The Uncertain Unit Root in Real GNP The impulse and propagation mecha- from a steadily growing trend (e.g., John Y. nisms of business cycles have long been Campbell and N. Gregory Mankiw, 1987; debated; however, until recently, economists Peter K. Clark, 1987). The long-run trend of were in fairly broad agreement that busi- these DS representations is not fixed, as in ness fluctuations could be studied sepa- a TS model, but stochastic; indeed, in the rately from the secular growth of the econ- typical DS model, almost all fluctuations in omy. This separation was justified because, output represent permanent shifts in trend to a first approximation, the factors underly- rather than transitory movements in cycle. ing trend growth were assumed to be stable From another perspective, the essential dif- at business-cycle frequencies. Indeed, the ference between the two models can be common practice of macroeconomists of all found in the persistence of their dynamic theoretical persuasions was to model move- responses to random shocks. In the DS ments in real GNP as stationary fluctuations model of output, the effect of a shock per- around a linear deterministic trend (e.g., sists forever because the disturbance Finn Kydland and Edward C. Prescott, 1980; changes the trend component and thus af- Olivier J. Blanchard, 1981). Such a trend- fects the level of output in all future peri- stationary (TS) model of real GNP was the ods. In contrast, the impact of a shock in canonical empirical representation of aggre- the TS model is transitory and is eliminated gate output until the early 1980's. quite quickly as output reverts to its steady In contrast to previous work, much of the trend. research of the last ten years has assumed a The widespread acceptance of a DS model unit root in the autoregressive representa- for aggregate output was based on evidence tion of real GNP, which is inconsistent with that the hypothesis of a unit root in real a TS model of output. A model with a unit GNP could not be rejected.' Although this root, commonly termed a "difference- evidence was extremely robust across vari- stationary" (DS) model, implies that any ous data samples and unit-root testing pro- stochastic shock to output contains an ele- cedures (see e.g., Charles R. Nelson and ment that represents a permanent shift in Charles I. Plosser, 1982; Rent2 M. Stulz and the level of the series. If real GNP is best Walter Wasserfallen, 1985; James H. Stock represented by a DS model, the traditional and Mark W. Watson, 1986; Pierre Perron separation between business cycles and and Peter C. B. Phillips, 1987; Perron, 1988; trend growth is incorrect. In the usual em- George W. Evans, 1989), the distinct con- pirical versions of the DS model estimated trast between traditional TS models of out- for real GNP, output behaves more like a put and recent DS models led many re- random walk than like transitory deviations searchers to challenge the unit-root tests. In particular, some questioned the power of these tests, that is, their ability to reject the unit-root null hypothesis when it is indeed *Board of Governors of the Federal Resenne Sys- tem, Division of Monetary Affairs, Washington. DC 20551. I thank two anonymous referees for comments as well as Bill Bell, Frank Diebold, Steve Durlauf, Spence Krane, Doug Steigenvald, and David Wilcox. h his acceptance was not generally shared by gov- Michelle Phillips and Rpberto Sella provided research ernment and business economists. For example, the assistance. The views expressed are my own and are official series on potential output, which is used in not necessarily shared by the Board of Governors or its policy and budget deliberations, has remained a smooth staff. nonstochastic trend. 265 VOL. 83 NO. 1 RUDEBUSCH: UNCERTAIN UNIT ROOT IN REAL GNP false.* However, the indictment of low tween these particular TS and DS represen- power against unit-root tests has not been a tations. Thus, the unit-root test has low decisive criticism. The fact that unit-root power against a plausible TS model that is tests may have low power against certain TS not local in economic terms to a plausible alternatives does not necessarily compro- DS model. Section I11 extends the basic mise the results from those tests. It would argument to account for small-sample bias not be at all surprising if unit-root tests had in the estimated model coefficients. little power against TS alternatives that mimicked a DS model and reverted to trend I. DS and TS Models of Real GNP extremely slowly; all statistical tests have low power against alternatives that are "lo- Two obvious candidates for plausible TS cal" to the null. The failure to reject near- and DS representations of the data generat- unit-root TS alternatives is of little eco- ing process for real GNP are simply the nomic importance, however, because these OLS estimates of these models from the alternatives are indistinguishable in eco- available data sample. The data consist of nomic terms from the DS null over the time quarterly observations on U.S. postwar log horizons of practical macroeconomic inter- real GNP per capita (denoted Y,) from est (say, shorter than 10 years). Critics of 1948:3 to 1988:4. Assuming second-order unit-root tests must instead make the dependen~e,~the sample OLS estimate of stronger claim that unit-root tests have low the TS model of aggregate output with a power against plausible TS alternatives that linear deterministic trend is display substantially different macroeco- nomic behavior than a plausible DS null model. Simple power studies are not well suited to answering this question. This note addresses the argument that unit-root tests have low power only against local alternatives. The goal is to select the most plausible DS and TS representations for output, determine whether these repre- (standard errors of the coefficients appear sentations have different short-run persis- in parentheses).4 I will refer to this specific tence properties, and then examine whether model estimate for the sample as the TSoLs unit-root tests can distinguish between these model. models. In the next section, I examine the Under the assumption of a unit root, the ordinary least-squares (OLS) estimates of a DS model for this data sample is estimated TS model and a DS model for the postwar in first differences as sample of real GNP. These models are plausible representations of the data gener- (2) AY, = 0.003 + 0.369AY,-, + D, ating process and yet imply very different (0.001) (0.074) economic dynamics at the horizons of eco- nomic relevance. In Section 11, I examine simulated data from these models and show that a unit-root test cannot distinguish be- This particular sample DS model will be denoted as the DS,,, model. '~mongmany others, Bennett T. McCallum (19861, Francis X. Diebold and Rudebusch (19911, and David N. DeJong et al. (1992) have warned of the low power 3~heanalysis was repeated assuming that the order of unit-root tests. Other challenges have been made by of the model was four, six, and eight. Similar persis- DeJong and Whiteman (1991) from a Bayesian per- tence and power results were obtained. spective and by Perron (1989) in a framework with 4~helast two quarters of 1948 were used as fixed structural breaks. initial conditions. 266 THE AMERICAN ECONOMIC REVIEW MARCH 1993 The estimated models (1) and (2) both DS models may be quite similar or quite appear to fit real GNP per capita fairly well; different depending on the values taken by the standard deviations of their residuals the parameters of the models. Thus, the are quite close, and plots of the residuals presence of a unit root determines whether suggest no obvious outliers. In addition, Q c, is positive or zero, but it does not deter- statistics computed from the fitted residuals mine all of the model properties of eco- provide little evidence against the null hy- nomic interest. It is in this sense that, as pothesis of no serial correlation at a variety noted in the introduction, focusing solely on of lags. the existence of unit roots and on the power However, the estimated TS,,, and DS,,, of unit-root tests against arbitrary TS alter- models have very different implications for natives is insufficient. What is of economic the persistence of the dynamic response of relevance is the ability of unit-root tests to output to a random disturbance. To mea- recognize when data have been generated sure this persistence, consider the moving- from TS models that differ substantially average representation for the first differ- from the DS model at short horizons (i.e., ence of output implied by a TS or DS the ability to identify economically nonlocal model: alternatives). Consequently, a comparison of the persistence properties of the estimated TS,,, and DS,, models at relevant hori- AY, = k + E~ + alEtPl+ a,&<-, + . (3) zons is required. The estimated model responses are shown where k is some constant and E< is the in Table 1, with standard errors in paren- innovation of the model. In this form, the these^.^ The impulse response of the DS,,, sum of the ai's measures the model re- model implies not only shock persistence sponse to a unit inn~vation.~A unit shock in but shock magnification. The effect of an period t affects AY,,, by a, and affects innovation is not reversed through time, and Y,,!, by c,=l+a,+ ... +a,. Thus, for it eventually increases the level of real GNP various horizons, the cumulative response by more than one and a half times the size c, answers the question: how does a shock of the innovation (c,, = 1.59).