# Business cycle asymmetry: a deeper look.

I. INTRODUCTION

The behavior of macroeconomic variables over phases of the business cycle has long been an object of interest to economists. A critical aspect of this is the symmetry or asymmetry of business cycles. An asymmetric cycle is one in which some phase of the cycle is different from the mirror image of the opposite phase; for example, contractions might be steeper, on average, than expansions. Although asymmetries were noted by early business cycle researchers, the issue has only recently been examined empirically.(1) This recent interest arises from a desire to carefully document the stylized facts of business fluctuations and because linear structural and time series models cannot represent asymmetric behavior under standard assumptions.

Neftci |1984~(2) and DeLong and Summers |1986~ reported evidence that increases in the unemployment rate are steeper than decreases. The evidence on the asymmetry of real GNP is less clear, however. Falk |1986~--using Neftci's procedure--found that real GNP does not exhibit this type of asymmetry. However, Hamilton |1989~ showed that a particular nonlinear (asymmetric) model for real GNP growth rates dominates linear models.

All of the above research has focused on asymmetries--or nonlinearities--in the rate of change of business cycle variables; that is, these researchers compare periods of increase to periods of decrease. Brock and Sayers |1988~ expanded the search for asymmetry by applying a test that identifies any form of nonlinearity or asymmetry. Brock and Sayers find evidence of nonlinear structure--in the postwar period--in employment, unemployment, and industrial production. One problem with their test, however, is that it cannot distinguish among different types of asymmetries.

This paper sharpens the asymmetry evidence by distinguishing two types of asymmetry that could exist separately or simultaneously.(3) The first type of asymmetry--which has been investigated in most of the research described above--occurs when contractions are steeper than expansions. I refer to this type of asymmetry as steepness. The second--and as yet not explicitly considered--type of asymmetry occurs when troughs are deeper than peaks are tall. I refer to this type of asymmetry as deepness.

These two types of asymmetry are illustrated in Figure 1 for a trendless time series. The first panel shows a symmetric cycle. The second panel shows a cycle exhibiting steepness, which pertains to relative slopes or rates of change and compares mirror images across imaginary vertical axes placed at peaks and troughs.(4) The third panel shows a cycle exhibiting deepness, which pertains to relative levels and compares mirror images across a horizontal axis (the dashed line in the figure). The final panel in the figure shows a cycle exhibiting both deepness and steepness.

In section II, I discuss the implications of asymmetry. Tests for deepness and steepness are presented in section III, and detrending is discussed. In section IV, I provide evidence of deepness in unemployment and industrial production; the evidence for real GNP is weaker. Previous evidence of steepness in unemployment is also confirmed. In section V, I discuss Monte Carlo simulations that demonstrate that these tests are actually able to identify different types of asymmetry. Section VI concludes.

II. IMPLICATIONS OF ASYMMETRY

Asymmetry in business cycle time series is important for macroeconometrics because linear and Gaussian models are incapable of generating asymmetric fluctuations.(5) Consider a variable |x.sub.t~, generated by a linear, Gaussian, and stationary autoregressive moving average (ARMA) process, and its infinite moving average representation:

(1a) A(L) ||chi~.sub.t~ = B(L) ||epsilon~.sub.t~

(1b) ||chi~.sub.t~ = |A.sup.-1~(L) B(L) ||epsilon~.sub.t~

where A(L) and B(L) are finite polynomials in the lag operator L and ||epsilon~.sub.t~ is an i.i.d. Gaussian disturbance. Since ||epsilon~.sub.t~ has a symmetric distribution, then the process ||chi~.sub.t~--a linear combination of these disturbances--is also symmetric. Therefore, a linear and Gaussian ARMA model cannot, asymptotically, represent asymmetric behavior.

Evidence of asymmetry, however, does not point toward any particular business cycle model. Rather, asymmetry simply provides evidence against the class of linear models with symmetric disturbances. Models outside of this class include nonlinear endogenous cycle models--as discussed in Boldrin and Woodford |1990~--and nonlinear stochastic models as in Hamilton |1989~.

Even though many models can generate deep and steep cycles, further intuition about these concepts is gained by considering simple models capable of generating deep and steep cycles. Deepness can be generated by a model with asymmetric price adjustment. For example, suppose prices rise rapidly above their expected level when output is above potential, but fall slowly when output is below potential. Then--starting from potential output--a positive nominal demand shock will push up prices, but will have a relatively small effect on output. In contrast, a negative nominal demand shock will have relatively larger impact on output than on prices.(6) Steepness can be generated by models with asymmetric costs of upward and downward adjustment. For example, Chetty and Heckman |1985~ and Baldwin and Krugman |1986~ present models in which exit from an industry is less costly than entry. Hence, production can fall rapidly, but it expands more slowly.

III. TESTING FOR DEEPNESS AND STEEPNESS

Tests

Consider a time series

(2) |y.sub.t~ = ||tau~.sub.t~ + |c.sub.t~

where ||tau~.sub.t~ is the nonstationary trend component and |c.sub.t~ is the stationary cyclical component. In this paper, I focus on whether |c.sub.t~--the cyclical component--is asymmetric, because secular growth is asymmetric a priori (almost always increasing). If |y.sub.t~ is nonstationary, analysis of |c.sub.t~ requires detrending, which is discussed at length in the following subsection. In this subsection, I describe tests for deepness and steepness that are to be applied to |c.sub.t~; that is, after the series |y.sub.t~ has been detrended.

If a time series exhibits deepness, then it should exhibit negative skewness relative to mean or trend; that is, it should have fewer observations below its mean or trend than above, but the average deviation of observations below the mean or trend should exceed the average deviation of observations above. To construct a test for deepness, I use the coefficient of skewness. Compute

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the mean of |c.sub.t~, |sigma~(c) is the standard deviation of |c.sub.t~, and T is the sample size.

Since the observations on |c.sub.t~ are sure to be serially correlated, the formula for the asymptotic standard error of the coefficient of skewness of an i.i.d. random variable is inapplicable. However, an asymptotic standard error of D(c) can be computed using a procedure suggested by Newey and West |1987~ as follows. Construct a variable with the tth observation equal to

|Mathematical Expression Omitted~

Regress this variable on a constant and compute the Newey-West standard error. The estimate of the constant in this regression is identical to D(c) in (3a), and the Newey-West standard error is consistent even in the presence of heteroscedasticity or serial correlation.(7) Further, the constant term divided by its standard error is asymptotically normal. Hence, conventional critical values can be used to test the significance of D(c).(8)

If a time series exhibits steepness, then its first differences should exhibit negative skewness. That is, the sharp decreases in the series should be larger, but less frequent, than the more moderate increases in the series. Hence, a test for steepness can be computed by using the coefficient of skewness for |delta~|c.sub.t~, the first difference of the cyclical component in equation (2):

|Mathematical Expression Omitted~

where, |Mathematical Expression Omitted~ and |sigma~(|delta~c) are the sample mean and standard deviation of |delta~c.(9) This test examines whether rates of change in |c.sub.t~ are asymmetric around their mean. An asymptotic standard error for the steepness test can be calculated analogously to the deepness test.

Detrending

For variables exhibiting secular growth, the asymmetry tests are applied to detrended series because secular growth is asymmetric a priori (almost always increasing). Although detrending has been quite controversial in macroeconomics,(10) I argue that any detrending filter that satisfies three requirements is appropriate for an analysis of asymmetry. The three requirements are sufficient to ensure that the detrending procedure itself is not inducing spurious asymmetry and that the asymmetry tests |D(c) and ST(|delta~c)~ have standard distributions and a natural interpretation.

The first requirement is that the detrending filter has a linear representation. Linearity is critical because a linear filter applied to a symmetric time series yields a symmetric time series.(11) Therefore, a linear filter cannot induce asymmetry if none is present to begin with. The second requirement is that the filter must induce stationarity. If the extracted cyclical component is nonstationary, then the asymmetry tests might have nonstandard distributions, substantially complicating inference. Finally, the detrending filter used for each test must extract the component appropriate for that asymmetry test; that is, a filter used for a deepness test must extract |c.sub.t~, while a filter used for a steepness test must extract |delta~|c.sub.t~. Otherwise, the interpretation of the test statistics is confounded; the deepness test might not correctly identify actual deepness, and the steepness test might not correctly identify actual steepness.

The first requirement--that the filters have a linear representation--is satisfied by many procedures that have been proposed for detrending. I consider four such procedures: simple linear detrending, first-differencing, the Beveridge-Nelson decomposition |1981~,(12) and the Hodrick-Prescott filter |1982~.(13) Since these procedures all have linear representations, none of these filters can induce asymmetry if none is present in the original series. However, these procedures would be appropriate for the deepness and steepness tests only if they satisfy the second and third requirements spelled out above. Namely, that they induce stationarity and extract the correct component, either |c.sub.t~ or |delta~|c.sub.t~. To determine whether these procedures satisfy these last two requirements, I consider the example of detrending real GNP.

Consider simple linear detrending. This procedure might not satisfy the second requirement spelled out above--that the filter induce stationarity. For example, if real GNP is difference stationary, simple linear detrending would not induce stationarity. Although it is an open question as to whether GNP is difference or trend stationary, the reasonable possibility that it is difference stationary raises questions about the use of simple linear detrending. Therefore, I do not use simple linear detrending for the deepness test or the steepness test.

Unlike simple linear detrending, first-differencing almost surely induces stationarity in real GNP, thus likely satisfying the second requirement. The third requirement is that the detrending procedure used for each test extract the appropriate component for that test. First-differencing extracts |delta~|c.sub.t~,(14) which is not the appropriate component for the deepness test, but is the appropriate component for the steepness test. Therefore, I do not use first-differencing for the deepness test, but I do use first-differencing for the steepness test.

Turning to the Beveridge-Nelson decomposition, if real GNP were difference stationary, this procedure would induce stationarity, thus satisfying the second requirement. As for the third requirement, this procedure extracts the component |c.sub.t~, which is the needed component for the deepness test. Therefore, I use the Beveridge-Nelson decomposition for the deepness test. For the steepness test, the needed component is |delta~|c.sub.t~. Although the Beveridge-Nelson decomposition does not directly extract |delta~|c.sub.t~, it would be possible to extract |c.sub.t~ with the Beveridge-Nelson decomposition and then first difference. This procedure, however, seems redundant, since just first-differencing extracts the desired component for the steepness test. Therefore, I do not report results for the steepness test using the Beveridge-Nelson decomposition.

Finally, consider the Hodrick-Prescott filter. Cogley and Nason |1991~ note that this filter would induce stationarity in a series whether it is trend or difference stationary. That is, even in the presence of a unit root, the Hodrick-Prescott filter would induce stationarity in real GNP, and thus the filter almost surely satisfies the second requirement. As to the third requirement, the appendix demonstrates that this filter yields an explicit expression for |c.sub.t~, the appropriate component for the deepness test. Therefore, I use it for the deepness test. For the steepness test, the Hodrick-Prescott filter suffers from the same redundancy as the Beveridge-Nelson decomposition. Therefore, I do not report results for the steepness test using this filter.

Before leaving the Hodrick-Prescott filter, it is necessary to discuss the recent critique of the filter leveled by Cogley and Nason. Using spectral analysis, Cogley and Nason demonstrate that the Hodrick-Prescott filter--when applied to a difference stationary series--strongly amplifies fluctuations in the series at business cycle frequencies. They point out that this can create serious problems for many types of business cycle analysis; for example, applying the filter to two series might induce a spurious correlation between the series at business cycle frequencies when, in fact, the series are uncorrelated at business cycle frequencies. However, this characteristic of the filter--amplification of fluctuations at business cycle frequencies--is actually desirable for an analysis of business cycle asymmetry. As a linear filter, the Hodrick-Prescott filter can not induce asymmetry in a series if none is present to begin with. Thus, if asymmetry is found, it must be present in the original series. Furthermore, since asymmetries at business cycle frequencies are of particular interest, a filter that amplifies fluctuations at these frequencies is ideal. So, while Cogley and Nason suggest that the Hodrick-Prescott filter probably should not be used for many types of analysis, the filter seems particularly well suited for the asymmetry tests in this paper.

To summarize the discussion in this section, I will use the Beveridge-Nelson decomposition and the Hodrick-Prescott filter for the deepness test. For the steepness test, I will detrend by first-differencing.

IV. EMPIRICAL RESULTS

The deepness test is computed for three U.S. post-war quarterly time series from 1949:I to 1989:IV--unemployment, real GNP in 1982 dollars, and industrial production. Values of the test statistics, standard errors, and one-sided p-values are shown in the top panel of Table I for both the Hodrick-Prescott filter and the Beveridge-Nelson decomposition.

Using the Hodrick-Prescott filter, unemployment, industrial production, and real GNP exhibit deepness at the .04, .06, and .12 significance level, respectively. I interpret this as fairly strong evidence of deepness for unemployment and industrial production, but as fairly weak evidence for real GNP.(15) Corroborating visual evidence of deepness is shown in Figures 2 to 4, which plot deviations from the Hodrick-Prescott trend for log real GNP, log industrial production, and the unemployment rate. Figure 2 confirms the fairly weak evidence of deepness for real GNP. While troughs do look somewhat deeper than peaks, the visual evidence is not overwhelming. Deepness is much more apparent for industrial production shown in Figure 3; troughs look notably deeper than peaks. Since the unemployment rate--shown in Figure 4--is countercyclical, deepness would appear as very high sharp peaks coinciding with troughs in the business cycle. Indeed, Figure 4 shows just such a pattern, visually confirming the evidence of deepness for the unemployment rate.(16)

Using the Beveridge-Nelson decomposition, unemployment, industrial production, and real GNP exhibit deepness at the .02, .35, and .27 significance level, respectively.(17) For unemployment, the results are similar to those using the Hodrick-Prescott filter. For industrial production and real GNP, the Beveridge-Nelson decomposition provides much weaker evidence of deepness than the Hodrick-Prescott filter.

While it would be comforting if the Hodrick-Prescott filter and the Beveridge-Nelson decomposition yielded the same evidence for or against deepness, it is not surprising that the evidence is weaker for industrial production and real GNP with the Beveridge-Nelson decomposition than with the Hodrick-Prescott filter. As discussed above, the filter amplifies fluctuations at business cycle frequencies. This amplification should make it easier to identify asymmetries at these frequencies with the Hodrick-Prescott filter than with many other filters. Therefore, I do not regard the weaker evidence for deepness from the Beveridge-Nelson decomposition as evidence against deepness, but rather as an indication that certain filters are more likely to highlight asymmetries at business cycle frequencies.

Finally, the following heuristic example helps shed some light on the magnitude of the deepness present in unemployment and industrial production. Consider a single sixteen observation deterministic cycle generated by the sinusoid in (5):

(5) ||chi~.sub.t~ = -A1 Cos |(|pi~/8)t~

t=1,2,3,4,13,14,15,16

||chi~.sub.t~ = -A2 Cos |(|pi~/8)t~

t=5, ..., 12.

If A2 and A1 equal unity, this sinusoid is symmetric and its maximum and minimum values are plus and minus unity, respectively. As the ratio A2/A1 increases, deepness is induced. Table II shows the value of D(|chi~) for these sixteen observations and the ratio of the maximum distance below the mean to the maximum distance above.(18) These values can be compared to empirically observed values of deepness shown in Table I. For example, the actual value of the deepness test for industrial production is -.73, when the Hodrick-Prescott filter is used. For this value, the trough of the sinusoid is about 1.8 times further below the mean than the peak is above the mean. This ratio seems large enough to warrant interest.

Returning to Table I, the second panel shows the steepness test results. Based on one-sided p-values, unemployment exhibits evidence of steepness, but real GNP and industrial production do not. This confirms results presented by DeLong and Summers |1986~ and Falk |1986~.(19)

V. ARE DEEPNESS AND STEEPNESS REALLY DIFFERENT?

Since more general tests for nonlinear structure--such as the test in Brock and Sayers |1988~--are available, a natural question is whether the deepness and steepness tests actually are identifying different types of asymmetry. Unless the deepness and steepness tests can make this distinction, more general tests might as well be used.

To investigate this question, I conducted a set of Monte Carlo experiments to see whether a deepness test would mistakenly identify steepness as deepness and vice-versa. First, I generated 500 replications of pseudo-data with nearly the amount of deepness found in the actual unemployment rate after application of the Hodrick-Prescott filter.(20) To each replication of pseudo-data, I applied the deepness and steepness tests. As it turned out, the deepness test identified deepness at the 5 percent significance level in more than one-third of the replications, while evidence of steepness was almost never found. Second, I generated 500 replications of pseudo-data with nearly the amount of steepness found in the actual unemployment rate. Again, I applied the deepness and steepness tests to each replication. For these data, the steepness test identified steepness at the 5 percent significance level in about two-thirds of the replications, while evidence of deepness was almost never found.

These simulations suggest that the deepness and steepness tests are able to identify different types of asymmetry and that asymmetry could be missed if only one test were applied. Therefore, distinguishing between these two types of asymmetry likely provides a sharper characterization of asymmetry than earlier work.

VI. CONCLUSION

This paper distinguishes two types of asymmetry in business cycles: deepness and steepness. Deepness characterizes fluctuations that have troughs deeper below trend than peaks are high, while steepness--which has been considered explicitly in some previous research--describes fluctuations with steeper contractions than expansions. A test for deepness is presented, which is extended in a natural way to provide a test for steepness. Monte Carlo simulations suggest that the deepness and steepness tests in this paper are actually able to identify different types of asymmetry.

The deepness test provides evidence that deepness is present in U.S. postwar quarterly unemployment and industrial production; that is, troughs are deeper than peaks. The evidence of deepness for real GNP is weaker. Consistent with some previous research, evidence of steepness is found only for unemployment.

The pattern of asymmetry across variables raises at least two interesting questions. First, the stronger presence of deepness in industrial production than in real GNP suggests that deepness is more prevalent in the production sector than elsewhere in the economy.(21) Second--focusing on the two variables covering the whole economy--the much stronger evidence of deepness and steepness in unemployment than in real GNP suggests that there is a nonlinearity in Okun's law.(22)

The presence of deepness and steepness in unemployment and deepness in industrial production suggests that standard linear structural or time series models with symmetric disturbances cannot represent the observed stylized facts for these variables. Nonlinear models or models with asymmetric disturbances might be more appropriate. Directions for future work include development of more general and more powerful tests of asymmetry and multivariate versions of these tests.(23) It is also important to consider further the types of theoretical models able to generate deepness and steepness.

APPENDIX

For a time series |y.sub.t~, the Hodrick-Prescott filter is obtained by finding the functional, ||delta~.sub.t~, that satisfies the penalized least squares minimization program,

|Mathematical Expression Omitted~

where L is the lag operator and |lambda~ can be interpreted as a Lagrange multiplier. The filter is obtained from this minimization by setting |lambda~ equal to 1600. When the Hodrick-Prescott filter is used in the paper, the values of the functional, ||delta~.sub.t~ are used as the values of the non-stationary trend component, ||tau~.sub.t~, that appears in equation (2). The cyclical component, |c.sub.t~, is then computed as |y.sub.t~ - ||delta~.sub.t~.

Note that if |lambda~ equals infinity, the sum of the squared second-differences of ||delta~.sub.t~ must be zero, meaning that ||delta~.sub.t~ must be the ordinary least squares linear trend. If |lambda~ equals zero, so that the smoothness constraint is non-binding, then ||delta~.sub.t~ perfectly interpolates the time series |y.sub.t~. As these extreme cases suggest, the choice of |lambda~ does affect the frequency of oscillations that pass through the filter. With |lambda~ = 1600, the filter is similar to a high-pass filter that removes oscillations with periodicity greater than thirty-two quarters or eight years; hence, the computed cyclical component (|y.sub.t~ - ||delta~.sub.t~) has been largely purged of these low frequency oscillations.

1. Keynes |1936~ and Burns and Mitchell |1946~ suggest that business cycles are asymmetric. Kaldor |1940~ and Hicks |1950~ provide examples of deterministic models yielding asymmetric cycles.

2. Sichel |1989~ identified a mistake in Neftci's empirical work that reversed his findings, and showed that Neftci's test has fairly low power. However, Rothman |1991~ showed that Neftci's evidence is resurrected if a first-order Markov process is used rather than the second-order Markov process used by Neftci.

3. A recent paper by Gerard Pfann also distinguishes different types of asymmetry.

4. The stylized pictures suggest a relationship between asymmetry and time deformation as in Stock |1987~. Stock's analysis of time deformation over the business cycle examines whether there is time deformation across expansions and contractions. This corresponds to steepness.

5. The argument is only outlined here. For a formal proof, see Blatt |1980~.

6. The asymmetric price adjustment example appears in DeLong and Summers |1988~. These authors also discuss other models capable of generating deepness. Ball and Mankiw |1991~ also present a model of asymmetric price adjustment that generates deepness.

7. In constructing the heteroscedasticity and serial correlation consistent covariance matrix, I allowed for serial correlation of up to order five. As loosely suggested by Newey and West, this order of correlation is set to equal the integer part of the sample size raised to the 1/3 power.

8. An earlier version of this paper used a bootstrap procedure to obtain a small-sample distribution of the deepness test statistic. That procedure, however, requires a variety of auxiliary assumptions and is notably more complicated to compute than the Newey-West standard error. Although the bootstrap distribution provided somewhat stronger evidence of deepness than the asymptotic distribution used in this paper, the simplicity and generality of the Newey-West procedure is quite advantageous.

9. This steepness test is the same as the test in DeLong and Summers |1986~, except for the computation of the standard error of the statistic.

10. See for example, Nelson and Plosser |1982~ and Perron |1990~.

11. This can be seen by applying some linear filter, say C(L), to ||chi~.sub.t~ in equation (1b). If ||chi~.sub.t~ is symmetric, then C(L)||chi~.sub.t~ is also symmetric because it is still a weighted sum of symmetric disturbances.

12. The Beveridge-Nelson decomposition is linear, conditional on the estimated

parameters of the ARIMA process used for the decomposition.

13. The derivation of the Hodrick-Prescott filter is outlined in the appendix. For the linear representation of the Hodrick-Prescott filter, see Cogley and Nason |1991~. Note also that this filter is nearly identical to certain cubic splines developed in the numerical analysis literature. For example, see Reinsch |1967~.

14. To see this, consider equation (2) with |c.sub.t~ a stationary process and ||tau~.sub.t~ either a random walk with drift or a simple linear trend. In either case, first-differencing extracts |delta~|c.sub.t~ plus the drift in ||tau~.sub.t~. Since the asymmetry tests focus on third moments around means, this drift term will not affect any of the asymmetry tests.

15. Neftci |1984~ used a confidence bound that corresponds to a .10 significance level for the test used here.

16. Figures 2-4 suggest that the trough in 1949:IV might be unduly influential and that the deepness evidence may be sensitive to the sample period. However, the strength of the deepness evidence only drops slightly for unemployment and industrial production over the sample 1952:I-1989:IV.

17. For each series, the Beveridge-Nelson decomposition is based on an ARIMA(2,1,2) estimated on the unemployment rate, log industrial production, and log real GNP.

18. As A2 increases, the mean of the sinusoid becomes negative. The ratios of maximum values in table II are relative to this mean.

19. In contrast to the weak evidence for steepness in real GNP in this paper, Hamilton's results do imply steepness. Hamilton's sample period extends from 1952:II to 1984:IV. Over that shorter sample period, the test in this paper finds more evidence of steepness in real GNP than in the longer sample.

20. The data for the Monte Carlo experiments were generated using Markov-switching models similar to the model in Hamilton |1989~. Details of the data generation are available from the author on request.

21. See Rothman |1991~ for some related evidence that supports this interpretation.

22. See Courtney |1989~ for some evidence that supports this interpretation.

23. For example, Cover |1989~ demonstrates that output responds more to negative money growth rate shocks than to positive money growth rate shocks.

REFERENCES

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Blatt, J. "On the Frisch Model of Business Cycles." Oxford Economic Papers, March 1980, 467-79.

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The behavior of macroeconomic variables over phases of the business cycle has long been an object of interest to economists. A critical aspect of this is the symmetry or asymmetry of business cycles. An asymmetric cycle is one in which some phase of the cycle is different from the mirror image of the opposite phase; for example, contractions might be steeper, on average, than expansions. Although asymmetries were noted by early business cycle researchers, the issue has only recently been examined empirically.(1) This recent interest arises from a desire to carefully document the stylized facts of business fluctuations and because linear structural and time series models cannot represent asymmetric behavior under standard assumptions.

Neftci |1984~(2) and DeLong and Summers |1986~ reported evidence that increases in the unemployment rate are steeper than decreases. The evidence on the asymmetry of real GNP is less clear, however. Falk |1986~--using Neftci's procedure--found that real GNP does not exhibit this type of asymmetry. However, Hamilton |1989~ showed that a particular nonlinear (asymmetric) model for real GNP growth rates dominates linear models.

All of the above research has focused on asymmetries--or nonlinearities--in the rate of change of business cycle variables; that is, these researchers compare periods of increase to periods of decrease. Brock and Sayers |1988~ expanded the search for asymmetry by applying a test that identifies any form of nonlinearity or asymmetry. Brock and Sayers find evidence of nonlinear structure--in the postwar period--in employment, unemployment, and industrial production. One problem with their test, however, is that it cannot distinguish among different types of asymmetries.

This paper sharpens the asymmetry evidence by distinguishing two types of asymmetry that could exist separately or simultaneously.(3) The first type of asymmetry--which has been investigated in most of the research described above--occurs when contractions are steeper than expansions. I refer to this type of asymmetry as steepness. The second--and as yet not explicitly considered--type of asymmetry occurs when troughs are deeper than peaks are tall. I refer to this type of asymmetry as deepness.

These two types of asymmetry are illustrated in Figure 1 for a trendless time series. The first panel shows a symmetric cycle. The second panel shows a cycle exhibiting steepness, which pertains to relative slopes or rates of change and compares mirror images across imaginary vertical axes placed at peaks and troughs.(4) The third panel shows a cycle exhibiting deepness, which pertains to relative levels and compares mirror images across a horizontal axis (the dashed line in the figure). The final panel in the figure shows a cycle exhibiting both deepness and steepness.

In section II, I discuss the implications of asymmetry. Tests for deepness and steepness are presented in section III, and detrending is discussed. In section IV, I provide evidence of deepness in unemployment and industrial production; the evidence for real GNP is weaker. Previous evidence of steepness in unemployment is also confirmed. In section V, I discuss Monte Carlo simulations that demonstrate that these tests are actually able to identify different types of asymmetry. Section VI concludes.

II. IMPLICATIONS OF ASYMMETRY

Asymmetry in business cycle time series is important for macroeconometrics because linear and Gaussian models are incapable of generating asymmetric fluctuations.(5) Consider a variable |x.sub.t~, generated by a linear, Gaussian, and stationary autoregressive moving average (ARMA) process, and its infinite moving average representation:

(1a) A(L) ||chi~.sub.t~ = B(L) ||epsilon~.sub.t~

(1b) ||chi~.sub.t~ = |A.sup.-1~(L) B(L) ||epsilon~.sub.t~

where A(L) and B(L) are finite polynomials in the lag operator L and ||epsilon~.sub.t~ is an i.i.d. Gaussian disturbance. Since ||epsilon~.sub.t~ has a symmetric distribution, then the process ||chi~.sub.t~--a linear combination of these disturbances--is also symmetric. Therefore, a linear and Gaussian ARMA model cannot, asymptotically, represent asymmetric behavior.

Evidence of asymmetry, however, does not point toward any particular business cycle model. Rather, asymmetry simply provides evidence against the class of linear models with symmetric disturbances. Models outside of this class include nonlinear endogenous cycle models--as discussed in Boldrin and Woodford |1990~--and nonlinear stochastic models as in Hamilton |1989~.

Even though many models can generate deep and steep cycles, further intuition about these concepts is gained by considering simple models capable of generating deep and steep cycles. Deepness can be generated by a model with asymmetric price adjustment. For example, suppose prices rise rapidly above their expected level when output is above potential, but fall slowly when output is below potential. Then--starting from potential output--a positive nominal demand shock will push up prices, but will have a relatively small effect on output. In contrast, a negative nominal demand shock will have relatively larger impact on output than on prices.(6) Steepness can be generated by models with asymmetric costs of upward and downward adjustment. For example, Chetty and Heckman |1985~ and Baldwin and Krugman |1986~ present models in which exit from an industry is less costly than entry. Hence, production can fall rapidly, but it expands more slowly.

III. TESTING FOR DEEPNESS AND STEEPNESS

Tests

Consider a time series

(2) |y.sub.t~ = ||tau~.sub.t~ + |c.sub.t~

where ||tau~.sub.t~ is the nonstationary trend component and |c.sub.t~ is the stationary cyclical component. In this paper, I focus on whether |c.sub.t~--the cyclical component--is asymmetric, because secular growth is asymmetric a priori (almost always increasing). If |y.sub.t~ is nonstationary, analysis of |c.sub.t~ requires detrending, which is discussed at length in the following subsection. In this subsection, I describe tests for deepness and steepness that are to be applied to |c.sub.t~; that is, after the series |y.sub.t~ has been detrended.

If a time series exhibits deepness, then it should exhibit negative skewness relative to mean or trend; that is, it should have fewer observations below its mean or trend than above, but the average deviation of observations below the mean or trend should exceed the average deviation of observations above. To construct a test for deepness, I use the coefficient of skewness. Compute

|Mathematical Expression Omitted~

where |Mathematical Expression Omitted~ is the mean of |c.sub.t~, |sigma~(c) is the standard deviation of |c.sub.t~, and T is the sample size.

Since the observations on |c.sub.t~ are sure to be serially correlated, the formula for the asymptotic standard error of the coefficient of skewness of an i.i.d. random variable is inapplicable. However, an asymptotic standard error of D(c) can be computed using a procedure suggested by Newey and West |1987~ as follows. Construct a variable with the tth observation equal to

|Mathematical Expression Omitted~

Regress this variable on a constant and compute the Newey-West standard error. The estimate of the constant in this regression is identical to D(c) in (3a), and the Newey-West standard error is consistent even in the presence of heteroscedasticity or serial correlation.(7) Further, the constant term divided by its standard error is asymptotically normal. Hence, conventional critical values can be used to test the significance of D(c).(8)

If a time series exhibits steepness, then its first differences should exhibit negative skewness. That is, the sharp decreases in the series should be larger, but less frequent, than the more moderate increases in the series. Hence, a test for steepness can be computed by using the coefficient of skewness for |delta~|c.sub.t~, the first difference of the cyclical component in equation (2):

|Mathematical Expression Omitted~

where, |Mathematical Expression Omitted~ and |sigma~(|delta~c) are the sample mean and standard deviation of |delta~c.(9) This test examines whether rates of change in |c.sub.t~ are asymmetric around their mean. An asymptotic standard error for the steepness test can be calculated analogously to the deepness test.

Detrending

For variables exhibiting secular growth, the asymmetry tests are applied to detrended series because secular growth is asymmetric a priori (almost always increasing). Although detrending has been quite controversial in macroeconomics,(10) I argue that any detrending filter that satisfies three requirements is appropriate for an analysis of asymmetry. The three requirements are sufficient to ensure that the detrending procedure itself is not inducing spurious asymmetry and that the asymmetry tests |D(c) and ST(|delta~c)~ have standard distributions and a natural interpretation.

The first requirement is that the detrending filter has a linear representation. Linearity is critical because a linear filter applied to a symmetric time series yields a symmetric time series.(11) Therefore, a linear filter cannot induce asymmetry if none is present to begin with. The second requirement is that the filter must induce stationarity. If the extracted cyclical component is nonstationary, then the asymmetry tests might have nonstandard distributions, substantially complicating inference. Finally, the detrending filter used for each test must extract the component appropriate for that asymmetry test; that is, a filter used for a deepness test must extract |c.sub.t~, while a filter used for a steepness test must extract |delta~|c.sub.t~. Otherwise, the interpretation of the test statistics is confounded; the deepness test might not correctly identify actual deepness, and the steepness test might not correctly identify actual steepness.

The first requirement--that the filters have a linear representation--is satisfied by many procedures that have been proposed for detrending. I consider four such procedures: simple linear detrending, first-differencing, the Beveridge-Nelson decomposition |1981~,(12) and the Hodrick-Prescott filter |1982~.(13) Since these procedures all have linear representations, none of these filters can induce asymmetry if none is present in the original series. However, these procedures would be appropriate for the deepness and steepness tests only if they satisfy the second and third requirements spelled out above. Namely, that they induce stationarity and extract the correct component, either |c.sub.t~ or |delta~|c.sub.t~. To determine whether these procedures satisfy these last two requirements, I consider the example of detrending real GNP.

Consider simple linear detrending. This procedure might not satisfy the second requirement spelled out above--that the filter induce stationarity. For example, if real GNP is difference stationary, simple linear detrending would not induce stationarity. Although it is an open question as to whether GNP is difference or trend stationary, the reasonable possibility that it is difference stationary raises questions about the use of simple linear detrending. Therefore, I do not use simple linear detrending for the deepness test or the steepness test.

Unlike simple linear detrending, first-differencing almost surely induces stationarity in real GNP, thus likely satisfying the second requirement. The third requirement is that the detrending procedure used for each test extract the appropriate component for that test. First-differencing extracts |delta~|c.sub.t~,(14) which is not the appropriate component for the deepness test, but is the appropriate component for the steepness test. Therefore, I do not use first-differencing for the deepness test, but I do use first-differencing for the steepness test.

Turning to the Beveridge-Nelson decomposition, if real GNP were difference stationary, this procedure would induce stationarity, thus satisfying the second requirement. As for the third requirement, this procedure extracts the component |c.sub.t~, which is the needed component for the deepness test. Therefore, I use the Beveridge-Nelson decomposition for the deepness test. For the steepness test, the needed component is |delta~|c.sub.t~. Although the Beveridge-Nelson decomposition does not directly extract |delta~|c.sub.t~, it would be possible to extract |c.sub.t~ with the Beveridge-Nelson decomposition and then first difference. This procedure, however, seems redundant, since just first-differencing extracts the desired component for the steepness test. Therefore, I do not report results for the steepness test using the Beveridge-Nelson decomposition.

Finally, consider the Hodrick-Prescott filter. Cogley and Nason |1991~ note that this filter would induce stationarity in a series whether it is trend or difference stationary. That is, even in the presence of a unit root, the Hodrick-Prescott filter would induce stationarity in real GNP, and thus the filter almost surely satisfies the second requirement. As to the third requirement, the appendix demonstrates that this filter yields an explicit expression for |c.sub.t~, the appropriate component for the deepness test. Therefore, I use it for the deepness test. For the steepness test, the Hodrick-Prescott filter suffers from the same redundancy as the Beveridge-Nelson decomposition. Therefore, I do not report results for the steepness test using this filter.

Before leaving the Hodrick-Prescott filter, it is necessary to discuss the recent critique of the filter leveled by Cogley and Nason. Using spectral analysis, Cogley and Nason demonstrate that the Hodrick-Prescott filter--when applied to a difference stationary series--strongly amplifies fluctuations in the series at business cycle frequencies. They point out that this can create serious problems for many types of business cycle analysis; for example, applying the filter to two series might induce a spurious correlation between the series at business cycle frequencies when, in fact, the series are uncorrelated at business cycle frequencies. However, this characteristic of the filter--amplification of fluctuations at business cycle frequencies--is actually desirable for an analysis of business cycle asymmetry. As a linear filter, the Hodrick-Prescott filter can not induce asymmetry in a series if none is present to begin with. Thus, if asymmetry is found, it must be present in the original series. Furthermore, since asymmetries at business cycle frequencies are of particular interest, a filter that amplifies fluctuations at these frequencies is ideal. So, while Cogley and Nason suggest that the Hodrick-Prescott filter probably should not be used for many types of analysis, the filter seems particularly well suited for the asymmetry tests in this paper.

To summarize the discussion in this section, I will use the Beveridge-Nelson decomposition and the Hodrick-Prescott filter for the deepness test. For the steepness test, I will detrend by first-differencing.

IV. EMPIRICAL RESULTS

The deepness test is computed for three U.S. post-war quarterly time series from 1949:I to 1989:IV--unemployment, real GNP in 1982 dollars, and industrial production. Values of the test statistics, standard errors, and one-sided p-values are shown in the top panel of Table I for both the Hodrick-Prescott filter and the Beveridge-Nelson decomposition.

Using the Hodrick-Prescott filter, unemployment, industrial production, and real GNP exhibit deepness at the .04, .06, and .12 significance level, respectively. I interpret this as fairly strong evidence of deepness for unemployment and industrial production, but as fairly weak evidence for real GNP.(15) Corroborating visual evidence of deepness is shown in Figures 2 to 4, which plot deviations from the Hodrick-Prescott trend for log real GNP, log industrial production, and the unemployment rate. Figure 2 confirms the fairly weak evidence of deepness for real GNP. While troughs do look somewhat deeper than peaks, the visual evidence is not overwhelming. Deepness is much more apparent for industrial production shown in Figure 3; troughs look notably deeper than peaks. Since the unemployment rate--shown in Figure 4--is countercyclical, deepness would appear as very high sharp peaks coinciding with troughs in the business cycle. Indeed, Figure 4 shows just such a pattern, visually confirming the evidence of deepness for the unemployment rate.(16)

Using the Beveridge-Nelson decomposition, unemployment, industrial production, and real GNP exhibit deepness at the .02, .35, and .27 significance level, respectively.(17) For unemployment, the results are similar to those using the Hodrick-Prescott filter. For industrial production and real GNP, the Beveridge-Nelson decomposition provides much weaker evidence of deepness than the Hodrick-Prescott filter.

While it would be comforting if the Hodrick-Prescott filter and the Beveridge-Nelson decomposition yielded the same evidence for or against deepness, it is not surprising that the evidence is weaker for industrial production and real GNP with the Beveridge-Nelson decomposition than with the Hodrick-Prescott filter. As discussed above, the filter amplifies fluctuations at business cycle frequencies. This amplification should make it easier to identify asymmetries at these frequencies with the Hodrick-Prescott filter than with many other filters. Therefore, I do not regard the weaker evidence for deepness from the Beveridge-Nelson decomposition as evidence against deepness, but rather as an indication that certain filters are more likely to highlight asymmetries at business cycle frequencies.

TABLE I Do Business Cycles Exhibit Deepness And Steepness?(a) (1949:I-1989:IV) Deepness Asymptotic Variable Trend D(c) Std. Err. p-value Unemployment HP .89 .50 .04 BN .96 .49 .02 NT .47 .65 .23 Industrial Production HP -.73 .46 .06 BN -.17 .44 .35 Real GNP HP -.62 .53 .12 BN -.16 .26 .27 Steepness Asymptotic Variable ST(|delta~c) Std. Err. p-value Unemployment .99 .57 .04 Industrial Production -.31 .56 .29 Real GNP -.19 .43 .33 a Real GNP and industrial production are analyzed in log form. HP is Hodrick-Prescott. BN is Beveridge-Nelson decomposition. NT is no trend removed. p-value is the one-sided significance level at which the null of D(c)=0 or ST(|delta~c)=0 can be rejected.

Finally, the following heuristic example helps shed some light on the magnitude of the deepness present in unemployment and industrial production. Consider a single sixteen observation deterministic cycle generated by the sinusoid in (5):

(5) ||chi~.sub.t~ = -A1 Cos |(|pi~/8)t~

t=1,2,3,4,13,14,15,16

||chi~.sub.t~ = -A2 Cos |(|pi~/8)t~

t=5, ..., 12.

If A2 and A1 equal unity, this sinusoid is symmetric and its maximum and minimum values are plus and minus unity, respectively. As the ratio A2/A1 increases, deepness is induced. Table II shows the value of D(|chi~) for these sixteen observations and the ratio of the maximum distance below the mean to the maximum distance above.(18) These values can be compared to empirically observed values of deepness shown in Table I. For example, the actual value of the deepness test for industrial production is -.73, when the Hodrick-Prescott filter is used. For this value, the trough of the sinusoid is about 1.8 times further below the mean than the peak is above the mean. This ratio seems large enough to warrant interest.

Returning to Table I, the second panel shows the steepness test results. Based on one-sided p-values, unemployment exhibits evidence of steepness, but real GNP and industrial production do not. This confirms results presented by DeLong and Summers |1986~ and Falk |1986~.(19)

V. ARE DEEPNESS AND STEEPNESS REALLY DIFFERENT?

Since more general tests for nonlinear structure--such as the test in Brock and Sayers |1988~--are available, a natural question is whether the deepness and steepness tests actually are identifying different types of asymmetry. Unless the deepness and steepness tests can make this distinction, more general tests might as well be used.

To investigate this question, I conducted a set of Monte Carlo experiments to see whether a deepness test would mistakenly identify steepness as deepness and vice-versa. First, I generated 500 replications of pseudo-data with nearly the amount of deepness found in the actual unemployment rate after application of the Hodrick-Prescott filter.(20) To each replication of pseudo-data, I applied the deepness and steepness tests. As it turned out, the deepness test identified deepness at the 5 percent significance level in more than one-third of the replications, while evidence of steepness was almost never found. Second, I generated 500 replications of pseudo-data with nearly the amount of steepness found in the actual unemployment rate. Again, I applied the deepness and steepness tests to each replication. For these data, the steepness test identified steepness at the 5 percent significance level in about two-thirds of the replications, while evidence of deepness was almost never found.

These simulations suggest that the deepness and steepness tests are able to identify different types of asymmetry and that asymmetry could be missed if only one test were applied. Therefore, distinguishing between these two types of asymmetry likely provides a sharper characterization of asymmetry than earlier work.

VI. CONCLUSION

This paper distinguishes two types of asymmetry in business cycles: deepness and steepness. Deepness characterizes fluctuations that have troughs deeper below trend than peaks are high, while steepness--which has been considered explicitly in some previous research--describes fluctuations with steeper contractions than expansions. A test for deepness is presented, which is extended in a natural way to provide a test for steepness. Monte Carlo simulations suggest that the deepness and steepness tests in this paper are actually able to identify different types of asymmetry.

The deepness test provides evidence that deepness is present in U.S. postwar quarterly unemployment and industrial production; that is, troughs are deeper than peaks. The evidence of deepness for real GNP is weaker. Consistent with some previous research, evidence of steepness is found only for unemployment.

The pattern of asymmetry across variables raises at least two interesting questions. First, the stronger presence of deepness in industrial production than in real GNP suggests that deepness is more prevalent in the production sector than elsewhere in the economy.(21) Second--focusing on the two variables covering the whole economy--the much stronger evidence of deepness and steepness in unemployment than in real GNP suggests that there is a nonlinearity in Okun's law.(22)

TABLE II Deepness And The Height And Depth Of Cycles(a) Ratio of Maximum Distance D(c) below and above Mean 0.00 1.00 -.24 1.20 -.44 1.51 -.60 1.60 -.74 1.79 a Computed for a sinusoid with deepness. See text for details of calculations.

The presence of deepness and steepness in unemployment and deepness in industrial production suggests that standard linear structural or time series models with symmetric disturbances cannot represent the observed stylized facts for these variables. Nonlinear models or models with asymmetric disturbances might be more appropriate. Directions for future work include development of more general and more powerful tests of asymmetry and multivariate versions of these tests.(23) It is also important to consider further the types of theoretical models able to generate deepness and steepness.

APPENDIX

For a time series |y.sub.t~, the Hodrick-Prescott filter is obtained by finding the functional, ||delta~.sub.t~, that satisfies the penalized least squares minimization program,

|Mathematical Expression Omitted~

where L is the lag operator and |lambda~ can be interpreted as a Lagrange multiplier. The filter is obtained from this minimization by setting |lambda~ equal to 1600. When the Hodrick-Prescott filter is used in the paper, the values of the functional, ||delta~.sub.t~ are used as the values of the non-stationary trend component, ||tau~.sub.t~, that appears in equation (2). The cyclical component, |c.sub.t~, is then computed as |y.sub.t~ - ||delta~.sub.t~.

Note that if |lambda~ equals infinity, the sum of the squared second-differences of ||delta~.sub.t~ must be zero, meaning that ||delta~.sub.t~ must be the ordinary least squares linear trend. If |lambda~ equals zero, so that the smoothness constraint is non-binding, then ||delta~.sub.t~ perfectly interpolates the time series |y.sub.t~. As these extreme cases suggest, the choice of |lambda~ does affect the frequency of oscillations that pass through the filter. With |lambda~ = 1600, the filter is similar to a high-pass filter that removes oscillations with periodicity greater than thirty-two quarters or eight years; hence, the computed cyclical component (|y.sub.t~ - ||delta~.sub.t~) has been largely purged of these low frequency oscillations.

1. Keynes |1936~ and Burns and Mitchell |1946~ suggest that business cycles are asymmetric. Kaldor |1940~ and Hicks |1950~ provide examples of deterministic models yielding asymmetric cycles.

2. Sichel |1989~ identified a mistake in Neftci's empirical work that reversed his findings, and showed that Neftci's test has fairly low power. However, Rothman |1991~ showed that Neftci's evidence is resurrected if a first-order Markov process is used rather than the second-order Markov process used by Neftci.

3. A recent paper by Gerard Pfann also distinguishes different types of asymmetry.

4. The stylized pictures suggest a relationship between asymmetry and time deformation as in Stock |1987~. Stock's analysis of time deformation over the business cycle examines whether there is time deformation across expansions and contractions. This corresponds to steepness.

5. The argument is only outlined here. For a formal proof, see Blatt |1980~.

6. The asymmetric price adjustment example appears in DeLong and Summers |1988~. These authors also discuss other models capable of generating deepness. Ball and Mankiw |1991~ also present a model of asymmetric price adjustment that generates deepness.

7. In constructing the heteroscedasticity and serial correlation consistent covariance matrix, I allowed for serial correlation of up to order five. As loosely suggested by Newey and West, this order of correlation is set to equal the integer part of the sample size raised to the 1/3 power.

8. An earlier version of this paper used a bootstrap procedure to obtain a small-sample distribution of the deepness test statistic. That procedure, however, requires a variety of auxiliary assumptions and is notably more complicated to compute than the Newey-West standard error. Although the bootstrap distribution provided somewhat stronger evidence of deepness than the asymptotic distribution used in this paper, the simplicity and generality of the Newey-West procedure is quite advantageous.

9. This steepness test is the same as the test in DeLong and Summers |1986~, except for the computation of the standard error of the statistic.

10. See for example, Nelson and Plosser |1982~ and Perron |1990~.

11. This can be seen by applying some linear filter, say C(L), to ||chi~.sub.t~ in equation (1b). If ||chi~.sub.t~ is symmetric, then C(L)||chi~.sub.t~ is also symmetric because it is still a weighted sum of symmetric disturbances.

12. The Beveridge-Nelson decomposition is linear, conditional on the estimated

parameters of the ARIMA process used for the decomposition.

13. The derivation of the Hodrick-Prescott filter is outlined in the appendix. For the linear representation of the Hodrick-Prescott filter, see Cogley and Nason |1991~. Note also that this filter is nearly identical to certain cubic splines developed in the numerical analysis literature. For example, see Reinsch |1967~.

14. To see this, consider equation (2) with |c.sub.t~ a stationary process and ||tau~.sub.t~ either a random walk with drift or a simple linear trend. In either case, first-differencing extracts |delta~|c.sub.t~ plus the drift in ||tau~.sub.t~. Since the asymmetry tests focus on third moments around means, this drift term will not affect any of the asymmetry tests.

15. Neftci |1984~ used a confidence bound that corresponds to a .10 significance level for the test used here.

16. Figures 2-4 suggest that the trough in 1949:IV might be unduly influential and that the deepness evidence may be sensitive to the sample period. However, the strength of the deepness evidence only drops slightly for unemployment and industrial production over the sample 1952:I-1989:IV.

17. For each series, the Beveridge-Nelson decomposition is based on an ARIMA(2,1,2) estimated on the unemployment rate, log industrial production, and log real GNP.

18. As A2 increases, the mean of the sinusoid becomes negative. The ratios of maximum values in table II are relative to this mean.

19. In contrast to the weak evidence for steepness in real GNP in this paper, Hamilton's results do imply steepness. Hamilton's sample period extends from 1952:II to 1984:IV. Over that shorter sample period, the test in this paper finds more evidence of steepness in real GNP than in the longer sample.

20. The data for the Monte Carlo experiments were generated using Markov-switching models similar to the model in Hamilton |1989~. Details of the data generation are available from the author on request.

21. See Rothman |1991~ for some related evidence that supports this interpretation.

22. See Courtney |1989~ for some evidence that supports this interpretation.

23. For example, Cover |1989~ demonstrates that output responds more to negative money growth rate shocks than to positive money growth rate shocks.

REFERENCES

Baldwin, R., and P. Krugman. "Persistent Trade Effects of Large Exchange Rate Shocks." Working Paper, Massachusetts Institute of Technology, August 1986.

Ball, L., and G. Mankiw. "Asymmetric Price Adjustment and Economic Fluctuations." Manuscript, December 1991.

Beveridge, S., and C. Nelson. "A New Approach to the Decomposition of Economic Time Series into Permanent and Transitory Components with particular Attention to the Measurement of the 'Business Cycle'." Journal of Monetary Economics, March 1981, 151-74.

Blatt, J. "On the Frisch Model of Business Cycles." Oxford Economic Papers, March 1980, 467-79.

Boldrin, M., and M. Woodford. "Equilibrium Models Displaying Endogenous Fluctuations and Chaos: A Survey." Journal of Monetary Economics 25, 1990, 189-222.

Brock, W., and C. Sayers. "Is the Business Cycle Characterized by Deterministic Chaos?" Journal of Monetary Economics 22, July 1988, 71-90.

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Chetty, V., and J. Heckman. "A Dynamic Model of Aggregate Output Supply, Factor Demand and Entry and Exit for a Competitive Industry with Heterogeneous Plants." Program in Quantitative Economic Analysis/NORC Discussion Paper 86-10, November 1985.

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DeLong, B., and L. Summers. "How does Macroeconomic Policy Affect Output?" Brookings Papers on Economic Activity 2, 1988, 433-94.

-----. "Are Business Cycles Symmetrical?" in American Business Cycle: Continuity and Change, edited by R. Gordon. Chicago: University of Chicago Press, 1986, 166-79.

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Hicks, J. A Contribution to the Theory of the Trade Cycle. Oxford: Clarendon Press, 1950.

Hodrick, R., and E. Prescott. "Post-war U.S. Business Cycles: An Empirical Investigation." Working Paper, Carnegie-Mellon University, 1982, 205.

Kaldor, N. "A Model of the Trade Cycle." Economic Journal, March 1940, 78-92.

Keynes, J. The General Theory of Employment, Interest, and Money. New York: Harcourt, Brace, 1936.

Neftci, S. "Are Economic Time Series Asymmetric over the Business Cycle?" Journal of Political Economy 92, April 1984, 307-28.

Nelson, C., and C. Plosser. "Trends and Random Walks in Macroeconomic Time Series." Journal of Monetary Economics 10, September 1982, 139-62.

Newey, W., and K. West. "A Simple Positive Semi-Definite Heteroskedasticity and Autocorrelation Consistent Covariance Matrix." Econometrica May 1987, 703-08.

Perron, P. "The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis." Econometrica November 1989, 1361-401.

Pfann, G. "Employment and Business Cycle Asymmetries: A Data Based Study." Institute for Empirical Macroeconomics Discussion Paper #39, March 1991.

Reinsch, D. "Smoothing by Spline Functions." Numerische Mathematik, 10, 1967, 177-83.

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Author: | Sichel, Daniel E. |
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Publication: | Economic Inquiry |

Date: | Apr 1, 1993 |

Words: | 5166 |

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