ROBUST TESTING
705
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Monahan 1992 and also report results for tests of the hypotheses: H0: 2s3s0, H0:
2s3s4s0, H0: 2 s3 s4s5 s0. We label the hypotheses according to the
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number of restrictions being tested, i.e. qs1, 2, 3, 4. The results for the AR 1 models
with a sample size of Ts128 are reported in Table III. Several patterns emerge from the
table. First, in nearly every case, null rejection probabilities of F* are less distorted and
closer to 0.05 than the QS or QS᎐PW tests. The differences become larger as q
increases. Although the F* test has less distortions, there are many cases in which null
rejection probabilities are much greater than 0.05. Nonetheless, the asymptotic approxi-
mation of the distribution of F* is substantially better compared to QS and QS᎐PW.
Second, as approaches one, distortions of the null rejection probabilities increase for
all the statistics. This is explained by the fact that the stationary asymptotic approxima-
tion becomes less accurate the closer the autoregressive root is to one. Third, for all
three statistics, as q increases, null rejection probabilities also increase, indicating the
asymptotic approximation is less precise when testing joint hypotheses compared to
testing simple hypotheses. This result suggests, in particular, that for joint hypotheses,
size distortions of HAC estimator tests can be substantial even when there is only modest
serial correlation in the errors.
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Results for the MA 1 models with Ts128 are given in Table IV. Similar patterns are
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seen as for the AR 1 models except that distortions overall are much less severe.
Rejection probabilities of F* are rarely above 0.10 while those of QS and QS᎐PW often
exceed 0.10 especially for large q.
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In Table V we report results for the AR 1 ᎐HOMO model for sample sizes Ts
256, 512. The table indicates that the asymptotic approximation improves substantially for
all the tests as T increases. For the most part, F* has rejection probabilities close to 0.05
for F0.5. For )0.5 rejection probabilities are inflated but by much less compared to
when Ts128. Rejection probabilities of QS and QS᎐PW are, for the most part, more
distorted than those of F*, especially for G0.9 and qG3.
7. FINITE SAMPLE POWER AND EMPIRICAL EXAMPLE
Using the DGPs from the previous section, we simulated size-adjusted power of the
statistics and found that power rankings of the statistics followed patterns qualitatively
similar to the local asymptotic power curve depicted in Figure 2. Therefore, we do not
report those simulations here and instead report results on finite sample power from
simulations based on the following empirical example. Let ⌬lre¨t denote the first
difference of the natural logarithm of real aggregate restaurant revenues for the United
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States, and let ⌬lgdpt denote the first difference of the natural logarithm of seasonally
.
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adjusted real gross domestic product GDP for the United States. We obtained
quarterly observations from 1971:1 to 1996:4 for the nominal versions of these series and
constructed the real series by dividing by the implicit GDP deflator. We seasonally
adjusted the nominal restaurant revenue series before constructing the real series. The
restaurant revenue series was obtained from the Current Business Reports published by
the Bureau of the Census, and the nominal GDP and deflator series were obtained from
the Sur¨ey of Current Business published by the Bureau of Economic Analysis, U.S.
Department of Commerce. The levels of the real revenue and real GDP series are clearly
trending over time and may have unit root errors. Therefore, the first differences of the
series are likely to be stationary and satisfy Assumptions A1 and A2, so we consider a
regression model in first differences of the data. For simplicity, we are ignoring the