Introduction to Intermediate Statistical Techniques

Abstract

This chapter introduces intermediate statistical techniques, which include pooled ordinary least squares (OLS) , fixed-effects, and random-effects regression . This chapter demonstrates how we can use these statistical techniques to analyze panel data. It shows how various tests can be conducted to determine the appropriate method that should be employed in correlational studies. The chapter also introduces how multivariate regression can be modified to infer causal effects by including difference-in-differences estimators.

7.7 Appendix

*Chapter 7 Stata syntax *Bivariate OLS Regression *use dataset from the previous chapter use "C:\Users\Marvin\Dropbox\Manuscripts\Book\Chapter 6\Stata files\Example 6.3.dta", clear *generate bivariate (one independent variable) OLS regression output regress netuit_fte stapr_fte if year ==2016 *create a new variable reflecting, say, the squared term /// (or quadratic) of another variable gen stapr_fte2 = stapr_fte*stapr_fte *include new variable in the regression model regress netuit_fte stapr_fte stapr_fte2 pc_income if year ==2016 *Multivariate Pooled OLS Regression reg netuit_fte stapr_fte stapr_fte2 pc_income *include the categorical variable (region_compact) in regression model reg netuit_fte stapr_fte stapr_fte2 pc_income i.region_compact *Multivariate Pooled OLS Regression with Interaction Terms reg netuit_fte stapr_fte i.region_compact##i.ugradmerit, allbaselevels *test to see if there is an interaction effects by quietly (qui) running the /// models and storing (est sto) the model results without (model1)and with and the *interaction terms (model2) qui reg netuit_fte stapr_fte i.region_compact est sto model1 qui reg netuit_fte stapr_fte i.region_compact##i.ugradmerit est sto model2 lrtest model1 model2 *Using the testparm command, the statistical significance of the interaction /// terms can also be checked. testparm i.region_compact#i.ugradmerit *if the interaction term is composed of one continuous (c) variable and one /// categorical (i) variable reg netuit_fte i.ugradmerit i.region_compact c.stapr_fte##i.tuitset testparm c.stapr_fte#i.tuitset *change working directory cd "C:\Users\Marvin\Dropbox\Manuscripts\Book\Chapter 7\Stata files" *use new dataset Use "Example 7.1.dta", clear *if two continuous variables are included reg netuit_fte i.region_compact c.stapr_fte##c.state_needFTE *using margins (with the vsquish option) margins, dydx(stapr_fte) at(state_needFTE=(0(3000)10000)) vsquish qui margins, at(stapr_fte=(0 10000) state_needFTE=(0(3000)10000)) vsquish *and marginsplot with different patterns marginsplot, noci x(stapr_fte) recast(line) xlabel(0(3000)10000) /// plot1opts(lpattern("...")) plot2opts(lpattern("-..-") color(black)) /// plot3opts(lpattern("---") color(black)) plot4opts(color(black)) *residual-versus-fitted plot rvfplot, mcolor(black) *comprehensive post-estimation estat imtest *POLS regression model using the robust option reg netuit_fte stapr_fte stapr_fte2 pc_income i.region_compact, robust *Levene test of homogeneity quietly: reg netuit_fte stapr_fte stapr_fte2 pc_income i.region_compact predict double eps, residual robvar eps, by(state) *To address this particular violation of the assumption of homoscedasticity , we /// use the cluster option, with state as the cluster variable, in our POLS *regression model. reg netuit_fte stapr_fte stapr_fte2 pc_income i.region_compact, cluster(state) *Fixed-Effects Regression * Unobserved Heterogeneity and Fixed-Effects Dummy Variable (FEDV) /// Regression Estimating FEDV Multivariate POLS Regression Models reg netuit_fte stapr_fte stapr_fte2 pc_income i.stateid, cluster(state) *we determine if the state fixed-effects as a whole are statistically /// significant, immediately after we run the above regression testparm i.stateid * alternative approach to producing the same results areg netuit_fte stapr_fte stapr_fte2 pc_income, cluster(stateid) absorb(stateid) *open another dataset use "Example 7.1.dta" areg eg statea tuition totfteiarep ftfac ptfac D, cluster(opeid5_new) *Unobserved Heterogeneity and Within-Group Estimator Fixed-Effects Regression /// using the xtreg command with the fe option xtreg eg statea tuition totfteiarep ftfac ptfac, fe cluster(opeid5_new) *Fixed-effects regression and difference-in-differences (DiD) *The DiD Estimator *Fixed-effects Regression-based DiD: An Example use "Example 7.1.dta", clear *We create the treatment variable (T). gen T=0 replace T=1 if state=="CO" *The post-treatment (P) is then created. gen P=0 replace P=1 if year>=2004 *Based on every state other than the treatment state (Colorado), we create the /// first control group. gen C1 = 0 replace C1=1 if state !="CO" *we create a second control group. gen C2 = 0 replace C2=1 if state !="CO" & region_compact==2 *we use the global command to create temporary variables reflecting the /// dependent variable net tuition revenue per FTE enrollment (y) global y "netuit_fte" *and the set of control variables state appropriations to higher education per /// FTE enrollment (stapr_fte) and state per capita income (pc_income). global controls "stapr_fte pc_income" *To take into account unobserved heterogeneity, we include the robust (rob) as /// an option in the syntax. reg $y i.T i.P T#P $controls i.year i.fips if year>=2000 & (C1==1 | T==1), rob *The within-group fixed-effects DiD regression model can also be employed. xtreg $y T##P $controls i.year if year>=2000 & (C1==1 | T==1) , fe rob *For comparison, we run the within-group fixed-effects model with the second /// control group (states in WICHE). xtreg $y T##P $controls i.year if year>=2000 & (C2==1 | T==1) , fe rob *DiD Placebo Tests gen placebo_2000 = 1 if year>=2000 recode placebo_2000 (.=0) xtreg $y T##placebo_2000 $controls if (year>1995 | year<2005) & (C2==1 | /// T==1), fe rob *Random-Effects Regression xtreg netuit_fte stapr_fte stapr_fte2 pc_income i.region_compact, /// re cluster(stateid) *Breusch and Pagan Lagrangian multiplier test for random effects xttest0 * Hausman test quietly: xtreg eg statea tuition totfteiarep ftfac ptfac, fe est sto fixed quietly: xtreg eg statea tuition totfteiarep ftfac ptfac, re est sto random hausman fixed random *log transform the variables and rerun the test xtreg lneg lnstatea lntuition lntotfteiarep lnftfac ptfac, fe est sto fixed xtreg lneg lnstatea lntuition lntotfteiarep lnftfac lnptfac, re est sto random hausman fixed random *install a Stata user-written Hausman routine by Kaiser (2015) ssc install rHausman *run rHausman qui: xtreg lneg lnstatea lntuition lntotfteiarep lnftfac ptfac, cluster(opeid5_new) fe est sto fixed qui: xtreg lneg lnstatea lntuition lntotfteiarep lnftfac ptfac, cluster(opeid5_new) re est sto random rhausman fixed random, reps(400) cluster *end