted November 2000.

(1.) This point is made by Durlauf and Quah (1999) in a recent survey article.

(2.) For a discussion of this trend and a list of seminal papers, see Ramey and Ramey (1995).

(3.) Beaudry and Koop (1993) fit their model to U.S. real GNP data over the period 1947:1 to 1989:4 and find that it performs better than a linear autoregressive model. Further, they construct impulse response functions and conclude that innovations to GNP are more persistent in expansions than in recessions. Bradley and Jansen (1997) fit the CDR model to the other G7 countries and find that the model is supported by the data, and Jansen and Oh (1999) also find that the CDR model outperforms a smooth transition regression model when fitted to a measure of U.S. industrial production. Bodman and Crosby (1998) find no evidence of CDR asymmetries for Australia.

(4.) The robustness of Caporale and McKiernan's (1996) results has been questioned by Speight (1999).

(5.) We are not aware of any previous study that has documented the effect of recessions on output volatility.

(6.) Because the GARCH(1, 1)I formulated in the squares [[member of].sub.t] of positive and negative shocks are treated in the same way. The relationship between [[member of].sup.2.sub.t] and [h.sub.t] is known as the news impact curve.

(7.) Our use of the terms "recession" and "nonrecession" differs from their normal usage. For exposition, we will restrict the use of the term "recession" to mean a period in which [CDR.sub.t] > 0.

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[Figure 1 omitted]

[Figure 2 Omitted]

[Figure 3 omitted]

Table 1

Parameter Estimates--CDR Models

[THETA](L)[DELTA][y.sub.t] = [micro] + {[PSI](L) - 1} [CDR.sub.t] +
[lambda] [square root of ([h.sub.t])] + [PHI](L)[[member of].sub.t]

 Model
 (1, 0, 0) (0, 2, 0) (2, 0, 1) (2, 0, 2)

[micro] 0.0052 0.0081 0.0018 0.0021
 (0.0007) (0.0010) (0.0007) (0.0007)
-[[THETA].sub.1] 0.3500 -- 0.4337 0.3486
 (0.0527) -- (0.0526) (0.0529)
-[[THETA].sub.2] -- -- 0.1818 0.2207
 -- -- (0.0532) (0.0530)
[[PHI].sub.1] -- 0.3045 -- --
 -- (0.0518) -- --
[[PHI].sub.2] -- 0.1855 -- --
 -- (0.0527) -- --
[[PHI].sub.3] -- -- -- --
 -- -- -- --
[[PSI].sub.1] -- -- 0.3339 0.1452
 -- -- (0.0694) (0.0685)
[[PSI].sub.2] -- -- -- 0.2111
 -- -- -- (0.0788)
[[PSI].sub.3] -- -- -- --
 -- -- -- --

 Model
 (2, 0, 3) (2, 3, 1)

[micro] 2.4940 (a) 0.0014
 (6.7567) (b) (0.0027)
-[[THETA].sub.1] 0.3683 1.2536
 (0.0529) (0.2927)
-[[THETA].sub.2] 0.1454 -0.4951
 (0.0538) (0.3862)
[[PHI].sub.1] -- -0.9227
 -- (0.3097)
[[PHI].sub.2] -- 0.3876
 -- (0.3639)
[[PHI].sub.3] -- -0.0005
 -- (0.1557)
[[PSI].sub.1] 0.2030 0.1548
 (0.0660) (0.1281)
[[PSI].sub.2] -0.0384 --
 (0.0742) --
[[PSI].sub.3] 0.2119 --
 (0.0705) --

Standard errors are in parentheses.

(a)X 10[e.sup.-3].

(b)X 10[e.sup.-4].
Table 2

Estimates of the CDR-GARCH(1, 1)-M Model

[THETA](L)[DLETA][y.sub.t] = [micro] + {[PSI][L] - 1} [CDR.sub.t] +
[lambda] [square root of ([h.sub.t])] + [PHI](L)[[member of].sub.t]
[[member of].sub.t] ~ N(0, [h.sub.t]); [h.sub.t] = [omega] +
[alpha](L)[h.sub.t-1] + [beta](L) [[member of].sup.2.sub.t-1] +
[gamma](L)[I.sub.t-1] [[member of].sup.2.sub.t-1]

where

[I.sub.t] = {1 if {max[{[y.sub.t-s]}.sup.t.sub.s=0] - [y.sub.t]} > 0, {0
otherwise.

[micro] -[[THETA].sub.1] -[[THETA].sub.2] [[PSI].sb.1] [lambda]

 0.0048 0.5898 -0.0243 0.2032 -0.2430
(0.0010) (0.0943) (0.0943) (0.0646) (0.1078)

[micro] [[PSI].sub.1] [[PSI].sub.2]

 0.0048 -0.1094 0.1157
(0.0010) (0.1068) (0.0756)
[omega] [alpha] [beta] [gamma] L

 0.00003 0.5030 0.0086 0.4958 853.0649
(0.00001) (0.1102) (0.0664) (0.1978)
 Q(4) [Q.sup.2](4) P - S N-sign N-size Joint

 0.7800 5.6853 2.6835 -0.3228 0.5906 0.8082
[0.9411] [0.2239] [0.2674] [0.7472] [0.5555] [0.8475]

Standard errors are in parentheses. Marginal significance levels are in
brackets.