Palgrave Handbook of Econometrics: Applied Econometrics

994 Testing the Martingale Hypothesis

–0.2

r(

j)

KS (

j)

–0.1

0 5 10 15 20 25 30 35

0

0.1

0.2

Autocorrelogram

Lag j

0 5 10 15 20 25 30 35

0

0.5

1

1.5

Nonlinear IPRF plot

Lag j

Figure 20.9 IPRF for the weekly Can
Top graph is the heteroskedasticity robust autocorrelation plot. Bottom graph is the IPRF plot.

which, after some algebra, boils down to:

Hw(λ,x)=γ0,w(x)λ+ 2

∑∞

j= 1

γj,w(x)

sinjπλ

jπ

. (20.10)

Tests can be based on the sample analogue of (20.10), i.e.:

Ĥw(λ,x)=̂γ0,w(x)λ+ 2

n∑− 1

j= 1

( 1 −

j

n

)

1

(^2) ̂γj,w(x)sinjπλ
jπ
,
where( 1 −nj)
1
(^2) is a finite sample correction factor. The effect of this correction
factor is to put less weight on very large lags, for which we have less sample infor-
mation. Note that under the MDH,Hw(λ,x)=γ 0 (x)λ, so that tests for MDH can be
constructed based on the discrepancy between̂Hw(λ,x)andĤ0,w(λ,x):=̂γ 0 (x)λ.
That is, we can consider the process:
Sn,w(λ,x)=
(n
2
)^12
{Ĥw(λ,x)−̂H0,w(λ,x)}=
n∑− 1
j= 1
(n−j)
1
(^2) ̂γj,w(x)
√
2 sinjπλ
jπ
, (20.11)
to test for the MDH.