Unit Root and Cointegration Regression
Unit Root and Cointegration Regression
The study of the evolution through time of a variable or group of variables has existed since the dawn of empirical analysis in the midseventeenth century. The formulation of explicit statistical models for a time series process {y_{t} } is typically in the form of an autoregressive moving average model
where the innovation or shock ε_{t} is assumed independently, identically distributed with mean zero and variance σ ^{2}.
When the roots of lie outside the unit circle, the process {y_{t} } is stationary in the sense that y_{t} has a fixed mean and variance for all t, and the covariance between any pair of values depends only on their distance apart in time, not on their absolute location. When the roots lie inside the unit circle, the process {y_{t} } is explosive in the sense that it quickly becomes unbounded as t increases. If there are d unit roots for the solution , and others are outside the unit circle, the process is called integrated of order d, I (d ), because the process can be interpreted as a d fold partial sum of stationary process for d = 1, … where y_{d1, t} = (1L )^{d}y_{t}, L denotes the backward shift operator, L^{s}y_{t} = y_{ts} (Box and Jenkins 1970). When there is only one unit root (d = 1), it is called an I(1) process or a unit root process ; thus when all the roots are outside the unit circle, it is also called an I(0) process.
Unit root process is intermediate between stationary and explosive process. Nelson and Plosser (1982) have found that many economic and financial time series are I (1) or I (2). There are many reasons that a time series of a variable is integrated. For instance, real businesscycle models predict that many real economic variables would contain a unit root. There is also the efficient market hypothesis in which the best predictor of tomorrow’s price y _{t +1} is today’s price y_{t} (Fama 1970).
When y _{t + 1} = y_{t} + ε_{t}, it is commonly referred to as a random walk process. Hall’s consumption function theory (1978) implies that, under a simple version of the permanent income hypothesis, future changes in consumption are unpredictable, so consumption follows a random walk. The predictions of these theories often extend to multivariate relations. For example, if income has a unit root, then consumption as a function of income will also have a unit root. The residuals that can be viewed as a linear combination of consumption and income will not have a unit root. When some linear combination of I (d ) variables has a lower order of integration, these variables are cointegrated, in Engle and Granger’s terminology (1987).
UNIT ROOT PROCESS
If a variable, say y_{t}, has a unit root, the shock to the variable never dies out. A unit root process can be written as the sum of past shocks plus an I (0) current shock. Because the sum of past shock is a dominant term in y_{t}, this permanent component constitutes a stochastic trend. Information on the degree of persistence in a time series and, in particular, on its order of integration, can help to guide the construction or testing of economic theories.
Box and Jenkins (1970) proposed an informal dataanalytic basis for the choice of d. Formal testing of a unit root is usually conducted by fitting a p th order autoregressive form
or its errorcorrection representation,
where δ = (1 L ), and The null that Σa_{j} = 1 is equivalent to the null ρ = 0.
Time series regressions that include integrated variables have statistical properties that are very different from the time series regressions with stationary variables. The least squares estimator of ρ when the regressor is I (1) converges to its true value 0 at the speed T, not at speed T ^{1/2}, and has a limiting distribution that is a functional of Brownian motion (Chan and Wei 1988). Dickey and Fuller (1979) derive the t statistics for the null hypothesis ρ = 0 by considering the limiting behavior of quadratic forms. The distribution of DickeyFuller t statistic is tabulated by Dickey (1976).
Applying DickeyFuller procedure to the first difference of y_{t} provides a test of I (2) against I (1). When the process {y_{t} } may possess two or more unit roots, Dickey and Pantula (1987) recommend a “downward,” sequential t test procedure starting with the greatest suspected number of unit roots, d (i.e., transform (2) into equivalent formulation (3) in terms of δ^{d}y_{t} ), because the “upward” testing procedure starting with a test for a single unit root is an inconsistent test. Pantula (1989) proves that the distribution of the relevant t statistic under each null has the standard DickeyFuller (1979) distribution.
From the perspective of empirical work, (2) may be overly restrictive. The process {y _{t}} may also be a function of trend and serially correlated I (0) errors. Moreover, {y _{t}} may be subject to trend breaks or structural breaks with unknown breakdays, or trend may be misspecified (Perron 1989). The bulk of the large literature on tests for unit roots has been to propose tests that (1) are asymptotically similar under the general I (1) null, in the sense that the null distribution depends neither on the parameters of the trend process nor the parameters describing the shortrun dynamics of ε_{t} (Phillips and Perron 1988); (2) have good power in large samples; and (3) exhibit small size distortions. Although the asymptotic size and power vary greatly across the proposed test statistics, the Monte Carlo findings for finite sample yield similar conclusions: The proposed tests have relatively low power against I (0) alternatives, and there are substantial size distortions for the tests. For instance, with 100 stationary observations, the onesided 5 percent significance level DickeyFuller t test has a power of only 0.19. On the other hand, processes that are I (1) but which have moderate negative autocorrelation in first difference are incorrectly rejected with high probability. (For detail, see Stock 1994).
COINTEGRATION REGRESSION
Multivariate time series methods are widely used by empirical economists. One important implication of economic theory is that certain ratios are stable. For instance, in response to growth in productivity and population, neoclassical growth models (King, Plosser, and Rebelo 1988) predict that output (y ), consumption (c ), and investment (i ) will grow in a balanced way. That is, even though y_{t}, c_{t}, and i_{t} increase permanently in response to increases in productivity and population, there are not permanent shifts in c_{t} – y_{t} or i_{t} – _{t}y. In other words, these variables are cointegrated.
For ease of exposition, here we shall only focus on the classical analysis of I (1) and I (0) systems. Let {w̰ _{t}} be a sequence of m × 1 vector of I (1) time series observations. Vector autoregressive model (VAR) of the form
provides a flexible and tractable framework for analyzing w̰ _{t} (as in Hsiao 1979 and Sims 1980). Transforming (4) into an errorcorrection form provides a framework for analyzing both longrun and shortrun economic relations
whereand j = 1, …, p, and I_{m} denotes
the m rowed identity matrix. If there are m unit roots, , each series in the system is governed by a different stochastic trend. When there are fewer than m unit roots in (4), say n, then rank (∏^{*}_{p}) = r, r = m– n, 1 ≤ n ≤ m. When rank , one can write where α̰′ and β̰ are m × r full column rank matrices. Since w̰_{t} is I (1), δ w̰ is I (0), β̰′w̰_{tp} must be I (0) to ensure compatibility between the lefthand and righthand side of (5). Therefore, the matrix, β̰’ ′is called the cointegrating matrix that describes “equilibrium” or “longrun” relations within a fully dynamic framework. The matrix α̰ transmits the deviations from the longrun relations, ’ḛ_{t} = β′̰’w̰_{t}, into respective elements of w̰_{t}. The matrix , j = 1, …, p –1, describes the shortrun adjustment pattern that provides information on how soon the “longrun equilibrium” is restored after any of the variables in the system are perturbed by a shock. In economics, the existence of longrun relations and the strength of attraction to such a state depends on the actions of a market or on government intervention. The concept of cointegration has been applied in a variety of economic models including the relationship between capital and output, real wages and labor productivity, and so on (King, Plosser, Stock, and Watson).
Several issues arise from the cointegration analysis. First, because β̰’w̰_{t} is I (0), w̰_{t} must be driven by some common trends. In other words, cointegration signifies comovements among trending variables. However, only the rank of cointegration can be uniquely determined. The cointegrating matrix β̰’ is not uniquely defined. Any linear combination of cointegrating vector is a cointegrating vector or for any r × r nonsingular constant matrix C. (Therefore, the maximum number of linearly independent cointegrating vectors in a system of m I (1) variable, is (m –1). Normalization rule for the unique determination of the cointegrating matrix has to be used.
Second, although the least squares estimator with integrated regressor is consistent, it has very different statistical properties from the least squares estimator with stationary regressors (Chan and Wei 1988; Phillips and Durlauf 1986). Some of the estimated coefficients of (4) converge to the true values at speed T ^{1/2} and are asymptotically normally distributed; some converge to the true values at speed T and have nonnormal asymptotic distributions (Sims, Stock, and Watson 1990). This raises the issue of statistical inference because in some instances the usual test statistics can be approximated by chisquare distributions while in other circumstances, they cannot.
Third, to overcome the miscentering and skewness of the usual test statistics because of the issues of unit root distribution, one can either condition on the innovations driving the common stochastic trends of the system (Phillips 1991) or use the reduced rank regression technique (Johansen 1988; 1991). In a system of m I (1) variables, the possible common trends vary between 1 and m. However, the knowledge of the rank of cointegration or nonstationarity (location of unit roots) is not generally known a priori. Johansen (1991) proposes likelihoodbased tests for the rank of cointegration by considering the likelihood function of (5), whereas Stock and Watson (1988) propose a rank test for based on the fact that if there are r linearly independent cointegrating relations, then w̰_{t} are driven by m – r common trends, hence the sum of crossproducts of w̰_{t} divided by T ^{2} can only have rank m – r. Unfortunately, neither method provides reliable inference in finite sample. Many Monte Carlo studies show that there are serious size and power distortions (Ho and Soresen 1996; Gonzalo and Pitarakis 1999).
The literature on unit root and cointegration analysis has greatly enhanced our understanding of dynamic econometric modeling of economic time series and provides a useful repertoire of tools for empirical analysis. However, unit root tests and cointegration analysis also raise serious finite sample issues. Tests with better finite sample size and power remain to be developed. More attention to economic theory and integrating economic analysis with time series analysis could also be fruitful.
SEE ALSO White Noise
BIBLIOGRAPHY
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Cheng Hsiao
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