INTRODUCTION TO ECONOMETRICS II
ECO 306
NOUN
59 that is, the set of all hypothetical values not contradicted by the experimental result. We shall also see how to test whether the goodness of fit of a regression equation is better than what might be expectedby pure chance.
2.2.3.1 The Random Components of the Regression Coefficients
The least squares regression coefficient is a special form of arandom variable whose properties depend on those of the disturbance term in the equation. This will be demonstrated first theoretically and then using a controlled experiment. In particular, we will investigate the implications for the regression coefficients of certain assumptions concerning the disturbance term. Throughout the discussion, we shall continue to work with the simple regression model where Y depends on X according to the relationship
And we fit the regression equation
̂
given a sample of n observations. We shall also continue to assume that X is a non-stochastic exogenous (not external randomly determined) variable that is, that its value in each observation maybe considered to be predetermined by factors unconnected with the present relationship. First, note that
has two components. It has nonrandom component (
), which owes nothing to the laws of chance (
maybe unknown, but nevertheless they are fixed constants) and it has the random component
. This implies that, when we calculate b
2
according to the usual formula
( )
( )
…[2.28]
b
2
would also have a random component
( ). ( depends on the values of Y, and the values of Y depend on the values of μ. If the values of the disturbance term had been different in the n observations, we would have obtained different values of Y, hence of Cov (X, Y), and hence of b
2
.Thus we have shown that the regression coefficient b
2
obtained from any sample consists of (1) a fixed component, equal to the true value
, and (2) a random component dependent on Cov(X, μ), which is responsible for its variations around this central tendency. Similarly, one may easily show that b
1
has a fixed component equal to the true value
, plus a random component that depends on the random factor μ.
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