Stata Tutorial Panel

Welcome to the eleventh issue of e-Tutorial. In this issue we introduce simultaneous dynamic equations and exogeneity (Hausman) tests. I would like to remark that the theoretical background given in class is essential to proceed with the computational exercise below. Thus, I recommend you to consult Prof. Koenker's Lectures Notes as you go through the tutorial.

The first thing you need is to download the data sets in ASCII format by clicking in the respecitive names: system1.dat and system2.dat. Save them in your preferred location (I'll save mine as "C:/system1.dat" and "C:/system2.dat"). Then I suggest you to open the files in Notepad (or another text editor) and type the name of the variable "year" in the first row, first column, i.e. before the variable "w". Use to separate the names of variables. Save both files in text format in your favorite directory (I will save mine as "C:/system1a.txt" and "C:/system2a.txt", respectively).

Part 1:

For the first part of the problem set, go to STATA and type: infile year z w p Q using "C:/system1a.txt"

Next you need to declare your data as time series: gen quarter=q(1947q1)+_n-1 tsset quarter

To obtain a flavor of the data, use the command summarize, detail. To work with time series functions, use previous tutorials.

(Supply) Qt= a1 + a2pt-1+ a3zt + ut (Demand) ps= b1 + b2Qt + b3wt + vtQuestion 1:

Here you just need to run the system above, using OLS: *Supply equation: regress Q L.p z *Demand equation: regress p Q w

For the graph, first consider the equations in the steady state. Then use the last observations values z and w. Finally plot those two equations in a single cartesian graph. In Matlab, you can do that using the following routine (substitute "yournumber" by the values from the equation system you have run):

Q=0:.1:20; p_supply=yournumber+yournumber*Q; p_demand=yournumber+yournumber*Q; plot(Q, p_supply, Q, p_demand, ':') legend('supply','demand') xlabel('Quantity') ylabel('Price') title('Dynamic Relationship Between Demand and Supply')

If you don't have access to Matlab, use any other graphical device or simply draw it by hand. Find the equilibrium price and quantity. Compare with your graph. Check if there's convergence in the graph (see Prof. Koenker class notes).

Question 2:

Here I suggest you to construct a table as follows:

n peridos aheadQpzw02.41686.71862.5089.11272712.5089.11272722.5089.112727(...)(...)(...)(...)(...)202.5089.112727The values at n=0 come from the last observation in the data set. After that, z and w stay fixed. To find p(n=1), you need first to calculate the forecast of Q(n=1). Use the system of equations you have estimated in question 1. Do the same for p(n=i), i=2,..., 20. Find prices and quantities of equilibrium.

Question 3:

A) Explaining the effects of autocorrelation:

The problem asks you to explain why L.p cannot be considered exogenous. Please also explain the consequences of applying OLS estimators in the presence of autocorrelated disturbances.

B) Testing for the presence of autocorrelation: see e-Tutorial 10.

C) Correcting the Model: see e-Tutorial 10.

Question 4:

Here you can use the Hausman specification test you have saw in Lecture 11. Make sure to explain the null and the alternative hypothesis, and describe how the test is computed. The choice of instruments is crucial: you need to select instruments that are exogenous, orthogonal to errors, but correlated with included variables. In STATA, you can calculate the Hausman test as follows:

*2SLS with full set of instruments: regress Q L.p z ( your instruments z ) matrix bFull=get(_b) matrix varFull=get(VCE)

*2SLS with reduced set of instruments: regress Q L.p z ( your instruments only ) matrix bRedux=get(_b) matrix varRedux=get(VCE)

*Hausman Specification Test: matrix Omega=varRedux-varFull matrix Omegainv=syminv(Omega) matrix q=bFull-bRedux matrix Delta=q*Omegainv*q' matrix list Delta

After you obtain your test statistic Delta - a scalar, you should compare it with a Chi-squared (1). The degrees of freedom correspond to the number of dubious variables, i.e., the number of variables included as instruments in the first equation, but not in the second equation. Compare your results with the command hausman in STATA.

Part 2:

For the second part of the problem set, go to STATA and type: infile year z w p Q p1 using "C:/system2a.txt"

Next you need to declare your data as time series: gen quarter=q(1947q1)+_n-1 tsset quarter

To obtain a flavor of the data, use the command summarize, detail. To work with time series functions, use previous tutorials.

(Supply) Qt= a1 + a2pt+ a3pt-1 + a4zt + ut (Demand) pt= b1 + b2Qt + b3wt + vtThe variable p1 in the data set was created to substitute the lag of p, so you don't have to loose any observation.

Question 5:

Here you just need to compare the OLS estimators with the 2SLS estimators for the whole system:

*Supply *OLS: regress Q p p1 z *2SLS: regress Q p p1 z ( your instruments )

*Demand: *OLS: regress p Q w *2SLS: regress p Q w ( your instruments )

As an econometrician, draw your analysis on the results above. Check signs, significance, and economic sense of the results.

Question 6.

This question is a simple hyptohesis testing exercise. Nevertheless, you don't know if you should test using OLS or 2SLS. The results might be different according to the model. So, I suggest you to implement a Hausman-Wu test, similar to the one you did in question 4, but considering a simple OLS regression versus a 2SLS. You decide your instruments. To implement the test, follow the commands in question 4, adjusting for the OLS equation. Based on your diagnostic, choose the best model (OLS or 2SLS), and type in STATA the following commands:

*Is the long-run supply response to change in price equals 1? test p+p1=1 *Are the first and second periods price effects the same? test p=p1

Optional: Checking if the errors are correlated in a system of equations:

Sometimes is useful to know whether the errors of the supply equation are correlated with the errors of the demand equation. For example, using 2SLS methods:

*Supply: regress Q p p1 z (z w p1) *Obtain the residuals of the supply: predict sres, res *Demand: regress p Q w (z w p1) *Obtain the residuals of the demand: predict dres, res *Regress the estimated residuals of the supply against the estimated residuals of the demand: regress sres dres, noconst

If you find the residuals are correlated, maybe it is because some of the variables are not exogenous. Then you can proceed with the Hausman test to verify who is not exogenous among the instruments.

Welcome to the twelfth issue of e-Tutorial. Here I will talk about the basic fundamentals of panel data estimation techniques: from the organization of your panel data sets to the tests of fixed effects versus random effects. In the example below I will use the theoretical background of Prof. Koenker's Lecture Note 13 (2004) to reproduce the results of Greene (1997). I insert STATA estimation techniques (plus some comments) whenever necessary. I also provide a short introduction to panel data in R. Have fun!!!

Example: Greene (1997) provides a small panel data set with information on costs and output of 6 different firms, in 4 different periods of time (1955, 1960,1965, and 1970). Your job is try to estimate a cost function using basic panel data techniques.

Stacking your data: The data is shown below in a stacked form, i.e., the first "T" lines (here T=4) regard the firm 1, then the second "T" lines regard firm 2, and so on. The columns are self-explanatory. To facilitate your work, I included firm specific dummy variables for each firm, represented by columns D1-D6. The data is described below and available here in ASCII format for download.

YearFirmCostOutputD1D2D3D4D5D6195513.154214100000196014.271419100000196514.584588100000197015.8491025100000195523.859696010000196025.535811010000196528.12716400100001970210.9662506010000(...)(...)(...)(...)(...)(...)(...)(...)(...)(...)1955673.050117960000011960698.8461555100000119656138.8802721800000119706191.56030958000001Save the data in your preferred path (I will save mine as "C:/econ508/greene.txt") and open your preferred software.

In R: The Appendix A contains a panel data session in R with the main results derived in this tutorial.

In STATA: The first step is to download your data into the software:

infile Year Firm Cost Output D1 D2 D3 D4 D5 D6 using "C:/econ508/greene14.txt"

Drop the first line of observations containing missing values (due to the labels of variables in the text file).

The next step is to generate the log values of costs and outputs: gen lnc=log(Cost) gen lny=log(Output)

Finally you declare your data set as panel: iis Firm tis Year where iis refers to the cross-sectional unit identification, and tis to the time series identification.

Theoretical Background:

Consider a simplified version of the equation (1) in Koenker's Lecture 13:

(1) yit = xitb + ai + uit

a) Pooled OLS:

The most basic estimator of panel data sets are the Pooled OLS (POLS). Johnston & DiNardo (1997) recall that the POLS estimators ignore the panel structure of the data, treat observations as being serially uncorrelated for a given individual, with homoscedastic errors across individuals and time periods:

(2) bPOLS = (X'X)-1X'y

In STATA, you can obtain the POLS as follows: regress lnc lny

Source | SS df MS Number of obs = 24 ---------+------------------------------ F( 1, 22) = 728.51 Model | 33.617333 1 33.617333 Prob > F = 0.0000 Residual | 1.01520396 22 .046145635 R-squared = 0.9707 ---------+------------------------------ Adj R-squared = 0.9694 Total | 34.6325369 23 1.50576248 Root MSE = .21482

------------------------------------------------------------------------------ lnc | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- lny | .8879868 .0328996 26.991 0.000 .8197573 .9562164 _cons | -4.174783 .2768684 -15.079 0.000 -4.748973 -3.600593 ------------------------------------------------------------------------------

scalar R2OLS=_result(7)

b) Fixed Effects (Within-Groups) Estimators:

In Koenker's Lecture 13 we examined the effects of applying the matrix P and Q to the data: P = D(D'D)-1D': transform data into individual means Q = I-P : transform data into deviation from individual means.

The within-groups (or fixed effects) estimator is then given by:

(3) bW = (X'QX)-1X'Qy

Given that Q is idempotent, this is equivalent to regressing Qy on QX, i.e., using data in the form of deviations from individuals means. In STATA, you can obtain the within-groups estimators using the built-in functionxtreg, fe:

xtreg lnc lny, fe Fixed-effects (within) regression Number of obs = 24 Group variable (i) : Firm Number of groups = 6

R-sq: within = 0.8774 Obs per group: min = 4 between = 0.9833 avg = 4.0 overall = 0.9707 max = 4

F(1,17) = 121.66 corr(u_i, Xb) = 0.8495 Prob > F = 0.0000

------------------------------------------------------------------------------ lnc | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- lny | .6742789 .0611307 11.030 0.000 .5453044 .8032534 _cons | -2.399009 .508593 -4.717 0.000 -3.472046 -1.325972 ------------------------------------------------------------------------------ sigma_u | .36730483 sigma_e | .12463167 rho | .89675322 (fraction of variance due to u_i) ------------------------------------------------------------------------------ F test that all u_i=0: F(5,17) = 9.67 Prob > F = 0.0002

matrix bW=get(_b) matrix VW=get(VCE)

Note: The intercept above shown is an average of individual intercepts. If you are interested in obtaining firm-specific intercepts, go to Appendix B.

Between-Groups Estimators: Another useful estimator is provided when you use only the group means, i.e., transforming your data by applying the matrix P to equation (1) above:

(4) bB = [X'PX]-1X'Py

In STATA, you can obtain the between-groups estimators using the built-in function xtreg, be:

xtreg lnc lny, be Between regression (regression on group means) Number of obs = 24 Group variable (i) : Firm Number of groups = 6


F(1,4) = 236.23 sd(u_i + avg(e_i.))= .1838474 Prob > F = 0.0001

------------------------------------------------------------------------------ lnc | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- lny | .9110734 .0592772 15.370 0.000 .7464935 1.075653 _cons | -4.366618 .4982409 -8.764 0.001 -5.749957 -2.983279 ------------------------------------------------------------------------------

matrix bB=get(_b) matrix VB=get(VCE)

c) Random Effects:

Following Koenker's Lecture 13, consider ai's as random. So, the model will be estimated via GLS:

(5) bGLS = [X'Omega-1X]-1X'Omega-1y

where Omega = (sigmau2*InT + T*sigmaa2*P)

You can obtain GLS estimators in STATA by using the built-in functionxtreg, re:

xtreg lnc lny, re Random-effects GLS regression Number of obs = 24 Group variable (i) : Firm Number of groups = 6


Random effects u_i ~ Gaussian Wald chi2(1) = 268.10 corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

------------------------------------------------------------------------------ lnc | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- lny | .7963203 .0486336 16.374 0.000 .7010002 .8916404 _cons | -3.413094 .4131166 -8.262 0.000 -4.222788 -2.6034 ---------+-------------------------------------------------------------------- sigma_u | .17296414 sigma_e | .12463167 rho | .65823599 (fraction of variance due to u_i) ------------------------------------------------------------------------------

GLS as a Combination of Within- and Between-Groups Estimators: You can recover GLS estimator from the combination of between and within estimators, as shown in Koenker's Lecture 13:

(5.a) bGLS = Delta* bB + (1-Delta)* bW

where Delta = VW / (VW + VB)

In STATA, you can recover random effects GLS estimators as follows:

matrix V=VW+VB matrix Vinv=syminv(V) matrix D=VW*Vinv matrix P1=D*bB' matrix I2=I(2) matrix RD=I2-D matrix P2=RD*bW' matrix bRE=P1+P2 matrix list bRE bRE[2,1] y1 lny .79632032 _cons -3.413094

What should I use: Fixed Effects or Random Effects? A Hausman (1978) Test Approach

Hausman (1978) suggested a test to check whether the individual effects (ai) are correlated with the regressors (Xit):

- Under the Null Hypothesis: Orthogonality, i.e., no correlation between individual effects and explanatory variables. Both random effects and fixed effects estimators are consistent, but the random effects estimator is efficient, while fixed effects is not.

- Under the Alternative Hypothesis: Individual effects are correlated with the X's. In this case, random effects estimator is inconsistent, while fixed effects estimator is consistent and efficient.

Greene (1997) recalls that, under the null, the estimates should not differ systematically. Thus, the test will be based on a contrast vecor H:

(6) H = [bGLS - bW]'[V(bW)-V(bGLS)]-1[bGLS - bW] ~ Chi-squared (k)

where k is the number of regressors in X (excluding constant). In STATA, you can obtain that as follows:

xtreg lnc lny, fe hausman, save xtreg lnc lny, re hausman

---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- lny | .6742789 .7963203 -.1220414 .0370369 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from xtreg. B = fully efficient estimates obtained from xtreg.

Test: Ho: difference in coefficients not systematic

chi2( 1) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 10.86 Prob>chi2 = 0.0010

So, based on the test above, we can see that the tests statistic (10.86) is greater than the critical value of a Chi-squared (1df, 5%) = 3.84. Therefore, we reject the null hypothesis. Given such result, the preferred model is the fixed effects.

Appendix A: Quick Session in R

The first thing to do is to download the data, save in your preferred directory (I will save mine as C:/econ508/greene14.txt), and infile the data into R:

greene14 F = 0.0000 Residual | 978.354138 9615 .1017529 R-squared = 0.7173 ---------+------------------------------ Adj R-squared = 0.7171 Total | 3461.02631 9622 .359699263 Root MSE = .31899

------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- y1 | .7788283 .0099088 78.600 0.000 .759405 .7982516 y2 | -.2373115 .0095526 -24.843 0.000 -.2560365 -.2185865 exp | .0837589 .0021972 38.120 0.000 .0794519 .088066 expsq | -.0016344 .0000441 -37.102 0.000 -.0017207 -.001548 ier | 3.407355 .1403287 24.281 0.000 3.132281 3.682429 d60 | .0204149 .0098334 2.076 0.038 .0011395 .0396904 sex | -.0189182 .0093459 -2.024 0.043 -.0372382 -.0005982 _cons | .4432838 .0192268 23.056 0.000 .4055953 .4809723 ------------------------------------------------------------------------------

. * WITHIN ESTIMATORS (FIXED EFFECTS) . xtreg y y1 y2 exp expsq ier d60 sex, fe Fixed-effects (within) regression Number of obs = 9623 Group variable (i) : id Number of groups = 500


F(5,9118) = 3438.90 corr(u_i, Xb) = 0.2409 Prob > F = 0.0000

------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- y1 | .6573427 .0098663 66.625 0.000 .6380026 .6766828 y2 | -.3369351 .0094686 -35.585 0.000 -.3554957 -.3183745 exp | .101091 .0023692 42.669 0.000 .0964469 .1057351 expsq | -.0020004 .000046 -43.470 0.000 -.0020906 -.0019102 ier | .2556901 .2487721 1.028 0.304 -.231959 .7433392 d60 | (dropped) sex | (dropped) _cons | 1.037784 .0282838 36.692 0.000 .9823415 1.093227 ------------------------------------------------------------------------------ sigma_u | .20105547 sigma_e | .3014717 rho | .30784989 (fraction of variance due to u_i) ------------------------------------------------------------------------------ F test that all u_i=0: F(499,9118) = 3.30 Prob > F = 0.0000

. hausman, save

. * BETWEEN ESTIMATORS . xtreg y y1 y2 exp expsq ier d60 sex, be Between regression (regression on group means) Number of obs = 9623 Group variable (i) : id Number of groups = 500



------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- y1 | 1.49359 .0292156 51.123 0.000 1.436188 1.550993 y2 | -.519665 .0294238 -17.661 0.000 -.5774767 -.4618533 exp | .0045644 .0027111 1.684 0.093 -.0007622 .0098911 expsq | -.0001547 .0000683 -2.264 0.024 -.0002889 -.0000204 ier | .0081957 .0830353 0.099 0.921 -.1549517 .1713432 d60 | -.0079103 .0113073 -0.700 0.485 -.0301268 .0143062 sex | -.0039254 .0042214 -0.930 0.353 -.0122197 .0043688 _cons | .0810791 .0155373 5.218 0.000 .0505515 .1116067 ------------------------------------------------------------------------------

. * GLS ESTIMATORS (RANDOM EFFECTS): . xtreg y y1 y2 exp expsq ier d60 sex, re Random-effects GLS regression Number of obs = 9623 Group variable (i) : id Number of groups = 500



------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- y1 | .7788283 .0099088 78.600 0.000 .7594074 .7982492 y2 | -.2373115 .0095526 -24.843 0.000 -.2560342 -.2185889 exp | .0837589 .0021972 38.120 0.000 .0794524 .0880654 expsq | -.0016344 .0000441 -37.102 0.000 -.0017207 -.001548 ier | 3.407355 .1403287 24.281 0.000 3.132316 3.682395 d60 | .0204149 .0098334 2.076 0.038 .0011419 .039688 sex | -.0189182 .0093459 -2.024 0.043 -.0372359 -.0006005 _cons | .4432838 .0192268 23.056 0.000 .4056001 .4809675 ---------+-------------------------------------------------------------------- sigma_u | 0 sigma_e | .3014717 rho | 0 (fraction of variance due to u_i) ------------------------------------------------------------------------------

. * HAUSMAN TEST: FIXED VS. RANDOM EFFECTS . hausman ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- y1 | .6573427 .7788283 -.1214856 . y2 | -.3369351 -.2373115 -.0996236 . exp | .101091 .0837589 .017332 .0008861 expsq | -.0020004 -.0016344 -.000366 .0000133 ier | .2556901 3.407355 -3.151665 .2054152 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from xtreg. B = fully efficient estimates obtained from xtreg.

Test: Ho: difference in coefficients not systematic chi2( 5) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 2114.69 Prob>chi2 = 0.0000

. * INSTRUMENTAL VARIABLES (1ST ROUND) . regress y y1 y2 exp expsq ier d60 sex (Pexp Qexp Pexpsq Qexpsq Qy1 Qy2 Qier Pd60 Psex) Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 9623 ---------+------------------------------ F( 7, 9615) = 2063.13 Model | 2274.04125 7 324.863035 Prob > F = 0.0000 Residual | 1186.98506 9615 .123451384 R-squared = 0.6570 ---------+------------------------------ Adj R-squared = 0.6568 Total | 3461.02631 9622 .359699263 Root MSE = .35136

------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- y1 | .6607854 .0114819 57.550 0.000 .6382785 .6832923 y2 | -.3351626 .0110306 -30.385 0.000 -.356785 -.3135403 exp | .1023073 .0027215 37.593 0.000 .0969726 .107642 expsq | -.0020375 .0000529 -38.501 0.000 -.0021412 -.0019338 ier | .3347733 .2895833 1.156 0.248 -.232871 .9024175 d60 | .0324825 .010838 2.997 0.003 .0112378 .0537272 sex | -.0198867 .010298 -1.931 0.053 -.0400729 .0002995 _cons | .9895473 .0332799 29.734 0.000 .9243117 1.054783 ------------------------------------------------------------------------------

. predict r,res . PQ r . gen Prsq=Pr^2 . quietly by id: gen mark=_n . *What does mark do? (see next regression) . quietly by id: gen T=_N . gen iT=1/T . regress Prsq iT if mark==1 Source | SS df MS Number of obs = 500 ---------+------------------------------ F( 1, 498) = 4.15 Model | .011654194 1 .011654194 Prob > F = 0.0420 Residual | 1.39687301 498 .002804966 R-squared = 0.0083 ---------+------------------------------ Adj R-squared = 0.0063 Total | 1.4085272 499 .0028227 Root MSE = .05296

------------------------------------------------------------------------------ Prsq | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- iT | .1525195 .0748252 2.038 0.042 .0055075 .2995315 _cons | .0291845 .0053924 5.412 0.000 .0185899 .0397791 ------------------------------------------------------------------------------

. matrix b=get(_b) . gen theta=sqrt(_b[iT]/(_b[iT]+_b[_cons]*T)) . *Now you need to transform the variables included in your model . replace y=y-(1-theta)*Py (9623 real changes made) . replace y1=y1-(1-theta)*Py1 (9623 real changes made) . replace y2=y2-(1-theta)*Py2 (9623 real changes made) . replace exp=exp-(1-theta)*Pexp (9623 real changes made) . replace expsq=expsq-(1-theta)*Pexpsq (9623 real changes made) . replace ier=ier-(1-theta)*Pier (9623 real changes made) . replace d60=d60-(1-theta)*Pd60 (7874 real changes made) . replace sex=sex-(1-theta)*Psex (1358 real changes made) . * INSTRUMENTAL VARIABLES (AFTER THETA CORRECTION) . regress y y1 y2 exp expsq ier d60 sex theta (Qy1 Qy2 Qier Pexp Qexp Pexpsq Qexpsq Pd60 Psex theta), noconstant Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 9623 ---------+------------------------------ F( 8, 9615) = . Model | 18502.9839 8 2312.87299 Prob > F = . Residual | 907.099508 9615 .094342123 R-squared = . ---------+------------------------------ Adj R-squared = . Total | 19410.0834 9623 2.01705117 Root MSE = .30715

------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- y1 | .6578081 .01005 65.454 0.000 .638108 .6775082 y2 | -.3367606 .0096466 -34.910 0.000 -.3556698 -.3178513 exp | .1012754 .0024055 42.101 0.000 .09656 .1059907 expsq | -.0020057 .0000468 -42.898 0.000 -.0020974 -.0019141 ier | .2589409 .2534411 1.022 0.307 -.237857 .7557388 d60 | .033342 .023551 1.416 0.157 -.0128229 .079507 sex | -.01201 .0203523 -0.590 0.555 -.0519049 .0278848 theta | 1.009543 .0361539 27.924 0.000 .9386737 1.080412 ------------------------------------------------------------------------------

. *Why do we have theta as a variable and no intercept here? . matrix list b b[1,2] iT _cons y1 .15251948 .02918446

. summarize theta Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- theta | 9623 .4463058 .0806179 .3194048 .7148795

. clear end of do-file

The Wages Equation

. do "C:\wages01.do" . use "C:\phuzics01.dta", clear . iis id . tis yr . sort id . quietly by id: gen s1=s[_n-1] . quietly by id: gen Y1=Y[_n-1] . *Shall I drop the missing variables here? . drop if s1==. (500 observations deleted) . replace s=log(s/s1) (10123 real changes made) . replace Y=log(Y/Y1) (10123 real changes made) . PQ s . PQ Y . PQ sex . PQ ru . * POOLED OLS . regress s Y ru sex Source | SS df MS Number of obs = 10123 ---------+------------------------------ F( 3, 10119) = 3335.56 Model | 7.67338445 3 2.55779482 Prob > F = 0.0000 Residual | 7.75952035 10119 .000766827 R-squared = 0.4972 ---------+------------------------------ Adj R-squared = 0.4971 Total | 15.4329048 10122 .001524689 Root MSE = .02769

------------------------------------------------------------------------------ s | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- Y | .1795768 .0019558 91.816 0.000 .175743 .1834107 ru | .0142779 .0006994 20.414 0.000 .012907 .0156489 sex | -.0082762 .0007914 -10.458 0.000 -.0098275 -.0067249 _cons | -.0026954 .0004037 -6.676 0.000 -.0034869 -.001904 ------------------------------------------------------------------------------

. * WITHIN ESTIMATORS (FIXED EFFECTS) . xtreg s Y ru sex, fe Fixed-effects (within) regression Number of obs = 10123 Group variable (i) : id Number of groups = 500


F(2,9621) = 4120.97 corr(u_i, Xb) = 0.0939 Prob > F = 0.0000

------------------------------------------------------------------------------ s | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- Y | .1793187 .0021088 85.035 0.000 .1751851 .1834523 ru | .0064123 .001264 5.073 0.000 .0039346 .0088899 sex | (dropped) _cons | -.0022472 .0004157 -5.406 0.000 -.0030619 -.0014324 ------------------------------------------------------------------------------ sigma_u | .01011922 sigma_e | .02691058 rho | .12388261 (fraction of variance due to u_i) ------------------------------------------------------------------------------ F test that all u_i=0: F(499,9621) = 2.19 Prob > F = 0.0000

. hausman, save

. * BETWEEN ESTIMATORS . xtreg s Y ru sex, be Between regression (regression on group means) Number of obs = 10123 Group variable (i) : id Number of groups = 500




. * GLS ESTIMATORS (RANDOM EFFECTS): . xtreg s Y ru sex, re Random-effects GLS regression Number of obs = 10123 Group variable (i) : id Number of groups = 500



------------------------------------------------------------------------------ s | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- Y | .1785767 .0019929 89.608 0.000 .1746707 .1824826 ru | .0123244 .0008565 14.390 0.000 .0106458 .014003 sex | -.0081517 .001123 -7.259 0.000 -.0103527 -.0059508 _cons | -.0022302 .0005162 -4.320 0.000 -.0032419 -.0012184 ---------+-------------------------------------------------------------------- sigma_u | .00597206 sigma_e | .02691058 rho | .04693787 (fraction of variance due to u_i) ------------------------------------------------------------------------------

. * HAUSMAN TEST: FIXED VS. RANDOM EFFECTS . hausman ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | Prior Current Difference S.E. ---------+------------------------------------------------------------- Y | .1793187 .1785767 .0007421 .0006894 ru | .0064123 .0123244 -.0059121 .0009296 ---------+------------------------------------------------------------- b = less efficient estimates obtained previously from xtreg. B = fully efficient estimates obtained from xtreg.

Test: Ho: difference in coefficients not systematic

chi2( 2) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 56.15 Prob>chi2 = 0.0000

. * INSTRUMENTAL VARIABLES (1ST ROUND) . regress s Y ru sex (PY QY Qru Psex) Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 10123 ---------+------------------------------ F( 3, 10119) = 3168.81 Model | 7.58349777 3 2.52783259 Prob > F = 0.0000 Residual | 7.84940703 10119 .00077571 R-squared = 0.4914 ---------+------------------------------ Adj R-squared = 0.4912 Total | 15.4329048 10122 .001524689 Root MSE = .02785


. predict r,res . PQ r . gen Prsq=Pr^2 . quietly by id: gen mark=_n . *What does mark do? (see next regression) . quietly by id: gen T=_N . gen iT=1/T . regress Prsq iT if mark==1 Source | SS df MS Number of obs = 500 ---------+------------------------------ F( 1, 498) = 4.46 Model | 7.4757e-08 1 7.4757e-08 Prob > F = 0.0352 Residual | 8.3462e-06 498 1.6759e-08 R-squared = 0.0089 ---------+------------------------------ Adj R-squared = 0.0069 Total | 8.4209e-06 499 1.6876e-08 Root MSE = .00013

------------------------------------------------------------------------------ Prsq | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- iT | .0004526 .0002143 2.112 0.035 .0000316 .0008736 _cons | .0000653 .0000141 4.630 0.000 .0000376 .000093 ------------------------------------------------------------------------------

. matrix b=get(_b) . gen theta=sqrt(_b[iT]/(_b[iT]+_b[_cons]*T)) . *Now you need to transform the variables included in your model . replace s=s-(1-theta)*Ps (10078 real changes made) . replace Y=Y-(1-theta)*PY (10123 real changes made) . replace sex=sex-(1-theta)*Psex (1426 real changes made) . replace ru=ru-(1-theta)*Pru (6318 real changes made) . * INSTRUMENTAL VARIABLES (AFTER THETA CORRECTION) . regress s Y ru sex theta (PY QY Qru Psex theta), noconstant Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 10123 ---------+------------------------------ F( 4, 10119) = . Model | 8.00935131 4 2.00233783 Prob > F = . Residual | 7.19637615 10119 .000711175 R-squared = . ---------+------------------------------ Adj R-squared = . Total | 15.2057275 10123 .001502097 Root MSE = .02667

------------------------------------------------------------------------------ s | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- Y | .1809964 .0020571 87.988 0.000 .1769642 .1850287 ru | .0065199 .0012524 5.206 0.000 .004065 .0089748 sex | -.0080601 .0015735 -5.122 0.000 -.0111445 -.0049757 theta | -.0014172 .00067 -2.115 0.034 -.0027306 -.0001039 ------------------------------------------------------------------------------

. *Why do we have theta as a variable and no intercept here? . matrix list b b[1,2] iT _cons y1 .00045259 .0000653 . summarize theta Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- theta | 10123 .4892002 .0797415 .3584998 .7321285

. clear end of do-file

Appendix A: Hausman-Taylor Instrumental Variables in R Here you can reproduce the results above using R. You should start running the functions PQ.R, tsls.R, and htiv.R available at the Econ 508 webpage (Routines, panel.R).:

"htiv"

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