The name is absent

Clearly, a Bierens weighting function can be added in Marmer’s test statistic to ensure
that the above is true i.e.

/ ^f (^s)Ψk(∙^s) exp(mΦ(s))ds = 0.

- -∞

4 Simulation Evidence

Next, we assess the finite sample properties of Bierens tests presented in the previ-
ous section. Our simulation experiment is based on 5000 replications. The data is
generated by the following I-regular model:

Vt = f (xt) + u_t, x_t = x_t-₁ + v_t with
( ”1-¹ ) -          (°, [ 0⁰ ]) -

and f is chosen as follows:

f₁ (x) = 1 {0 < x < 1}

f2(x) = (1 — 0.5x) 1 {0 < x < 2}

f₃(x) = x²1 {0 < x < 3¹∕³}

f₄(x) = 2φ(x + 0.25) — φ(x — 0.75) (φ is the standard Gaussian density)

f₅ (x) = x

f₆ (x) = ln(0.1 + ∣x∣)

f₇ (x) = exp(x)(1 + exp(x))^-1

The functions f₁-f₄ are the same as those used in the simulation experiment of Marmer
(2005). In addition, we consider f₅-f₇ in order to investigate the power properties of
our tests against non-integrable alternatives. The fitted specification is:

Vt = c + Ut                                    (7)

Our test statistic utilises the following weight functions: w(s) = (1 + s²)^-1 and
Φ(s) = tan^-1(s∕10). The set M is the interval [—15,15]. Finally, m_o (Lemma 7) is
chosen from a uniform distribution over M.

Table 1 shows the finite sample properties of the Bierens modified test (B(rn))
and the Bierens randomised test (B(m_o)). For the construction of the B(m) statistic
we have used three different penalty terms. For n = 500, the randomised test has
size very close to the nominal one. The modified test has good size in most cases,
but overrejects the null hypothesis, when the penalty term is small. Clearly, the size
properties of the tests improve, when sample size increases. Both Bierens tests have
good power properties. In many cases however, the modified test has significantly
superior power than that of the randomised test. Table 2 shows the finite sample
properties of Marmer’s RESET test. This test has good finite sample properties as

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