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Use the subset of 401 $\mathrm{KSUBS}$ with fsize$=1 ;$ this restricts the analysis to single-person households; see also Computer Exercise $\mathrm{C} 8$ in Chapter 4(i) What is the youngest age of people in this sample? How many people are at that age?(ii) In the modelnetfa$=\beta_{0}+\beta_{1} i n c+\beta_{2} a g e+\beta_{3} a g e^{2}+u$what is the literal interpretation of $\beta_{2} ?$ By itself, is it of much interest?(iii) Estimate the model from part (ii) and report the results in standard form. Are you concerned that the coefficient on $a g e$ is negative? Explain.(iv) Because the youngest people in the sample are $25,$ it makes sense to think that, for a givenlevel of income, the lowest average amount of net total financial assets is at age $25 .$ Recall that the partial effect of age on nettfa is $\beta_{2}+2 \beta_{3} a g e,$ so the partial effect at age 25 is $\beta_{2}+2 \beta_{3}(25)=\beta_{2}+50 \beta_{3} ;$ call this $\theta_{2} .$ Find $\hat{\theta}_{2}$ and obtain the two-sided $p$ -value for testing $\mathrm{H}_{0} : \theta_{2}=0 .$ You should conclude that $\hat{\theta}_{2}$ is small and very statistically insignificant. [Hint: One way to do this is to estimate the model netta $=\alpha_{0}+\beta_{1}$ inc $+\theta_{2} a g e+\beta_{3}(a g e-25)^{2}+u$ where the intercept, $\alpha_{0}$ is different from $\beta_{0}$ . There are other ways, too.](v) Because the evidence against $\mathrm{H}_{0} : \theta_{2}=0$ is very weak, set it to zero and estimate the model nettfa $=\alpha_{0}+\beta_{1} i n c+\beta_{3}(a g e-25)^{2}+u$ In terms of goodness-of-fit, does this model fit better than that in part (ii)?(vi) For the estimated equation in part (v), set inc $=30$ (roughly, the average value) and graph therelationship between nettfa and age, but only for $a g e \geq 25 .$ Describe what you see.(vii) Check to see whether including a quadratic in $i n c$ is necessary.

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Chapter 6

Multiple Regression Analysis: Further Issues

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part one. The youngest age is 25 there are 99 people of this age in the simple with family size of one part two, we can interpret beta two in two ways. Waiter to Is Thean Crease in Net. Total financial assets When age increases by one year, holding fixed income and age square off course. We cannot change age without changing age Square. So a better, um, alternative interpretation is. Consider their partial derivative of Net F A. With respect to age, it is beta two plus two times beta, three times age. So wait a two is the approximate increase in Net F A when age increases from 0 to 1. But in this application, the partial effect starting at age equals zero is not interesting, because in the simple, the youngest people are at least are 25 years old. Heart tree. We have the old LS estimates as follows. Net total financial assets is a function of income, age, age square, and we have an estimation for intercept or the constant term. Okay, we have 0.8 to 5 as the coefficient of income minus 1.3 to 2 for age and 0.2 56 for age square and the intercept is minus 1.2. I'm also filling in the standard error for each coefficient. Well, we have over 2000 observations, and our square is 12 point 3%. Should right 0.12 three. So you may see that the coefficient on H is negative. It may seem counterintuitive. Okay, The estimated relationship is a U shape. Because you have a negative coefficient for age but a positive coefficient for H Square. To make sense of it, we need to find the turning point in the quadratic. Yeah, we find the turning point. By taking derivative of this equation, we will find the partial derivative of net F A with respect to H. The turning point is the value of age that makes this ratio zero. And we have this value from Part two, which is beta two plus two beta three age equals zero. So you can find the value of age that makes up the turning point. This value of age would be minus beta two over to beta three, and you will use the value of beta two and beta three in the result above you will have 1.322 divided by two times point 0256 and so the turning point is each equal to 25.8. This is very close to the youngest age in the sample. In other words, starting at roughly age equal to 25 the relationship between Net F A and H is positive, as we might expect. So in this case, the negative coefficient on age makes sense when we compute the partial effect in part, for you can run a new regression. Following the hint, you inform the new regress er age minus 25 square and run the regression Net F A on income age and age minus 25 square. The results are okay again. Net F A hat equal 0.8 to 5 times income minus 0.4 27 times age plus point 0256 times age minus 25 square minus 17.2 Which is the intercept, but I should have left space for the standard error for income. The standard error is 0.6 for H is 0.767 for H minus 25 square is very small number 250.9 and for the intercept is 9.97 We have our square equal 2.1 229 It's the same value of our square. You can see that the estimated partial effect starting at age equal to 25 is on Li minus 250.44 and it is very statistically insignificant. Okay, The T value is minus 0.13 and the P value is 0.89 in part five. We will repeat this regression, but we will drop age from the right hand side. We get net f A equals minus 18.49 plus 0.8 to 4 times income plus 0.244 times age minus 25 square. The standard errors are 2.18 0.60 and 0.25 The Oscar still the same 0.1229 The actual change is in the A trusted our square. So in this regression, the adjusted our square following the notation of the textbook is our square bar. In this regression, it is one 0.1 2 to 0 and in part four adjusted. Our square is 0.1 to 16 So the adjusted our square is slightly higher when we drop age variable. But the real reason for dropping age is that it's t statistic is quite small, and the model without it has a straightforward interpretation. Last part part six. We fixed income at 30. Yeah, and we plot the relationship between net F A and age. You will see a line like this is not linear. It's convicts. So you have a change on the X axis and predicted net f A on the Y axis. You will start at age equal to 25 and up to 65 Mhm. Overall, the slope of this line is increasing, so there is an increasing marginal effect. The model is constructed so that the slope is zero at age equal to 25. From there the slope increases. Let me write that down. So increasing Martian on effect

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