under the null hypothesis that yt is I(do) process as in( 3) Theorem 1 demonstrates that the limiting distribution of the two statistics Z(do) and Zu(do)are invariant within a very wide class of weakly dependent and possible heterogeneously distributed innovation Et Table 1 gives the critical values of vao 5o [Bdo(r Pa and-2Vdo(o[Bdo (r)12dr-[o Bdo(r)dr 2) calculated via a direct simulation from a simple transformation of 8 )and(10), using a sample size of 500, and 10000 replication. The calculations were done in GAUSS using the normal random number. Observation on a I(d) process for d E(0.5, 0) were generated using the Durbin-Levinson algorithm See for example, Brockwell and Davis(1991), Section 5.2
under the null hypothesis that yt is I( ˜d0) process as in (3). Theorem 1 demonstrates that the limiting distribution of the two statistics Z(d0) and Zµ(d0) are invariant within a very wide class of weakly dependent and possible heterogeneously distributed innovation εt . Table 1 gives the critical values of − 1 2 1 Vd0 R 1 0 [Bd0 (r)]2dr and − 1 2 1 Vd0 (R 1 0 [Bd0 (r)]2dr−[R 1 0 Bd0 (r)dr] 2) , calculated via a direct simulation from a simple transformation of (8) and (10), using a sample size of 500, and 10000 replication. The calculations were done in GAUSS using the normal random number. Observation on a I(d) process for d ∈ (−0.5, 0) were generated using the Durbin-Levinson algorithm. 4 4See for example, Brockwell and Davis (1991), Section 5.2. 10
Table 1 Critical values for the Z(d) and Zu(d) statistics Percentiles of the distribution of T=500 5% 95%97.5%99% 0.059.2387.238-5.736-4.2120.2440.4060.5400.708 6912-5.552-4.444-3.336-0.0360.0720.145 0.220 0.048 3.825-3.201 0.061 0.25-2601-2.229-1.893-1.552-0.153-0.104-0.075-0.053 0.30-1.840-1.5831.380-1.169-0.143-0.100-0.080-0.058 -0.35-1.229 1.093-0.981-0.843-0.118-0.088 0.069 0.052 0.40-0.8310.736-0.663-0.582-0.093-0.070-0.0550.043 0.45-0.522 0.473 0.436-0.388 0.072 0.053 0.043 0.032 Percentiles of the distribution of 2 Va( Ba(r)2dr-Uo Ba(r)dr)2) T=500 90%95%975%99% 0.05-14.591-12.281-10.423-8538-1.064-0.636-0.3030.013 0.10-10.622-8977 0.772 0.560 0.15-7.575-6.487-5.734-4.860-0.976-0.755-0.592-0.452 0.20 5.144-4.487-3.964-3.429-0.822-0.650 0.536-0.426 3.548-3.125-2.778-2.442-0.702-0.573-0.481 0.385 0.30-2.347-2.113-1.917-1.723-0.569-0.470-0.398 0.327 0.35-1.533-1.394-1.289-1.172-0.454-0.384-0.334 0.976-0.8990.8350.769-0.342-0.297-0.261 0.45-0.603-0.5650.535-0.496-0.253 0.221-0.196 0.173
Table 1. Critical values for the Z(d) and Zµ(d) statistics Percentiles of the distribution of − 1 2 1 Vd R 1 0 [Bd(r)]2dr T = 500 d 1% 2.5% 5% 10% 90% 95% 97.5% 99% −0.05 −9.238 −7.238 −5.736 −4.212 0.244 0.406 0.540 0.708 −0.10 −6.912 −5.552 −4.444 −3.336 −0.036 0.072 0.145 0.220 −0.15 −5.198 −4.238 −3.461 −2.637 −0.124 −0.048 0.001 0.044 −0.20 −3.825 −3.201 −2.688 −2.132 −0.157 −0.093 −0.061 −0.028 −0.25 −2.601 −2.229 −1.893 −1.552 −0.153 −0.104 −0.075 −0.053 −0.30 −1.840 −1.583 −1.380 −1.169 −0.143 −0.100 −0.080 −0.058 −0.35 −1.229 −1.093 −0.981 −0.843 −0.118 −0.088 −0.069 −0.052 −0.40 −0.831 −0.736 −0.663 −0.582 −0.093 −0.070 −0.055 −0.043 −0.45 −0.522 −0.473 −0.436 −0.388 −0.072 −0.053 −0.043 −0.032 Percentiles of the distribution of − 1 2 1 Vd(R 1 0 [Bd(r)]2dr−[R 1 0 Bd(r)dr] 2) T = 500 d 1% 2.5% 5% 10% 90% 95% 97.5% 99% −0.05 −14.591 −12.281 −10.423 −8.538 −1.064 −0.636 −0.303 0.013 −0.10 −10.622 −8.977 −7.843 −6.484 −1.083 −0.772 −0.560 −0.361 −0.15 −7.575 −6.487 −5.734 −4.860 −0.976 −0.755 −0.592 −0.452 −0.20 −5.144 −4.487 −3.964 −3.429 −0.822 −0.650 −0.536 −0.426 −0.25 −3.548 −3.125 −2.778 −2.442 −0.702 −0.573 −0.481 −0.385 −0.30 −2.347 −2.113 −1.917 −1.723 −0.569 −0.470 −0.398 −0.327 −0.35 −1.533 −1.394 −1.289 −1.172 −0.454 −0.384 −0.334 −0.282 −0.40 −0.976 −0.899 −0.835 −0.769 −0.342 −0.297 −0.261 −0.221 −0.45 −0.603 −0.565 −0.535 −0.496 −0.253 −0.221 −0.196 −0.173 11