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Use the data from JTRAIN for this exercise.(i) Consider the simple regression model$$\log (s c r a p)=\beta_{0}+\beta_{1} g r a n t+u$$where scrap is the firm scrap rate and grant is a dummy variable indicating whether a firmreceived a job training grant. Can you think of some reasons why the unobserved factors in umight be correlated with grant?(ii) Estimate the simple regression model using the data for 1988. (You should have 54observations.) Does receiving a job training grant significantly lower a firm's scrap rate?(iii) Now, add as an explanatory variable log $\left(s c r a p_{87}\right) .$ How does this change the estimated effect of grant? Interpret the coefficient on grant. Is it statistically significant at the 5$\%$ level against the one-sided alternative $\mathrm{H}_{1} : \beta_{g r a n t}<0 ?$(iv) Test the null hypothesis that the parameter on $\log \left(\operatorname{scrap}_{87}\right)$ is one against the two-sided alternative. Report the $p$ -value for the test.(v) Repeat parts (iii) and (iv), using heteroskedasticity-robust standard errors, and briefly discussany notable differences.

(i) u contains omitted factors that affect both grant and scrap such as the ability and education of the employees and other firm characteristics (ii) See video. (iii) Reject $H_0: \beta =0 $ in favor of $H_a: \beta <0$ at the 5$\%$ level (iv) Reject $H_0$ (v) t-stats change but still significant

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December 1, 2020

(5 points) Estimate the simple regression model using the data for only 1988 (You should have 54 observations.) Does receiving a job training grant significantly lower a firm’s scrap rate?

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part one. The grants could be awarded to firms based on firm or workers characteristics, so it could be correlated with these factors. If these factors are not controlled for in the regression, they wouldn't be contained in the error term. You mhm part two. We have the simple regression estimates using the 1988 data as follows. The coefficient on grant is positive, but it is not statistically significant, and you can tell that based on their T statistic, the T start for the coefficient of grant is measured by taking 0.57 The coefficient estimate of grant divided by the standard error 0.406 This is a very small number, much smaller then one. And with the T value smaller than than one, we can conclude that this coefficient is not mhm statistically, statistically different from zero part three. We have a new regression equation where we regret lock of scrap rate on grant and lock of scrap rate in the year before. Given this result, we can run a hypothesis test where the not hypothesis is beta grant is zero, and the alternative hypothesis is one started. Beta grant is negative. Women calculate the T statistic again. We will take beta grand hat, which is minus 0.254 divided by its standard error, which is 0.147 And this result ISS roughly minus 1.73 We we need to compare this value with the 5% critical value of the T statistic. At 40 of degree of freedom, this is one sided. The value we look for is minus point minus 1.68 Because the T value we get for the non hypothesis is minus 1.73 Smaller than this critical value, we are able to reject the null hypotheses in favor of the alternative hypotheses. At the 5% level, we have another hypothesis test in part four. We have a two sided tests here. Again, we will calculate the t start. You take the value of beta lock of scrap hat, which is Hm? Oh, so here we Actually, we are testing whether beta locks crab is one, not zero. And so the T statistic would be by the head of lock of scrap rate minus one, divided by the standard error of this beta. And we will get 0.831 minus one over 10.44 And the T statistic is minus 3.8. And for this one, we will use a critical value for T statistic off 40 degrees of freedom at the 5% level. Andi is a two sided value. Uh huh. The value is two points. 02 So this is what I got from the table. And you can compare the absolute value of the T. Start with this critical value. Okay, The absolute value of the T start is greater than the critical value. So we are able to reject the novel hypotheses at the 5% level in part five. Yeah. We will re estimate the equation in part three. On day, we will estimate robbers standard error. Even the new estimation result. We will redo part three and four again. Four. Part three, the T Stott. The new T start is minus point 254 You can see that it is the same beta estimate. Same beta had the only difference between the old ls estimate and the robbers estimate is in the standard error. Usually, the standard error for the robust option is much larger and we were in half 0.142 here. Eso in this k is it has a smaller standard error that is minus 1.79 So the coefficient is even more significantly less than zero when we use the robots than that error option. Before we have minus 1.73 and again, we can still reject the not hypothesis at the 5% level. We weren't finished with redoing part four and the not have policies is beta of locks. Crab rate is one and it is a two sided test. The T statistic here Yeah, we have again the same denominator a numerator and you can see that the standard error of beta l scrapped is much larger. When we use the robust option, we get minus 2.38 and recall that the critical value for the T start we use earlier is 2.2 Mhm. So the absolute value of the T start for this null hypotheses is 2.38 greater than the critical value. 2.2 This T start is smaller in terms of absolute value compared to what we get in part four. But it is still significant. Were still able to reject the novel had policies at the 5% level

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