Estimation of Panel Data Models with Random Interactive Effects and Multiple Structural Breaks when T is Fixed

Figures & data

Table 1 Simulation results for the case with stationary factors ($\varphi =0.8$) and low error dependence ($\pi =0.4$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000
MSE	0.086	0.077	0.041	0.044	0.018	0.083	0.038	0.033	0.011	0.010	0.035	0.026	0.009	0.012	0.001
$m^0=1$
NB	0.019	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BP	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.983	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
MSE	0.387	0.120	0.106	0.062	0.041	0.142	0.090	0.061	0.019	0.018	0.105	0.040	0.025	0.020	0.011
$m^0=2$
NB	0.151	0.045	0.005	0.000	0.000	0.003	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.840	1.955	1.995	2.000	2.000	1.997	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000
MSE	1.571	0.519	0.144	0.053	0.036	0.188	0.075	0.048	0.044	0.016	0.106	0.068	0.031	0.016	0.011

Table 2 Simulation results for the case with nonstationary factors ($\varphi =1$) and low error dependence ($\pi =0.4$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.001	0.000	0.000	0.000	0.000	0.004	0.000	0.000	0.000	0.000	0.038	0.002	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.001	0.000	0.000	0.000	0.000	0.004	0.000	0.000	0.000	0.000	0.038	0.002	0.000	0.000	0.000
MSE	0.119	0.108	0.035	0.020	0.022	0.056	0.060	0.039	0.027	0.016	0.077	0.052	0.031	0.012	0.011
$m^0=1$
NB	0.015	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	3.000	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.985	0.990	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.003	1.000	1.000	1.000	1.000
MSE	0.351	0.096	0.076	0.081	0.030	0.131	0.043	0.039	0.041	0.021	0.050	0.053	0.0139	0.024	0.014
$m^0=2$
NB	0.153	0.043	0.012	0.000	0.000	0.002	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.847	1.957	1.988	2.000	2.000	1.998	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000	2.000
MSE	1.596	0.457	0.191	0.039	0.029	0.124	0.073	0.068	0.024	0.016	0.099	0.058	0.027	0.026	0.000

Table 3 Simulation results for the case with stationary factors ($\varphi =0.8$) and high error dependence ($\pi =0.8$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.011	0.050	0.062	0.000	0.001	0.017	0.047	0.060	0.000	0.001	0.014	0.057	0.043	0.014	0.001
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.011	0.001	0.068	0.000	0.001	0.018	0.051	0.000	0.000	0.001	0.014	0.069	0.047	0.015	0.001
MSE	0.128	0.102	0.132	0.023	0.044	0.076	0.074	0.071	0.035	0.025	0.049	0.038	0.011	0.028	0.000
$m^0=1$
NB	0.011	0.024	0.030	0.007	0.001	0.003	0.018	0.010	0.002	0.000	0.015	0.019	0.004	0.003	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.005	1.024	1.033	1.007	1.001	1.003	1.020	1.010	1.002	1.000	1.015	1.023	1.005	1.003	1.000
MSE	0.313	0.329	0.177	0.123	0.043	0.173	0.144	0.133	0.051	0.041	0.054	0.120	0.019	0.031	0.015
$m^0=2$
NB	0.098	0.037	0.008	0.003	0.000	0.003	0.013	0.004	0.002	0.000	0.005	0.013	0.006	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.909	1.992	2.007	2.003	2.000	2.003	2.012	2.004	2.002	2.000	2.005	2.013	2.006	2.000	2.000
MSE	1.182	0.485	0.238	0.113	0.057	0.208	0.181	0.143	0.062	0.023	0.148	0.127	0.076	0.050	0.020

Table 4 Simulation results for the case with nonstationary factors ($\varphi =1$) and high error dependence ($\pi =0.8$).

	T = 5					T = 10					T = 20
N	25	50	100	200	600	25	50	100	200	600	25	50	100	200	600
$m^0=0$
NB	0.015	0.049	0.048	0.019	0.000	0.028	0.057	0.051	0.020	0.001	0.057	0.068	0.034	0.007	0.003
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	0.017	0.051	0.052	0.02	0.000	0.029	0.068	0.056	0.02	0.001	0.061	0.075	0.037	0.007	0.005
MSE	0.179	0.069	0.074	0.073	0.040	0.104	0.049	0.071	0.041	0.025	0.038	0.034	0.032	0.014	0.012
$m^0=1$
NB	0.003	0.021	0.021	0.006	0.000	0.002	0.012	0.009	0.001	0.000	0.005	0.016	0.007	0.002	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.003	1.024	1.023	1.006	1.000	1.002	1.015	1.009	1.001	1.000	1.005	1.016	1.007	1.002	1.000
MSE	0.270	0.238	0.150	0.061	0.056	0.170	0.151	0.051	0.063	0.053	0.102	0.059	0.049	0.035	0.013
$m^0=2$
NB	0.103	0.026	0.008	0.001	0.000	0.002	0.005	0.004	0.000	0.000	0.003	0.003	0.006	0.000	0.000
BP	0.000	0.000	0.000	0.000	0.000	0.000	0.001	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
AVE	1.898	1.998	2.007	2.001	2.000	2.002	2.005	2.004	2.003	2.000	2.003	2.004	2.006	2.000	2.000
MSE	1.208	0.412	0.238	0.130	0.048	0.115	0.121	0.143	0.062	0.032	0.100	0.065	0.076	0.024	0.015

NOTE: “NB”, “BP”, “AVE” and “MSE” refer to the frequency of false detection of the estimated number of breaks, the frequency of false detection of the estimated breakpoints given that the number of breaks is selected correctly, the average number of estimated breaks, and the MSE of PDL2S estimator times 100, computed as the average $100·\left|\right|{\widehat{\mathit{A}}}_{\widehat{m}}-{\mathit{A}}_{m^0}^0\left|\right|/(\widehat{m}+1)$ p across the Monte Carlo replications, respectively. m⁰, $\varphi$, and π refer to the number of breaks, the persistence of the factors, and the serial and cross-sectional correlation of the errors in the equations for $y_{i,t}$ and $x_{i,t}$, respectively.

Table 5 Empirical estimation results.

	Breaks allowed (PDL2S)			No breaks
Regressor	1982–1984	1985	1986–1987	PDL2S	FE
Controls included in break estimation
PRBARR	–0.459***	–0.613***	–0.301***	–0.414***	–0.354***
PRBCONV	–0.271***	–0.422***	–0.222***	–0.297***	–0.282***
PRBPRIS	–0.196***	–0.196	–0.128	–0.168***	–0.173***
AVGSEN	0.057	–0.084	–0.134**	–0.022	–0.003
POLPC	0.483***	0.548***	0.414***	0.456***	0.413**
DENSITY	–0.005	–0.149	0.293	–0.568	0.414
WCON	0.017	0.490**	0.358**	–0.004	–0.038
WTUC	0.026	0.778*	0.084	0.056**	0.046***
WTRD	0.040	–0.103	–0.038	0.055	–0.021
WFIR	–0.002	0.080	0.025	–0.023	–0.004
WSER	–0.182	0.584	–0.217	0.010	0.009
WMGF	–0.047	–0.080	–0.080	–0.166	–0.360***
WFED	–0.120	–0.124	0.023	–0.155	–0.309*
WSTA	–0.030	–0.738	0.293	–0.038	0.053
WLOC	–1.705***	–1.989**	–0.271	0.147	0.182*
PCTYMLE	2.504***	1.608	1.217**	1.408***	0.627*
Controls projected out prior to break estimation
PRBARR	–0.444***			–	–0.720***
PRBCONV	–0.307***			–	–0.546***
PRBPRIS	–0.118**			–	0.248***
AVGSEN	–0.017			–	–0.087
POLPC	0.449***			–	0.366***

NOTE: The dependent variable is the crime rate (CRMRTE). The deterrence regressors are cases the probability of arrest (PRBARR), the probability of conviction given arrest (PRBCONV), and the probability of a prison sentence given a conviction (PRBPRIS). The included control variables are average prison sentence in days (AVGSEN), the number of police per capita (POLPC), population density (DENSITY), percent young male (PCTYMLE), and the average weekly wage by industry, where the included industries are construction (WCON), transportation, utilities and communication (WTUC), wholesale and retail trade (WTRD), finance, insurance and real estate (WFIR), services (WSER), manufacturing (WMFG), federal government (WFED), state government (WSTA), and local government (WLOC). All variables are expressed in logs. “FE” refers to the usual two-way fixed effects LS estimator. The PDL2S estimator allows for county fixed effects using the initial value adjustment discussed in Section 3.1. When no breaks are allowed PDL2S is simply LS with initial value adjustment and cross-county demeaning. The symbols *, ** and *** denote statistical significance at the 10%, 5% and 1% levels, respectively. In the top panel the controls are included in the PDL2S procedure, whereas in the bottom panel they are projected out prior to applying PDL2S, which means that they do not affect the break estimation procedure. When the controls are included we estimate two breaks, while when they are projected out we estimate no breaks.