Is the endogenous business cycle dead?

I. Introduction

A clearly defined dichotomy exists in the business cycle literature between endogenous and exogenous cycles. Exogenous cycles are either temporary, heavily damped random deviations from a stable long-run growth path or permanent stochastic fluctuations in the growth path which both require repeated stochastic impulses to generate typically observed recurrent and irregular fluctuations. In contrast, endogenous cycles are systematic (deterministic), self-generating recurrent cycles that result from the inherent instability (structure) of the underlying economy.

The most recent and most severe critiques of endogenous theory are empirical in nature and stem from the unit root debates which contrast trend stationary (TS) and difference stationary (DS) models. Despite this critique, the evolution of this methodology has produced conflicting results with respect to the most appropriate model. More importantly this approach implicitly rejects, through the use of an overly restrictive specification, endogenous cycles in favor of stochastic cycles. In this light, the purpose of this paper is to justify and apply an alternative, more general, estimation framework that includes DS, TS and endogenous cycles as nested alternatives.

In particular, I employ a structural time series (STS) or unobserved components methodology which allows for a direct empirical test of endogenous cycle theory against stochastic alternatives and/or mixed stochastic-endogenous models. The integration of secular regime shifts into the basic STS model effectively introduces nonlinearities and thus moves the analysis one step beyond simple linear models. This general approach which relies on economic theory for model specification is superior to the ARIMA-based unit root methodology which relies solely on the data to identify the structure of macro time series.

Using this approach, I estimate STS models for seven relevant U.S. macroeconomic time series and find that endogenous cycles play a fundamental role in characterizing the data generation process.

The remainder of this paper is organized in the following manner. Section II reviews the restrictive nature of the unit root-ARIMA methodology. Section III offers an alternative approach. Section IV presents estimation results and section V contains my conclusions.

II. The Unit Root-ARIMA Methodology

While the early work of Nelson and Plosser [6] sparked interest in the subject of unit roots, it also severely limited the scope of inquiry through a restrictive specification of endogenous business cycles. The unit root debate contrasts two variants of new classical stochastic business cycle theory - real business cycles versus equilibrium business cycles based on incomplete information and rational expectations.

More formally, a TS process can be represented as follows

[Y.sub.t] = [[Alpha].sub.0] + [[Alpha].sub.1]t + [Theta](L)[[Phi].sup.-1](L)[[Epsilon].sub.t] (1)

where L is the lag operator, [[Alpha].sub.0] and [[Alpha].sub.1] are constants, t is a time trend, and [Theta](L)[[Phi].sup.-1](L)[[Epsilon].sub.t] is a stationary ARMA (p, q) process. While stationarity does not preclude complex conjugate roots for the AR polynomial and thus systematic cyclical behavior, it limits that behavior to damped fluctuations. Thus constant amplitude (self-generating) cycles are not readily considered. While this approach subsumes endogenous cycles as a special case - complex conjugate roots with a modulus statistically indistinguishable from one, the vast majority of unit root tests accept DS over TS. In the DS case, the treatment of endogenous cycles is even more restrictive.

Equation (2) represents a DS process:

[Delta][Y.sub.t] = [Beta] + [Theta](L)[[Phi].sup.-1](L)[[Epsilon].sub.t] or [Y.sub.t] = [Y.sub.t-1] + [Beta] + [Theta](L)[[Phi].sup.-1](L)[[Epsilon].sub.t] (2)

where [Beta] is a constant and [Delta] is the difference operator. Even though [Delta]Y can follow an AR(p) process and thus include systematic cycles implying cycles in [Y.sub.t], the coefficient restrictions implied by the I (1) structure make it extremely difficult to find evidence of constant amplitude behavior.

In particular a DS plus AR(p) or ARIMA (p, 1, 0) results in a potential cycle characterized by a p + 1 order difference equation for [Y.sub.t] (in levels) with p independent coefficients.(1) In addition, recent evidence of smooth stochastic trends - a special case of a local linear trend where the intercept is not stochastic but the slope is mildly stochastic - in macro time series, found by Harvey and Jaeger [4] suggests that the DS and TS models are misspecified.(2)

On the practical side, the treatment of autocorrelation in unit root equations neglects the importance of structural cycles by treating the MA (q) process as an infinite AR process and thus conflating the AR and MA components.(3) Finally, the common finding of a unit root and a significant time trend in unit root tests casts doubt on the validity of such tests. Technically, the acceptance of the unit root hypothesis requires that both the coefficient on [Y.sub.t-1] and t be insignificant in a [Delta][Y.sub.t] equation. Yet practitioners interpret this common result as support for the DS hypothesis. In contrast, these results, as are the smooth trend findings, may be indicative of more complex behavior where [Delta][Y.sub.t] is nonstationary and thus suggest the need for a more flexible modelling approach.

In summary, the unit root-ARIMA methodology neglects, on the practical level, the existence of structural cycles and, on the theoretical level, treats these cycles through a restrictive specification. Thus, this approach is seriously flawed with respect to cross-paradigm or mixed tests of business cycle theories.

III. Statistical Methodology

In this section, I describe an alternative statistical methodology - structural time series (STS) or unobserved components models - which includes a less restrictive specification of cycles and effectively considers relevant stochastic and deterministic trend and cycle models as nested alternatives or mixed models which can be decomposed.

In a formal specification of an STS model,(4) the trend, [Mu], is modelled with a random level (intercept), [Alpha], and a random slope, [Beta], as:

[[Mu].sub.t] = [[Mu].sub.t-1] + [[Beta].sub.t-1] + [[Eta].sub.t] (3a)

[[Beta].sub.t] = [[Beta].sub.t-1] + [[Zeta].sub.t] (3b)

where [Mathematical Expression Omitted], [Mathematical Expression Omitted], [[Eta].sub.t] and [[Zeta].sub.t] are mutually uncorrelated and [[Mu].sub.0] = [Alpha]. The equations in (3) characterize a local linear trend. In the case where [Mathematical Expression Omitted] a special case of an I(2) trend, a smooth stochastic trend results, when [Mathematical Expression Omitted] and [Mathematical Expression Omitted] a DS model results, and when [Mathematical Expression Omitted], [[Mu].sub.t] = [Alpha] + [Beta]t is a deterministic trend or TS.

Consistent with the general solution of a difference equation that can exhibit a constant amplitude cycle, the cyclical component is modelled as

[[Psi].sub.t] = [Gamma]cos[[Lambda].sub.c]t + [Delta] sin [[Lambda].sub.c]t (4)

where [Gamma] and [Delta] are unknown parameters and [[Lambda].sub.c] is the unknown frequency of the cycle measured in radians. The cycle period is thus 2[Pi]/[[Lambda].sub.c]. An appropriate stochastic variant of equation (4) which includes both a damping factor, [Rho], and random walk-type evolution of [Gamma] and [Delta] is generated from a two equation recursion.(5) In order to form [[Psi].sub.t], the recursion requires the use of a constructed variable, [Mathematical Expression Omitted].

Thus the cycle can be expressed as:

[Mathematical Expression Omitted] (5)

where 0 [less than or equal to] [Rho] [less than or equal to] 1 is a damping factor and [[Kappa].sub.t] and [Mathematical Expression Omitted] are two white noise disturbances. This vector AR(1) model is identifiable if either [Mathematical Expression Omitted] or [[Kappa].sub.t] and [Mathematical Expression Omitted] are uncorrelated. For parsimony both of these assumptions are imposed.

The reduced form of equation (5) allows for the decomposition of the cycle into deterministic and stochastic components:

[Mathematical Expression Omitted] (6)

where [Y.sub.t] is the observed series. The LHS of equation (6) represents the deterministic cycle, while the RHS the stochastic cycle. The deterministic cycle in (6) will have complex conjugate roots under the condition that 0 [less than] [[Lambda].sub.c] [less than] [Pi]. Here two independent parameters, [[Lambda].sub.c] and [Rho], determine the nature of the cycle, thus this representation is less restrictive than the cycle in the unit root-ARIMA approach. The modulus associated with the cycle is [Rho]. Thus for [Rho] [less than] 1, the structural cycle is damped and for [Rho] = 1 the cycle is endogenous (self-sustaining) by virtue of its constant amplitude. Finally, when [Mathematical Expression Omitted], the cycle is deterministic (nonstochastic).

The damping factor, [Rho], is at the heart of statistical tests for an endogenous cycle. The stationarity conditions on the AR(p) polynomial in the TS and DS specifications require that the estimation techniques employed restrict [Rho] such that [Rho] [less than or equal to] 1. Thus a tradeoff exists between the specification of an all encompassing estimation framework that includes the three key hypotheses in a nested format and the equal statistical treatment of the competing hypotheses. In particular the exclusion of values of [Rho] [greater than] 1 makes, on a priori grounds, the hypothesis that [Rho] = 1 more difficult to accept. Thus the key issue in testing for an endogenous cycle is restricted to not whether [Rho] = 1 or [Rho] [not equal to] 1, but is rather the degree of damping. Point estimates of [Rho] close to, but less than, one imply that the cycle is only mildly damped with a virtually nonexistent dependence on random shocks for propagation. As a result, STS estimation can only effectively, rather than definitively, treat competing hypotheses as nested alternatives. Given that the macroeconomic theory that underlies both TS and DS specifications argues for heavily damped AR(p) components, it is not difficult to distinguish this behavior from a heavily undamped, but stationary, process.

Finally, the irregular component, [Epsilon], is modelled as:

[Mathematical Expression Omitted]. (7)

Thus the full STS model can be written as

[y.sub.t] = [[Mu].sub.1] + [[Psi].sub.t] + [[Epsilon].sub.t] (8)

where y, is the dependent variable, [[Mu].sub.t], [[Psi].sub.t] and [[Epsilon].sub.t] are as defined in equations (3), (5), and (7) and all distributance terms, [[Eta].sub.t], [[Zeta].sub.t], [[Kappa].sub.t], [Mathematical Expression Omitted] and [[Epsilon].sub.t], are assumed to be mutually uncorrelated.

The three main business cycle hypotheses can be distinguished on the basis of the [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Rho] coefficients: (1) DS requires [Mathematical Expression Omitted] and [Rho] [less than] 1; (2) TS requires [Mathematical Expression Omitted] and [Rho] [less than] 1; and (3) endogenous cycles necessitate [Rho] = 1. Thus [Rho] = 1 is a sufficient condition to distinguish endogenous cycles from the other specifications. A deterministic trend or a stochastic trend plus an endogenous cycle are evidence against the TS and DS hypotheses. In addition, a smooth stochastic trend also suggests the same result.

A non-nested alternative to the trend plus cycle specification is a cyclical trend (level) specification. This is achieved by modifying equation (3a) to be [[Mu].sub.t] = [[Mu].sub.t-1] + [[Psi].sub.t-1] + [[Beta].sub.t-1] + [[Eta].sub.t] and substituting this modification along with equations (3b), (5) and (7) into equation (8). This alternative is considered below.

Statistical treatment of STS models requires that the parameters ([Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Rho], [[Lambda].sub.c], [Mathematical Expression Omitted]) governing the evolution of the unobserved components (state variables), referred to as hyperparameters, be estimated. The Kalman filter is used to decompose the likelihood function into one-step ahead prediction errors, thus allowing for maximum likelihood (ML) estimates of the hyper-parameters to be generated. After these parameters are estimated the Kalman filter is used again to generate optimal forecasts and to generate optimal estimates of the entire state (trend, cycle) trajectories via a smoothing algorithm.(6)

IV. Estimation Results

I consider the estimation of the STS trend and cycle model for seven key quarterly macroeconomic time series. The variables cover four major areas considered in endogenous theories of the cycle - production and employment, aggregate demand, profitability, and financial conditions - and are grouped into these four categories: (1) real gross domestic product (GDP) and the civilian unemployment rate (UN); 2) real consumption (CON) and real net nonresidential investment (INV); 3) profit's share of national income (PS); and 4) the real BAA industrial bond rate (BAA) and the debt-equity ratio for the manufacturing sector (DER). A full description of each variable with sources is contained in Appendix A. All STS models are estimated over the sample range from 1949:1 to 1995:2 - a period that includes eight complete cycles and a portion of the ninth cycle as determined by the NBER cycle dating system.(7)

Before estimation techniques are employed, I describe the level and first difference of these variables. With the exception of UN, the production and demand series show a noticeable increase in volatility in the post-1970 period. There is also a possible upward shift in the trend of UN and CON in this same subperiod. The profitability and financial series reveal an increase in volatility in the DER variable and a possible shift in the trend for BAA upward. The latter shift does not occur until 1979 and is most likely associated with the regime shift in the conduct of monetary policy.

These findings have important implications for the specification of STS models. The heteroscedasticity and trend shifts can be theorized in one of four ways as a result of: 1) the normal evolution of the stochastic components of the STS model; 2) a structural change in the data generation process; 3) in the case of increased volatility only, a (significant) random shock which increases variances and levels in an autocorrelated (damped, but persistent) manner; or 4) in the case of increased volatility only, a simple heteroscedastic pattern (as a continuous function of time). Each of these four perspectives respectively supports a different statistical specification: 1) one STS model for the entire sample; 2) two STS models, one prior to the structural break the other after; 3) an autoregressive conditional heteroscedasticity (ARCH) model integrated with a single period STS model; and 4) a single period STS model for log ([Y.sub.t]) rather than [Y.sub.t].

The heteroscedastic pattern in the data and the possible trend shifts cast serious doubt on the appropriateness of a single uncorrected STS model with assumed homoscedastic error variances for the entire sample range. The only exception is the PS series which does not experience a shift or increased volatility. The existence of a heteroscedastic pattern which is a positive step-function, rather than a continuous function, of time suggests that a log transformation may induce a reverse heteroscedastic (negative function of time) pattern and thus not improve the properties of the statistical estimates and tests.

The ARCH and structural change models are competing perspectives on the evolution of a heteroscedastic pattern. The former is a stochastic explanation whereas the latter is structural/deterministic. An ARCH interpretation implies that the damping process is slow enough such that significant supply shocks in the 1970s have resulted in increased volatility into the 1990s without a return of [Mathematical Expression Omitted], to its steady state. In contrast a structural break argument focuses on a major regime shift, circa 1970, associated with a rapid deterioration in U.S. industrial relations, a significantly lower rate of productivity growth, increased levels of indebtedness, the intensification of international competition, a decline in profitability, and the breakdown of the international monetary system. These fundamental changes in both the economic and institutional structure created a permanently more uncertain and volatile environment.

Based upon a model selection strategy and my strong theoretical prior for the structural shift argument, I report estimation results below for an STS model estimated over two distinct periods, 1949:1-1970:4 and 1970:4-1995:2, for all series with the exceptions, discussed above, of PS and BAA.(8) The results reported below are robust for alternative break points between 1969-1975. In addition, selective results for alternative specifications - models (1) and (4) above, model (4) estimated over two time periods and a cyclical trend variant, discussed above, of model (4) over the same periods(9) - are also reported. Irrespective of model specification, the results concerning the endogenous nature of the business cycle are robust.

One last pre-estimation diagnostic, the sample autocorrelation and partial autocorrelation coefficients for the first differences of the seven series are reported in Table I for the period 1949:1-1995:2.(10) Examining Table I, in all series except BAA the weak condition for the existence of a cyclical component - a positive first order autocorrelation coefficient for the first difference along with higher order coefficients that are not strictly zero - is clearly met. The stronger condition for a cycle - existence of an AR(2) pattern - is met by the GDP, INV, PS, and UN series where a clear wave-like pattern exits, while in the CON, and DER series there is less strong, yet observable, evidence of an AR(2) pattern. The existence of a cyclical component is further supported in the GDP and INV time series by the appearance of two positive spikes in the sample partial autocorrelation function. The patterns observed for PS, DER, CON and UN are more indicative of mixed trend and cycle models. In summary, besides the strong theoretical support, there is strong pre-estimation evidence for the inclusion of a cyclical component in the modelling of these time series. Thus I now turn to the estimation of the STS model in (8).

Table I. Sample Autocorrelation and Partial Autocorrelation
Coefficients of First Differences, 1949:1-1995:2

Lag              DGDP   DINV   DCONS   DUN    DPS    DDER   DBAA(+)

Autocorrelations

1                 .36    .57     .22    .61    .14    .12    -.33
2                 .21    .43     .25     24    .02    .07    -.04
3                 .07    .28     .30   -.07   -.04    .13     .03
4                 .05    .13     .06   -.21   -.15    .18     .07
5                -.01   -.03     .10   -.21   -.14    .11    -.18
6                 .01   -.04     .09   -.10   -.02   -.06     .03
7                -.06   -.11     .02   -.10    .00    .06     .01
8                -.18   -.22    -.03   -.13   -.08    .03     .03
9                -.04   -.13    -.01   -.06    .03    .13     .02
10                .05   -.08     .05   -.04    .03    .04     .04

Partial Autocorrelations

1                 .36    .57     .22    .61    .14    .12    -.33
2                 .10    .16     .21   -.21    .00    .06    -.17
3                -.04   -.02     .24   -.20    .05    .11    -.05
4                 .02   -.10    -.08   -.05    .14    .15     .07

Notes: Standard error of all autocorrelation coefficients is .078.

+ Sample: 1951:2 to 1995:2.

The ML estimation results for the early subperiod (1949:1-1970:4) model and the later subperiod (1970:4-1995:2) model are respectively reported in Tables II and III. The estimates for PS are for the entire period (1949:1-1995:2) and only appear in Table II. The tables are organized by the estimates of the four hyperparameters ([Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted]), other key parameters - [Rho], the damping factor, [[Lambda].sub.c], the cycle frequency, and the cycle period (2[Pi]/[[Lambda].sub.c]) - and a series of test statistics - PEV, the prediction error variance for one step-ahead predictions, Q (lag length), the Box-Ljung test for serial correlation of the residuals, N, the Jarque-Bera test for normality of the residuals, H, a standard split sample test for heteroscedasticity, [R.sup.2] and [Mathematical Expression Omitted], the percentage improvement in fit over a random walk plus drift model.(11)

Given that it is common either for some STS parameters to lie on the boundary of the parameter space or to formulate hypotheses for parameter values that lie on the boundary, a violation of one of the regularity conditions, standard distribution theory cannot be relied upon to specify appropriate hypothesis tests. Alternatively, I rely on a series of nonstandard tests which are valid as long as all of the other regularity conditions are met (see n. 6). A most powerful invariant (MPI) test is used to test for a deterministic trend [Mathematical Expression Omitted], a likelihood ratio (LR) test is used to test [Mathematical Expression Omitted], and a modified Lagrange multiplier (LM) test is employed to test [Rho] = 0 and [Rho] = 1.(12)

Examining the results in Tables II and III, the overwhelming majority of the series in both periods are characterized by a smooth stochastic trend ([Mathematical Expression Omitted] and [Mathematical Expression Omitted]) and an additive cyclical component ([Rho] [greater than] 0). The exceptions are UN in the early period and CON in the late period [TABULAR DATA FOR TABLE III OMITTED] which exhibit a local linear trend plus cycle, and BAA in the early period and PS over the whole period which include cycles, but for which tests cannot distinguish between a local linear and a smooth trend. The dominant smooth trend result does not support either the TS or DS hypotheses, but is consistent with the results found by Harvey and Jaeger [4] for macro time series.

Turning to the important damping factor, [Rho], the results reveal several point estimates greater than or equal to .90 implying that the cyclical components are dramatically undamped, but still are stationary. A 95% (99%) two-tailed confidence interval for [Rho] contains 1.0 in 11 (12) out of 13 cycles. The exceptions being CON in the late period (95% level only) and BAA in the late period (95% and 99%). Yet the [Rho] confidence interval for BAA contains .99 at the .01 level of significance. If a stricter one tail interval is constructed at the .01 (.05) significance level, in 12 (10) out of 13 cycles this interval contains a value of one.(13) The exception in the .01 case is late period BAA. Thus the early and late period cycles in GDP, UN, INV, and DER and the early period cycles in CON and BAA are statistically indistinguishable from constant amplitude cycles. This finding provides strong evidence, in spite of the effectively more stringent intervals employed, for the existence of self-generating, endogenous cycles in these key macroeconomic time series. For the severely restricted number of cases for which the LM tests of [Rho] = 1 are valid (regularity conditions hold), these results are confirmed. In particular, [Rho] = 1 for early period INV, CON and DER and full period PS are not rejected.

The combined findings of constant amplitude cycles coexisting with a smooth, rather than erratic, stochastic trend in all periods heavily favors the hypothesis that cycles are endogenous phenomena with little or no dependence on random shocks in contrast to the alternative that cycles are shock dependent stochastic/exogenous occurrences. These findings also suggest that both the DS and TS models are inappropriate and that the unit root (TS-DS) debate is unnecessarily limited in scope.

Other results support the appropriateness of the statistical specification. The estimated period of the cycles range from 10.44 quarters for UN to 20.91 quarters for DER in the early period and from 10.51 quarters for UN to 34.13 quarters for CON in the late period. This periodicity, averaging 15.11 and 21.79 quarters respectively in the early and late subperiods, is eminently reasonable particularly in light of readily available explanations for the more extreme values associated with early and late period UN and GDP.(14) In addition the diagnostics for the 13 models reported in Tables II and III suggest for the most part that the overall STS specifications are acceptable. If tests for normality, homoscedasticity and autocorrelation are performed at a .01 level of significance, then all models with the exception of the UN model in both periods and BAA in the late period pass all diagnostic tests. While the violations associated with the UN model can be explained,(15) no explanation is readily available in the BAA case. Thus all models with the exception of BAA in the late period can be argued to exhibit acceptable diagnostics. In addition the results reported below show that the trend plus cycle model with a structural break, used here, is superior in the vast majority of cases to the alternative specifications considered.

The [Mathematical Expression Omitted] statistics which report the percentage improvement in fit over the random walk with drift model - a special case of the DS formulation and the basic representation of the dominant stochastic cycle theory - range from .03 to .37 and thus provide further evidence of the importance of the undamped (endogenous) cyclical component found in the majority of models.

The estimated cycle and trend components for selected series are depicted in Figure 1. In all four cycles the underlying constant amplitude (endogenous) nature of the cycle is quite apparent. The depiction of trend components is in increasing order of stochastic influences with the late INV and GDP trends being quite smooth and the full period PS and early period UN trends being quite erratic.

I now consider selective results, reported in Table IV, from the alternative specifications discussed above. The PEVs for the log ([Y.sub.t]) equations have been multiplied by the factor [e.sup.[(2/T)[Sigma]Log [Y.sub.t]]] to make them comparable to PEVs from [Y.sub.t] models. The BAA results are excluded in the log cases due to the existence of negative real interest rates in some periods. As expected by the structural break hypothesis, the full period [Y.sub.t] and log ([Y.sub.t]) models exhibit severe violations of statistical assumptions. One major problem in the log ([Y.sub.t]) models is the overcorrection for heteroscedasticity exhibited in the H statistics. This result is most consistent with a step-wise increase in volatility, as a result of a structural break.

[TABULAR DATA FOR TABLE IV OMITTED]

While the subperiod log ([Y.sub.t]) models with an additive cycle have similar diagnostic problems in the early period, they have somewhat less problems in the late period. The INV and CON model have acceptable diagnostics. In all cases with three exceptions the log ([Y.sub.t]) subperiod models fit the data less well than their untransformed counterparts reported above.(16) The results for the subperiod log ([Y.sub.t]) models with a trend in cycle are similar to the log ([Y.sub.t]) additive cycle models. In the late period both the UN and INV models outperform the respective models reported above.

Thus with few exceptions, the models reported above are superior to the rival specifications. Turning to the important damping factor, the alternative models produce qualitatively and quantitatively similar results to those reported above. Thus, the results that endogenous cycles are prevalent in these key seven macroeconomic variables is robust.

Finally, I assess the relative importance of the endogenous cycles reported in Tables II and III. To make such an assessment operational, it is necessary to decompose the estimated cycle, [Mathematical Expression Omitted], in each time period into three components: 1) the pure endogenous cycle, EC; 2) the pure stochastic cycle, SC; and 3) the mixed endogenous-stochastic cycle, MC. Equation (6) allows this decomposition to be carried out where the independent evolution of the LHS of the equation determines EC and the independent evolution of the RHS generates SC. The MC which is calculated as a residual ([Mathematical Expression Omitted]) represents the effect of past stochastic shocks after they have been incorporated into the structural/endogenous propagation (cycle) mechanism.

In order to assess the relative importance of these three components, I simulate [Mathematical Expression Omitted] (equation 6) and its three components and then calculate the average correlation coefficient between [Mathematical Expression Omitted] and EC, SC, (EC + MC) and (SC + MC) generated from 1,000 simulations.(17) The average correlation coefficients are reported in Table V for two sets of simulations: (1) [Mathematical Expression Omitted] from Tables III and IV; and (2) [Rho] = 1.

The treatment of the mixed cycle component, MC, is crucial for the interpretation of the results in Table V. Endogenous and exogenous cycle theorists disagree and consider MC respectively as an endogenous and stochastic element. While I report results consistent with both interpretations, here I focus on the endogenous approach. In particular, a stochastic shock irrespective of the complexity of its dynamic structure, such as RHS of equation (6), has no subsequent effect [TABULAR DATA FOR TABLE V OMITTED] of its own unless it is absorbed into the systematic/endogenous propagation mechanism. At that point, it becomes part of the endogenous cycle and loses its independent existence.

In this light the results in Table V, strongly corroborate the earlier results on the relevance of endogenous cycles. In particular there exists a strong, almost perfect when [Rho] = 1, linear association between the total endogenous cyclical components and the overall cycle ([Mathematical Expression Omitted], (EC + MC)). In contrast, the association between the pure stochastic component and the cycle ([Psi], SC) is much weaker. In addition, the relative strength of the total endogenous components is strengthened as [Rho] [approaches] 1.(18) If we confine our analysis to the relative strength of the pure endogenous and stochastic elements, the two have correlation coefficients of similar magnitude in the [Mathematical Expression Omitted] case, but the EC dominates in the [Rho] = 1 case. This result further establishes the relative importance of the endogenous cycle.

V. Conclusion

The empirical debates over alternative business cycle theories and the accompanying theoretical literatures have predominantly focused on the relative merits of two variants of stochastic business cycles - trend stationary versus difference stationary models. As a result endogenous theories of the business cycle have been dismissed, either by restrictive specifications or by exclusion, as being empirically uninteresting. Given this situation, I suggest a statistical methodology, structural time series modelling, which includes trend stationary, difference stationary and endogenous cycles as nested alternatives. This framework is applied to seven U.S. macroeconomic time series typically considered in endogenous business cycle theories. A pre and post-1970 regime shift is also integrated into the structural model.

Using this approach, I find strong evidence that endogenous cycles play a fundamental role in characterizing the data generation process. In particular, in both the early and late periods the vast majority of the seven series are characterized by smooth stochastic trends and a constant amplitude additive cyclical component. The combination of a smooth trend and a constant amplitude cycle emphasizes the importance of permanent endogenous cycles. These results are shown to be robust with respect to alternative specifications.

Appendix A: Definition of Variables

BAA

The real interest rate series is defined as Moody's Industrial BAA bond rate minus the general rate of inflation. The inflation rate is the percentage change in the GDP deflator.

Mean = 3.81 Standard Deviation = 3.34

CON

Personal consumption expenditures in constant (1987) dollars. Source: NIPA (U.S. Department of Commerce).

Mean = 2015.55 Standard Deviation = 846.85

DER

The DER series is complied from the debt-equity ratio for manufacturing in various issues of the Quarterly Financial Report for Manufacturing Corporations, 1947-86 U.S. Department of Commerce). The debt component of the ratio is based on the market value of the current stock of debt, while the equity component is based on book value.

Mean = 41.03 Standard Deviation = 18.29

GDP

Gross Domestic Product in constant (1987) dollars. Source: NIPA (U.S. Department of Commerce).

Mean = 3135.97 Standard Deviation = 1190.02

INV

Fixed nonresidential investment in constant (1987) dollars. Source: NIPA (U.S. Department of Commerce).

Mean = 333.25 Standard Deviation = 157.95

PS

Corporate profits with inventory valuation adjustment and capital consumption adjustment as a percent of national income. Source: NIPA (U.S. Department of Commerce).

Mean = 10.77 Standard Deviation = 2.22

UN

The civilian unemployment rate. Source: Department of Labor, Bureau of Labor Statistics.

Mean = 5.79 Standard Deviation = 1.60

Means and standard deviations are for the period 1949:1 to 1995:2.

1. In the popular ARIMA (1, 1, 0) case the additive cycle is represented by [Y.sub.t] = [Rho][Y.sub.t-1] - [Rho][Y.sub.t-2]. While a cycle is possible if 4[Rho] [greater than] [[Rho].sup.2], a self-perpetuating cycle only occurs when [Rho] = 1 implying that the frequency (period) of the cycle is constrained to be exactly 1.045 radians (6.00 time periods).

In contrast an AR(p) allows for a damped, cycle in [Delta][Y.sub.t]. Given that most endogenous cycle theories are cast in terms of levels rather than growth rates, the DS plus cycle model fails to adequately incorporate endogenous cycles. Even in the case of a growth cycle, the stationarity requirement in the DS specification rules out endogenous growth cycles.

2. It can readily be shown that treating the stochastic slope of a time trend as exogenous results in positive autocorrelation.

As Harvey and Jaeger [4] demonstrate via Monte Carlo simulations, these I(2) stochastic trends are not readily detected by tests for multiple unit roots or ARIMA modelling methods. Thus the AR corrections typically used in single unit root tests fail to completely correct for autocorrelation and result in inappropriate test statistics for the unit root hypothesis.

Further, simulations reveal a tendency for t statistics to be biased upward (towards accepting a single unit root) when a smooth stochastic trend underlies the data but is not specified. In particular, data was generated from the local linear trend model so as to emulate GDP behavior. In addition, data was generated from a random walk with drift with the same innovations. Standard unit root tests were performed on both variables for a sample size of 150 and with the inclusion of from one to eight lagged difference terms. On the basis of 500 samples per set of parameter values, the average t statistic associated with the unit root coefficient was .31 to .45 larger (less likely to reject a unit root) in the cases where the local linear trend was the true data generating process.

3. While this problem can be resolved by the estimation of an ARIMA model after the issue of a unit root is resolved, Harvey [2; 3, 90-93] and Harvey and Jaeger [4, 238-39] have convincingly argued that ARIMA models can be misleading, particularly in the case of uncovering cycles, when such models are chosen on grounds of parsimony.

In addition to this methodological critique of unit root tests, there is an operational critique. See Enders [1, 239-58], McCallum [5] and Stadler [8, 1772-73]. In particular, the power of unit root tests is notoriously weak, the tests are biased by the failure to consider structural breaks, heteroscadastic and autocorrelated errors and the correct specification of deterministic regressors. Most recently there has been a turnaround in the assessment of the empirical evidence. After years of interpreting the evidence in favor of DS models, tests that correct for the weaknesses of the original unit roots tests now find that the TS model is better supported. See McCallum [5, 16-17], Simkins [7, 977-78], and Stadler [8, 1773].

4. The remainder of this section relies heavily on Harvey [3].

5. The recursion is used to avoid discontinuities in the specification and to ensure that the associated forecasting function retains the property of discounting. See Harvey [3, 27-9].

6. In general desirable properties of the ML estimates hold when a set of regularity conditions are met. In the case of the model in equation (8), this requires that all disturbances are distributed NID, are mutually uncorrelated, [Mathematical Expression Omitted], [Mathematical Expression Omitted], 0 [less than] [Rho] [less than] 1, and 0 [less than] [[Lambda].sub.c] [less than] [Pi]. Under these conditions, the estimates of [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Rho], [[Lambda].sub.c], [Mathematical Expression Omitted] are distributed asymptotically normal, and are asymptotically unbiased with asymptotic standard errors.

7. Alternatively, estimation results were generated over the period which includes eight complete cycles (1949:1 to 1991:1). The results for this sample are qualitatively similar to the results reported in the text.

8. A one period model is fit for PS, 1949:1 to 1995:2. The break in the BAA series occurs around 1979:2. Eliminating the outlying data points associated with the Korean conflict, the early period for BAA is 1951:2 to 1979:2 and the late period 1979:2 to 1990:2.

9. The discussion in the text justifies the inclusion of the first three alternative specifications. The addition of the fourth is motivated by Harvey's [2] findings that annual data on the natural log of GNP, industrial production, and UN are best modeled by a cyclical trend, rather than an additive cycle, for the pre-1948 period. The same finding also holds for GNP in the 1948-1970 period. In addition, the inclusion of all log models is justified by the reliance on log transformed data in the variable trend and unit root literatures.

While the integration of an ARCH process with an STS model is beyond the scope of this paper, diagnostic evidence on the relevance of an ARCH specification - Q statistics for the autocorrelation function of the squared residuals associated with the models in Tables II and III, not reported - do not support a full period ARCH model in favor of a structural break model. In particular based on Q(20) statistics, an ARCH process is not found to exist in both subperiods for any one variable.

10. The correlograms for the pre and post-1970 period are qualitatively similar and are thus not reported.

11. [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and H [similar to] F(h, h) where m is the number of nonzero estimated parameters (hyperparameters plus [Mathematical Expression Omitted] and [[Lambda].sub.c]) minus one and h is the nearest integer to (T - 4)/3 and T is the number of observations.

12. These tests are developed in Harvey [3, 236-39, 242-46, 248-54]. The MPI test statistic is (T - 2)[1 - S([[Psi]*.sub.1])/S ([[Psi]*.sub.0])]/375.1 where S is the sum of squares of the standardized innovations, [[Psi]*.sub.1] is the parameter set with [Mathematical Expression Omitted] set equal to 375.1/[(T - 2).sup.2], [[Psi]*.sub.0] is the parameter set with [Mathematical Expression Omitted] (a deterministic trend), T is the number of observations. The ratio of [Mathematical Expression Omitted] to [Mathematical Expression Omitted] is chosen on the basis of Pitman efficiency. Critical values for the test ([Alpha] = .05) are given in Harvey [3, 254].

The LR test of [Mathematical Expression Omitted] is standard with the exception that its asymptotic distribution is equivalent to a [Mathematical Expression Omitted] with 2[Alpha] level of significance.

The LM tests for p take the form of a weighted regression of V(j) on elements of the vector Z where V(j) is a function of the sample spectrum and spectral generating function (s.g.f.) for the STS model in equation (8) with [Mathematical Expression Omitted] eliminated and Z is a vector of partial derivatives (evaluated at the restricted parameter values) of the s.g.f. with respect to the parameters of the model (with the exception of [[Lambda].sub.c] - the LM test is not sensitive to different values of [[Lambda].sub.c]). The weights are the s.g.f. The test statistic is 1/2[TR.sup.2] where [R.sup.2] is the coefficient of determination from the regression. [Mathematical Expression Omitted] under [H.sub.0]. When [Rho] = 0, [Mathematical Expression Omitted] and [Mathematical Expression Omitted] cannot be identified from one another and [[Lambda].sub.c] is not identified. Dropping [[Epsilon].sub.t] from the model allows [Mathematical Expression Omitted] to be identified and using a test which is insensitive to [[Lambda].sub.c] overcomes its under identification. Technically [Rho] = 1 violates one of the regularity conditions and invalidates the test and results in perfect multicollinearity in the weighted regression. To avoid this problem I test [H.sub.0]: [Rho] = .9999. While this avoids the unit root problem and ensures desirable large sample properties of the test, the small sample properties may be compromised.

13. A one-tailed interval is not necessarily the appropriate construction. Values of [Rho] [greater than] 1 are typically dismissed as being unrealistic and thus lead to a one tail interval. Alternatively, [Rho] [greater than] 1 could be interpreted as a sign of misspecification implying that more complex (nonlinear) behavior may exist. In such a situation, [Rho] = 1 - cyclical behavior from a linear structure - should be rejected, but lacking a nested nonlinear alternative this result can be considered to support the null hypothesis.

14. UN exhibits minor setbacks and recoveries within the course of a normal business cycle. This leads to an underestimation of the period of the UN cycle. Fitting an STS model to a slightly smoothed variant of UN m a 4 or 6 period moving average - leads to more respectable estimates of the period (13.93 and 17.43 in the early and late periods).

The estimated cycle periods for GDP are explained by a fitted model that breaks the long 1961-70 cycle into two shorter cycles and includes the short 1980-82 cycle in the 1975-80 cycle (See Figure 1).

15. See previous note. The STS model for smoothed UN leads to N and H statistics that are accepted at the .01 level of significance. The [R.sup.2] statistics suggest that the BAA model in both periods is problematic. The very erratic nature of the series suggests that these models should be interpreted with caution.

16. In the late DER and early UN cases the rejection of the normality and homoscedasticity hypotheses with only a marginal improvement in fit imply that the untransformed models are superior.

17. In each trial, the initial conditions for the recursion in equation (5) are taken from the first two values of [Mathematical Expression Omitted] generated by the smoothing algorithm when applied to the estimated models in Tables II and III. The initial values for [Kappa] and [[Kappa].sup.*] are assumed to be zero. [[Kappa].sub.t] and [Mathematical Expression Omitted] are generated as NID (0, [Mathematical Expression Omitted]). [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are used for [[Lambda].sub.c] and [Rho], except for the case where [Rho] is set to one.

18. Correlations for [Psi], MC are not reported because the coefficients are all less than .01.

References

1. Enders, Walter. Applied Econometric Time Series. New York: John Wiley, 1995.

2. Harvey, Andrew C., "Trends and Cycles in Macroeconomic Time Series." Journal of Business and Economic Statistics, July 1985, 216-27.

3. -----. Forecasting, Structural Time Series Models and the Kalman Filter. New York: Cambridge University Press, 1989.

4. Harvey, A. C. and A. Jaeger, "Detrending, Stylized Facts and the Business Cycle." Journal of Applied Econometrics, May 1993, 231-47.

5. McCallum, Bennett T. "Unit Roots in Macroeconomic Time Series: Some Critical Issues." NBER Working Paper No. 4368, 1993.

6. Nelson, Charles R. and Charles I. Plosser, "Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications." Journal of Monetary Economics, September 1982, 139-62.

7. Simkins, Scott P., "Business Cycles, Trends, and Random Walks in Macroeconomic Time Series." Southern Economic Journal, April 1994, 977-88.

8. Stadler, G. W., "Real Business Cycles." Journal of Economic Literature, December 1994, 1750-83.