The Sources and Nature of Long-Term Memory in Aggregate Output Joseph G

15 http://clevelandfed.org/research/review/ Economic Review 2001 Q2 The Sources and Nature of Long-Term Memory in Aggregate Output Joseph G. Haubrich is an economic advisor at the Federal Reserve Bank by Joseph G. Haubrich and Andrew W. Lo of Cleveland and Andrew W. Lo is the Harris & Harris Group Professor at the MIT Sloan School of Manage- ment. This research was partially supported by the MIT Laboratory for Financial Engineering, the National Science Foundation (SES–8520054, SES–8821583, SBR–9709976), the Rodney L. White Fellowship at the Wharton School (Lo), the John M. Olin Fellowship at the NBER (Lo), and the University of Pennsylvania Research Foundation (Haubrich). The authors thank Don Andrews, Pierre Perron, Fallaw Sowell, and seminar participants at Columbia University, the NBER Summer Insti- tute, the Penn Macro Lunch Group, and the University of Rochester for helpful comments, and Arzu Ilhan for research assistance. Introduction One promising technique (Geweke and Porter-Hudak [1983] and Diebold and Rudebusch [1989]) also has Questions about the persistence of economic shocks serious small-sample bias, which limits its usefulness currently occupy an important place in economics. (Agiakloglou, Newbold, and Wohar [1993]). Much of the recent controversy has centered on “unit Although both Deibold and Rudebusch (1989) and roots,” determining whether aggregate time series are Sowell (1992) find point estimates that suggest long- better approximated by fluctuations around a deter- term dependence, they cannot reject either extreme of ministic trend or by a random walk plus a stationary finite-order ARMA or a random walk. The estimation component. The mixed empirical results reflect the problems raise again the question posed by Christiano general difficulties in measuring low-frequency and Eichenbaum (1990) about unit roots: do we components. Persistence, however, has richer and know and do we care? In this paper we provide an more relevant facets than the asymptotic behavior at affirmative answer to both questions. the heart of the unit root debate. In particular, frac- Do we know? Applying the modified rescaled tionally differenced stochastic processes parsimo- range (R/S) statistic confronts the data with a test niously capture an important type of long-range at once both more precise and more robust than dependence midway between the quick decay of an previous estimation techniques. The R/S statistic ARMA process and the infinite persistence of a has shown its versatility and usefulness in a variety random walk. Fractional differencing allows some- of different contexts (Lo [1991] and Haubrich thing of a return to the classical NBER business cycle [1993]). Operationally, we can determine what we program exemplified by Wesley Claire Mitchell, who know about our tests for long-range dependence by urged examinations of stylized facts at all frequencies. using Monte Carlo simulations of their size and Though useful in areas such as international power. Not surprisingly, with typical macroeco- finance (Diebold, Husted, and Rush [1991] and nomic sample sizes we cannot distinguish between Baillie, Bollerslev, and Mikkelsen [1993]), fractionally fractional exponents of 1.000 and 0.999, but we differenced processes have had less success in macro- can distinguish between exponents of 0 and 0.333. economics. For GDP at least, it is hard to estimate the Do we care? Persistence matters directly for mak- appropriate fractional parameter with any precision. ing predictions and forecasts. It matters in a more 16 http://clevelandfed.org/research/review/ Economic Review 2001 Q2 subtle way when making econometric inferences. Perhaps the most intuitive exposition of This is especially true for fractionally differenced fractionally differenced time series is via their processes, which though stationary, are not “strong- infinite-order autoregressive and moving-average mixing” and so in a well-defined probabilistic sense representations. Let Xt satisfy: behave very differently than more standard ARMA processes. Furthermore, because the optimal filter d ε (1) (1 – L) Xt = t , depends on the characteristics of the underlying process (Christiano and Fitzgerald [1999] and Baxter ε where t is white noise, d is the degree of differenc- and King [1999]), long-term persistence (sometimes ing, and L denotes the lag operator. If d = 0, then called long memory) matters even for estimates of Xt is white noise, whereas Xt is a random walk if d = higher frequency objects such as business cycle 1. However, as Granger and Joyeux (1980) and properties. While for some purposes persistence is less Hosking (1981) have shown, d need not be an inte- important than how agents decompose shocks into ger. Using the binomial theorem, the AR represen- permanent and temporary components (Quah tation of Xt becomes: [1990]), the results still depend on the persistence of ∞ the time series decomposed and on the persistence of ∑ (2) A(L) Xt = Ak the temporary component. In addition, the univari- k=0 ate approach, in contrast to Quah, has the advantage ∞ k ∑ ε that it does not assume agents observe more than the (3) L Xt = Ak Xt–k = t , econometrician. k=0 Finally, beyond the value of purely statistical ≡ k d where Ak (–1) ()k . The AR coefficients are often explorations, a compelling reason to search for re-expressed more directly in terms of the gamma fractional differencing comes from economic the- function: ory. We show how fractional differencing arises as a natural consequence of aggregation in real business Γ (4) Ak ≡ (–1)k d = ( k – d ) . cycle models. This holds out the promise of more ()k Γ(– d )Γ (k + 1) specific guidance in looking for long-range depen- By manipulating equation (1) mechanically, X may dence, and conversely, adds another criterion to t also be viewed as an infinite-order MA process judge and calibrate macroeconomic models. since: This paper examines the stochastic properties of aggregate output from the standpoint of fraction- (5) X = (1 – L)–d εε= B(L) ally integrated models. We introduce this type of t t t process in section I, reviewing its main properties, Γ B = ( k + d ) . advantages, and weaknesses. Section II develops a k Γ (d )Γ (k + 1) simple macroeconomic model that exhibits long- The particular time-series properties of X range dependence. Section III employs the modi- t depend intimately on the value of the differencing fied rescaled range statistic to search for long-range parameter d. For example, Granger and Joyeux dependence in the data. We conclude in section IV. (1980) and Hosking (1981) show that when d is less than 1 , X is stationary; when d is greater than 2 t 1 I. Review of Fractional Techniques – 2 , Xt is invertible. Although the specification in in Statistics equation (1) is a fractional integral of pure white noise, the extension to fractional ARIMA models Macroeconomic time series look like neither a ran- is clear. dom walk nor white noise, suggesting that some The AR and MA representations of fractionally compromise or hybrid between white noise and its differenced time series have many applications and integral may be useful. Such a concept has been given illustrate the central properties of fractional content through the development of the fractional processes, particularly long-range dependence. calculus, that is, differentiation and integration to The MA coefficients, Bk, give the effect of a shock noninteger orders.1 The fractional integral of order k periods ahead and indicate the extent to which between 0 and 1 may be viewed as a filter that current levels of the process depend on past values. smooths white noise to a lesser degree than the ordi- nary integral; it yields a series that is rougher than a ■ 1 The idea of fractional differentiation is an old one, dating back to random walk but smoother than white noise. an oblique reference by Leibniz in 1695, but the subject lay dormant until Granger and Joyeux (1980) and Hosking (1981) the nineteenth century when Abel, Liouville, and Riemann developed it develop the time-series implications of fractional dif- more fully. Extensive applications have only arisen in this century; see, for example, Oldham and Spanier (1974), who also present an extensive ferencing in discrete time. For expositional purposes historical discussion. Kolmogorov (1940) was apparently the first to we review the more relevant properties in this section. notice its applications in probability and statistics. 17 http://clevelandfed.org/research/review/ Economic Review 2001 Q2 F I G U R E 1 it is immediate that for 0 < d < 1, limk ← ∞ Bk = 0, Autocorrelation Functions and asymptotically there is no persistence in a –.475 ε fractionally differenced series, even though the pj for (1 – L) t autocorrelations die out very slowly.2 1 Autocorrelation This holds true not only for d < 2 (the stationary 1.0 1 case), but also for 2 < d < 1 (the nonstationary case). From these calculations, it is apparent that the –d e Fractionally differenced seriesXdt = (1 – L)t , = 0.475 long-run dependence of fractional processes relates 0.8 to the slow decay of the autocorrelations, not to any permanent effect. This distinction is important; an IMA(1,1) can have small but positive 0.6 persistence, but the coefficients will never mimic the slow decay of a fractional process. The spectrum, or spectral density (denoted 0.4 f (ω )) of a fractionally differenced process reflects these properties. It exhibits a peak at 0 (unlike the 0.2 flat spectrum of an ARMA process), but one not as –d ε sharp as the random walk’s. Given Xt = (1 – L) t, AR(1) with coefficient 0.90 the series is clearly the output of a linear system 0 with a white-noise input, so that the spectrum of 0 30 60 90 120 Xt is: Lag σ 2 SOURCE: Authors’ calculations.

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