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Article Excerpt
Abstract
The issue of the Permanent Income Hypothesis (PIH) is revisited in this paper by examining the relationship between U.S. consumption and income through new statistical techniques based on fractional integration and cointegration. Using a procedure by Robinson [1994a] that permits the testing of I(d) statistical models, the results show that both individual series are I(1). However, the differences seem to be I(d), with d being smaller than 1 in some cases. Also, when performing different regressions of consumption on income, the estimated residuals from the cointegrating regressions appear to be mean reverting. This implies that consumption and income may be fractionally cointegrated, so that deviations from equilibrium are highly persistent. Thus, the results provide further evidence regarding the validity of the PIH for the U.S. (JEL C22)
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Introduction
The relationship between consumption and income is arguably one of the most important in macroeconomics. The most influential and perhaps most widely tested view of this relationship is the Permanent Income Hypothesis (PIH) [Hall, 1978, 1989]. According to this theory, consumption and income should be related in the long run, and standard econometric techniques have been used to test this hypothesis using cointegrated models. However, the observation that aggregate consumption is both excessively smooth and excessively sensitive, the so-called Deaton Paradox, has led researchers to question many of the assumptions of the PIH. Explanations have emphasized costly adjustments [Attfield et al., 1992], imperfect capital markets [Campbell and Mankiw, 1989], finite planning horizons [Gali, 1990], and precautionary savings [Zeldes, 1989; Deaton, 1991]. In this paper, the issue of the PIH is revisited by examining the relationship between consumption and income through new statistical techniques based on fractional integration and cointegration.
These two variables were analyzed from an error correction point of view in Davidson, Hendry, Srba, and Yeo (DHSY) [1978] and from a time series viewpoint in Hall [1978]. In the first of these studies, evidence was presented for the error correction model of consumption behavior from both theoretical and empirical points of view. Consumers make plans which may be frustrated, so they adjust next period's plans to recoup a portion of the error between consumption and income. Hall [1978] found evidence that U.S. consumption was a random walk and that past values of income had no explanatory power, which implied that income and consumption were not cointegrated. Neither of these studies modeled income itself and it was taken to be exogenous in DHSY [1978]. Engle and Granger [1987] performed the first tests of Fuller [1976] and Dickey and Fuller [1979] to check if both individual variables were in fact I(1). Then, they performed several cointegration tests, concluding that they were cointegrated, though income may be exogenous in view of the error correction representation. Using the same dataset, DeJong [1992] used a Bayesian approach to analyze the cointegration inference in these variables. He concluded that when trend-stationary was given zero prior probability, the cointegration inference was often supported. However, when this prior probability was relaxed, the data supported the trend stationary representations.
All of these previous works test the PIH by means of using classical cointegrating techniques, in the sense that they implicitly assume (or test in an a priori step) that the individual series are I(1) while the cointegrating relationship is I(0) stationary. This paper claims that consumption and income may be fractionally cointegrated. In other words, the error correction term might exhibit long memory, so that deviations from equilibrium are highly persistent. Under these circumstances, a fractional cointegrating relationship provides a much better understanding of the dynamic behavior of the series and may overcome the mixed evidence found when using classical techniques.
This paper is organized as follows. The next section briefly reviews the concepts of fractional integration and cointegration, summarizing some of the techniques for estimating and testing the long-memory parameters. Then, these techniques are applied to the U.S. consumption and income series. The final section contains some concluding remarks.
Fractional Integration and Cointegration
Unit roots or I(1) processes are extremely specialized models used to describe the nonstationary character of many macroeconomic time series. They became popular after Nelson and Plosser's [1982] influential paper, following Box and Jenkins [1970], which argued that the fluctuations in the level of many economic time series were better explained in terms of stochastic, rather than deterministic, models. Commonly, the unit root behavior is nested in autoregressive (AR) alternatives. However, any number of mathematical forms can be constructed that nest a unit root. One of these models is the class that allows for a fractional degree of integration. To illustrate this in case of a scalar time series [x.sub.t], t = 1, 2,..., suppose [u.sub.t] is an unobservable covariance stationary sequence with spectral density that is bounded and bounded away from zero at the origin, and:
(1 - L)[.sup.d] [x.sub.t] = [u.sub.t], t = 1, 2,...,, (1)
where L is the lag operator. The process [u.sub.t] could be a stationary and invertible autoregressive moving-average (ARMA) sequence with an exponentially decaying autocovariance. This property can characterize a weakly autocorrelated process. When d = 0, [x.sub.t] = [u.sub.t], so a weakly autocorrelated [x.sub.t] is allowed for. When d = 1, [x.sub.t] has a unit root, while for a general integer d, [x.sub.t] has d unit roots. However, d need not be an integer, a case analyzed by Adenstedt [1974], Taqqu [1975], and numerous subsequent authors. For < d < 0.5, [x.sub.t] is still stationary, but its lag-j autocovariance [[gamma].sub.j] decreases very slowly, like the power law [j.sup.2d-1] as j [right arrow] [infinity], and so the [[gamma].sub.j] are non-summable. If d [member of] [0.5, 1), the series is no longer covariance stationary, but it is still mean reverting, with the effect of the shocks dying away in the long run. Finally, if d [greater than or equal to] 1, [x.sub.t] is nonstationary and non-mean reverting. Thus, the fractional differencing parameter d plays a crucial role in describing the persistence in the time series behavior. The higher d is, the higher the association will...
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