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1、JournalofAppliedStatistics,Vol.30,No.6,July,2003,635667The impact of fat-tailed distributions on some leading unit roots testsK. D. PATTERSON11360-0532online/03/060635-332003 Taylor hence, the t(2)distribution does not have a finite variance. To assess finite sample issues we alsoconsider t(3) and t
2、(4) distributions, which are fat-tailed but have a finite variance. We provide an evaluation of a number of unit root test statistics in the context of innovations generated from these fat-tailed distributions. The test statisticsinclude the widely applied DF q type tests; that is, the t ratio on th
3、e coefficient of the lagged dependent variable in a reparameterized AR model (the pivotal statistic), and the DF normalized biasor o typetests (Fuller, 1976). In addition, we include the weighted-symmetric (WS) versions of the q and o tests due to Park and, second, by considering innovations from so
4、me fat-tailed distributions.Impact of fat-tailed distributions637In addition, we develop Vasiceks (1976) entropy-based test to discriminate amongst some possible fat-tailed distributions. The test statistics are applied toshare prices for two financial series where there is prima-facie evidence that
5、 they have been generated from a fat-tailed distribution. The remainder of this paper is organized as follows. Section 2 gives a brief outline of the test statistics, relevant theoretical results and the particular non- normal distributions to be simulated; it also includes development of an entropy
6、- based test to assist discrimination between fat-tailed distributions. In Section 3,we consider whether we can distinguish between finite sample distributions of the test statistics, and whether we can assess the errors in inference if the assumptionof normally distributed innovations is incorrect.
7、 The effect of non-normality on power and size is addressed in Section 4; and two applications are reported in Section 5. Section 6 contains some concluding remarks, and an appendix gives relevant response functions.2 The test statistics and fat-tailed distributions2.1 Test statisticsThe DickeyFulle
8、r procedure (Fuller, 1976) is well known and hence an outline is all that is needed (see, for example, MacKinnon, 1991, for the set-up in a simulation context). The three possible maintained regressions and DF and WS test statistics considered here are, for a sample of t1,.,T observations:Specificat
9、ionDF and WS test statistics Model 1:*ytcyt1etq , o , q ws, o ws(1)Model 2:*ytkcyt1etq k, o k, q wsk, o ws k(2)Model 3:*ytkcyt1btetq b, o b, q ws b, o ws b(3)In each case, the null hypothesis is H0:c0, with one-sided alternative Ha:c0, where c1. In the standard simulation set-up to obtain critical v
10、alues, etis a series of niid innovations (shocks) with mean 0 and variance p2, normalized at unity. The tabulated empirical distribution, see Fuller (1976) and McKinnon (1991), assumes that, in the DGP, k0 in Model 2, and k0,b0 in Model 3. Thus, for example, if it is suspected in a practical situati
11、on that the DGP is a random walk with drift, k0, invariance with respect to the generally unknown value of k is achieved by using Model 3. Models 2 and 3 are practically more important for economic time series. The q type tests, q , qkand q b, are simply the respective t ratios (pivotal statistics)
12、on the LS estimators c of c; similarly, the o type tests are based on o Tc T(1), with an appropriate subscript for different models, and are sometimes referred to as the normalized bias test statistics; is the least squares (LS) estimator of . A ws superscript indicates that the test statistic is ba
13、sed on the weighted-symmetric estimator of Park Tt2wt(YtoYt1)2;T1t1(1wt1)(YtoYt1)2(4)where wt(t1)/T. In Model 1, Ytyt; in Model 2, Ytyty , where y pj1ajytjet(5a)The corresponding ADF is:*ytcyt1;pj2hj*ytj1et(5b)where ch1, h1 for a derivation of these results and references to relevant earlier work se
14、e Phillips (1987) and Chan and U a(r) is the left-hand limit of Ua(r). When a2, U2(r) is standard Brownian motion as in equation (6). Phillips (1990) has also extended equation (7b) for the case of weakly dependent errors. An index of a1 corre- sponds to the Cauchy (t(1) distribution. Although t(,)
15、distributions are not generally in the stable distribution family, their extreme value behaviour declines as the degrees of freedom index increases (see DuMouchel, 1983); thus, a t(1) distribution has fatter tails than a t(2) distribution. Hence, consistency of and convergence in distribution of T(1
16、) and q for innovations drawn from t(2) are captured in the bounds provided by these properties in the more extreme t(1) and less extreme t(3) cases. Drifts and polynomial time trends are accommodated by prior regression of the time series on these terms, and then using the residuals from such regressions rather than the original series. Equivalently, the deterministic terms can be
