Lecture 7 –Regression Models with Serially
Correleated Disturbances
1. Asymptotic Normality of the OLS Estimator with
Serially Correlated Errors
Proposition 2.1 in Hayashi uses the conditions that
{xt,t} is a jointly stationary and ergodic process and
{xtt} is an m.d.s. in order to establish the asymptotic
normality of the OLS estimator. The m.d.s. restriction
rules out serial correlation in the ’s.
It turns out that the m.d.s. restriction can be replaced
with the weaker condition that {xtt} is a “mixingale”
and the OLS estimator will be consistent and
asymptotically normal.
The mixingale condition is essentially the condition
that the sequence behaves asymptotically like an
m.d.s. in the sense that E(xt+st+s │t,t-1,…,xt,xt-1,…)
goes to 0 as s goes to ∞. (See White, 2001, for more
on this.) This assumption does allow for serial
correlation in the disturbances.
However, if the disturbances are serially correlated,
then the orthogonality condition (A.3) will, for
example, rule out lagged dependent variables among
the regressors:
yt = β0 + β1yt-1 + εt and εt = ρεt-1 + ut , ut ~ i.i.d.
In this case, yt-1 will correlated with εt because both
are determined by εt-1.
2.
The GLS and FGLS Estimator
In the linear regression model with strictly exogenous
regressors and heteroskedastic or serially correlated,
normally-distributed disturbances, the GLS estimator
is the BUE, MLE, and efficient estimator of the ’s.
What are the large sample properties of the GLS
estimator in time series regressions?
If we assume that the regressors are predetermined
but not necessarily strictly exogenous, then in general
the GLS estimator will be inconsistent!
The idea –
Suppose that we have a linear model with a single
regressor, which is predetermined, so that E(t │
xt, xt-1,…) = 0 for all t.
Let var(1,…,T) = 2ΩT
Suppose, that ΩT is known for each T.
Let CTCT’ = ΩT-1
Now, recall that GLS amounts to transforming the
original model to:
~
yt  ~
xt   ~t
where
~
yt  ct1 y1  ct 2 y2  ...  ctT yT
~
xt  ct1 x1  ct 2 x2  ...  ctT xT
~t  ct1 1  ct 2 2  ...  ctT  T
then fitting the model by OLS. But note that unless x
is strictly exogenous,
E(~
xt ~t )  0
and the GLS estimator will be inconsistent!
Note: The GLS will be consistent and asymptotically
superior to OLS in the important special case where
the disturbances follow a “finite-order autoregressive
process.” However, unless the regressors are strictly
exogenous, the FGLS estimator will be inconsistent
regardless of the form of the error process. We will
return to this case in the next section of the course.
3. The Linear Trend Model
A commonly encountered time series regression
model is the linear trend model –
yt = 0 + 1t + t
where t is a zero-mean, covariance stationary
process. That is, yt is the sum of a deterministic linear
trend and a stationary process. In this case, we say
that yt is a trend stationary process.
This is a very simple and appealing way to think
about many trending economic time series, like the
log of real GDP.
In this case, it is natural to consider estimating the
linear trend:
 the estimated trend function, ˆ  ˆ t , provides
us with an estimate of the long-run path of yt
 the residuals, ˆ  y  ˆ  ˆ t , provide us with an
estimate of the short-run or cyclical
behavior of yt
0
t
t
0
1
1
 we can use the residuals in regression
models that require stationary variables
If t is an i.i.d. N(0,2) process, then for any fixed
sample size T, this model satifies all of the
assumptions of the classical normal linear regression
model: linearity, strictly exogenous regressors, no
multicollinearity, spherical normal disturbances. OLS
is the BUE and MLE of the ’s; the simple OLS t and
F statistics can be applied in the “usual” way.
If t is i.i.d. (0,2), the OLS estimator is consistent,
asymptotically normal, and asymptotically efficient.
The simple OLS t and F statistics can be applied in
the “usual” way, though the justification will be
asymptotic (since the disturbances are not assumed to
be normal). (More on this shortly.)
Suppose that t is a serially correlated, stationary
process. Now we are in the situation of the linear
regression model with strictly exogenous regressors
and non-spherical disturbances, which we discussed
on the first day of this part of the course. (See the
notes to Lecture 1.)
There is a very interesting fact about the asymptotic
behavior of the OLS estimator of the linear trend
model with serially correlated disturbances: The OLS
estimator is asymptotically equivalent to the GLS
(and FGLS) estimator of the ’s! There is no
asymptotic gain to knowing the form of the serial
correlation and taking it into account with regard to
estimating and drawing inferences about the ’s. (See
Grenander and Rosenblatt, 1957. The OLS estimator
of the linear trend model is consistent, asymptotically
normal, and asymptotically efficient. Standard OLS t
and F test, which assume spherical disturbances, can
be applied and will be valid asymptotically.