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Use the data in SLEEP 75 for this exercise. The equation of interest is$$=\beta_{0}+\beta_{1} \text { totwrk }+\beta_{2} e d u c+\beta_{3} a g e+\beta_{4} a g e^{2}+\beta_{5} \text { yngkid }+u$$$$\begin{array}{l}{\text { (i) Estimate this equation separately for men and women and report the results in the usual form. }} \\ {\text { Are there notable differences in the two estimated equations? }}\end{array}$$$$\begin{array}{l}{\text { (ii) Compute the Chow test for equality of the parameters in the sleep equation for men and women. }} \\ {\text { Use the form of the test that adds male and the interaction terms male.totwrk,\ldots., male-yngkid }}\end{array}$$ $$\begin{array}{l}{\text { and uses the full set of observations. What are the relevant } d f \text { for the test? Should you reject the }} \\ {\text { null at the } 5 \% \text { level? }}\end{array}$$$$\begin{array}{l}{\text { (iii) Now, allow for a different intercept for males and females and determine whether the interaction }} \\ {\text { terms involving male are jointly significant. }} \\ {\text { (iv) Given the results from parts (ii) and (iii), what would be your final model? }}\end{array}$$

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Chapter 7

Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables

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Olivier M.

November 29, 2021

can you show the command you entered into STATA to get the regressions for part 1

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Consider the following mod…

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Using the data in SLEEP75 …

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In a study relating colleg…

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The following model is a s…

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The following equations we…

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An experimenter has prepar…

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(a) identify the claim and…

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Test the given claim. Iden…

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Please provide the followi…

you an estimate. An equation where sleep depends on total working our education H H Square and young kid dummy for men and women separately. These are the regression results for both genders. You can compare and contrast the results between two regressions. The first thing you could noticed is the variable total work total working our has a negative sign and is highly significant in both regressions. The meaning of this variable is when Children working hours increased sleeping hours decrease. The second variable of interest is education. Education has a negative signs in both the regressions. However, it is significant Onley for men, not for women. The next variable is Age H and H Square have different and opposite signs between two regressions. The age variable has a positive sign for mill, which means when men get older, they can sleep more. But H square in the regression for men is a negative estimation and it means that they're positive effect of age on sleeping hours to men decreases over time. Okay, the opposite happens for women. When women become older, they sleep less, but the negative effect of age on women sleeping hours decreases over time. Young kid is another variable that has different signs between two regressions. Having a young kid increases man's sleeping hours but decreases women sleeping hours. However, these variables age, age square and young kids are not statistically significant. The last difference between the two regressions you can notice is the fit of the model. The model fit women's data worse, and the intercept in the regression for women is much larger compared to the regression for men. In part one of this problem, we have shown the differences in the regressions between two genders. However, we need a test to see whether such differences are statistically significant for that purpose. In Part two, we will conduct a child test, a test that based on the F distribution or F statistic, the F statistic has two degrees of freedom. The first degree of freedom is the number of debate house we want to test. So this test has a not hypotheses where each beta in the regression equation equal across two groups. You can say that under the not hypotheses, the intercept must equal between men and women. The estimation of the coefficients of the total work variable must equal between two genders and the same for education, Age H Square and young kids. They're five exponentially variables and one intercept. So six factors and their two equations the degree of freedom, the first one of the F statistic is six. Because we have six factors in each equations. The second degrees of freedom is calculated by something, the number of observation of each equation and subtract from it twice the number of factors or betas. We have 400 men and 306 women in the sample, so the second degree of freedom of this F statistic is 700 and six minus 12, which is 694. We can fill in six and 694 year our F statistic. If you calculate correctly, you can get 2.12 and the P value is approximately 0.5 Even this p value we are able to reject the null hypotheses at the 10% level. When we reject the null hypothesis like this, the test implies that there's at least one pair of betas that are not equal across two groups. We still cannot conclude whether the explanatory variables have different effects on sleeping hours between men and women. That is because the child has has a limitation. It allows for intercept difference. So it could be the case that the explanatory variables have the same effect on sleeping hours of men and women. But the two regression equation differ by only the intercept. For that reason, we must do another test. We will regress. A new equation. We're asleep depends on the same set of explanatory variable, but we include on the right hand side Mel Dummy and its interaction terms with total work education, H, H Square and Young Kid. We will use the full sample, including both men and women in our Equation F test, where the non hypothesis is the beta of the interaction terms all equal zero. This is an F test, and again the F statistic has two degree of freedom. The first degree of freedom is the number of constraint on number of betas. We want to test they're five betas here, So the first degree of freedom of the F statistic is five. The second degree of freedom for this F test is the number of observations or the size of the sample which is 706 minus the number of Betas. We estimate in the regression equation, they're actually 12 if you add the former, um, the former set of explanatory variable there. Five. We have one intercept. So six we have mail. So seven and we have the interaction of male with the former set of explanatory variables, which is another fight that is 12 in total. So the second degree of freedom of the F statistic is 694 the same as the F statistic before we will get the value of the F F test is one point 26 and the P value is roughly 0.28 Given this p value, we are unable to reject the novel hypotheses. Yeah, the test results from Part Three shows that there is no strong evidence of slow differences between men and women. If we still want to test whether there are differences in slopes, we need a larger sample size for the purposes of studying the sleep work. Trade off the original model with mill added as an exponent very variable is sufficient

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