Recessions and recoveries in real business cycle models.

Balke, Nathan S. ; Wynne, Mark A.

I. INTRODUCTION

Do general equilibrium business cycle models generate business cycles that look like the business cycles observed in a modern industrial economy? The early real business cycle literature as exemplified by Kydland and Prescott [1982], Hansen [1985], and the papers in the March/May 1988 issue of the Journal of Monetary Economics evaluated the ability of dynamic general equilibrium models to generate business cycles by comparing selected second moments of the data generated by the models with those generated by actual (detrended) data. Specifically, this literature considers the deviations of various macroeconomic aggregates from trend as the cyclical movement that needs to be explained by a model of the business cycle. A model is judged to be successful if the second moments generated by the model are "close" (in some informal sense) to the second moments found in the data.

In this paper, rather than examine whether real business cycle models can mimic the autocorrelation and cross-correlation properties of the data, we ask whether a representative model of this class is able to generate cyclical behavior consistent with the traditional NBER conception of business cycles. The traditional NBER definition of a business cycle defines the cycle in terms of periods of absolute increases and decreases in economic activity rather than fluctuations about some trend. Below we examine whether a real business cycle model can produce business cycles in this more traditional sense that are of appropriate duration and shape.

There are several reasons for considering this alternative method of evaluating real business cycle models. There is a large and old literature, including, among many others, Mitchell [1927], Burns and Mitchell [1946], Eckstein and Sinai [1986], and Zarnowitz [1992], that examines fluctuations in economic activity in terms of a NBER-type chronology of business cycle peaks and troughs. Furthermore, it is business cycles in the traditional NBER sense that policymakers, the press, and the general public are typically concerned with. While periods of sub-par growth do receive attention and are a source of concern to policymakers, it is nonetheless the case that the qualitative nature of policy deliberations seems to change when the economy experiences absolute declines in economic activity. Finally, by examining the nature of business cycles in light of a NBER-type chronology one may be able to uncover relationships that otherwise would be undetected if only auto- or cross-correlations of the data were used.

Using the concept of business cycle time introduced by Burns and Mitchell [1946], we divide recessions and expansions in calendar time into separate business cycle "phases." Based on this framework, we show that for real GNP expansions tend to be concave and recessions linear. That is, growth tends to be faster earlier in the expansion and slower later in the expansion, while the rate of decline is not significantly different over the course of the recession. Furthermore, output tends to have "rounded" peaks and "pointed" troughs. In addition to aggregate output, we consider the "shape" of consumption of nondurables and services, fixed investment, hours, and real wages. While not every variable displays concave expansions, "round" peaks and "pointed" troughs are present in most of these variables.

A simple real business cycle model considered in King, Plosser, and Rebelo [1988a] - a variant of the Hansen [1985] and Rogerson [1988] models - is used to generate artificial economic histories. The algorithm developed by Bry and Boschan [1971] to mimic the business cycle dating procedure of the NBER is used to date business cycle peaks and troughs in the artificial data. We then compare the nature of business cycles implied by these simple real business cycle models to that of actual business cycles. We find that while these models are adequate at capturing the duration and the amplitude of actual business cycles, they do not capture the entire shape of the business cycle. In particular, they fail to produce concave expansions for aggregate output and investment, and peaks tend to be "pointed" rather than "round." On the other hand, the real business cycle models do a much better job of mimicking the actual growth cycles, or cycles in detrended data.

The work reported below is closely related to work by King and Plosser [1989] on the approach to evaluating business cycle models first proposed by Adelman and Adelman [1959]. The objective of the King and Plosser paper is to evaluate a simple real business cycle model using the methods developed by Burns and Mitchell to characterize the business cycle. To this end they date peaks and troughs in economic activity in the artificial economy they study using the Bry-Boschan algorithm; the cycles thus obtained are further divided into various phases. King and Plosser then compare the qualitative features of the cycles found in the real business cycle model with corresponding features of the data and conclude that they are unable to distinguish between the two. This is the sense in which they conclude that the real business cycle model passes the Adelmans' test, although they note reservations about the power of this type of test in their conclusion.

While King and Plosser stay as close as possible to the letter of the Burns and Mitchell methods in evaluating a real business cycle model, we simply use the Burns and Mitchell phases as a point of departure for our analysis, combining their phase classification with formal statistical tests to evaluate our artificial economies. We do this by taking the real business cycle model as the "null" model and generate a distribution of artificial business cycle shapes as implied by a real business cycle model and compare the actual business cycle shape with this distribution.(1) Thus while King and Plosser are unable to reject, statistically, the possibility that the actual business cycle behavior is generated by a simple real business cycle model, we are able to reject this possibility. We do this by showing that various statistics of interest calculated for the actual data lie in the tails of the implied Monte Carlo distributions generated by the real business cycle model.(2)

II. BUSINESS CYCLES VERSUS GROWTH CYCLES

In this paper we consider two alternative definitions of the business cycle. The standard practice in the contemporary literature on business cycles is to consider the deviations of output from some secular trend as the cyclical movement that needs to be explained by a business cycle model. This is in contrast to the definition of business cycles employed by the NBER, where recessions and expansions are periods of absolute decreases and increases in economic activity. Turning points are then peaks and troughs in the level of economic activity. A definition of the cyclical component of economic activity as consisting of fluctuations about trend is more akin to the NBER's growth-cycle concept, where turning points are peaks and troughs in economic activity relative to trend.(3)

The distinction between business cycles and growth cycles might be important for several reasons. First, as we pointed out above and Zarnowitz [1992] discusses in more detail, policymakers and the general public may view absolute declines and increases in economic activity differently from relative declines and increases. Second, the problem of defining and measuring trend growth provides a practical reason for why one might want to consider the traditional notion of the business cycle rather than the growth cycle. Is the trend level of output at a point in time simply the extrapolation of past values of output? Or is it the maximum "sustainable" level of GNP, something more akin to potential output? How one removes the trend from economic activity, whether by differencing, detrending with a linear time trend, or employing the Hodrick-Prescott filter, can influence the cyclical nature properties of the data. For example, Falk [1986] provides an interesting example of how trend removal can bias tests of business cycle symmetry.

Furthermore, the traditional conception of the business cycle underlies many recent statistical models of economic fluctuations. For example, the two-state Markov switching model of Hamilton [1989] when applied to real GNP characterizes output as switching between positive growth states and negative growth states - expansions and contractions. In fact, Hamilton obtains regime switch dates that are similar to the NBER business cycle chronology. Recent extensions of Hamilton's regime switching model by Boldin [1990] have tended to confirm the presence of traditional business cycles. We evaluate real business cycle models in terms of both the traditional notion of business cycles as well as in terms of growth cycles.

III. THE SHAPE OF THE BUSINESS CYCLE

In their monumental study of business cycles, Burns and Mitchell [1946] divided the business cycle into nine phases. The first phase in this classification is defined as the three months centered on the initial trough, while the fifth phase is defined as the three months centered on the subsequent peak. The ninth phase is defined as the three months centered on the trough marking the end of the recession, and is also the first phase of the next business cycle. The second, third and fourth phases break the expansion into three time intervals of equal length, while the sixth, seventh and eighth phases break the subsequent recession into three time intervals of equal length. The Burns and Mitchell phases allow for the possibility that business cycles evolve according to economic or business cycle time rather than calendar time.(4)

It is possible to think of alternative ways to characterize the business cycle, such as a multistate Markov switching model, that are not as ad-hoc or as ex post in nature as the Burns and Mitchell phase characterization.(5) However, the Burns and Mitchell characterization does have a long history of use in business cycle analysis and, while ex post, does have the advantage of being a relatively simple way of describing certain features of the business cycle. Furthermore, the computational ease of calculating the Burns and Mitchell phases is attractive when we try to evaluate the ability of artificial economies to replicate features of actual business cycles.

We use the Burns and Mitchell phase classification to characterize two features of the business cycle. First, we examine the overall "shape" of the business cycle as reflected in certain key aggregate variables. By the shape of the cycle we mean the pattern of variation in growth rates of the key aggregates over the course of expansions and recessions? Second we use the Burns and Mitchell phase characterization to examine the question of business cycle symmetry.(7) Specifically, we will consider the extent to which recessions are simply negative expansions.

The Shape of Post-World War II Business Cycles

In our analysis of post-World War II business cycles, we follow Burns and Mitchell [1946] and divide the business cycle into eight phases. Since we use quarterly data, the first phase is defined as the quarter of the initial trough, while the fifth phase is defined as the quarter of the subsequent peak. The second, third and fourth phases break the expansion into three time intervals of equal length, while the sixth, seventh and eighth phases break the subsequent recession into three time intervals of equal length.

To determine the shape of the business cycle, we simply regress the growth rate of each series against dummy variables that break the business cycle into the different phases described above.(8) The coefficient estimates then represent the average growth rate of the series (per quarter) during the different phases of the business cycle. To keep things manageable we restrict our attention to five key real macroeconomic aggregates: GNP, consumption of nondurable goods and services, fixed investment, and hours worked (all in per capita terms), and real wages.

The peak and trough dates that make up the official NBER business cycle chronology are determined by the business cycle dating committee of the NBER.(9) To evaluate the ability of an artificial economy to mimic certain features of actual world business cycles, we need to be able to replicate the NBER business cycle dating procedure using time series generated in the artificial economy. To this end we employ the business cycle dating algorithm devised by Bry and Boschan [1971] to automate the rules used by the business cycle dating committee in picking peak and trough dates. The structure of this algorithm is described in detail in Appendix B of Balke and Wynne [1993]. Essentially, the Bry-Boschan algorithm involves finding local maximums and minimums of a smoothed version of a time series subject to restrictions on the length of the entire cycle and on the length of expansion and recession phases. An obvious and important question is how well does the Bry-Boschan algorithm mimic the procedures used by the NBER committee. The algorithm tends to perform best in terms of picking trough dates, matching seven of the nine in the NBER chronology. One trough is dated one quarter after the NBER date, and one is dated three quarters after the NBER date. The algorithm is a little less successful in picking peaks, matching only three of the NBER dates perfectly. Two peaks are dated one quarter before the NBER dates, and four are dated two quarters before the NBER dates. Recessions tend to be slightly longer in the Bry-Boschan chronology (4.9 quarters versus 3.6 quarters for the NBER dates) and expansions slightly shorter (16.1 quarters versus 17.1 quarters for NBER dates).(10)

Table I presents the results of regressing the growth rates of the various series against the Burns and Mitchell phase dummies as determined by the Bry-Boschan algorithm over the 1948-1992 period.(11) Table II presents p-values for various hypotheses about the nature of business cycle phases. The first set of hypotheses test whether various phases have the same growth rates and is designed to show whether the business cycle has a distinctive shape. The second set of hypotheses considers whether business cycles are symmetric. The symmetry hypothesis implies that the average rate of growth in a recession phase is just the negative of the growth rate in the corresponding expansion phase (correcting for trend growth). Thus, for example, symmetry implies

[Delta][y.sub.Expansion] + [Delta][y.sub.Contraction] = 2 (Trend growth rate)

where [Delta]y denotes the average rate of growth of Y during the indicated business cycle stage. We can make the symmetry tests more elaborate using the Burns and Mitchell phases:

[Delta][y.sub.Phase1] + [Delta][y.sub.Phase5] = 2(Trend growth rate),

[Delta][y.sub.Phase2] + [Delta][y.sub.Phase6] = 2(Trend growth rate),

[Delta][y.sub.Phase3] + [Delta][y.sub.Phase7] = 2(Trend growth rate),

[Delta][y.sub.Phase4] + [Delta][y.sub.Phase8] = 2(Trend growth rate).

If the trend rate of growth is zero, then symmetry implies that growth in recessions is just the negative of growth in expansions.(12)

[TABULAR DATA FOR TABLE I OMITTED]

From Tables I and II, we find that all the series with the exception of real wages reject the simple two-phase characterization of the business cycle (that is, we can reject the null hypothesis: phase 2=3=4=5 and phase 1=6=7=8). Output, investment, and, to a lesser extent, consumption display concave-shaped expansions (that is, phase 2 is greater than phase 3 which is in turn greater than phase 4); growth in these series is highest in the early phases of the expansion and falls as the expansion progresses. The behavior of output and investment is consistent with a "recovery" effect.(13) If the economy does indeed "recover" from a recession, we would expect the growth rate to be greater in the second phase of the business cycle (the first third of the recovery) than in the third and fourth phases. No recovery effect is apparent in either hours or real wages. In contrast to the concavity of expansions, recessions appear to be more or less linear. Aside from consumption, there is little evidence that the recession phases are statistically different. For consumption, the trough phase is quite different from the rest of the recession phases. Thus, the Burns and Mitchell phase regressions seem to imply that for aggregate output and investment the "shape" of the business cycle is characterized by concave expansions and linear recessions. The combination of a concave expansion and a linear recession supports a characterization of traditional business cycles as having "rounded" peaks and "pointed" troughs. There is also strong evidence against symmetric business cycles. With the exception of real wages, all of the series reject symmetry hypotheses either individually or jointly.(14)

TABLE II

p-Values for Tests of Shape and Symmetry of the Business Cycle

Hypothesis Output Consumption Investment Hours Wages

Tests of Shape

Phase 0.030 0.113 0.135 0.010 0.365
2=3=4=5
and
Phase
6=7=8=1
Phase
2=3=4=5 0.011 0.125 0.056 0.142 0.949
Phase 0.011 0.482 0.066 0.260 0.859
2=3=4
Phase 0.104 0.946 0.054 0.362 0.989
2=3
Phase 0.003 0.282 0.035 0.470 0.639
2=4
Phase
3=4 0.125 0.310 0.843 0.102 0.628
Phase 0.403 0.203 0.528 0.009 0.105
6=7=8=1
Phase 0.450 0.594 0.738 0.004 0.524
6=7=8
Phase 6=7 0.209 0.481 0.558 0.014 0.431
Phase 6=8 0.591 0.321 0.459 0.001 0.269
Phase 7=8 0.479 0.754 0.863 0.405 0.729

Tests of Symmetry

Recession = 0.000 0.006 0.000 0.001 0.087
Expansion
Phase 5 = 0.000 0.166 0.135 0.006 0.520
Phase 1
Phase 6 = 0.410 0.404 0.627 0.359 0.051
Phase 2
Phase 7 = 0.001 0.170 0.012 0.100 0.309
Phase 3
Phase 8 = 0.001 0.006 0.006 0.000 0.393
Phase 4
Joint 0.000 0.010 0.003 0.000 0.201

The Shape of Growth Cycles

For the sake of comparison we also decided to examine the shape and symmetry of the growth cycles experienced by the U.S. economy in the postwar period, and consider the ability of a prototypical real business cycle model to replicate these features of the actual data.

Tables III and IV present results from the Burns and Mitchell phase regressions for the growth cycle characterization of business cycles. Unlike the traditional business cycle conception, the growth cycle seems to be adequately characterized by two phases: recessions and expansions. There is little evidence of concavity (or convexity) for either expansions or recessions. Furthermore, the hypothesis of symmetry is not rejected for any of the variables. Thus, in contrast to traditional business cycles, growth cycles appear to be characterized by "linear" expansions and contractions, and by symmetry. For nearly every variable, there is less negative growth in the first recession phase (phase 6) than in the other recession phases, but it is not statistically significant.

IV. BUSINESS AND GROWTH CYCLES IN A SIMPLE ARTIFICIAL ECONOMY

The question we are concerned with in this paper is to what extent can simple artificial economies of the type developed in the current real business cycle literature generate traditional business cycles. What is required for such an exercise is a fully articulated dynamic general equilibrium model that can be calibrated to the data and simulated to generate time paths for the various economic aggregates of interest. Prototypical versions of models of this type that are driven by real shocks are studied in some detail in King, Plosser and Rebelo [1988a; 1988b]. We will consider one of the basic models discussed in that paper that is driven by transitory shocks to productivity, specifically a variant of the Rogerson [1988] and Hansen [1985] models with indivisible labor.

A Simple Model Economy

The structure of the model we examine is as follows. Household preferences are assumed to be defined over consumption, [C.sub.t], and leisure, [L.sub.t], and to have the standard time-separable form:

[summation of] [[Beta].sup.t]U([C.sup.t], [L.sup.t]) where t = 0 to [infinity]

We assume that the point-in-time utility function takes the logarithmic form:

U([C.sub.t], [L.sub.t]) = log([C.sub.t]) + [Theta]log(1 - [L.sub.t]).

In the Hansen-Rogerson economy, households can work either some fixed number of hours, [Mathematical Expression Omitted], or not at all. Optimal allocations in this economy involve trading in [TABULAR DATA FOR TABLE III OMITTED] lotteries over consumption and leisure, which in turn yield a preference specification for the representative household that is linear in hours worked:

U([C.sub.t], [L.sub.t]) = log([C.sub.t]) + [Psi][N.sub.t]

where [Mathematical Expression Omitted] as in Hansen [1985, 315-318].

Output, [Y.sub.t], is produced with capital, [K.sub.t], and labor, [N.sub.t], by means of standard constant returns to scale technology:

[Y.sub.t] = [A.sub.t]F([K.sub.t], [X.sub.t][N.sub.t])

where [A.sub.t] denotes a transitory productivity shock and [X.sub.t] represents labor augmenting technological progress. We further restrict ourselves to a Cobb-Douglas specification of the production function: [Mathematical Expression Omitted]. We assume that technological progress occurs at some exogenously determined rate, [[Gamma].sub.x] = [X.sub.t+1]/[X.sub.t], as the simplest way to induce nonstationarity in our model. Capital accumulation follows the standard process:

[K.sub.t+1] = (1 - [[Delta].sub.K])[K.sub.t] + [I.sub.t]

where [I.sub.t] denotes investment and [[Delta].sub.K] denotes the rate of depreciation of the capital stock. Resource constraints on time and output are specified as follows:

[L.sub.t] + [N.sub.t] = 1

and

[C.sub.t] + [I.sub.t] = [Y.sub.t].

TABLE IV

p-Values for Tests of Shape and Symmetry of the Growth Cycle

Hypothesis Output Consumption Investment Hours Wages

Tests of Shape

Phase 0.196 0.311 0.760 0.182 0.896
2=3=4=5
and
Phase
6=7=8=1
Phase 0.429 0.480 0.998 0.214 0.908
2=3=4=5
Phase 0.264 0.794 0.988 0.216 0.790
2=3=4
Phase 2=3 0.634 0.503 0.925 0.119 0.605
Phase 2=4 0.112 0.667 0.950 0.144 0.891
Phase 3=4 0.271 0.807 0.876 0.908 0.514
Phase 0.118 0.200 0.347 0.221 0.642
6=7=8=1
Phase 0.231 0.252 0.193 0.154 0.891
6=7=8
Phase 6=7 0.866 0.331 0.103 0.489 0.854
Phase 6=8 0.124 0.098 0.127 0.058 0.635
Phase 7=8 0.150 0.447 0.968 0.194 0.755

Tests of Symmetry

Recession = 0.934 0.549 0.988 0.766 0.764
Expansion
Phase 5 = 0.347 0.855 0.846 0.703 0.257
Phase 1
Phase 6 = 0.152 0.397 0.247 0.957 0.767
Phase 2
Phase 7 = 0.402 0.590 0.687 0.367 0.730
Phase 3
Phase 8 = 0.091 0.634 0.619 0.587 0.987
Phase 4
Joint 0.167 0.865 0.772 0.867 0.819

The models were calibrated using parameter values from King, Plosser and Rebelo [1988a]. We set [Beta] equal to 0.988, [Alpha] equal to 0.58, [[Gamma].sub.x] equal to 1.004, [Delta] equal to 0.025 and N, the fraction of hours worked in the steady state, equal to 0.2.

The Shape of the Business Cycle in the Model Economy

We simulated a linearized version of the model described above to generate a series for (detrended) output lasting 180 periods (which corresponds to the number of quarters in the post-World War II sample). To examine traditional business cycles, we then restored the trend to the output series to generate the path of the level of output. The Bry-Boschan algorithm was then used to pick peak and trough dates in the output series to obtain an NBER-style business cycle chronology. Using these dates we broke the expansions and contractions up into the Burns and Mitchell phases, and ran the phase regressions and symmetry tests for the generated data. We also considered the ability of these models to generate growth cycles. For these tests we applied the Bry-Boschan algorithm to the detrended output series generated by the artificial economies, and proceeded as before. These exercises were repeated 1000 times to arrive at Monte Carlo distributions of the various statistics of interest.

Table V provides some summary statistics for the business cycles generated in the real business cycle model. The mean and standard deviation of the Monte Carlo distribution is presented as well as the percentile of the Monte Carlo distribution in which the actual business cycle statistic is placed. The Hansen-Rogerson model tends to generate more cycles (whether traditional business cycles or growth cycles) than were actually present in the post-World War II sample. However, because the actual business cycle statistics are not in the extreme tails of the Monte Carlo distributions implied by the model (with the possible exception of the expansion duration), the model does a reasonable job of producing recessions and expansions with durations and amplitudes consistent with actual business cycles.(15)

Monte Carlo Phase Results

To determine whether a standard real business cycle model can replicate the "shape" of the business cycle, we ran the Burns and Mitchell phase regressions for each replication and tabulated the Monte Carlo distribution. Table VI presents the mean and the standard deviation of the Monte Carlo distribution of the phase growth rates implied by the Hansen-Rogerson model. The percentile of the Monte Carlo distribution in which the actual phase growth rate is present is presented as well. Table VII tabulates the percentage of replications in which the various hypotheses about the shape and symmetry of the business cycle are rejected at the 0.05 significance level in the model economy.

From Tables VI and VII, we see that the Hansen-Rogerson real business cycle model does not match the phase behavior of actual business cycles. Actual phase growth rates are often in the tails of the Monte Carlo distribution. For most of the variables, the actual growth rates in phases 1, 2, 5, and 6 are in the extreme tails of the Monte Carlo distribution. Typically, the rates of decline in the trough (phase 1) are too high in the model economies relative to actual growth rates, but the growth rates early in the expansion (phase 2) implied by the model economies are too low. Similarly, the peak (phase 5) growth rates implied by the model economies are too high, while for output, investment, and hours the model economies imply a much sharper decline at the onset of a recession (phase 6) than is the case for actual recessions. Furthermore, actual consumption is in the tails for almost every [TABULAR DATA FOR TABLE V OMITTED] phase for the model; the growth rate of actual consumption is higher in expansions and lower in recessions than is implied by the model economy. This is consistent with the well-known result that consumption is not volatile enough in these classes of models. In contrast, the business cycle shape of investment suggests greater volatility in these models than in actual investment; another well-known characteristic of these models.

In addition, the model does not mimic the overall shape of business cycles particularly well. This is illustrated in Figure 1. For ease of comparison, the figures have been normalized so that both the actual and model-generated business cycles have the same cyclical duration and trend. For output, investment, and hours, the model economy implies "spiked" peaks and troughs while in actual business cycles the peaks are more "round." Focusing on aggregate output, concave expansions are not a feature of the model. From Table VII, the proportion of replications in which the hypothesis of linear expansions (phase 2 = 3 = 4) is rejected is close to the nominal size of the test. Similarly, investment in the model does not show a concave business cycle shape. Consumption, on the other hand, is actually convex over the expansion as the growth rate of consumption increases as the expansion progresses. For all the variables, aside from perhaps the trough phase, linear recessions appear to be a characteristic of these models. In fact, evidence against the basic two-phase cycle with just expansions and contractions is due primarily to the "spiked" nature of peaks and troughs; the greatest rates of growth and decline occur at peaks and troughs.

TABLE VI

Average Growth Rates of Selected Aggregates during Business Cycle
Phases in an Artificial Economy

 Output Consumption Investment Hours Wages

Phase 1 -5.07 0.03 -17.23 -5.10 0.03
(Trough) (1.22) (0.22) (3.44) (0.98) (0.22)
 [0.72] [1.00] [1.00] [0.97] [1.00]

Phase 2 3.68 1.51 8.86 2.17 1.51
 (0.73) (0.18) (2.19) (0.62) (0.18)
 [0.85] [1.00] [0.43] [0.03] [1.00]

Phase 3 2.95 1.88 5.46 1.06 1.88
 (0.72) (0.20) (2.13) (0.60) (0.20)
 [0.72] [1.00] [0.17] [0.85] [0.99]

Phase 4 3.12 2.19 5.34 0.93 2.19
 (0.72) (0.20) (2.14) (0.60) (0.20)
 [0.07] [0.18] [0.12] [0.27] [0.21]

Phase 5 5.59 2.64 12.64 2.95 2.64
(Peak) (1.20) (0.24) (3.65) (1.03) (0.24)
 [0.00] [0.00] [0.00] [0.00] [0.01]

Phase 6 -4.08 1.08 -16.38 -5.16 1.08
 (1.12) (0.25) (3.42) (0.97) (0.25)
 [0.98] [0.00] [0.99] [1.00] [0.00]

Phase 7 -2.46 0.91 -10.50 -3.37 0.91
 (1.34) (0.27) (4.10) (1.16) (0.27)
 [0.14] [0.00] [0.52] [0.45] [0.01]

Phase 8 -2.04 0.64 -8.42 -2.67 0.64
 (1.27) (0.28) (3.86) (1.09) (0.28)
 [0.36] [0.00] [0.25] [0.04] [0.68]

Notes: Standard deviation of the Monte Carlo distribution is in
parentheses. Percentile of Monte Carlo distribution in which the
actual datum lies is reported in square brackets.

Tables VIII and IX present phase results for growth cycle phases for the Hansen-Rogerson model. The model seems to mimic the shape of the growth cycle better than the shape of the business cycle. Like the actual growth cycle, the modeled growth cycle is symmetric.(16) However, there appears to be more evidence against the two-phase characterization of the growth cycle in the model than in the actual data. This is primarily due to the behavior of the model around peak and trough phases. As was the case above, phases 1, 2, 5, and 6 from the actual growth cycle are typically in the tails of the Monte Carlo distribution. Again, the model economies generate peaks and troughs that are "spiked." The model also generates a consumption series that is not volatile enough while investment is too volatile. The performance of the model in terms of its ability to mimic the shape of the growth cycle is shown in Figure 2.

TABLE VII

Tests of Shape and Symmetry of Business Cycles in an Artificial
Economy Percentage of Replications which Reject the Null
Hypothesis at the 5 Percent Level

Hypothesis Output Consumption Investment Hours Wages

Tests of Shape

Phase 0.21 0.95 0.22 0.25 0.95
2=3=4=5
and
Phase
6=7=8=1

Phase 0.16 0.91 0.17 0.20 0.91
2=3=4=5

Phase 0.04 0.72 0.10 0.16 0.72
2=3=4

Phase 0.07 0.37 0.12 0.16 0.37
2=3

Phase 0.05 0.81 0.12 0.19 0.81
2=4

Phase 0.03 0.30 0.03 0.03 0.30
3=4

Phase 0.16 0.62 0.17 0.18 0.62
6=7=8=1

Phase 0.08 0.11 0.13 0.16 0.11
6=7=8

Phase 6=7 0.09 0.03 0.13 0.14 0.03

Phase 6=8 0.11 0.17 0.18 0.22 0.17

Phase 7=8 0.03 0.07 0.04 0.05 0.07

Tests of Symmetry

Recession = 0.98 0.89 0.98 0.97 0.89
Expansion

Phase 5 = 0.17 0.20 0.14 0.13 0.20
Phase 1

Phase 6 = 0.55 0.47 0.53 0.51 0.47
Phase 2

Phase 7 = 0.08 0.23 0.31 0.31 0.23
Phase 3

Phase 8 = 0.18 0.21 0.17 0.17 0.21
Phase 4

Joint 0.74 0.55 0.72 0.70 0.53

V. CONCLUSION

What have we learned from this exercise? Our objective in this paper was to evaluate the ability of a simple real business cycle model to generate classical business cycles (absolute increases and decreases in output) by breaking the cycle up into the expansion and contraction phases used by Burns and Mitchell in their monumental empirical study. We have shown that this way of looking at the cycle yields interesting insights about fluctuations in economic activity and illuminates some shortcomings of basic real business cycle models. In some respects, the basic neo-classical model does quite well in capturing actual business cycles. It generates cycles with expansion and recession durations and growth rates that are not too different (in a statistical sense) from actual business cycles. However, with respect to the subtle shape of the business cycle, this model fails to generate the concave expansions and "rounded" peaks typical of actual business cycles. Second, the difference between business cycles and growth cycles matters in the evaluation of the adequacy of the real business cycle model: the real business cycle model does a better job matching the shape of growth cycles than of business cycles. It is apparent that the business cycle is not just a growth cycle with a (deterministic) trend tacked on.

The fact that the simple real business cycle model examined here was better able to replicate the shape of growth cycles than of business cycles suggests that there may be an interaction between the trend and cycle that the artificial economy considered above is unable to capture. Two extensions to the above analysis readily suggest themselves. The first is to allow for a common stochastic trend rather than a common deterministic trend as in King, Plosser and Rebelo [1988b]. While the additional persistence implied by this model is likely to lengthen the duration of the cycle, it not clear that this extension would generate concave expansions or "round off" peaks. The second is to allow for external increasing returns to scale as in Baxter and King [1991]. They have found that productive externalities improved the ability of the neoclassical model to match actual economic data.(17) However, increasing returns to scale may not be capable of generating concave expansions and rounded peaks. The effect of the externality will be largest when aggregate output is high, such as during the late stages of the expansion, which may actually work against generating concave expansions.

TABLE VIII

Average Growth Rates of Selected Aggregates During Growth Cycle
Phases in an Artificial Economy

 Output Consumption Investment Hours Wages

Phase 1 -5.31 -1.28 -14.93 -4.03 -1.28
(Trough) (0.98) (0.20) (2.98) (0.84) (0.20)
 [0.80] [1.00] [1.00] [0.99] [1.00]

Phase 2 3.70 0.20 12.07 3.51 0.20
 (0.86) (0.19) (2.62) (0.74) (0.19)
 [0.42] [1.00] [0.01] [0.00] [0.99]

Phase 3 2.39 0.44 7.03 1.95 0.44
 (0.91) (0.20) (2.77) (0.79) (0.20)
 [0.77] [1.00] [0.35] [0.49] [0.10]

Phase 4 2.54 0.78 6.75 1.76 0.78
 (0.87) (0.20) (2.64) (0.75) (0.20)
 [0.21] [0.98] [0.27] [0.53] [0.58]

Phase 5 5.36 1.31 15.02 4.05 1.31
(Peak) (1.01) (0.19) (3.10) (0.88) (0.19)
 [0.01] [0.00] [0.01] [0.04] [0.01]

Phase 6 -3.82 -0.20 -12.44 -3.61 -0.20
 (0.82) (0.18) (2.50) (0.71) (0.18)
 [0.99] [0.34] [1.00] [1.00] [0.21]

Phase 7 -2.41 -0.45 -7.09 -1.96 -0.45
 (0.92) (0.20) (2.81) (0.80) (0.20)
 [0.60] [0.01] [0.46] [0.87] [0.36]

Phase 8 -2.40 -0.77 -6.29 -1.63 -0.77
 (0.90) (0.22) (2.73) (0.77) (0.22)
 [0.08] [0.00] [0.36] [0.17] [0.36]

Notes: Standard deviation of Monte Carlo distribution is in
parentheses. Percentile of Monte Carlo distribution in which the
actual datum lies is reported in square brackets.
TABLE IX

Tests of Shape and Symmetry of Growth Cycles in an Artificial
Economy Percentage of Replications which Reject Null Hypothesis at
the 5 Percent Level

Hypothesis Output Consumption Investment Hours Wages

Tests of Shape

Phase 0.42 0.99 0.44 0.48 0.99
2=3=4=5
and
Phase
6=7=8=1

Phase 0.28 0.92 0.28 0.30 0.45
2=3=4=5

Phase 0.09 0.45 0.16 0.21 0.45
2=3=4

Phase 0.13 0.14 0.19 0.23 0.14
2=3

Phase 0.08 0.58 0.18 0.26 0.58
2=4

Phase 0.04 0.24 0.04 0.04 0.24
3=4

Phase 0.30 0.90 0.32 0.34 0.90
6=7=8=1

Phase 0.13 0.42 0.23 0.29 0.42
6=7=8

Phase 6=7 0.15 0.13 0.22 0.26 0.13

Phase 6=8 0.13 0.54 0.25 0.34 0.54

Phase 7=8 0.04 0.18 0.04 0.05 0.18

Tests of Symmetry

Recession = 0.01 0.08 0.01 0.01 0.08
Expansion

Phase 5 = 0.01 0.01 0.01 0.01 0.01
Phase 1

Phase 6 = 0.02 0.03 0.02 0.02 0.03
Phase 2

Phase 7 = 0.03 0.04 0.03 0.03 0.04
Phase 3

Phase 8 = 0.02 0.05 0.02 0.02 0.05
Phase 4

Joint 0.01 0.03 0.01 0.01 0.03

A further extension would be to add sectors to the basic real business cycle model to reflect government spending, net exports, inventory investment, and consumer durables. While actual government spending and net exports have business cycle shapes that would, if anything, make it more difficult to generate concave expansions (Balke and Wynne [1992]), they would introduce an alternative source of shocks into the real business cycle model that has in other contexts improved "the fit" of the basic neoclassical model (Christiano and Eichenbaum [1992]). Inventory investment and expenditures on consumer durables, on the other hand, have very strong Burns and Mitchell phase behavior - they both have very strong recovery effects and rounded peaks (see Balke and Wynne [1992] and Sichel [1992]). Nonetheless given the tendency of the basic real business cycle model to generate "spiked" peaks and troughs, it is not clear that adding these sectors will generate the desired business cycle shape.

The typically concave shape of expansions and the linear shape of recessions suggest an asymmetry over the business cycle that is inconsistent with linear models of the business cycle. Nonlinear models of the cycle that utilize the concept of a ceiling on output, such as Friedman's [1993] "plucking model" of business cycles and Hicks's [1950] model of the trade cycle (which also places a floor on output), may be consistent with the concavity of output over expansions. So too may neo-classical models with irreversible investment as analyzed by Sargent [1980] and Dow and Olson [1992].

1. This approach is very much in the spirit of Gregory and Smith [1991].

2. After writing the original version of this paper, we became aware of a paper by Simkins [1993], who like King and Plosser [1989] examines business cycles in terms of the Burns and Mitchell methodology but also conducts a statistical examination similar to ours. The primary difference between our paper and that of Simkins is his focus is on growth cycles while ours is primarily on business cycles. Also, the null real business cycle model studied by Simkins is different to the one studied in this paper.

3. Moore and Zarnowitz [1986] discuss the distinction between business cycles and growth cycles.

4. Stock [1987] is a more detailed analysis of the possibility that economic variables evolve according to economic rather than calendar time.

5. See for example Hamilton [1989] and Boldin [1990; 1992].

6. The shape of the business cycle has been studied by Neftci [1993] and Sichel [1993] among others.

7. The issue of asymmetry in business cycles has previously been addressed by Blatt [1980], DeLong and Summers [1986], Neftci [1984], Falk [1986], and Sichel [1989]. Blatt [1980] explicitly points out the implications of asymmetric (growth) cycle for "Frisch-type" models of the sort we will consider below.

8. See Appendix A of Balke and Wynne [1993] for details of how the phase dummies were set.

9. For a description of the business cycle dating process see Moore and Zarnowitz [1986] and Hall [1992].

10. The accuracy of the Bry-Boschan algorithm is also discussed in King and Plosser [1989].

11. We also ran versions employing the White [1980] consistent covariance matrix estimator with the Newey-West [1987] correction for serial correlation. The results were qualitatively similar.

12. The trend rate of growth for output, consumption, and investment was estimated by fitting a common linear time trend to these series as in King, Plosser, and Rebelo [1988a]. The trend growth rate for wages was calculated separately using a linear time trend.

13. See also Sichel [1992]. Elsewhere we have examined the relationship between the rate of growth during the recovery period and various measures of the severity of the prior recession. See Wynne and Balke [1992; 1993].

14. When the official NBER dates are used we obtained essentially the same results. If anything, output and investment display even more concavity over official NBER expansions.

15. King and Plosser [1989] show essentially the same result - see their figures 7 and 8.

16. This is not surprising given that the variables were generated by a linear approximation around the steady state.

17. They also include a government sector and preference shocks in the model.

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NATHAN S. BALKE and MARK A. WYNNE, Associate Professor, Southern Methodist University and Research Associate, Federal Reserve Bank of Dallas, and Senior Economist, Federal Reserve Bank of Dallas respectively. We are grateful to Anton Braun, Scott Freeman, Greg Huffman and an anonymous referee for comments. We also thank participants in the October 1992 Federal Reserve Business Analysis System Committee Meeting in Philadelphia, the 1993 Western Economic Association Meetings, the Summer 1993 North American Meetings of the Econometric Society and seminar participants at the University of Texas-Austin and Ohio University for suggestions. Shengyi Guo provided expert research assistance. This paper is an extensive revision of parts of an earlier paper entitled "The Dynamics of Recoveries." The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.