*! Version 11

help for gmmcovearn(update August 2010)

Title

GMM estimator for covariance structure of earnings

Syntax

gmmcovearn
earningsvar [if] [in], required options
[other options]

required_opts

modeln specifies the type of model to be estimated. The default is modeln(1)

AR, no heterogeneity: modeln(1)
ARMA, no heterogeneity: modeln(2)
AR, random growth: modeln(3)
ARMA, random growth: modeln(4)
AR, random walk: modeln(5)
ARMA, random walk: modeln(6)
AR, combined random growth and random walk: modeln(7)
ARMA, combined random growth and random walk: modeln(8)

All models include time factor loadings on the permanent and transitory components.

yearn specifies the number of years used for the analysis

other_opts

expvar specifies the name of the experience variable to be used for models that allow for heterogeneity in the life-cycle earnings profile ( modeln(3) to modeln(8)).

firstyr specifies the numeric year indicator attached to the first wave of earnings. The default value is 1. earningsvar and expvar are assumed to be indexed in consecutive integers from firstyr to (firstyr + yearn -1).

cohortn specifies the number of cohorts used for the analysis. The default is 1.

cohortvar specifies the name of the cohort indicator variable. The default is cohort

firstcohort specifies the numeric indicator of the first cohort. The default value is 1. Cohorts are assumed to be coded in consecutive integers from firstcohort to (firstcohort + cohortn -1). [e.g. In a model with four cohorts cohortvar could contain values such as 1 to 4 or 1994 to 1997. However values such as 1960, 1970, 1980 and 1990 would have to be recoded before being used.]

stvalue specifies the starting values for the estimation. For T years of data and C cohorts, values are entered in the following order, separated by commas: sigalpha, rho, sigv1, sige, l2-lT, p2-pT, q2-qC, s2-sC, sigbeta, covalphabeta, sigw, theta. The user should specify starting values only for the parameters estimated in the chosen model.The default values for the l’s, p’s, q’s and s’s are 1; for sigalpha and rho they are 0.5; for sigv1 and sige they are 0.1; for sigbeta, covalphabeta and sigw they are 0; and for theta it is -0.5.

newdataname allows the user to create a dataset called newdataname containing the sample moments used in the estimation and the number of observations used in calculating each of these moments. If a heterogeneous model is specified the dataset will also contain the average of expvar and the average of squared expvar.

graph 1 if the user wants a graphical display of the estimated variance decomposition, 0 otherwise. The default value is 0.

Description

gmmcovearn provides GMM estimates of the parameters of the covariance structure of earnings using the earnings variable specified in earningsvar.

The general earnings dynamics model (modeln(8)) for individual i, belonging to cohort c, with x years of experience, at time t, y_icxt, is specified as

y_icxt=q_c*p_t*(alpha_ix)+s_c*l_t*v_it

alpha_ix=alpha_i(x-1)+b_i+w_ix

And

v_it=rho*v_i(t-1)+theta*e_i(t-1)+e_it

The parameters estimated are sigalpha, rho, sigv1, sige, l2-lT,
p2-pT, q2-qC, s2-sC, sigbeta, covalphabeta, sigw, theta.

The model is described in more detail in Doris et al (2010a).

The command makes use of an additional program nlgmmcovearn that must be downloaded along with gmmcovearn.
A detailed analysis of this approach to estimating the covariance structure of earnings can be found in Doris et al (2010b).

The program requires that the data be in wide format. It must contain an earnings (or earnings residual) variable. If using a model with heterogeneous profiles, it must also include a labour market experience variable. If using a model with cohort effects, it must also include a cohort indicator variable.

The Identity matrix is used as the GMM weighting matrix and standard errors are adjusted for unbalanced data using the approach reported in Haider(2001)

Examples

Example 1: NLS Earnings Data

In this example we make use of the NLS panel data set used in Wooldridge (2002) and available for download from within Stata. The dataset provides an unbalanced panel of data on earnings, schooling, and demographic information for 530 individuals from the National Longitudinal Survey for the years 1981-1987.

. use http://www.stata.com/data/jwooldridge/eacsap/nls81_87.dta

Since the data is in long format we must first reshape it prior to using gmmcovearn.

. keep id year exper lwage

. reshape wide lwage exper, i(id) j(year)

We estimate a earnings covariance structure model on log wages without heterogeneous profiles and an AR(1) model for the transitory component as follows:

. gmmcovearn lwage, yearn(7) modeln(1) cohortn(1) firstyr(81)
(obs = 28)

Iteration 0: residual SS = .1643298
Iteration 1: residual SS = .05161
Iteration 2: residual SS = .0017779
Iteration 3: residual SS = .0016158
Iteration 4: residual SS = .001615
Iteration 5: residual SS = .001615
Iteration 6: residual SS = .001615
Iteration 7: residual SS = .001615
Iteration 8: residual SS = .001615
Iteration 9: residual SS = .001615

Source | SS df MS
-------------+------------------------------ Number of obs = 28
Model | .796840451 16 .049802528 R-squared = 0.9980
Residual | .001614961 12 .00013458 Adj R-squared = 0.9953
-------------+------------------------------ Root MSE = .0116009
Total | .798455413 28 .028516265 Res. dev. = -193.8376

------------------------------------------------------------------------------
moment | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
/sigalpha | .0683058 .0088071 7.76 0.000 .0491168 .0874948
/rho | .3130349 .0608686 5.14 0.000 .1804137 .4456561
/sigv1 | .201089 .0144893 13.88 0.000 .1695194 .2326586
/sige | .0588356 .0375667 1.57 0.143 -.0230152 .1406864
/l2 | 1.209775 .3380594 3.58 0.004 .4732069 1.946343
/l3 | 1.497133 .496198 3.02 0.011 .4160105 2.578256
/l4 | 1.142064 .3835471 2.98 0.012 .3063868 1.977742
/l5 | 1.317238 .4160183 3.17 0.008 .4108118 2.223664
/l6 | 1.438042 .464473 3.10 0.009 .4260424 2.450042
/l7 | 1.706241 .5657103 3.02 0.011 .4736643 2.938818
/p2 | .9159306 .0803285 11.40 0.000 .7409098 1.090951
/p3 | 1.112308 .0998032 11.15 0.000 .894856 1.329761
/p4 | 1.307378 .1214745 10.76 0.000 1.042708 1.572048
/p5 | 1.449588 .1411007 10.27 0.000 1.142156 1.75702
/p6 | 1.466273 .1388268 10.56 0.000 1.163796 1.768751
/p7 | 1.470464 .1267276 11.60 0.000 1.194348 1.74658
------------------------------------------------------------------------------
coefficients and corrected standard errors below
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
sigalpha | .0683058 .024722 2.76 0.006 .0198516 .1167599
rho | .3130349 .077011 4.06 0.000 .1620961 .4639736
sigv1 | .201089 .0402344 5.00 0.000 .122231 .279947
sige | .0588356 .0456725 1.29 0.198 -.0306809 .1483521
l2 | 1.209775 .4187484 2.89 0.004 .3890433 2.030507
l3 | 1.497133 .6406559 2.34 0.019 .2414706 2.752796
l4 | 1.142064 .4497494 2.54 0.011 .2605716 2.023557
l5 | 1.317238 .5380952 2.45 0.014 .2625906 2.371885
l6 | 1.438042 .5672408 2.54 0.011 .3262706 2.549814
l7 | 1.706241 .6850775 2.49 0.013 .3635139 3.048968
p2 | .9159306 .1452523 6.31 0.000 .6312413 1.20062
p3 | 1.112308 .2170894 5.12 0.000 .6868211 1.537796
p4 | 1.307378 .237724 5.50 0.000 .8414475 1.773308
p5 | 1.449588 .2657075 5.46 0.000 .9288111 1.970366
p6 | 1.466273 .270121 5.43 0.000 .9368458 1.995701
p7 | 1.470464 .3122724 4.71 0.000 .8584214 2.082507
-------------+----------------------------------------------------------------

Hypothesis tests can be carried out using test after running the gmmcovearn command. For example a test that the permanent factor loadings, the p_t’s, are constant over time can be carried out using a Wald test as follows:

. test _b[p2]=_b[p3]=_b[p4]=_b[p5]=_b[p6]=_b[p7]=1

( 1) p2 - p3 = 0
( 2) p2 - p4 = 0
( 3) p2 - p5 = 0
( 4) p2 - p6 = 0
( 5) p2 - p7 = 0
( 6) p2 = 1

chi2( 6) = 12.14
Prob > chi2 = 0.0588

In this example we reject constant permanent factor loadings at the 10% significance level.

Example 2: German Earnings Data

To illustrate the use of gmmcovearn for a more complicated model including cohort effects we use data taken from the eight waves of the European Community Household Panel for Germany. This data set has earnings for 8 years denoted yi1994 to yi2001, a potential experience variable denoted potexp1994 to potexp2001 and 4 cohorts labeled 1 to 4. To estimate a random growth model of the covariance structure with these data we type:

gmmcovearn yi, yearn(8) modeln(3) cohortn(4) expvar(potexp) firstyr(1994)

Iteration 0: residual SS = .4483834
Iteration 1: residual SS = .0186083
Iteration 2: residual SS = .0086681
Iteration 3: residual SS = .0070441
Iteration 4: residual SS = .0070062
Iteration 5: residual SS = .0070061
Iteration 6: residual SS = .0070061
Iteration 7: residual SS = .0070061
Iteration 8: residual SS = .0070061

Source | SS df MS
-------------+------------------------------ Number of obs = 144
Model | 2.16612446 26 .083312479 R-squared = 0.9968
Residual | .007006143 118 .000059374 Adj R-squared = 0.9961
-------------+------------------------------ Root MSE = .0077055
Total | 2.17313061 144 .015091185 Res. dev. = -1021.378

------------------------------------------------------------------------------
moment | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
/sigalpha | .4386704 .0605217 7.25 0.000 .318821 .5585198
/rho | .3417541 .0361788 9.45 0.000 .2701102 .4133979
/sigv1 | .0742456 .0062163 11.94 0.000 .0619355 .0865556
/sige | .0301694 .0112735 2.68 0.009 .0078448 .0524939
/l2 | 1.498371 .2427683 6.17 0.000 1.017624 1.979118
/l3 | 1.312379 .2569202 5.11 0.000 .8036072 1.821151
/l4 | 1.162531 .2301328 5.05 0.000 .7068056 1.618257
/l5 | 1.193413 .2324563 5.13 0.000 .7330861 1.65374
/l6 | 1.297056 .2507765 5.17 0.000 .8004506 1.793662
/l7 | 1.279366 .2507241 5.10 0.000 .7828636 1.775868
/l8 | 1.428739 .2791012 5.12 0.000 .8760424 1.981435
/p2 | .9842998 .0222086 44.32 0.000 .9403208 1.028279
/p3 | 1.07534 .0254406 42.27 0.000 1.024961 1.125719
/p4 | 1.08418 .0279634 38.77 0.000 1.028804 1.139555
/p5 | 1.148706 .0318299 36.09 0.000 1.085674 1.211738
/p6 | 1.168077 .033754 34.61 0.000 1.101235 1.234919
/p7 | 1.205651 .0364914 33.04 0.000 1.133388 1.277914
/p8 | 1.219382 .0372949 32.70 0.000 1.145528 1.293236
/q2 | .989772 .0419597 23.59 0.000 .9066803 1.072864
/q3 | .7322496 .0536754 13.64 0.000 .6259578 .8385415
/q4 | .4866869 .0387748 12.55 0.000 .4099022 .5634715
/s2 | .6293472 .0447309 14.07 0.000 .5407679 .7179265
/s3 | .8336288 .0377501 22.08 0.000 .7588734 .9083842
/s4 | 1.214287 .0390211 31.12 0.000 1.137014 1.291559
/sigbeta | .0003872 .0000489 7.92 0.000 .0002904 .0004841
/covalphab~a | -.012158 .001729 -7.03 0.000 -.0155819 -.0087341
------------------------------------------------------------------------------
coefficients and corrected standard errors below
------------------------------------------------------------------------------
| Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
sigalpha | .4386704 .1706614 2.57 0.010 .1041801 .7731606
rho | .3417541 .0361377 9.46 0.000 .2709254 .4125827
sigv1 | .0742456 .0118533 6.26 0.000 .0510135 .0974776
sige | .0301694 .0104324 2.89 0.004 .0097223 .0506164
l2 | 1.498371 .2208264 6.79 0.000 1.065559 1.931183
l3 | 1.312379 .2312379 5.68 0.000 .8591613 1.765597
l4 | 1.162531 .1944341 5.98 0.000 .7814474 1.543615
l5 | 1.193413 .1942618 6.14 0.000 .8126667 1.574159
l6 | 1.297056 .2167269 5.98 0.000 .8722795 1.721833
l7 | 1.279366 .2119609 6.04 0.000 .8639299 1.694801
l8 | 1.428739 .2584156 5.53 0.000 .9222535 1.935224
p2 | .9842998 .0313717 31.38 0.000 .9228123 1.045787
p3 | 1.07534 .0433522 24.80 0.000 .9903711 1.160309
p4 | 1.08418 .050535 21.45 0.000 .9851328 1.183226
p5 | 1.148706 .0694794 16.53 0.000 1.012529 1.284883
p6 | 1.168077 .0789378 14.80 0.000 1.013362 1.322793
p7 | 1.205651 .0887034 13.59 0.000 1.031796 1.379507
p8 | 1.219382 .0922591 13.22 0.000 1.038558 1.400207
q2 | .989772 .111584 8.87 0.000 .7710713 1.208473
q3 | .7322496 .1311763 5.58 0.000 .4751487 .9893505
q4 | .4866869 .1032265 4.71 0.000 .2843667 .689007
s2 | .6293472 .0628975 10.01 0.000 .5060703 .7526241
s3 | .8336288 .0684399 12.18 0.000 .6994891 .9677684
s4 | 1.214287 .0917179 13.24 0.000 1.034523 1.394051
sigbeta | .0003872 .0001758 2.20 0.028 .0000426 .0007319
covalphabeta | -.012158 .005614 -2.17 0.030 -.0231613 -.0011547
------------------------------------------------------------------------------

Saved results

gmmcovearn saves the following in e():

Scalars

e(nummomnent) number of moment conditions used in estimation

Vectors and Matrices

e(b) coefficient vector

e(V) variance-covariance matrix of the estimators

e(momentc) sample moments for earnings variable for cohort c, c=1 to cohortn

e(permc) predicted permanent component of earnings variance for cohort c, c=1.. cohortn

e(tempc) predicted transitory component of earnings variance for cohortc, c=1.. cohortn

References

Doris, A, D. O’Neill and O.Sweetman (2010a) “GMMCOVEARN: A Stata Module for GMM Estimation of the Covariance Structure of Earnings.,” NUIM Economics Working paper No.212-10.

Doris, A, D. O’Neill and O.Sweetman (2010b) “Identification of the Covariance Structure of Earnings using the GMM Estimator,” IZA Working paper No. 4952.

Haider, S. (2001), “Earnings Instability and Earnings Inequality of Males in the United States: 1967-1991’, Journal of Labor Economics, Vol. 19(4), pp. 799-836.

Wooldridge, J. (2002), Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambdridge, Massachusetts.

Authors

Aedin Doris, Donal O’Neill and Olive Sweetman, Economics, NUI Maynooth, Ireland.