Introduction to econometrics II eco 356 faculty of social sciences course guide course Developers: Dr. Adesina-Uthman

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Introduction to Econometrics ECO 356 Course Guide and Course Material

4.1.2.0 OBJECTIVE
4.1.3.0 MAIN CONTENTS 4.1.3.1 Stochastic Regressors
INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN

4.1.1.0 INTRODUCTION
The least squares regression model assumed that the explanatory variables arenonstochastic, that is, that they do not have random components. Although relaxing this assumption does not in itself undermine the OLS regression technique, it is typically an unrealistic assumption, so it is important you know the consequences of relaxing it. We shall see that in some contexts we can continue to use OLS, but in others, for example when one or more explanatory variables are subject to measurement error, it is a biased and inconsistent estimator.
4.1.2.0 OBJECTIVE The main objective of this unit is to provide a general understanding of the topic stochastic regressors and measurement errors and point out that random element in a regression model is not the only disturbance term but that the variables themselves do have random components.
4.1.3.0 MAIN CONTENTS
4.1.3.1 Stochastic Regressors
Based on the adopted assumption that the regressors, which is the explanatory variables in the regression model are nonstochastic, their values in the sample are therefore fixed and unaffected by the way the sample is generated. Perhaps the best example of a nonstochastic variable is time, which, as we will see when we come to time series analysis, is sometimes included in the regression model as a proxy for

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
112 variables that are difficult to measure, such as technical progress or changes in tastes.
Nonstochastic explanatory variables are unusual in regression analysis. A rationale for making the nonstochastic assumption has been one of simplifying the analysis of the properties of the regression estimators. For example, we saw that in the regression model
…[4.01] the OLS estimator of the slope coefficient maybe decomposed as follows
( )
( )
( )
( )
…[4.02]
Here, if X is nonstochastic, so is
( ) and the expected value of the error term can be written, ( )- ( ). Also if X is nonstochastic,
, ( )- is 0. Which easily helps us to prove that b
2
is an unbiased estimator of The desirable properties of the OLS estimators remain unchanged even if the descriptive variables have stochastic components, provided that these components are distributed independently of the disturbance term, and provided that their distributions do not depend on the parameters
u

Let us demonstrate the unbiasedness and consistency properties and as typical, taking an efficient approach.

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