Business cycles, trends, and random walks in macroeconomic time series.

• Author: Simkins, Scott P.

1. Introduction

   The debate concerning the proper characterization of the nonstationary component of macroeconomic time series began over a decade ago when Nelson and Plosser |12~, using tests for autoregressive unit roots developed by Dickey and Fuller |8~, argued persuasively that a wide variety of macroeconomic time series favored a difference-stationary representation over a simple trend-stationary alternative.(1) In recent years, however, a number of authors |14; 15; 16; 20; 17; 4; 1; 23~ have shown that the characterization of the trend alternative used in unit-root tests can significantly affect the conclusions regarding stationarity resulting from those tests. Perron |14~, for example, found that when a more flexible trend specification that allows a single break in the trend slope or intercept at a predetermined date is imposed under both the null and alternative hypotheses, Nelson and Plosser's results are reversed. With this alternative trend specification most of the time series in the Nelson and Plosser data set reject the null hypothesis of a unit root. Zivot and Andrews |23~, examining the same time series but treating the break dates as endogenous, find less support for the trend-stationary alternative Perron proposed, but strongly reject the unit-root null hypothesis for U.S. nominal and real output series.(2)

   Taking a Bayesian approach, DeJong and Whiteman |7~ show that support for Nelson and Plosser's conclusions is dependent on a very sharp prior that the time series are not trend-stationary. When more general priors are allowed they find that the time series analyzed by Nelson and Plosser are more likely to have been generated from a trend-stationary process.(3) DeJong, Nankervis, Savin, and Whiteman |6~ have also shown that the power of standard unit-root tests is extremely low against plausible time series representations of the Nelson-Plosser data, suggesting that Nelson and Plosser's widespread failure to reject the unit-root hypothesis may be due to low power rather than the presence of unit roots. Overall, this lack of consensus on the proper treatment of nonstationarity in macroeconomic time series suggests that it may be profitable to examine other trend specifications to determine if they can provide a meaningful representation of economic growth.(4)

   This paper examines the issue of nonstationarity in macroeconomic time series by considering a trend specification developed from the business cycle characteristics of macroeconomic data. The business cycle approach to the trend specification tests the null hypothesis of an auto-regressive data-generating process containing a unit root against the alternative hypothesis of a trend-stationary process undergoing structural change in its mean behavior from one business cycle to the next. Rejection of the null hypothesis implies that economic variables grow in a step-wise fashion, with changes in the level of cyclical means occurring during the expansion phase of each business cycle.(5) This step-wise trend model of economic growth is related to the breaking-trend work of Perron |14~ and Rappaport and Reichlin |20~, who consider the possibility of a single break in a linear trend line, and is also related to work by Perron |15~, who analyzes series with a single shift in their mean behavior. The step-wise trend model proposed here extends their work by allowing for multiple breaks in the underlying trend of macroeconomic time series.

2. Historical Motivation

   The step-wise growth pattern implied by the alternative hypothesis is motivated by business cycle research during the early part of the twentieth century which sought to link the cyclical and growth processes of economic time series.(6) Rostow |21~ emphasizes that business cycle researchers at that time viewed economic growth and cyclical fluctuations as integral parts of the same dynamic process, with the forces driving long-term economic growth also responsible for cyclical fluctuations in the economy. According to this view, exogenous increases in investment spending play a particularly important role, lifting the economy out of recessions and also contributing to the secular growth of the economy. As Rostow |22~ notes, this model of economic growth and cycles appears to characterize nineteenth-century business cycle fluctuations:

   . . . |in the 19th century~ each major business expansion began with investment in new technological innovations: e.g. cotton textile machinery, railroads, steel, electricity, chemicals. Such investment expanded jobs and consumption.

   At the same time, these technological innovations permanently increased the productive capacity of the economy, leading to long-term economic growth. This characterization of economic activity suggests that macroeconomic time series experience permanent changes in their behavior from one business cycle to the next as new technological innovations occur. Business cycle expansions are driven by changes in technology or increases in investment spending that are accompanied by increases in the capital stock, leading to a permanently higher level of output in the economy.(7)

   Consistent with this explanation of economic growth is the notion that growth proceeds in a step-wise fashion, with exogenous events affecting the level of a time series at the beginning of each business cycle. This view of economic growth also underlies the pioneering methods of business cycle analysis developed by Arthur Burns and Wesley C. Mitchell at the NBER during the 1930s and 1940s. In the preface to their seminal work, Measuring Business Cycles, Burns and Mitchell |2~ outlined the empirical methods they developed to analyze the business cycle characteristics of economic time series and described the characterization of economic growth inherent in their methods:

   The simplest plan is to express the original data as percentages of their average value during a specific cycle, and determine in terms of these percentages the rise from trough to peak, the fall from peak to trough, and the change from one stage to another into which the phases of expansion and contraction may be broken. That plan has the further advantage of eliminating in step-wise fashion the secular trend of a series. |italics mine~

   By expressing each observation in a series as a percentage of the cyclical mean of that series during a particular business cycle, Burns and Mitchell's methods remove the inter-cycle trend in the series but retain any intra-cycle trend that may be present in the data. That is, their methods are equivalent to step-wise detrending a time series across business cycles.

   Rostow's description of economic growth in the nineteenth century is also consistent with a structural change occurring in the trend growth rate at each cycle in addition to a change in the level of the trend. However, Burns and Mitchell, analyzing the behavior of seven aggregate series over a sixty-year period, found little evidence to suggest this shifting trend growth-rate pattern of behavior. Rather, they concluded that the essential characteristics of business cycle behavior had remained the same over time and that the step-wise pattern of growth was adequate to describe past business cycle behavior.(8)

   The analysis presented here tests whether the characterization of economic growth motivated by Rostow's description of nineteenth century cyclical fluctuations and Burns and Mitchell's methods of business cycle analysis is consistent with the time series behavior of U.S. macroeconomic data over the past century. In particular, I test whether the fourteen macroeconomic time series analyzed by Nelson and Plosser |12~ are better represented as difference-stationary with no structural breaks or as step-wise trend-stationary, with break dates predetermined by NBER business cycle dates. To determine which characterization better represents...