14.4 UNIT ROOTS 581

P2, , P4 will have means 0 and variances K, X, X", respectively. Since X < 1, as the lag length increases, the variance decreases; that is, we are more and more sure that the coefficient is zero. With the second equation, the priors are similar. The coefficient of y,.,-! will have a prior with mean 1. All other coefficients will have prior 0, with the coefficients of the distant lags having priors more tightly concentrated around zero.

The example of priors above is just meant to illustrate the flavor of the VAR method. Other priors can be incorporated using the RATS program. The practical experience with the BVAR model has been very good. It has produced better forecasts than many structural simultaneous equation models." It has, however, been criticized as being "a-theoretical econometrics" because it just does not use any economic theory. Sims criticized the traditional simultaneous equations models on the ground that they relied on ad hoc restrictions on the parameters to achieve identification. However, the BVAR model brings in some restrictions through the back door. The question is: What interpretation can be given to these restrictions? Estimation of the VAR and BVAR models is left as an exercise using some of the data sets at the end of Chapter 13.

14.4 Unit Roots

The single topic in the 1980s that attracted the most attention and to which most econometricians have devoted their energies is that of testing for unit roots. The number of papers on this topic runs into the hundreds. We have given a brief introduction to this in Section 6.10, where we discussed the difference between trend stationary (TS) and difference stationary (DS) time series and the Dickey-Fuller test. We review the problems in greater detail here.

The issue of whether a time series is TS or DS has both economic and statistical implications. If a series is DS, the effect of any shock is permanent. For instance, consider the model

y, = y,-i + e,

where e, is a zero-mean stationary process. Suppose that in some time period, say, yj, there is a jump in e. Then yj, yj+,, +2 all increase by C. Thus the effect of the shock is permanent. On the other hand, if we have the model

y, = ay,-i + e, H < 1

"R. B. Litterman, "Forecasting with Bayesian Vector Autoregression; Five Years of Experience," Journal of Business and Economic Statistics, Vol. 4, 1986, pp. 25-38. There have been other procedures of imposing constraints on the coefficients in the vector autoregressions that give them a structural interpretation. An example is O. J. Blanchard, "A Traditional Interpretation of Macroeconomic Fluctuations," American Economic Review, Vol. 79, 1989, pp. 1146-1164.

See F. X. Diebold and M. Nerlove, "Unit Roots in Economic Time Series: A Selective Survey," in T. Fomby and G. Rhodes (eds.). Advances in Econometrics, Vol. 8 (Greenwich, Conn.: JAI Press, 1990). This "selective" survey lists more than 200 papers in the 1980s.

the effect of the shock fades away over time. Starting with y, which will jump by C, successive values of y, will increase by Ca, Ca, Ca .... Since monetary shocks probably do not have a permanent effect on GNP, if real GNP is DS, fluctuations in real GNP have to be explained by real shocks, not monetary shocks. Thus the issue of whether in the autoregression, y, = ay, , + e„ the root ct is equal to 1 or < 1, that is, whether there is a unit root or not, is very important for macroeconomists.

On the statistical side, there are two issues: The first is about the trend removal methods used (by regression or by differencing). As pointed out by Nelson and Kang (1981) and as discussed in Section 6.10, spurious autocorrelation results whenever a DS series is de-trended or a TS series is differenced.

The second statistical problem is that the distribution of the least squares estimate of the autoregressive parameter a has a nonstandard distribution (not the usual normal, t or F) when there is a unit root. This distribution has to be computed numerically on a case-by-case basis, depending on what other variables are included in the regression (constant term, trend, other lags, and so on). This accounts for the proHferation of the unit root tests and the associated tables.

Returning to the economic issue, it does not really make sense to hinge an economic theory on a point estimate of a parameter, on whether or not there is a unit route (i.e., a = 1). A model with a = 0.95 is really not statistically distinguishable from one with a = 1 in small sample. The relevant question is not whether a = 1 or not, but how big the autoregressive parameter is or how long it takes for shocks in GNP to die out.* Cochrane argued that GNP does revert toward a "trend" following a shock, but that this reversion occurs over a time horizon characteristic of business cycles-several years at least. Yet another point is that as we noted earlier, the effect of a shock is permanent if a = 1 and goes to zero progressively if a < 1. However, in many economic problems, what we are concerned with is the present value of future streams of y,. If the discount factor is p, the present value of the effect of a shock is C/(l - Pa), Without discounting (p = 1) this is finite if a < 1 and infinite with a = 1. Thus the unit root makes a difference. But if p < 1 (i.e., with discounting), the effect is finite for all a < (1/p). Thus the existence of a unit root is not important.

14.5 Unit Root Tests

Consider first the model

y, = oty, i + e,

7. H. Cochrane, "A Critique of the Application of the Unit Root Tests," Journal of Economic Dynamics and Control, Vol. 15, No. 2, 1991, pp. 275-284.

*J. H. Cochrane, "How Big Is the Random Walk in GNP?" Journal of Political Economy, Vol. 96, 1988, pp. 893-920.

See S. R. Blough, "Unit Roots, Stationarity and Persistence in Finite Sample Macroeconometrics," Discussion Paper, Johns Hopkins University, 1990.

where e, is white noise. In the random walk case (a = 1) it is well known that the OLS estimation of this equation produces an estimate of a that is biased toward zero. However, the OLS estimate is also biased toward zero when a is less than but near to 1. Evans and Savin (1981, 1984)" provide Monte Carlo evidence on the bias and other aspects of the distributions. To discuss the Dickey-Fuller tests, consider the model

y, = Po + Pi/ + «, u, = au, i + 8,

where e, is a covariance stationary process with zero mean. The reduced form for this model is

y, = + 6r + ay, , + e, (14.3)

where = Po(l - a) + p,a and 6 = p,(l - a). This equation is said to have a unit root if a = 1 (in which case 6 = 0).

Dickey-Fuller Test

The Dickey-Fuller tests are based on testing the hypothesis a = 1 in (14.3) under the assumption that e, are white noise errors. There are three test statistics

m) = Tia - 1) t{\) = F(0, 1)

SE(a)

where a is the OLS estimate of a from (14.3), SE(a) is the standard error of a, and F(0, 1) is the usual F-statistic for testing the joint hypothesis 6 = 0 and a = 1 in (14.3). These statistics do not have the standard normal, /, and F distributions. The critical values for K{\) and r(l) are tabulated for 6 = 0 in Fuller (1976) and the critical values for the F(0, 1) statistic are tabulated in Dickey and Fuller (1981)."

The Serial Correlation Problem

Dickey and Fuller, Said and Dickey (1984), Phillips (1987), Phillips and Perron (1988), and others developed modifications of the Dickey-Fuller tests when 8, is not white noise. These tests, called the "augmented" Dickey-Fuller (ADF) tests, involve estimating the equation

, = -I- 8/ + ay, , -I- 2 Jy,~i + e, (14.4)

"G. Evans and N. E. Savin, "Testing for Unit Roots: I," Econometrica, Vol. 49, 1981, pp. 753-779, and "Testing for Unit Roots: II," Econometrica, Vol. 52, 1984, pp. 1241-1269. "W. A. Fuller, Introduction to Statistical Time Series (New York: Wiley, 1976), Table 8.5.2. The reference to Dickey and Fuller and excerpts from their tables are shown in Section 6.10. S. E. Said and D. A. Dickey, "Testing for Unit Roots in ARMA Models of Unknown Order," Biometrilia, Vol. 71, 1980, pp. 599-607; R C. B. Phillips, "Time Series Regression with a Unit Root." Econometrica. Vol. 55, 1987, pp. 277-302; P. C. B. Phillips and P. Perron, "Testing for a Unit Root in Time Series Regression," Biometrilia, Vol. 75, 1988, pp. 335-346.