1. Let's study whether the presence of 401(k) pension plans increase net savings. The dataset 401KSUBS contains information on net financial assets (netfa), family income (inc), a binary variable for eligibility in a 401(k) plan (e401k), and several other vari- ables (which are labeled). 3 (i) What fraction of the families in the sample are eligible for participation in a 401(k) plan? (ii) Estimate a linear probability model explaining 401(k) eligibility in terms of in- come, age, and gender. Include income and age in quadratic form, and report the results in a table with standard errors and asterisks on the coefficients indicating the level of statistical significance. (iii) Obtain fitted values from the linear probability model estimated in part (ii). Are there any fitted values negative or greater than 1? Discuss your findings as they relate to the model and the laws of probability. (iv) Using the fitted values e401k; from part (iv), define e401k; = 1 if e401k;2.5 and e401 ki = 0 if e401 ki < .5. Out of 9,275 families, how many are predicted to be eligible for a 401(k) plan? (vi) For the 5,638 families not eligible for a 401(k), what percentage of these are pre- dicted not to have a 401(k), using the predictor e401k;? For the 3,637 families eligible for a 401(k) plan, what percentage are predicted to have one? (Stata's tab- ulate command will be useful here) (vi) The overall percent correctly predicted is about 64.9%. Do you think this is a com- plete description of how well the model does, given your answers in part (vi)? (vii) Add the variable pira (a binary variable equal to 1 if an individual has an individ- ual retirement account (IRA)) as an explanatory variable to the linear probability model. Other things equal, if a family has someone with an IRA, how much higher is the estimated probability that the family is eligible for a 401(k) plan? is it statis- tically different from zero at the 10% level? 2. Use the dataset in 401KSUBS for this exercise. (i) Use OLS estimate a linear probability model for e401k, using as explanatory vari- ables, inc, inc, age, age, and male. Obtain both the usual OLS standard errors and the heteroskedasticity-robust versions. Are there any important differences? (ii) In the special case of the White test for heteroskedasticity, where we regress the squared OLS residuals on a quadratic in the OLS fitted values, u on yi, y,i= 1,..., n, argue that the probability limit of the coefficient on ýi should be one, the probability limit of the coefficient on y should be - 1, and the probability limit of the intercept should be zero. Hint: remember that Var(y|x1,...,xk) = p(x)[1 – p(x)], where p(x) = Bo + B1x1 +...+ Bx** (iii) For the model estimated from part (i), obtain the White test, interpret the result, and discuss whether the coefficient estimates roughly correspond to the theoret- ical values described in part (ii). (iv) After verifying that the fitted values from part(i) are all between zero and one, obtain the weighted least squares esimates of the linear probability model. Do they differ in important ways from the OLS estimates? Discuss. ES AB Edit mode TI Sidebar Save Find inc[6] 15 e401k marr ma le age fsize nettfa p401k pira incse Y 1 1 9 9 1 1 Variables 6 6 9 1 1 1 1 0 40 35 44 44 2 3 Name 1 8 0 0 4 1 2 2 2 2 173.4489 3749.113 165.3282 9777.254 511.393 225 4 ages 1600 1225 1936 1936 2809 3609 9 1 0 4.575 154 0 21.8 18.45 9 3.483 -2.1 3 5 e401k inc marr male 8 9 inc 13.17 61.23 12.858 98.88 22.614 15 37.155 31.896 47.295 29.1 23.457 53 1 9 6 1 9 60 3 9 Label =1 if eligble for 401(k) annual income, $10... =1 if married =1 if male respondent in years family size net total fin. assets,... =1 if participate in 4... =1 if have IRA 7 9 1 0 49 5 0 1 2401 age 19 fsize 8 0 1 0 38 5 1380.494 1017.355 2236.817 0 1444 9 8 1 0 52 nettfa 2 5.29 1 9 2704 1 1 1 45 1 1 10 10 11 12 12 0 1 1 2025 3721 6 9 3 3 9 0 61 40 29.6 0 18.149 .695 p401k pira incs ages inca2 9 1 0 6 0 0 age 2 13 0 1 0 48 0 0 3 2 846.81 550.2309 1010.286 1220.874 596.9226 631.5672 363.8174 1600 2304 3600 14 8 0 1 9 66 .2 0 0 1 9 43 5 3 1849 15 16 6 6 1 43 1 9 3 1849 2209 1 1 0 2 1 3 17 18 31.785 34.941 24.432 25. 131 19.074 38.772 12.48 45.39 39.861 102.6 39.579 40.194 25.254 47 27 1 1 0 A 0 9 1 1 1 1 2 2 -4.25 0 4.15 -10 122.5 1.6 40.999 12.175 729 3249 19 1 0 0 9 @ 0 1 A 0 2 1 1225 57 35 53 36 1 1 1 5 5 1 2809 20 21 22 23 24 Q- 0 1 0 4 0 3 1296 0 1 0 40 3 8.3 0 1 1503.268 155.7504 2060.252 1588.899 10526.76 1566.497 1615.558 637.7645 116.64 729 318.8368 1516.323 149.8176 324 1600 961 8 0 1 1 31 48 9.687 .5 9 A 3 1 a 0 8 2304 25 23 26 20 27 10.8 27 0 6 1 0 1 1 9 2 1 5 inc annual income, $1000s float %9.09 % . 13 3 1764 Properties V Variables Name Label Type Format Value label Notes Data Frame Filename 0 0 0 0 1764 42 42 35 32 28 17.856 38.94 12.24 1 1 0 2 -15.495 .2 -2.5 1 1225 1024 8 0 2 9 1 29 30 1 9 9 18 6 46 2 2 0 9 1 2116 default 401ksubs.