Introduction to econometrics II eco 356 faculty of social sciences course guide course Developers: Dr. Adesina-Uthman
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| INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 68 Or you could conclude that the regression result contradicts the hypothesis. You are not convinced by the explanation in (1) because the probability is so small and you think that a much more likely explanation is that is not really equal to 1. In other words, you adopt the alternative hypothesis H 1 instead. We can summarize this decision rule mathematically by saying that we will reject the null hypothesis if …[2.43] wherez is the number of standard deviations between the regression estimate and the hypothetical value of : ( ) …[2.44] The null hypothesis will not be rejected if This condition can be expressedregardingb2 and by substituting for z from ( ) …[2.45] Multiplying through by the standard deviation of b 2 , one obtains ( ) ( ) …[2.46] from which one obtains ( ) ( ) …[2.47] [2.47] gives the set of values of which will not lead to the rejection of a specific null hypothesis . It is known as the acceptance regionfor , at the 5 percent significance level. 2.2.3.8 What Happens if the Standard Deviation of is Not Known 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 denotet crit , 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 H 0 : is ( ) …[2.49] Hence we have the decision rule |