Although a t value is a test statistics of the null

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Although a t-value is a test statistics of the null hypothesis that the corresponding coefficient in the regression model is zero, it is quite easy to rebuild the t-value for testing other null hypotheses, as follows. Suppose you want to test the null hypothesis that where β ' β 0 , β 0 is a given number, for example Then β 0 ' 1.
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13 $ $ & $ 0 $ F $ $ ' $ $ $ F $ $ & $ 0 $ F $ $ ' $ $ $ F $ $ & $ 0 $ $ $ $ $ F $ $ ' $ $ $ F $ $ 1 & $ 0 $ $ ' $ $ & $ 0 $ $ . $ t $ $ , (21) so that by Proposition 5, $ t $ $ , $ ' $ 0 ' $ $ & $ 0 $ $ . $ t $ $ - t n & 2 . (22) For example, suppose that in the ice cream case we want to test the null hypothesis . H 0 : β ' 1 Then $ t $ $ , $ ' 1 ' $ $ & 1 $ $ . $ t $ $ ' 1.5 & 1 1.5 ×4.597 . 1.532, (23) which under the null hypothesis is a random drawing from the t distribution with 6 H 0 : β ' 1 degrees of freedom. Note that the value of this test statistic is in the acceptance regions in Figures 2 and 3. This trick is useful if the econometric software you are using only reports the t-values but not the standard errors. If the standard errors are reported, you can compute directly as ˆ t ˆ β , β ' β 0 Of course, if only the standard errors are reported and not the t-values you ˆ t ˆ β , β ' β 0 ' ( ˆ β & β 0 )/ˆ σ ˆ β . can compute the t-value of as ˆ β ˆ t ˆ β ' ˆ β / ˆ σ ˆ β . 6. The R 2 The R 2 of a regression model compares the sum of squared residuals ( SSR ) of the model with the SSR of a “regression model” without regressors: Y j ' " % U j , j ' 1,2, ..... , n . (24) It is easy to verify that the OLS estimator of α is just the sample mean of the ‘s: ˜ α Y j # " ' ¯ Y ' 1 n j n j ' 1 Y j .
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14 Therefore, the SSR of “regression model” (24) is which is called the Total Sum of ' n j ' 1 ( Y j & ¯ Y ) 2 , Squares ( TSS ), is TSS ' j n j ' 1 ( Y j & ¯ Y ) 2 . (26) The R 2 is now defined as: R 2 ' def . 1 & SSR TSS . (27) The R 2 is always between zero and one, because SSR # TSS . ( Exercise : Why?) If SSR = TSS , so that R 2 = 0, then model (24) explains the dependent variable ‘s equally well as model (2). In Y j other words, the explanatory variables in (2) do not matter: β = 0. The other extreme case is X j where R 2 = 1, which corresponds to SSR = 0. Then the dependent variable in model (2) is Y j completely explained by , without error: / Thus, the R 2 measures how well the X j Y j α % β X j . explanatory variables are able to explain the corresponding dependent variables For X j Y j . example, in the ice cream case, SSR = 11.5 and TSS = 52, hence R 2 = 0.778846. Loosely speaking, this means that about 78% of the variation of ice cream sales can be explained by the variation in temperature. 7. Presenting regression results When you need to report regression results you should include, next to the OLS estimates of course, either the corresponding t-values or the standard errors, the sample size n , the standard error of the residuals (SER), and the R 2 , because this information will enable the reader to judge your results. For example, our ice cream estimation results should be displayed as either Sales ' & 0.25 % 1.5 Temp ., n ' 8, SER ' 1.384437, R 2 ' 0.778846 ( & 0.100) (4.597) ( t & values between brackets ) or
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15 Sales ' & 0.25 % 1.5 Temp ., n ' 8, SER ' 1.384437, R 2 ' 0.778846 (2.49583) (0.32632) ( standard errors between brackets ) It is helpful to the reader if you would indicate whether you have displayed the t-values between brackets or the standard errors, but you only need to mention this once.
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