Econ 371 Problem Set #6 Answer Sheet In this first question, you are asked to consider the following equation:

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1 Econ 37 Problem Set #6 Answer Sheet 0. In this first question, you are asked to consider the following equation: Y it = β 0 + β X it + β 3 S t + u it. () You are asked how you might time-demean the data so as to estimate β. To answer this question, let s start much like when we had entity-effects by gathering the nuisance variable S t with the intercept; i.e., by rewriting () as: Y it = β 0 + β 3 S t + β X it + u it (2) = φ t + β X it + u it. (3) where φ t β 0 + β 3 S t becomes our time-specific intercept term. We can then consider averaging the equation over all individuals in given time period, yielding: Ȳ t n = n = n Y it (φ t + β X it + u it ) (φ t ) + n = φ t + β Xt + ū t (β X it ) + n (u it ) where X t n n (X it) and ū t n n (u it). Subtracting the above equation from equation (3) yields: Ỹ it Y it Ȳt (4) = [φ t + β X it + u it ] [ φ t + β Xt + ū t ] (5) = β [ Xit X t ] + [uit ū t ] (6) = β Xit + ũ it (7) where X it X it X t and ũ it u it ū t. Notice that equation (7) no longer contains the time-specific intercepts. We have time-demeaned the data. We can then estimate β by simply estimating the model Ỹ it = β Xit + ũ it. 0.2 The second question asks that you again consider a model with only time effects (as in equation ()). The question notes that the transition from equation () above to equation (3). Finally, the question suggests that the equation can also be written in the form: Y it = β 0 + δ 2 B2 t + + δ T BT t + β X it + u it (8) where Bs t = if t = s and = 0 otherwise. You are asked to determine the relationship between the parameters in equation (3) and in equation (8). One way to see the relationships between the parameters is to note that we can use the Bs ts to write equation (3) as: Y it = φ B t + φ 2 B2 t + + φ T BT t + β X it + u it (9) However, since all the Bs ts together sum to, we have that B t = ( B2 t BT t ). Substituting this into equation (9) gives us: Y it = φ ( B2 t BT t ) + φ 2 B2 t + + φ T BT t + β X it + u it (0) = φ φ B2 t φ BT t + φ 2 B2 t + + φ T BT t + β X it + u it () = φ + (φ 2 φ )B2 t + + (φ T φ )BT t + β X it + u it. (2)

2 Comparing equations (2) and (8), we can see that: β 0 = φ and δ t = (φ t φ ) for t = 2,..., T. In words, β 0 is just the time-specific intercept for period. The δ t s measure the difference between the time-specific intercept for period t and the time specific intercept for period. Finally, the problem with adding a δ B t term to equation (8) is that we would have perfect multicollinearity between the intercept and the Bs ts (since they sum to one). The OLS estimates could not be computed in this case. Dropping the overall intercept would solve this problem. 0.3 In this next question, you are asked to consider a simple panel data model of the form: Y it = β 0 + β X it + u it. (3) To this model, you are asked to add both entity-fixed effects and time-fixed-effects for several cases and to comment on the total number of parameters involved. The resulting model is of the form in equation (0.20) in the text. Note that, just under the equation, it lists the resulting parameters, with a total of + + (n ) + (T ) = n + T parameters (i.e., the overall intercept, slope coefficient, (n ) entity-fixed effects, and (T ) time-fixed effects). To answer, the question all you need to know is n and T. a. In this case, we have n = 9 and T = 7, so there are 6 parameters. b. In this case, we have n = 5 and T = 38, so there are 43 parameters. c. In this case, we have n = 04 and T = 3, so there are 07 parameters. d. In this case, we have n = 30 and T = 2, so there are 32 parameters. 0.4 You are asked to consider the following logit regression: P r(y = X) = Λ( X). (4) You are then asked to calculate the change in probability for X increasing by 0 for X = 40 and X = 60 and then explain why is there such a large difference in the change in probabilities. Remember from class (and the book) that Λ(z) = ez + e z =. + e z (5) Thus: P r(y = 40) = Λ( ) = Λ(5.7) = = e 5.7 (6) P r(y = 50) = Λ( ) = Λ(3.3) = = e 3.3 (7) P r(y = 60) = Λ( ) = Λ(0.9) = = e0.9 (8) P r(y = 70) = Λ( ) = Λ(.5) = = e.5 (9) Using these results, we have: and P r(y = 50) P r(y = 40) = = (20) P r(y = 70) P r(y = 60) = = (2) The large differences happens as a result of the non-linearity of the function, and the points at which they are calculated. 0.5 This question focuses on the three discrete choice models and the relationship among their parameters. a. In the first case, you are asked to approximate the probit and LPM slope coefficients based on the coefficient estimates from the logit model. Using the formulas provided one gets: and ˆβ probit = ˆβ logit = 0.625( 0.236) = (22) ˆβ linear = 0.25 ˆβ logit = 0.25( 0.236) = (23) Both approximations are relatively close to the estimated coefficients in their respective models, though the linear approximation is off by a larger amount in percentage terms. 2

3 b. In this second question, you are given the formulas for transforming the intercepts. These yield: and ˆβ 0,probit = ˆβ 0,logit = 0.625(5.297) = 9.56 (24) ˆβ 0,linear = 0.25 ˆβ 0,logit = 0.25(5.297) = (25) You are then asked to provide the fitted probabilities for various temperatures: LPM Probit Logit Temp. (X) Actual Approx. Actual Approx. Actual In terms of calculated probabilities, the approximation is closer for the probit model than for the linear probability model. Also, the logit and probit models yield very similar fitted probabilities. E.3 The empirical exercise in this homework uses the dataset: Insurance. A program that carries all of the tasks for this problem is appended to this answer sheet. Note that the questions in the homework set are slightly different from those in the book. a. In this first question, you are asked to estimate a probit model of the individual s health insurance status (insured) as a function of the discrete variable (selfemp). Then you are asked if your results suggest that self-employed individuals are less likely to have health insurance and whether the difference large. From the attached Stata output, we can see that the coefficient on selfemp that the effect of employment type is statistically significant. The program also uses the scalar command to compute the fitted probability for individuals for are self-employed (P=.689) versus for individuals who are not self-employed (P2=.867). The difference is That is, self-employed individuals are 2.76% less likely to have insurances, which is a fairly substantial difference. b. In the second question, you are asked to add additional explanatory variables, including the individual s age, race, and whether or not they have a college degree. Notice that in my code, I assumed that they have a college degree if they have a masters, PhD or other post-graduate degree. i. You are first asked why you might include these additional variables. Well, as in the linear regression model, we are also worried about omitted variables bias in the probit model. The problem is that, if we exclude important other factors and those factors are correlated with the included variable (in this case selfemp), then the coefficient on selfemp will end up being biased, capturing not only its own impact but also reflecting the impact of the excluded variables. For example, self-employed individuals may tend to be older (since it might take time to come up with an idea for a new company and the money to back its start-up). But older individuals may also be more likely to have insurance (in part because they may now be able to afford it and may have dependents who need to be covered). All else equal, self-employed may have higher insurance rates than we might expect simply because they tend to be older. ii. Next, you are asked which coefficients are the most important factors. Notice that all of the factors are statistically significant at any reasonable significance level. This is first step in determining whether a factor is important. Second, as we discussed in class, we need to consider both the size of the coefficient and the standard deviation of the corresponding variable. The table below provides this information: Variable β k σ Xk β k σ Xk selfemp age race bl race ot colgrad The results suggest that an individual s age has the biggest impact, followed by the individual s degree, self-employment status and then their racial background. iii. Finally, you are asked whether self-employment status still matters in this more complex model. The answer is yes. In fact, the self-employment effect is computed in the code for each individual and then summarized. We see that the mean change in the impact of self-employment is now 8.% 3

4 Problem Set #6 # delimit ; clear; cap log close; cd "R:\users\jaherrig\My Documents\Classes\Economics 37\Problem Sets"; Specify the output file log using Problemset6.log,replace; set more off; Read in and summarize the data use Insurance.dta; describe; summarize; Estimate the model for question E.3a probit insured selfemp,r; scalar P = normal(_b[_cons] + _b[selfemp]); scalar P2 = normal(_b[_cons]); scalar Diff = P-P2; scalar list; Estimate the model for question E.3b generate colgrad = deg_ba + deg_ma + deg_phd + deg_oth; probit insured selfemp age race_bl race_ot colgrad,r; generate PC = normal(_b[_cons] + _b[selfemp] + _b[age]age + _b[race_bl]race_bl + _b[race_ot]race_ot + _b[colgrad]colgrad); generate P2C = normal(_b[_cons] + _b[age]age + _b[race_bl]race_bl + _b[race_ot]race_ot + _b[colgrad]colgrad); generate DiffC = PC-P2C; summarize PC P2C DiffC; summarize selfemp age race_bl race_ot colgrad; Estimate the model for question E.3c

5 generate ageself = ageselfemp; probit insured selfemp ageself age race_bl race_ot colgrad,r; log close; clear; exit;

6 log: R:\users\jaherrig\My Documents\Classes\Economics 37\Problem Sets \Problemset6.log log type: text opened on: 3 Nov 2009, 2:06:47. set more off;.. > Read in and summarize the data > >. use Insurance.dta;. describe; Contains data from Insurance.dta obs: 8,802 vars: Dec :50 size: 809,784 (90.3% of memory free) storage display value variable name type format label variable label healthy age anylim male insured deg_nd deg_ged deg_hs deg_ba deg_ma deg_phd deg_oth married selfemp familysz reg_ne reg_mw reg_so reg_we race_bl race_ot race_wht Sorted by:. summarize; Variable Obs Mean Std. Dev. Min Max healthy age

7 anylim male insured deg_nd deg_ged deg_hs deg_ba deg_ma deg_phd deg_oth married selfemp familysz reg_ne reg_mw reg_so reg_we race_bl race_ot race_wht > Estimate the model for question E.3a > >. probit insured selfemp,r; Iteration 0: log pseudolikelihood = Iteration : log pseudolikelihood = Iteration 2: log pseudolikelihood = Probit regression Number of obs = 8802 Wald chi2() = Prob > chi2 = Log pseudolikelihood = Pseudo R2 = Robust insured Coef. Std. Err. z P> z [95% Conf. Interval] selfemp _cons scalar P = normal(_b[_cons] + _b[selfemp]);. scalar P2 = normal(_b[_cons]);. scalar Diff = P-P2;. scalar list; Diff = P2 = P =

8 .. > Estimate the model for question E.3b > >. generate colgrad = deg_ba + deg_ma + deg_phd + deg_oth;. probit insured selfemp age race_bl race_ot colgrad,r; Iteration 0: log pseudolikelihood = Iteration : log pseudolikelihood = Iteration 2: log pseudolikelihood = Iteration 3: log pseudolikelihood = Iteration 4: log pseudolikelihood = Probit regression Number of obs = 8802 Wald chi2(5) = Prob > chi2 = Log pseudolikelihood = Pseudo R2 = Robust insured Coef. Std. Err. z P> z [95% Conf. Interval] selfemp age race_bl race_ot colgrad _cons generate PC = normal(_b[_cons] + _b[selfemp] + _b[age]age > + _b[race_bl]race_bl + _b[race_ot]race_ot > + _b[colgrad]colgrad);. generate P2C = normal(_b[_cons] + _b[age]age > + _b[race_bl]race_bl + _b[race_ot]race_ot > + _b[colgrad]colgrad);. generate DiffC = PC-P2C;. summarize PC P2C DiffC; Variable Obs Mean Std. Dev. Min Max PC P2C DiffC summarize selfemp age race_bl race_ot colgrad; Variable Obs Mean Std. Dev. Min Max selfemp age race_bl race_ot colgrad

9 .. > Estimate the model for question E.3c > >. generate ageself = ageselfemp;. probit insured selfemp ageself age race_bl race_ot colgrad,r; Iteration 0: log pseudolikelihood = Iteration : log pseudolikelihood = Iteration 2: log pseudolikelihood = Iteration 3: log pseudolikelihood = Probit regression Number of obs = 8802 Wald chi2(6) = Prob > chi2 = Log pseudolikelihood = Pseudo R2 = Robust insured Coef. Std. Err. z P> z [95% Conf. Interval] selfemp ageself age race_bl race_ot colgrad _cons log close; log: R:\users\jaherrig\My Documents\Classes\Economics 37\Problem Sets \Problemset6.log log type: text closed on: 3 Nov 2009, 2:06:47