Lecture 1 – Introduction
(Reference – Chapter 1, Hayashi)
This final part of your core training in
econometrics will introduce you to a
variety of topics that are relevant for
regression analysis involving time
series data.
Time series data arise in virtually
every area of applied economic
research, making up nearly all of the
data that we study in macroeconomics
and financial economics.
More comprehensive and advanced
treatments of time series analysis are
provided in Economics 674 and
Statistics 551.
Begin by recalling the assumptions
that define the classical linear
regression model.
The Classical Normal Linear
Regression Model:
Let
Xi = [x1i … xki], i = 1,…,n
be a collection of kdimensional random vectors
X = the nxk random matrix
whose i-th row is Xi.
Assume that the random
variables y1,…,yn are related
to X1,…,Xn according to:
• A.1. (Linearity)
yi = β1x1i + … + βkxki + εi , i = 1,…,n
where β1,…, βk are constants
• A.2. (Strict Exogeneity)
E[εi │X ] = 0, i = 1,…,n
• A.3. (No Multicollinearity)
Prob(rank(X) = k) = 1
• A.4. (Spherical Disturbances)
A.4.1.(Conditional Homoskedasticity)
E[εi2 │X ] = σ2 > 0 , i = 1,…,n
A.4.2. (Serially Uncorrelated Errors)
E[εiεj │X ] = 0, i,j= 1,…,n, i≠j
A.4.3. (Normally Distributed Errors),
εi│X ~ N(0, σ2) , i = 1,…,n
Under these assumptions,the
ˆ

OLS estimator of β, ,
• is unbiased
• exists and is unique
• has variance σ2(X’X)-1
• is conditionally normally
distributed
• is the BUE of β
• is the MLE of β
In addition,
ˆ   ) / se(ˆ )
(

 ti = i
~ t(n-k)
i
i
where
se(ˆi )  ˆ 2 [( X ' X ) 1 ]ii
,
ˆ 2  SSR /( n  k )
 if Rβ = r, where R is qxk and r is qx1
then
F=
2
ˆ
ˆ
-1
-1
ˆ

{(R  -r)’[R(X’X) R’] (R  -)/q}/
~ F(q,n-k)
That is, the “OLS” t-statistic is drawn from a tdistribution and the “OLS” F-statistic is drawn
from an F-distribution, providing the basis for
the application of classical hypothesis test
procedures and the construction of confidence
intervals/regions.
We will assume throughout this
course that assumptions A.1 and A.3
hold. However, we will relax
assumptions A.2 and A.4.
Note: With regard to A.1, nonlinear
models, e.g., threshold models, play
an increasingly important role in the
analysis of economic time series.
Nonlinear time series models are
considered in Economics 674.
The assumptions that the errors are
serially uncorrelated and conditionally
homoskedastic are often implausible in
regressions with time series data.
Instead, it is typically the case that the
errors are positively autocorrelated and it
is often the case, particularly in
applications with financial time series
data, that they are conditionally
heteroskedastic.
Suppose we relax A.4.1 and/or A.4.2 but
retain all of the other assumptions. We
replace these assumptions with:
E(εε’) = σ2Ω, Ω a p.d. nxn matrix
How do the properities of the OLS
estimator and the t and F statistics
change?
Under these assumptions the OLS
estimator of β,
• is unbiased ---- YES
• exists and is unique ---- YES
• has variance σ2(X’X)-1--- NO
• is conditionally normally
distributed --- YES, if the ε’s are
normally distributed
• is the BUE of β ---- NO
• is the MLE of β ---- NO
In addition,
 ti = (ˆi   i ) / se(ˆi ) ~ t(n-k) ---- NO
where
se(ˆi )  ˆ 2 [( X ' X ) 1 ]ii
2
, ˆ  SSR /( n  k )
and
 if Rβ = r, where R is qxk and r is qx1
then
F=
{(R ˆ -r)’[R(X’X)-1R’]-1(R ˆ -r)/q}/ ˆ 2
~ F(q,n-k) --- NO
The OLS estimator is still unbiased, but it
does not have any optimality properties, and
inference based on the OLS t and F statistics
as though they were drawn from t and F
distributions is not valid.
Suppose, however, that the matrix Ω
is known. Then the GLS estimator of
β, ̂GLS ,
• is unbiased
• exists and is unique
• has variance σ2(X’ Ω-1 X)-1
• is conditionally normally
distributed
• is the BUE of β
• is the MLE of β
In addition,
 ti = (ˆGLS , i   i ) / se(ˆGLS , i ) ~ t(n-k)
where
se(ˆGLS , i )  ˆ 2 [( X '  1 X ) 1 ]ii , ˆ 2  SSR /( n  k )
and
 if Rβ = r, where R is qxk and r is qx1
then
F=
2
-1
-1
-1
ˆ

(R ̂ -r)’[R(X’Ω X) R’] (R ̂ -r)/(q/ )
~ F(q,n-k)
GLS
GLS
That is, the “GLS” t-statistic is drawn
from a t-distribution and the “GLS” Fstatistic is drawn from an F-distribution,
providing the basis for the application of
classical hypothesis test procedures and
the construction of confidence
intervals/regions.
But, what if Ω is unknown, which would
be the typical situation in practice? We
could replace it with an estimate, ̂ , to
construct the FGLS estimator of β, ̂ F GLS ,
and the corresponding t and F statistics.
What can we say about the properties of
the FGLS estimator?
• is unbiased --- NO
• exists and is unique --- NO (depends
on how we estimate Ω)
• has variance σ2(X’ Ω -1X)-1 --- NO
• is conditionally normally distributed -- NO
• is the BUE of β --- NO
• is the MLE of β --- NO
In addition,
 ti = ( ˆ F GLS , i   i ) / se( ˆ F GLS , i ) ~ t(n-k)--- NO
where
ˆ 1 X ) 1 ]
se( ˆ F GLS , i )  ˆ 2 [( X ' 
ii ,
ˆ 2  SSR /( n  k )
and
 if Rβ = r, where R is qxk and r is qx1
then
F=
{(R ̂ -r)’[R(X’
F GLS
̂
-1
X)-1R’]-1(R ̂ -r)/q}/ ˆ
2
F GLS
~ F(q,n-k) --- NO
That is, the “FGLS” t-statistic is not
drawn from a t-distribution and the
“FGLS” F-statistic is not drawn from an
F-distribution.
So, it appears that if the errors are
heteroskedastic or serial correlated and
we have incomplete information about
the form of the heteroskedasticity or
serial correlation (i.e., Ω is unknown),
we are in a very bad situation.
• The OLS estimator is unbiased, the
FGLS is biased; neither has any
optimal properties
• Neither the OLS nor FGLS
estimators can be used as the basis
for constructing valid hypothesis tests
or confidence intervals!
Now, suppose we relax A.2 (strict
exogeneity). This assumption is rarely
plausible in time series regressions
because the i-th observation of the
dependent variable y will frequently be
correlated with observations i+1,i+2,…
of the explanatory variables, X:
 lagged dependent variables (yi = β1 +
β2xi + β3yi-1 + εi)
 feedback effects between x and y (xi
is determined by yi-1)
Instead, the best we might hope for is
that the x’s are predetermined:
E(εi │x11,…,xk1,…, x1i,…,xki) = 0, all i
In this case, the OLS estimator of β,
•
•
•
•
is unbiased ---- NO
exists and is unique ---- YES
has variance σ2(X’X)-1--- NO
is normally distributed --- YES, but
…
• is the BUE of β ---- NO
• is the MLE of β ---- NO
In addition,
 ti = (ˆi   i ) / se(ˆi ) ~ t(n-k) ---- NO
where
se(ˆi )  ˆ 2 [( X ' X ) 1 ]ii , ˆ 2  SSR /( n  k )
 if Rβ = r, where R is qxk and r is qx1
then
2
-1
-1
ˆ
ˆ
ˆ

F = {(R  -r)’[R(X’X) R’] (R  -r)/q}/
~ F(q,n-k) --- NO
So, the OLS estimator is not unbiased,
does not retain any optimality properties,
and inference based on the OLS t and F
statistics are not longer valid.
This is an even worse situation than
when A.4 fails since, in that case, at least
OLS is unbiased.
Summary –
 In time series regressions A.2 and /or
A.4 are often implausible assumptions.
 When these assumptions fail, the OLS
estimator may or may not be unbiased
(depending upon whether A.2 is or is not
correct). The OLS t and F statistics will
not be valid starting points for testing or
interval construction.
 If A.4 fails but A.2 holds, the GLS
procedure provides a simple solution to
this problem, but relies on our knowing
more about the variance of the error
vector than is reasonable in most
applications. That is, GLS will generally
not be a feasible approach. The FGLS
estimator is a biased estimator and the
FGLS t and F statistics will not be valid
starting points for hypothesis testing or
interval construction.
This would seem to leave us in a very bad
position! However, there is a way around
these problems - Formulate an alternative
set of assumptions that will
 accommodate weakening of assumptions
A.2 and A.4 to allow for predetermined
(rather than strictly exogenous)
regressors and for non-spherical
disturbances
 allow us to apply asymptotic distribution
theory to construct estimation and test
procedures that will have desirable
properties in applications with
sufficiently large sample sizes.
Note: In Econ 671 you were introduced to
the application of asymptotic distribution
theory to study large sample properties of
OLS and FGLS under a particular set of
assumptions. However those assumptions
are not adequate for our purposes. We will
formulate a set of assumptions that turn out
to be more useful in time series regressions.