Problem Set 10: Panel Data

Transcription

1 Problem Set 10: Panel Data 1. Read in the data set, e11panel1.dta from the course website. This contains data on a sample or 1252 men and women who were asked about their hourly wage in two years, 2005 & Some of the variables in the data set change over time (experience, age, job tenure) and other variables do not (gender, years of education). The variable pid is the cross-section (individual) identifier. The variable year is the time series identifier (so each pid is observed twice) Sort the data by pid and year and list the values for hourly pay years of education, experience and gender. What do you see? Estimate the returns to experience by OLS based on the following equation Ln(hourpay) = b 0 + b 1 exper i + +b 2 exper i 2 (1) Interpret the effect of years of experience on hourly pay Now see if these results are affected by omitted variable bias by comparing the OLS estimates with fixed effects and random effects regressions. What do you find? Do the Hausman test to help you decide whether to prefer the fixed or random effects estimates. Now repeat the exercise adding years of education to the model ie estimate Ln(hourpay) = b 0 + b 1 exper i + +b 2 exper i 2 + b 3 yearsed (2) What do you see? Good idea to sort data by individual and year to get a feel of how the data look. sort pid year. l pid year hourpay yearsed sex pid year hourpay yearsed sex male male male male male male female female female female female Can see each individual is observed twice and there are two measures of hourly pay years of education and experience for each person

2 To estimate the model you will need to create the log of hourly pay and generate the square of experience. g lhw=log(hourpay). g exper2=experience^2 then do OLS. reg lhw experience exper2 Source SS df MS Number of obs = F( 2, 2501) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = experience exper _cons semi-log model and the effect of years of expereince is non-linear so dlnw/dexperience =.041-2(.001)Experience which means the effect varies depending on the level of experience, but reaches a maximum at 0=.041-2(.001)Experience ie Experience max =.049/.002 = 20.5 years If there are unobserved characteristics of individuals however then we may suspect that these estimates will be biased (omitted variable bias) So try fixed effects. xtreg lhw experience exper2, fe i(pid) Fixed-effects (within) regression Number of obs = 2504 R-sq: within = Obs per group: min = 2 between = avg = 2.0 overall = max = 2 F(2,1250) = corr(u_i, Xb) = Prob > F = experience exper _cons sigma_u rho (fraction of variance due to u_i) F test that all u_i=0: F(1251, 1250) = Prob > F = and random effects

3 . xtreg lhw experience exper2, re i(pid) Random-effects GLS regression Number of obs = 2504 R-sq: within = Obs per group: min = 2 between = avg = 2.0 overall = max = 2 Random effects u_i ~ Gaussian Wald chi2(2) = corr(u_i, X) = 0 (assumed) Prob > chi2 = lhw Coef. Std. Err. z P> z [95% Conf. Interval] experience exper _cons sigma_u rho (fraction of variance due to u_i) The random effects estimates are quite close to the original OLS estimates, but the fixed effects are somewhat different. Sometimes this can be because the assumptions needed for random effects estimation (that the unobservables are not correlated with the observed X variables) are not met. So try the Hausman test xtreg lhw experience exper2, fe i(pid) est store fixed xtreg lhw experience exper2, re i(pid) hausman fixed ---- Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) fixed. Difference S.E. experience exper b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg Test: Ho: difference in coefficients not systematic chi2(2) = (b-b)'[(v_b-v_b)^(-1)](b-b) = Prob>chi2 = The Hausman test looks to see whether the estimates from the fixed and random effects models are significantly different from each other. If they are (the Chi-squared value will be above the critical value) then conclude fixed effects is to be preferred In the example above with 2 degrees of freedom (since there are 2 explanatory variables), the test value is above the critical Chi-squared value for 2 degrees of freedom at the 95% level (5.99). So go with the fixed effects estimates. The true returns to experience are now dlnw/dexperience = (.00001)Experience

4 The returns to experience are much higher with fixed effects estimation than with OLS. In the fixed effects case the coefficient on the quadratic is positive, so pay rises continuously with experience rather than reaching a maximum. Now adding years of education to the model OLS reg lhw experience exper2 yearsed Source SS df MS Number of obs = F( 3, 2500) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = experience exper yearsed _cons semi-log model so coefficients suggest that an extra year of education raises pay by 6.8% The effect of years of experience is now dlnw/dexperience =.048-2(.001)Experience, which means the effect varies depending on the level of experience, but reaches a maximum at 0=.048-2(.001)Experience ie Experience max =.048/.002 = 24 years Now fixed effects. xtreg lhw experience exper2 yearsed, fe i(pid) Fixed-effects (within) regression Number of obs = 2504 R-sq: within = Obs per group: min = 2 between = avg = 2.0 overall = max = 2 F(2,1250) = corr(u_i, Xb) = Prob > F = experience exper yearsed (dropped) _cons sigma_u rho (fraction of variance due to u_i) F test that all u_i=0: F(1251, 1250) = 9.98 Prob > F = Notice that Stata gives a message that the variable yearsed has been dropped from the estimation. This is no accident since the fixed effect methiod involves first differencing (or within-group deviation) which means that any variable which is constant (fixed) will be dropped. This applies equally to unobservables and observed variables.

5 You can see that the variable yearsed does not change over time in the list command above The remaining coefficients are however net of the influence of both the unobserved and observed fixed variables. Random effects, because it is a different estimation method and treats the unobserved variables as part of the residual, does allow an explicit estimation of any variable that remains constant. xtreg lhw experience exper2 yearsed, re i(pid) Random-effects GLS regression Number of obs = 2504 R-sq: within = Obs per group: min = 2 between = avg = 2.0 overall = max = 2 Random effects u_i ~ Gaussian Wald chi2(3) = corr(u_i, X) = 0 (assumed) Prob > chi2 = lhw Coef. Std. Err. z P> z [95% Conf. Interval] experience exper yearsed _cons sigma_u rho (fraction of variance due to u_i) Again the random effects estimates are not very different from the OLS ones. The Hausman test continues to suggest that the two estimation strategies produce significantly different estimates and so the fixed effects is still preferred. xtreg lhw experience exper2 yearsed, fe i(pid) est store fixed xtreg lhw experience exper2 yearsed, re i(pid) hausman fixed ---- Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) fixed. Difference S.E. experience exper b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg Test: Ho: difference in coefficients not systematic chi2(2) = (b-b)'[(v_b-v_b)^(-1)](b-b) = Prob>chi2 = C it = b 0 +X it B + γdummy2 + u it i = 1, t = 1, 2

6 Can not include age as an explanatory variable in the original model. Each person in the panel data set is exactly one year older in the second wave of the panel. This means that age i = 1 for all i. But the equation we would estimate is of the form C i = γ + β Age i +, where γ is the coefficient the year dummy for year 2 in the original model. If we difference the year 2 dummy variable then it will always have the value 1 for everyone in the data set (remember we lose the 1 st year if we 1 st difference this is what the dots show below) Eg Individual Time Dummy2 Dummy So a differenced year dummy is like having a constant in the differenced regression When we have an intercept in the model we cannot include any other explanatory variable that is constant across i. This is pure multicolinearity. Intuitively, since age changes by the same amount for everyone, we cannot distinguish the effect of age from the aggregate time effect. However it is possible to include a variable like income, since this is likely to vary over time by differing amounts across individuals So a model like C i = γ + β Income i +, Will not suffer from perfect multicolinearity 3. a) Economists tend to think that variables like ability or motivation are difficult to measure and therefore more likely to be unobserveable. Yet these variables are also likely to be correlated with education. This means that if ability were not included in the model then part of its effect would be picked up by the education variable (see lecture notes on omitted variable bias) and so tthe OLS estimate of education would be biased. One way to get round this is to net out the influence of unobserveables using either fixed or random effects estimation b) Macro economic shocks such as recessions may affect wages Also if there are any trends in education or wages (eg wages tend to grow in line with aggregate productivity, participation in teriary education is on a rising trend). These sort of

7 macroeconomic events can be modelled by including year dummy variables in a panel data model Note that when T >2 there is no longer the same issue of multicolinearity as in question 2 Eg FOR t=3 Individual Time Dummy2 Dummy2 Dummy3 Dummy So it is OK to difference out the individual effects and retain the year dummies