980 Testing the Martingale Hypothesis
have recently considered a large class of directional tests based on linear combina-
tions of autocorrelations. Their tests are shown to be optimal in certain known local
alternative directions and are asymptotically equivalent to Lagrange multiplier
tests. Finally, we mention Kuan and Lee (2004), who propose a correlation-based
test for the MDH that, instead of using lagged values ofYtas the functionw(·),
employs some other arbitraryw(·). This test shares with all the tests analyzed in
this section the problem of inconsistency, derived from not using a whole family
of functionsw(·).
20.3.2 Tests based on an infinite-dimensional conditioning set
The approach presented in the previous sub-section lays naturally in the time
domain since a finite number of autocorrelations are tested. However, when the
infinite past is considered, the natural framework for performing inference is the
frequency domain. The advantage of the frequency domain is the existence of one
object, namely the spectral density, that contains the information contained in all
the autocovariances. Hence, in the frequency domain, the role previously taken by
autocorrelations is now carried by the spectral density function. Define the spectral
densityf(λ)implicitly by:
γk=∫f(λ)exp(ikλ)dλ k=0, 1, 2,...,where=[−π,π]. Define also the periodogram asI(λ)=|w(λ)|^2 , wherew(λ)=
n−^1 /^2
∑n
t= 1 xtexp(itλ). Although the periodogram is an inconsistent estimator of
the spectral density, it can be employed as a building block to construct a consistent
estimator. The integral of the spectral density is called the spectral distribution,
which, under the MDH, is linear inλ.
For this infinite lag case, the MDH implies as the null hypothesis of interest that
γk=0 for allk =0, and equivalently, in terms of the spectral density, the null
hypothesis states thatf(λ)=γ 0 / 2 πfor allλ∈.
The advantage of the frequency domain is that the problem of selectingp,
which was present in the previous sub-section, does not appear because the null
hypothesis is stated in terms ofallautocorrelations, as summarized by the spectral
density or distribution. The classical approach in the frequency domain involves
the standardized cumulative periodogram, i.e.,
Zn(λ)=√
T⎛
⎝∑[λT/π]
j= 1 I(λj)
∑T
j= 1 I(λj)−λ
π⎞
⎠,whereλj = 2 πj/n,j =1, 2, ...,n/2, are called the Fourier frequencies. Based on
Zn(λ), the two classical test statistics are the Kolmogorov–Smirnov:
max
j=1,...,T∣∣
∣Zn(λj)∣∣
∣,and the Cramér–von Mises: