times Lj (1, O) that relate to the covariates of the model. Notice, however, that no
local time features in the limit, when the empirical model involves a single covariate.
In this instance the limit distribution is Gaussian rather than mixed Gaussian and
the covariance function Γ(m1,m2) depends only on the regression function and the
weighting employed. Under H1 the test is consistent. In particular, the test statistic
diverges with rate ʌ/n which is slower than the rate attained for stationary data (n).
Theorem 3 suggests that the truth of H0 implies certain properties for the asymp-
totic moments of the sup-statistic. The following result is analogous to Corollary 1
of Bierens (1984). It demonstrates that there is one-to-one correspondence between
the truth of H0 and the asymptotic behaviour of the sup-statistic.
Lemma 6: H1 holds if and only if supmeM c(m) > O a.s.
The limit distribution of the sup-statistic is not pivotal. Bierens (1990) suggests a
modification of the sup-statistic in order to obtain a tractable limit distribution under
the null hypothesis. The modified test statistic has a chi-square limit distribution.
This approach is applicable to our models as well.
Lemma 7: Let Assumptions A-B hold. Choose independently of the data generating
process ∙y > O, p E (0,1/2) and some mo E M. Let m = argmaxmeMB(m) and let
m = mo
∙rA∕^∖ A/ ∖ о о ^∙rA∕^∖ A/ ∖ . о
if B(m) — B(mo) < pnp and m = m if B(m) — B(mo) > γnp.
Then as n → ∞ we have:
(i) Under H0, B(m) → χf,
(ii) Under H1, B(m)∕^n → supmeM c(m).
In view of Lemma 6, the modified statistic B (m) has a pivotal limit distribution and
yields a consistent test. It is reasonable to expect that the penalty term qnp affects
the properties of the test in finite samples. In fact, a small penalty term should result
in size larger than the nominal one. The sensitivity of the test on the choice of the
penalty term is explored in the simulation experiment of Section 4.
It follows from Bierens (1990) that any test based on conditional moment con-
ditions can be converted into a fully consistent (under strict stationarity). This is
true for I-regular family as well. Marmer (2005) proposes a RESET type of test for
I-regular models. In particular, the regression residuals are regressed on integrable
polynomials (basis functions) of a unit root covariate. The aim of the polynomials is
to approximate neglected nonlinear components. The significance of the basis func-
tions is checked with the aid of an F-test. Marmer’s test is not a fully consistent one,
but it can be converted into a fully consistent test. Under misspecification Marmer’s
test has power as long as the inner product of f (f is a neglected I-regular term) and
at least one of the basis functions (φk’s) employed is non-zero i.e.
f f(s)φ⅛(s)ds =
J -∞
0,
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