Is the endogenous business cycle dead?

• Author: Goldstein, Jonathan P.

1. 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.

2. 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.

3. 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...