INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 69 So far we have assumed that the standard deviation of
is known, which is most unlikely in practice. It has to be estimated
by the standard error of . This causes two modifications to the test procedure. First,
z is now defined using s.e.(
) instead of s.d.(
) and it is referred to as the
t statistic (
)
…[2.48]
Second, the critical levels of
t depend on upon what is known as a
tdistribution instead of a normal distribution. We will not
go into the reasons for this, or even describe the
t distribution mathematically. But enough to say that it is a partner of the normal distribution. Its exact shape depends on the number of degrees of freedom in the regression and approximates the normal distribution increasingly closely as the number of degrees of freedom increases. You will certainly
have encountered the t distribution in your introductory statistics course. The estimation of each parameter in a regression equation consumes one degree of freedom in the sample. Hence the number of degrees of freedom is equal to the number of observations in the sample minus the number of parameters estimated. The parameters are constant (assuming that this is specified in the regression model) and the coefficients of the explanatory variables. In the present case of simple regression analysis,
only two parameters,
, are estimated and hence the number of degrees of freedom is
n – 2. It should be emphasized that a more general expression will be required when we come to multiple regression analysis.
The critical value of t, which we will denote
tcrit, replaces the number 1.96 in [2.43], so the condition that a regression estimate should not lead to the rejection
of a null hypothesis H0:
is
(
)
…[2.49] Hence we have the decision rule
