In order to derive an expression for the standard error used in the Wald test statistic, we rely on the sandwich estimator of White (1982). From the log-likelihood equation defined
in (10), the variance covariance matrix of ˜θ is
V˜(˜θ) = ˜J(˜θ)−1H(˜˜ θ)[ ˜J(˜θ)−1]′,
where
H(θ) =˜ X
i
1{0< yi+ < T}s˜i(θ) ˜si(θ)′
and ˜J(θ) is the information matrix defined in (11). Once the matrix ˜V(˜θ) has been computed as above, the standard error for ˜ψ may be obtained in the usual way from the main diagonal of this matrix.
References
Bartolucci, F. and Farcomeni, A. (2009). A multivariate extension of the dynamic logit model for longitudinal data based on a latent markov heterogeneity structure. Journal of the American Statistical Association, 104(486):816–831.
Bartolucci, F. and Nigro, V. (2010). A dynamic model for binary panel data with unob-served heterogeneity admitting a √
n-consistent conditional estimator. Econometrica, 78:719–733.
Bartolucci, F. and Nigro, V. (2012). Pseudo conditional maximum likelihood estimation of the dynamic logit model for binary panel data. Journal of Econometrics, 170:102–116.
Carro, J. (2007). Estimating dynamic panel data discrete choice models with fixed effects.
Journal of Econometrics, 140:503–528.
Chamberlain, G. (1985). Heterogeneity, omitted variable bias, and duration dependence.
In Heckman, J. J. and Singer, B., editors, Longitudinal analysis of labor market data.
Cambridge University Press: Cambridge.
Chamberlain, G. (1993). Feedback in panel data models. Technical report, Department of Economics, Harvard University.
Cox, D. (1972). The analysis of multivariate binary data. Applied statistics, 21:113–120.
Feller, W. (1943). On a general class of ”contagious” distributions.Annals of Mathematical Statistics, 14:389–400.
Fernandez-Val, I. (2009). Fixed effects estimation of structural parameters and marginal effects in panel probit model. Journal of Econometrics, 150:71–85.
Hahn, J. and Kuersteiner, G. (2011). Bias reduction for dynamic nonlinear panel models with fixed effects. Econometric Theory, 27:1152–1191.
Hahn, J. and Newey, W. (2004). Jackknife and analytical bias reduction for nonlinear panel models. Econometrica, 72:1295–1319.
Halliday, T. H. (2007). Testing for state dependence with time-variant transition proba-bilities. Econometric Reviews, 26:685–703.
Heckman, J. (1981). Heterogeneity and state dependence. In Rosen, S., editor, Studies in labor markets, pages 91–140. Chicago University Press: Chicago.
Hochberg, Y. and Tamhane, A. (1987). Multiple comparison procedures. Wiley series in probability and mathematical statistics: Applied probability and statistics. Wiley.
Honor´e, B. E. and Kyriazidou, E. (2000). Panel data discrete choice models with lagged dependent variables. Econometrica, 68:839–874.
Hsiao, C. (2005). Analysis of Panel Data. Cambridge University Press, New York, 2nd edition.
Hyslop, D. R. (1999). State dependence, serial correlation and heterogeneity in intertem-poral labor force participation of married women. Econometrica, 67(6):1255–1294.
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica,
Figure 1: Power plots for Wald (QE2) and Halliday’s tests: bidirectional (H1 :γ 6= 0)
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ n=1000, T=5, β = 1
test_cov test_nocov Halliday
(“test cov” refers to the case in which the covariate xit is included in the QE2 model; “test nocov” is referred to the case in which the covariate is not included; power curves are smoothed by means of sixth–order polynomials)
Figure 2: Power plots for Wald (QE2) and Halliday’s tests: lower tailed (H1 :γ <0)
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ n=1000, T=5, β = 1
test_cov test_nocov Halliday
(“test cov” refers to the case in which the covariate xit is included in the QE2 model; “test nocov” is referred to the case in which the covariate is not included; power curves are smoothed by means of sixth–order polynomials)
Figure 3: Power plots for Wald (QE2) and Halliday’s tests: upper tailed (H1 :γ >0)
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ
Power for nominal size = 5%
γ n=1000, T=5, β = 1
test_cov test_nocov Halliday
(“test cov” refers to the case in which the covariate xit is included in the QE2 model; “test nocov” is referred to the case in which the covariate is not included; power curves are smoothed by means of sixth–order polynomials)
Table 1: Simulation results for Wald (QE2) and Halliday’s test statistics: bidirectional
n = 500 n= 1000
γ test cov test nocov Halliday test cov test nocov Halliday
β = 0 -1.0 0.854 0.855 0.846 0.989 0.988 0.987
T = 2 -0.5 0.313 0.314 0.312 0.557 0.550 0.556
0.0 0.048 0.044 0.052 0.054 0.058 0.059
0.5 0.261 0.262 0.270 0.474 0.471 0.474
1.0 0.684 0.690 0.682 0.943 0.946 0.930
β = 0 -1.0 1.000 1.000 0.990 1.000 1.000 1.000
T = 5 -0.5 0.983 0.982 0.524 0.998 0.998 0.849
0.0 0.054 0.055 0.053 0.060 0.061 0.058
0.5 0.971 0.972 0.413 0.998 0.998 0.719
1.0 1.000 1.000 0.929 1.000 1.000 1.000
β = 1 -1.0 0.523 0.122 0.122 0.815 0.172 0.171
T = 2 -0.5 0.158 0.059 0.064 0.270 0.076 0.078
0.0 0.052 0.226 0.233 0.051 0.430 0.441
0.5 0.137 0.573 0.583 0.243 0.848 0.850
1.0 0.368 0.832 0.832 0.658 0.983 0.980
β = 1 -1.0 1.000 0.530 0.452 1.000 0.800 0.745
T = 5 -0.5 0.810 0.123 0.073 0.980 0.146 0.113
0.0 0.056 0.951 0.283 0.043 1.000 0.417
0.5 0.761 1.000 0.748 0.971 1.000 0.980
1.0 1.000 1.000 0.981 1.000 1.000 1.000
(“test cov” refers to the case in which the covariate xit is included in the QE2 model; “test nocov” is referred to the case in which the covariate is not included)
Table 2: Simulation results for Wald test (QE1) test statistic: bidirectional
n = 500 n= 1000
γ test cov test nocov test cov test nocov
β = 0 -1.0 0.709 0.710 0.935 0.933
T = 2 -0.5 0.195 0.191 0.351 0.354
0.0 0.045 0.050 0.050 0.056
0.5 0.130 0.130 0.220 0.222
1.0 0.268 0.268 0.483 0.480
β = 1 -1.0 0.347 0.102 0.600 0.149
T = 2 -0.5 0.110 0.058 0.169 0.062
0.0 0.039 0.148 0.045 0.246
0.5 0.091 0.311 0.136 0.525
1.0 0.158 0.448 0.308 0.772
(“test cov” refers to the case in which the covariate xit is included in the QE1 model; “test nocov” is referred to the case in which the covariate is not included)
Table 3: Tests for state dependence (H1 : γ 6= 0): proposed Wald test (QE2) and Halli-day’s test statistics for the overall PSID dataset
Employment Fertility stat. p-value stat. p-value Proposed Wald test
W 13.58 0.00 −6.80 0.00
Halliday’s test
S1 (1st triple) 5.75 0.00 −4.74 0.00 S2 (2nd triple) 4.80 0.00 −4.97 0.00 S3 (3rd triple) 4.02 0.00 −1.09 0.27 S4 (4th triple) 4.27 0.00 −5.10 0.00
Sample size 1446 1446
(QE2 model is estimated with covariates; Bonferroni corrected nominal size: 0.010206)
Table 4: Estimation results based on the PCML approach (Bartolucci and Nigro, 2012):
overall PSID dataset
Employment Fertility
coeff. s.e. Wald-stat. p-value coeff. s.e. Wald-stat. p-value
Child 1–2 -0.675 0.13 -5.10 0.00 -0.719 0.15 -4.72 0.00
Child 3–5 -0.312 0.12 -2.52 0.01 -1.085 0.21 -5.05 0.00
Child 6–13 -0.032 0.12 -0.25 0.40 -1.055 0.26 -4.08 0.00
Child 14– -0.010 0.14 -0.07 0.47 -0.800 0.43 -1.86 0.03
Income/1000 -0.007 0.00 -1.68 0.05 -0.000 0.00 -0.13 0.45
1989 0.089 0.14 -1.12 0.13 0.402 0.15 4.64 0.00
1990 0.317 0.13 0.65 0.26 0.445 0.19 2.64 0.00
1991 0.089 0.13 2.49 0.01 0.397 0.24 2.31 0.01
1992 0.001 0.13 0.67 0.25 0.448 0.29 1.66 0.05
Lag fertility -0.185 0.17 0.01 0.50 -0.906 0.21 1.56 0.06 Lag employment 1.550 0.11 13.93 0.00 0.801 0.17 -4.35 0.00
Table 5: Tests for state dependence (H1 : γ 6= 0): proposed Wald test (QE2) and Halli-day’s test statistics for the PSID dataset
Years of schooling ≤12 Years of schooling >12 Employment Fertility Employment Fertility stat. p-value stat. p-value stat. p-value stat. p-value Proposed Wald test
W 10.18 0.00 −2.70 0.01 9.04 0.00 −6.25 0.00
Halliday’s test
S1 (1st triple) 3.04 0.00 −1.60 0.11 5.27 0.00 −5.79 0.00 S2 (2nd triple) 3.49 0.00 −0.81 0.41 3.30 0.00 −6.28 0.00 S3 (3rd triple) 2.83 0.00 −0.79 0.43 2.75 0.01 −0.78 0.00 S4 (4th triple) 4.40 0.00 −1.18 0.24 1.68 0.09 −5.45 0.00
Sample size 773 673
(QE2 model is estimated with covariates; Bonferroni corrected nominal size: 0.010206)
Table 6: Estimation results based on the PCML approach (Bartolucci and Nigro, 2012):
PSID dataset for the subsample with years of schooling ≤12
Employment Fertility
coeff. s.e. Wald-stat. p-value coeff. s.e. Wald-stat. p-value
Child 1–2 -0.419 0.20 -2.12 0.02 -0.726 0.24 -2.96 0.00
Child 3–5 0.055 0.17 0.32 0.33 -0.750 0.33 -2.26 0.02
Child 6–13 0.127 0.16 0.77 0.22 -0.524 0.38 -1.36 0.17
Child 14– 0.067 0.17 0.38 0.35 -0.757 0.69 -1.10 0.27
Income/1000 -0.018 0.01 -2.47 0.01 -0.004 0.01 -0.27 0.78
1989 0.153 0.19 0.13 0.45 0.328 0.23 3.99 0.00
1990 0.193 0.17 0.82 0.20 0.189 0.29 1.41 0.16
1991 0.341 0.18 1.13 0.13 0.004 0.36 0.66 0.51
1992 -0.102 0.18 1.92 0.03 -0.516 0.46 0.01 0.99
Lag fertility 0.031 0.25 -0.57 0.28 -0.145 0.32 -1.13 0.26 Lag employment 1.529 0.15 10.10 0.00 1.090 0.27 -0.45 0.65
Table 7: Estimation results based on the PCML approach (Bartolucci and Nigro, 2012):
PSID dataset for the subsample with years of schooling >12
Employment Fertility
coeff. s.e. Wald-stat. p-value coeff. s.e. Wald-stat. p-value
Child 1–2 -0.932 0.19 -0.86 0.19 -0.763 0.20 -3.89 0.00
Child 3–5 -0.690 0.19 -0.64 0.19 -1.350 0.29 -4.68 0.00
Child 6–13 -0.220 0.21 -4.95 0.00 -1.438 0.36 -3.99 0.00
Child 14– -0.154 0.25 -3.59 0.00 -0.811 0.61 -1.34 0.09
Income/1000 -0.001 0.00 -1.05 0.15 -0.000 0.00 -0.02 0.49
1989 0.038 0.29 -0.61 0.27 0.529 0.21 2.99 0.00
1990 0.505 0.20 -0.24 0.41 0.670 0.26 2.57 0.01
1991 -0.139 0.21 -1.43 0.08 0.694 0.33 2.53 0.01
1992 0.208 0.21 0.18 0.43 1.071 0.39 2.10 0.02
Lag fertility -0.325 0.23 2.57 0.01 -1.275 0.28 2.71 0.00 Lag employment 1.591 0.17 -0.68 0.25 0.684 0.23 -4.57 0.00