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Use the data in ECONMATH to answer this question.(i) Logically, what are the smallest and largest values that can be taken on by the variable score?What are the smallest and largest values in the sample?(ii) Consider the linear model$\quad$ score $=\beta_{0}+\beta_{1}$ colgpa $+\beta_{2} a c t m t h+\beta_{3} a c t e n g+u .$Why cannot Assumption MLR.6 hold for the error term $u ?$ What consequences does this havefor using the usual $t$ statistic to test $\mathrm{H}_{0} : \beta_{3}=0 ?$(iii) Estimate the model from part (ii) and obtain the $t$ statistic and as $\mathrm{H}_{0} : \beta_{3}=0 .$ How would you defend your findings to someone w ment: "You cannot trust that p-value because clearly the error term in the equation cannot have a normal distribution."

(i) 0 and 100 (ii) because the dependent variable is bounded (iii) t won't be valid (iv) this would be totally correct if it had the word "asymptotic". See video

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Chapter 5

Multiple Regression Analysis: OLS Asymptotics

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part one logically the minimum value for score Is zero and the maximum value is 100. This is not the actual minimum and maximum of this variable in our sample. Okay. Part two. In the given yeah, linear model, the dependent variable is score a variable that is we can show to be that is positive and found it Between zero and 100. In this model assumption. Multiple linear regression number six cannot hold because the dependent variable is bounded. So there's no way the error terms can be normally distributed. And the consequence of that for the T statistic is this statistic will be invalid. That is the answer for part three and part four. You need to comment on this statement. You cannot trust that P value because clearly the error term in the equation cannot have a normal distribution. So for that we don't quite agree. Okay, this comment is Not 100% correct. You can trust the P values even when the error terms are not normally distributed. So it seems conflict with what I just said in part three but actually we we are dealing with um fight night sample properties of L. S. That is the subject of this chapter. The main idea behind this chapter is that when the simple size gross with them bound or to infinity the distribution of the arrow terms would become as symptomatically normal. This is the key word. So when this is true the influence is still valid and we can trust the P value of the T. Start. Even in our simple the error terms are not normally distributed. If we have strong enough reason to believe the error terms will be normally distributed. If we can collect more data points then we can trust the p value of the T statistic. However, in the case of this model we have a bounded dependent variable, the score variable. And even when we collect more score points, our ERA terms will never be normally distributed, and we have no reason to believe on there p value of the T. Statistics.

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