Please attach Stata output, including graphs. Submit all your work in ONE file.
Total points = 25.
Q1. ( 8 = 3 + 3 + 2 points)
We continue the analysis of the data set in wealth.dta from Question 3 in Assignment 3.
(a) Estimate the model
nettfa = 0 + 1inc + 2inc2 + 3age + 4age2 + 5marr + 6fsize + 7e401k + u
using OLS with default standard errors:
reg nettfa c.inc##c.inc c.age##c.age marr fsize e401k
Note: the specification with # or ## is less straightforward than "reg nettfa inc incsq age agesq marr fsize e401k" but it simplifies the estimation of predicted values and marginal effects, since Stata will take into account that the same variable is present in two or more terms. If you generate a separate variable incsq instead, Stata will treat inc and incsq as unrelated regressors when computing marginal effects. Term "c.inc##c.inc" denotes both inc and inc squared; you can also denote these terms separately as "inc" and "c.inc#c.inc". Prefix c. indicates a continuous variable. If you do not know how to refer to a particular factor variable or an interaction term in further hypothesis testing, run the regression command with option "coeflegend"; it will report the Stata name for each coefficient. Try "reg nettfa c.inc##c.inc c.age##c.age marr fsize e401k, coeflegend" and then "test inc c.inc#c.inc".
Perform the misspecification tests.
RESET test:
estat ovtest
Q. Briefly interpret the result. Do you need to modify the model? If yes, how can you modify the model?
Re-estimate the model adding a cubic term for inc and interaction term age*inc:
reg nettfa c.inc##c.inc##c.inc c.age##c.age marr fsize e401 c.inc#c.age
Q. Repeat the RESET test and interpret the result (compare the test statistics for
the two specifications).
Now run the Breusch-Pagan test for heteroskedasticity for the second specification:
estat hettest, fstat
and White’s test for heteroskedasticity:
estat imtest, white
Q. Briefly interpret the results.
(b) Re-estimate the second model using robust errors:
reg nettfa c.inc##c.inc##c.inc c.age##c.age marr fsize
e401 c.inc#c.age, r
Q. Briefly discuss any changes in individual significance levels when using robust standard errors.
Q. Write down the expression for the fitted value of nettfa using the estimated coefficients and standard notation for regressors: inc, inc2, inc3, etc.
Q. Write down the partial derivative of the fitted value with respect to inc; it estimates the effect of a one-unit increase in income.
(c) Plot the predicted average wealth as a function of income for 35-year-old married respondents who are eligible for 401k plan, with 3 persons in the household:
margins, at (inc=(10(20)190) marr=1 fsize=3 e401k=1 age=35) marginsplot
Q. Explain the shape of the graph.
Note: you can save the graphs as png files and insert them in the text. Plot the marginal effects of income on wealth:
margins, dydx(inc) at (inc=(10(20)190) marr=1 fsize=3 e401k=1 age=35)
marginsplot
Q. Explain the shape of the graph (this is the partial derivative of the fitted value with respect to income).
2. ( 17 = 2 + 4 + 2 + 2 + 3 + 3 + 1 points)
Linear probability model (LPM)
We will analyze the labour supply of married women using a data set from Mroz (1987). Source: 1976 Panel Study of Income Dynamics; 753 observations.
lfp = A dummy variable: 1 if woman worked in 1975, otherwise 0 hrs = Wife’s hours of work in 1975
k6 = Number of children less than 6 years old in the household k18 = Number of children between ages 6 and 18 in the household wa = Wife’s age
we = Wife’s educational attainment, in years
ww = Wife’s average hourly earnings, in 1975 dollars faminc = Family income, in 1975 dollars
age_int = Wife’s age in intervals; age_int = 1 if 30≤age≤ 35, age_int = 2 if 35<age≤ 40, age_int = 3 if 40<age≤ 45, age_int = 4 if 45<age≤ 50, age_int
= 5 if 50<age≤ 55 age_int = 6 if 55<age≤ 60.
(a) Download the data and generate the following variables:
wife’s non-wage income (in $1000), determined as "family income - wife’s earnings":
generate nwi=(faminc-hrs*ww)/1000
a dummy variable for young children in the household:
replace k6=1 if k6>0 replace k18=1 if k18>0
Estimate LPM for lfp. The regressors include women’s age, age2, education, a dummy for young children, a dummy for older children, her non-wage income, and a square of non-wage income. Use robust standard errors.
reg lfp wa c.wa#c.wa we k6 k18 nwi c.nwi#c.nwi, r
Q. Briefly explain why we should use robust standard errors.
(b) Discuss each coefficient (its statistical significance and economic interpretation).
(c) For the model in (a), find the fitted values for the dependent variable, report their detailed statistics, and plot a histogram:
predict yhat
summarize yhat, detail hist yhat
Q. How do you interpret the fitted values in this regression? Is this interpretation
appropriate for the lowest and the largest fitted values?
Obtain the residuals
predict uhat, residuals
and plot a scatterplot of uhat (vertical axis) versus yhat (horizontal axis).
twoway (scatter uhat yhat)
Q. Briefly explain why the scatterplot of residuals exhibits this particular pattern.
(d) For the model in (a), use Stata to estimate the probability of being in the labour force as a function of age for a woman without children under 18, with 12 years of education and 20,000 dollars of other income:
margins, at (wa=(30(5)60)we=12 nwi=20 k6=0 k18=0) marginsplot
Q. Explain the shape of the graph.
Estimate the impact of a one-year increase in age on the probability of being in the labour force as a function of age for a woman without children under 18, with 12 years of education and 20,000 dollars of other income:
margins, dydx(wa)at (wa=(30(5)60)we=12 nwi=20 k6=0 k18=0) marginsplot
Q. Explain the shape of the graph.
(e) At the 5% level, test for joint significance all coefficients that were individually insignificant at the 5% using command "test". Note: if you wish to test both age coefficients, run "test wa c.wa#c.wa".
If the null is rejected, repeat the test for joint significance of a smaller group of variables.
When the null hypothesis is not rejected, re-estimate the model without the regressors with jointly insignificant coefficients.
Q. Briefly discuss the changes in the magnitude and statistical significance of the remaining coefficients (compared to the model in (a)).
(f) There may exist a non-smooth relationship between Prlfp = 1|regressors and age.
Reestimate the model in (a), using dummy variables for age_int instead of continuous wa:
reg lfp i.age_int we k6 k18 nwi c.nwi#c.nwi, r
Q. How will you interpret the coefficients for age dummies?
Q. Are the dummy coefficients individually statistically significant?
Q. Test the dummy coefficients for joint significance and interpret the result.
testparm i.age_int
(g) Using the model in (f), sketch by hand a function that shows the dependence of the estimated probability of working on age (age between 30 and 60). Assume no children under 18, 12 years of education and 20,000 dollars of other income.
Use the following command for predicted probabilities:
margins age_int, at (we=12 nwi=20 k6=0 k18=0)
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