3
t
t
t
Y
=
x
+
u
t = 1,2,...,T
′
β
For the simple k=2 variable case, this can be written as
t
1
t
2
t
t
Y
=
X
+
X
+
u
t = 1,2,...,T
β
β
1
2
,
,
If
X
1
is an intercept term, then we can more compactly write this as
t
1
2
t
t
Y
=
+
X
+
u
t = 1,2,...,T
β
β
In matrix terminology, the assumptions in Table 4.1 would
be re-expressed as in Table 4.2.
TABLE 4.2
(1)
The dependent variable is a linear function of the set of possibly stochastic but stationary
regressor variables and a random disturbance term as specified in (A). No variables which
influence
Y
are omitted from the regressor set
X
, nor are any variables which do not
influence
Y
included in the regressor set. In other words, the model specification is correct.
(2)
Lack of perfect collinearity (the T*k matrix
X
has rank k)
(3)
The error process has zero mean (
E(
u
) = 0)
(4)
The errors terms, u
t
, are serially uncorrelated
( E(u
t
,u
s
) = 0 for all s not equal to t).
(5)
The errors have a constant variance ( E(u
t
2
) =
σ
2
) for all t.
In matrix terms, (4) and (5) are written as Var(
u
) =
σ
2
I
T
(6)
plim(1/T{X
/
u}) = 0
and if we wish to use it:
(7)
The errors are normally distributed.
THE LINEAR REGRESSION MODEL WITH STOCHASTIC REGRESSORS
A variable is stochastic if it is a random variable and so has a probability distribution; it is non-stochastic if it
not a random variable. Some variables are non-stochastic; for example intercept, quarterly dummy, dummies
for special events and time trends are all non-random. In any period, they take one value known with
certainty.
However, many economic variables are stochastic. Consider the case of a lagged dependent variable. In the
following regression model, the regressor Y
t-1
is a lagged dependent variable (LDV):
t
1
2
t
3
t-1
4
t-1
t
Y
=
+
X
+
X
+
Y
+
u
, t = 1,...,T
β
β
β
β
Clearly, Y
t
= f(u
t
), and so is a random variable. But, by the same logic, Y
t-1
= f(u
t-1
), and so Y
t-1
is a random or
stochastic variable. Any LDV must be a stochastic or random variable. If our regression model includes one,
the assumptions of the