National open university of nigeria introduction to econometrics II eco 356

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Introduction to Econometrics ECO 356 Course Guide and Course Material
Introduction to Econometrics ECO 356 Course Guide and Course Material

NOUN
56

2
var(
|
)
[
(
|
)]
i
i
i
i
i
X
E
E
X






2
(
|
)
i
i
E
X


(due to assumption 3)

2


…[03]
where,

var is variance.
Assumption 5: No autocorrelation between the disturbances.
If given any two X values,
i
X
and
(
),
j
X i
j

the autocorrelation between any two
i

and
(
)
j
i
j


is zero, as shown in.

cov( ,
|
,
)
{[
( )] |
}{[
(
)] |
}
i
j
i
j
i
i
i
j
j
j
X X
E
E
X
E
X
 








(
|
)(
|
) (

)
i
i
j
j
i
j
E
X
X
shows
and
are uncorrelated





…[04]

0

where,
(
)
i and j
are different observations and cov is covariance.
Assumption 6: Zero covariance between
(
)
i
i
and X

. As earlier expressed;

cov( ,
)
[
( )]
(
)]
i
i
i
i
i
i
X
E
E
X
E X







sin
( )
0
i
ce E



[ (
(
))]
i
i
i
E
X
E X




(
)
(
) ( )
i
i
i
i
E
X
E XE ib

sin
(
)
( )
0
i
i
ce EX is nonstochastic and E



INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
57

(
)
0
i
i
E
X



…[05]

Assumption 7: The number of observations n must be greater
than the number of parameters to be estimated. On the other hand, the number of observations n must be greater than the number of descriptive (explanatory) variables.
Assumption 8: Variability in X values. The X values in a given
sample must not all be the same. That is, var(X) must be a finite positive number.
Assumption 9: The regression model is correctly speciﬁed. On the other hand, there is no error in the model used in observed analysis.
Assumption 10: There is no perfect multicollinearity. That is, there are no perfect linear relationships among the descriptive variables.
Multicollinearity as a problem associated with CLRM is discussed in unit 3.
- Properties of Ordinary Least Square (OLS) estimator The following properties are associated with OLS estimators
1. Linearity
2. Unbiasedness
3. Efficient it has the minimum variance
4. Consistency
5. Asymptotic Unbiasedness

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
58 The OLS estimator is sometimes referred to as the CLRM and in data analysis the best estimator is refer to as BLUE (best linear unbiased estimator. Therefore, the OLS estimator requires that the descriptive variables are received outside a data group and there is no perfect multicollinearity. Also, OLS is best in the class of linear unbiased estimators when the errors are vector of random variables and successively uncorrelated. Within these conditions, the OLS offers minimum-variance mean- unbiased estimation when the errors have fixed variances. Again, the OLS is a maximum likelihood estimator under the additional assumption that the errors are normally distributed. So, whenever students are planning to use a linear regression model by means of OLS, each time check for the OLS assumptions. Inasmuch as the
OLS assumptions are satisfied, the analysis becomes simpler. Through the Gauss-
Markov theorem (as will be seen later in this unit) students can directly use OLS for the best results. When the OLS estimator is asymptotically normal and a consistent estimator of the asymptotic covariance matrix is available to carryout hypothesis tests on the coefficients of a linear regression model.

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