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$$\begin{array}{l}{\text { Use the data in } 401 \mathrm{KSUBS} \text { for this exercise. }} \\ {\text { (i) Compute the average, standard deviation, minimum, and maximum values of netfa in the }} \\ {\text { sample. }}\end{array}$$$$\begin{array}{l}{\text { (ii) Test the hypothesis that average nettfa does not differ by 401 (k) eligibility status; use a two- }} \\ {\text { sided alternative. What is the dollar amount of the estimated difference? }}\end{array}$$$$\begin{array}{l}{\text { (iii) From part (ii) of Computer Exercise } \mathrm{C} 9, \text { it is clear that } e 401 k \text { is not exogenous in a simple }} \\ {\text { regression model; at a minimum, it changes by income and age. Estimate a multiple linear }} \\ {\text { regression model for netfa that includes income, age, and } e 401 k \text { as explanatory variables. The }}\end{array}$$ $$\begin{array}{l}{\text { income and age variables should appear as quadratics. Now, what is the estimated dollar effect }} \\ {\text { of } 401(\mathrm{k}) \text { eligibility? }}\end{array}$$$$\begin{array}{l}{\text { (iv) To the model estimated in part (iii), add the interactions } e 401 k \cdot(a g e-41) \text { and }} \\ {e 401 k \cdot(a g e-41)^{2} . \text { Note that the average age in the sample is about } 41, \text { so that in the new }} \\ {\text { model, the coefficient on } e 401 k \text { is the estimated effect of } 401(\mathrm{k}) \text { eligibility at the average age. }} \\ {\text { Which interaction term is significant? }}\end{array}$$$$\begin{array}{l}{\text { (v) Comparing the estimates from parts (iii) and (iv), do the estimated effects of } 401(\mathrm{k}) \text { eligibility at }} \\ {\text { age 41 differ much? Explain. }} \\ {\text { (vi) } \text { Now, drop the interaction terms from the model, but define five family size dummy variables: }}\end{array}$$ $$2, \text { fsize } 3, \text { fsize } 4, \text { and } f \text { size } 5$$ The variable fsize 5 is unity for families with five or more members. Include the family size dummies in the model estimated from part (iii); be sure tochoose a base group. Are the family dummies significant at the 1$\%$ level?(vii) Now, do a Chow test for the model$$=\beta_{0}+\beta_{1} i n c+\beta_{2} i n c^{2}+\beta_{3} a g e+\beta_{4} a g e^{2}+\beta_{5 e} 40 I k+u$$$$\begin{array}{l}{\text { across the five family size categories, allowing for intercept differences. The restricted sum of }} \\ {\text { squared residuals, } S S R_{r}, \text { is obtained from part (vi) because that regression assumes all slopes are }}\end{array}$$ the same. The unrestricted sum of squared residuals is SSR $_{w r}=S S R_{1}+\ldots+$ SSR $_{5}$ where $S S R_{f}$ is the sum of squared residuals for the equation estimated using only familysize $f .$ You should convince yourself that there are 30 parameters in the unrestricted model$(5$ intercepts plus 25 slopes) and 10 parameters in the restricted model ( 5 intercepts plus 5 slopes). Therefore, the number of restrictions being tested is $q=20$ , and the $d f$ for theunrestricted model is $9,275-30=9,245 .$

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in this part, you are going to show some statistics of the variable net F. A thistle variable indicates the Net total financial assets in thousands of dollars. This variable has a mean of 19 up seven. The standard deviation is 63 going nice six. The minimum level is minus 502.3 and the maximum level is 1000 536 18 Remember, these numbers are in 1000. In the second part, you are going to test whether the average net total asset differs by 401 k eligibility studies and you can use a two sided alternative. The null hypothesis of this question is the average of net S A for their group that is eligible for for one cave equal that f A of the group that is not eligible for 41 K. We would do a T test and the T test. We will get a value of 13 11 The P value off the test is very small is way smaller than 0.1 So we are able to reject the non hypotheses that two groups have equal net total financial asset. The difference between the two groups in that total financial asset is, well intake. The value of Met Toto A set the average one for the group that is eligible for for a one K, which is 30 points 54 These numbers are generated by their statistical software. Then we subtract from it their average net F A of the group that is not eligible for for a one K, which is 11 going six eight What we get. ISS roughly 18 0.86 mhm. So the group that is eligible for for one K has a larger average net total financial asset, and it is larger by the average of the groups ineligible for 41 K by 18.8 $6000. We will estimate a mentally near regression model for net total financial assets. That includes income age eligibility for 401 K, and we also have age and income included in the regression As quadratic form. We will look at the coefficient of the variable E for a one K as an indicator of the estimated dollar effect of the eligibility status. Can you talk? The coefficient of E 41 K is nine point 705 It means that if the family is Elizabeth 4401 k, their net total financial asset when increased by $9705 we will add to the model estimated in Part three, the interaction terms of a legibility status and H minus 41. And another term is the interaction of eligibility status and age minus 41 square. This is the regression result. When we look at the interaction terms, we see that only this term it's significant. The term with Asian minus 41 the term with Asia minus 41 square, is not significant. In future models, we can safely drop the last term. Women compare the estimated effect of the eligibility status between model in Part three and model in Part four. The left panel show the regression results from the model. In part, we and the right panel show the results from the model. In Part four, you can tell the difference of the coefficient of E 401 K between two models. In monetary, it is 9.705 A model for it is 9.960 You should note that the meaning of the coefficient of e 401 k in first between two models in model three, it is the effect for all ages. A model for it is the effect when age equals 40. 1 way will include family size dummies In the regression equation, you can see that in our equation we now have F s two s, three s four and FS five. These are their family side dummies. Efforts to takes a value of one If the family has to people fs two equals zero Otherwise, after three takes a value of one is the family has three people and takes a value of zero otherwise, ever as far is for family of four. And Verse five is for family with more than five people. So you don't cfs one in the equation. That is because Fs one or a family of one is the base group. We have the based group so all the coefficient of the dummies should be interpreted as the difference in that total asset compared to you the based group. So among the four Dummies, you can see that Fs two is not significant F s 32 Fs five are significant, and the level of significance of these dummies increases with the size of the families so ever sweet. It's significant at the 5% level, but s four and s wide are significant at the 1% level. Even the P value you can say that compared to the base group, a family of a single person having three or more people in the family associate with a greater average value of net total financial assets. We can also conduct a F test to see the joint significance of their family size dummies. The null hypothesis is the coefficient of the family size dummy equals each other and equals zero. We will get an F statistic with two degrees of freedom four and NYT housing 265. The statistic is 5.44 and the P value it's very, very small is way smaller than 0.1 So we are able to reject the null hypothesis. In other words, the family size dummies are jointly significant. At the 1% level, we will do a child test forward coefficient equality across five family sized categories for the test statistic we we need some of square residual from the unrestricted models and re restricted model. The restricted model comes from Part six and we use the foreign sample to estimated The sum of square residual for the respective model is 30 million. Yeah, 213,000 and 115 the for the and restricted models. They are five of them. They are estimated separately for each category of family size. The sum of square residual for and restricted models is the sum of the sum of square residual of all five models. I think them up. You wouldn't get a total of 29 million, 980,000 and 959 now, given that we have there some of square residual for the restricted model and unrestricted models, we can calculate the F statistic or the child test statistic the child has is based on the F distribution. That's why we have an F statistic. This is their formula for the F statistic. We will take the sum of square residual of the restricted model minus the sum of square of residual of the unrestricted model divided by their sum of square of residual of the unrestricted model altogether multiply with the ratio of the second degree of freedom divided by the first degree of freedom. These degrees of freedom are given from their problem. We have D F one. The first degree of freedom is the number of constraints were going to test, which is 20 and the second degrees of freedom is given as NYT housing and 245. The F statistic would be three point 57 and we can back up the P value using your statistical software. The P value for a F start with thes two degrees of freedom is very, very small. It's almost zero. So given the P value, we are able to reject the null hypotheses at the 1% level. In other words, the coefficients are not equal across family sized categories.

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