The eight Dickey-Fuller test statistics that we have discussed have distributions that tend to eight different asymptotic distributions as the sample size tends to infinity. These asymptotic distributions are referred to as nonstandard distributions or as Dickey-Fuller distributions.

We will analyze only the simplest case, that of the znc statistic, which is applicable only for the model (14.16) with y0 = 0. For DGPs in that model, the test statistic (14.17) simplifies to

Et=i wt

We begin by considering the numerator of this expression. By (14.02), we have that n n t-l

Since E(£t £s) = 0 for s < t, it is clear that the expectation of this quantity is zero. The right-hand side of (14.24) has ^n=1 (t — 1) = n(n — 1)/2 terms; recall the result used in (14.11). It is easy to see that the covariance of any two different terms of the double sum is zero, while the variance of each term is just 1. Consequently, the variance of (14.24) is n(n — 1)/2. The variance of (14.24) divided by n is therefore (1 — 1/n)/2, which tends to one half as n —> to. We conclude that n-1 times (14.24) is 0(1) as n — to.

We saw in the last section, in equation (14.11), that the expectation of ET=1 w2 is n(n + 1)/2. Thus the expectation of the denominator of (14.23) is n(n — 1)/2, since the last term of the sum is missing. It can be checked by a somewhat longer calculation (see Exercise 14.11) that the variance of the denominator is 0(n4) as n — to, and so both the expectation and variance of the denominator divided by n2 are 0(1). We may therefore write (14.23) as

where everything is of order unity. This explains why ( — 1 is multiplied by n, rather than by n1/2 or some other power of n, to obtain the z statistic.

In order to have convenient expressions for the probability limits of the random variables in the numerator and denominator of expression (14.25), we can make use of a continuous-time stochastic process called the standardized Wiener process, or sometimes Brownian motion. This process, denoted W(r) for 0 < r < 1, can be interpreted as the limit of the standardized random walk wt as the length of each interval becomes infinitesimally small. It is defined as

where [rn] means the integer part of the quantity rn, which is a number between 0 and n. Intuitively, a Wiener process is like a continuous random walk defined on the 0-1 interval. Even though it is continuous, it varies erratically on any subinterval. Since et is white noise, it follows from the central limit theorem that W(r) is normally distributed for each r € [0,1]. Clearly, E(W(r)) = 0, and, since Var(wt) = t, it can be seen that Var(W(r)) = r. Thus W(r) follows the N(0,r) distribution. For further properties of the Wiener process, see Exercise 14.12.

We can now express the limit as n ^ to of the numerator of the right-hand side of equation (14.25) in terms of the Wiener process W(r). Note first that, since wt+i — wt = et+i, n n—1 n—1 n—1 n—1

Yw2 = £ w + (wt+l — wt))2 = £ w2 + 2 £ wt et+l + £ e2+l • t=l t=0 t=0 t=0 t=0

Since w0 =0, the term on the left-hand side above is the same as the first term of the rightmost expression, except for the term wJ. Thus we find that n— 1 n n wt et+1 = £ wt—1et = — w t=0 t=1 t=1

Dividing by n and taking the limit as n ^ to gives n plim — E wt—1et = — (W2(1) — 1), (14.27)

where we have used the law of large numbers to see that plim n—1J2 e2 = 1. For the denominator of the right-hand side of equation (14.25), we see that n—1 n—1

If f is an ordinary nonrandom function defined on [0,1], the Riemann integral of f on that interval can be defined as the following limit:

It turns out to be possible to extend this definition to random integrands in a natural way. We may therefore write n—1 n 1

which, combined with equation (14.27), gives

nc n

A similar calculation (see Exercise 14.13) shows that plim Tnc = 21W (1)—^ - (14.30)

nc n

More formal proofs of these results can be found in many places, including Banerjee, Dolado, Galbraith, and Hendry (1993, Chapter 4), Hamilton (1994, Chapter 17), Fuller (1996), Hayashi (2000, Chapter 9), and Bierens (2001).

Results for the other six test statistics are more complicated. For zc and tc , the limiting random variables can be expressed in terms of a centered Wiener process. Similarly, for zct and Tct, one needs a Wiener process that has been centered and detrended, and so on. For details, see Phillips and Perron (1988) and Bierens (2001). Exercise 14.14 looks in more detail at the limit of zc.

Unfortunately, although the quantities (14.29) and (14.30) and their analogs for the other test statistics have well-defined distributions, there are no simple, analytical expressions for them.2 In practice, therefore, these distributions are always evaluated by simulation methods. Published critical values are based on a very large number of simulations of either the actual test statistics or of quantities, based on simulated random walks, that approximate the expressions to which the statistics converge asymptotically under the null hypothesis. For example, in the case of (14.30), the quantity to which Tnc tends asymptotically, such an approximation is given by

where the wt are generated by the standardized random walk process (14.01).

Various critical values for unit root and related tests have been reported in the literature. Not all of these are particularly accurate. Some authors fail to use a sufficiently large number of replications, and many report results based on a single finite value of n instead of using more sophisticated techniques in order to estimate the asymptotic distributions of interest. See MacKinnon (1991, 1994, 1996). The last of these papers probably gives the most accurate estimates of Dickey-Fuller distributions that have been published. It also provides programs, which are freely available, that make it easy to calculate critical values and P values for all of the test statistics discussed here.

2 Abadir (1995) does provide an analytical expression for the distribution of Tnc, but it is certainly not simple.

The asymptotic densities of the Tnc, tc , Tct, and Tctt statistics are shown in Figure 14.2. For purposes of comparison, the standard normal density is also shown. The differences between it and the four Dickey-Fuller T distributions are striking. The critical values for one-tail tests at the .05 level based on the Dickey-Fuller distributions are also marked on the figure. These critical values become more negative as the number of deterministic regressors in the test regression increases. For the standard normal distribution, the corresponding critical value would be —1.645.

The asymptotic densities of the znc, zc, zct, and zctt statistics are shown in Figure 14.3. These are much more spread out than the densities of the corresponding T statistics, and the critical values are much larger in absolute value. Once again, these critical values become more negative as the number of deterministic regressors in the test regression increases. Since the test statistics are equal to n(3 — 1), it is easy to see how these critical values are related to 3 for any given sample size. For example, when n = 100, the zc test rejects the null hypothesis of a unit root whenever 3 < 0.859, and the zct test rejects the null whenever 3 < 0.783. Evidently, these tests have little power if the data are actually generated by a stationary AR(1) process with 3 reasonably close to unity.

Of course, the finite-sample distributions of Dickey-Fuller test statistics are not the same as their asymptotic distributions, although the latter generally provide reasonable approximations for samples of moderate size. The programs in MacKinnon (1996) actually provide finite-sample critical values and

P values as well as asymptotic ones, but only under the strong assumptions that the error terms are normally and identically distributed. Neither of these assumptions is required for the asymptotic distributions to be valid. However, the assumption that the error terms are serially independent, which is often not at all plausible in practice, is required.

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