Lecture 6 - Testing Restrictions on the
Disturbance Process
(References – Sections 2.7 and 2.10, Hayashi)
We have developed sufficient conditions for the
consistency and asymptotic normality of the OLS
estimator that allow for
 conditionally heteroskedastic and serially
correlated regressors
 conditionally heteroskedastic, serially
uncorrelated disturbances.
In this lecture we will consider testing the
disturbances for conditional heterokedasticity and
serial correlation.
It is clear why we would want to test for serial
correlation in the distrubances: one of the maintained
assumptions of the model is that there is no serial
correlation.
The case for testing conditional heteroskedasticity is
less compelling since our theory provides procedures
that do not depend on whether the disturbances are
conditionally homoskedastic or conditionally
heteroskedastic. But, such tests still might be useful.
 Under assumptions A.1-A.5, if the disturbances
are conditionally homoskedastic we showed last
time that the inference procedures that are
appropriate in the classical linear regression
model with strictly exogenous regressors and
spherical disturbances can be applied for testing
and interval construction and these procedures
will be asymptotically valid.
 If the disturbances are conditionally
homskedastic, procedures that take this into
account may perform better in finite samples.
(Monte Carlo studies might be helpful in this
regard.)
 If the disturbances are conditionally
heteroskedastic, the OLS estimator is not
asymptotically efficient. If the disturbances are
conditionally heteroskedastic the FGLS
estimator will be asymptotically efficient. (To
apply the FGLS estimator we need to be able to
model the conditional heteroskedasticity and
have a consistent estimator of the parameters of
that model.)
 The most popular nonparametric tests for serial
correlation (Box-Pierce and Ljung-Box tests)
rely on the assumption that the disturbances are
conditionally homoskedastic.
There are two ways to approach these testing
problems: parametric tests and nonparametric tests.
A parametric test specifies a model for the serial
correlation or the conditional heteroskedasticity and
then tests a set of restrictions on the parameters of
that model.
For instance, we might think that if the disturbances
are conditionally heteroskedastic it is because the
disturbances follow the first-order ARCH process:
 t  vt  0  1 t 1
In this case, conditional homoskedasticity is the
restriction that 1 = 0 and so a natural way to test for
conditional homoskedasticity would be to construct a
test of the null hypothesis, H0: 1=0.
A nonparametric test does not require you to
formulate a specific model of the conditional
heteroskedasticity or the serial correlation under the
alternative hypothesis.
Testing for Conditional Heteroskedasticity
Most tests for conditional heteroskedasticity in time
series regressions are parametric tests that formulate
the possible heteroskedasticity in terms of an ARCHtype model. These models will be discussed in more
detail in Econ 674.
Note Engle’s TR2 test here
Section 2.7 of Hayashi’s textbook outlines a
nonparametric test of heteroskedasticity developed by
White (1980). This test relies on the assumption of
i.i.d regressors and so it is not particularly useful in
time series settings.
The idea is that when regression disturbances are
conditionally homoskedastic, there are (at least) two
consistent estimators of the matrix S that appears in
the asymptotic variance formula for the OLS
2
'
E
(

x
x
estimator, where S =
t t t ):
1 T 2 '
ˆ
S   ˆt xt xt
T 1
and
1 T
2
2  xt xt'
s Sxx = s T
, where s2 = SSR/T.
1
So, if the disturbances are homoskedastic, the
difference between these two estimators should be
getting “small” as sample size increases. In fact, the
difference will converge in probability to zero if the
disturbances are homoskedastic. White constructed a
statistic based on this difference that converges in
distribution to a χ2 random variable if the
disturbances are conditionally homoskedastic.
Testing for Serial Correlation
Both parametric and nonparametric tests are widely
used in time series to test for serial correlation in the
regression disturbances. The next major section of
the course will be concerned with “time series
models” and one of the applications of time series
will be to parameterize serial correlation in regression
models: that will allow to test for serial correlation
and, if it appears to be present, apply the FGLS
estimator and test procedures.
Our focus today will be on nonparametric tests. The
advantage of these tests is that they do not require us
to formulate a specific model of serial correlation.
The disadvantages are that 1) they will rely on the
assumption of conditionally homoskedastic
disturbances and 2) if we find evidence of serial
correlation, then what do we do?
Note: nonparametric tests for serial correlation in the
presence of conditional heteroskedasticity do exist
and nonparamteric adjustment procedures that
account for potential serial correlation in regression
models do exist. (HAC robust testing; HAC robust
standard errors.)
Let’s begin by assuming that {zt} is a stationary and
ergodic process with finite variance (i.e., it is
covariance stationary,too). Then the j-th
autocovariance of the process is:
cov( zt , zt  j )  E[( zt   z )( zt  j   z )]   j
for all t and j, where  z  E ( z t )
The sample j-th autocovariance is defined by:
1 T
ˆ j   ( zt  zT )( zt  j  zT )
T t  j 1
where
zT
is the sample mean of z1,…,zT
Corresponding to these are the j-th autocorrelations
and the sample j-th autocorrelations:
 j  corr ( z t , z t  j )  cov( z t , z t  j ) / var( zt )   j /  0
and
ˆ j  ˆ j / ˆ0
We have already claimed as a fact that under the
assumption that zt is a stationary and ergodic process,
the sample autocovariances and autocorrelations are
consistent estimators of the population
autocovariances and autocorrelations.So, if zt is a
serially uncorrelated process, the sample
autocorrelations will converge almost surely to zero
for all j > 0. (Note: Since the autocovariance and
autocorrelation functions are symmetric around 0, we
only have to consider j > 0.)
Assume that {zt – μz} is a conditionally
homoskedastic m.d.s. That is:
zt = μz + t
where
E(t │ t-1,…) = 0 and E(t2 │ t-1,…) = 2 for all t.
Then, for any positive integer p,
T ˆ  N (0, I p ) , ˆ  [ ˆ1 ...ˆ p ]'
d
and, since uncorrelated normally distributed random
variables are independent random variables, { T ̂ j }
is asymptotically an i.i.d. N(0,1) sequence.
To test for first-order serial correlation
(H0: ρ1=0, vs. HA: ρ1≠0) use
  ˆ1 (1 / T )  N (0,1)
d
To test for j-th order serial correlation
(H0: ρj=0, vs. HA: ρj≠0) use
  ˆ j (1 / T )  N (0,1)
d
To test H0: 1=0,2=0,…,p=0 vs. HA: ρ1≠0 or ρ2≠0…
p
Q  T  ˆ 2j   2 ( p)
1
d
which is the Box-Pierce Q statistic. (This result
follows from the fact that the T ̂ j ’s are
asymptotically i.i.d. N(0,1).)
The Ljung-Box Q statistic, QLB,
p
QLB =
T (T  2)
1
ˆ 2j
Tj
is asymptotically equivalent to the B-P Q statistic
2
Q


( p) ), but seems to
(i.e., Q-QLB  0 and LB
p
d
work a little better in practice.
These tests can be applied to any stationary time
series to test for serial correlation.
We are interested in applying them to test for serial
correlation in regression disturbances. These
disturbances are unobservable! It would seem natural
to consider whether we can apply these tests using
the residuals ( ˆ' s) from the fitted regression in place
of the unobservable regression disturbances (’s).
Let
1 T
~
~
~
~
 j   j /  0 , where  j    t  t  j , for j = 0,1,2,…
T j 1
and
ˆ j  ˆ j / ˆ0 ,
1 T
ˆ j   ˆt ˆt  j for j = 0,1,2,…
T j 1
Recall, we have already modified our assumptions
A.1-A.5 to restrict the disturbances to be
conditionally homoskedastic. If, in addition, the
regressors are strictly exogenous,
T ˆ  N (0, I p ) , ˆ  [ ˆ1 ...ˆ p ]'
d
i.e., the limiting distribution of the t and Q statistics
do not depend on whether we construct the sample
autocorrelations using the ~ ’s or the ̂ ’s.
Suppose the regressors are predetermined but not
strictly exogenous. In particular, suppose:
E(t │t-1,t-2,…,xt,xt-1,…) =0
and
E(t2 │t-1,t-2,…,xt,xt-1,…) =2
for all t.
N (0, I p  ) , ˆ  [ ˆ1 ...ˆ p ]'
In this case, T ˆ 
d
where Φ is a pxp matrix whose ij-th element is:
Φij = E(xtt-i)’E(xtxt’)-1E(xtt-j)/2
This leads to the modified Box-Pierce Q statistic
ˆ ) 1 ˆ
QMBP = T ˆ ' ( I p  
where ̂ is the consistent estimator of
equation (2.10.19) in Hayashi.
 , given by
Under the null hypothesis of no serial correlation,
QMBP  Χ2(p)
d