Monte Carlo simulation

In this Section we conduct a small-scaled Monte Carlo experiment to assess the finite sample properties of our adjusted BCFE estimator compared to several other estimators.

Design

The data generating process (DGP) is chosen such that the properties of the sim-ulated data match with those of the observed data as much as possible:

• The sample size of the simulated data equals the one available for estima-tion. This implies running separate simulations for advanced (T = 9, N = 247) and developing (T = 9, N = 388) origin countries.

• Data for the endogenous variables migration flow M_dot and migration stock MST_dot are drawn from their data generating process (DGP) in equations (11) and (8) respectively, using the observed values in the first year of the sample as initialisation.

• The parameter values for θ in the DGP for M_dot in equation (11) are set equal to the BCFE estimates from Table 3 below while δ_do in equation (8) is set equal to the value observed in the sample data.

• Error terms εdot are generated from a normal distribution with estimated variance from the residuals of the BCFE regressions in Table 3.

• The observed values for the exogenous variables X_dot−1and ∆X_dotare treated as fixed in each MC iteration.

We generate data both for the full model and for a partial model with only stocks and lagged flows as explanatory variables (θ₃= θ4= 0 in equation (2.21)). This results in four experiments with the coefficients for lagged flows, θ1, and stocks, θ₂, respectively set to 0.61 and 0.46 (0.64 and 0.49) for the complete (partial) model using the advanced dataset, and 0.75 and 0.23 (0.74 and 0.23) for the com-plete (partial) model using the developing dataset. In each experiment, we perform 1000 replications.

Estimators

We compare the performance of the BCFE estimator with (i) FE, the standard fixed effects estimator, (ii) GMMd, the first-difference GMM estimator proposed by Arellano and Bond (1991) and (iii) GMMs, the system GMM estimator pro-posed by Arellano and Bover (1995) and Blundell and Bond (1998). For the GMMd estimator, at least one period lagged values (ln M_dot−1−sand ln MST_dot−s with s ≥ 1) are available as instruments for the predetermined variables ln M_dot−1 and ln MST_dot⁸in each period. For the exogenous variables X_dot−1 and ∆X_dot, the available instruments set is (X_do1, . . . , X_doT−1, ∆X_do2, . . . , ∆X_doT) in each period.

GMMs has the same instrument set as GMMd in the first difference part of the system and has ∆ ln M_dot−2, ∆X_dot−1 and ∆²X_dot as additional instruments in the levels part of the system. Note that the first-differenced stock ∆ ln MST_dot−1 can not be used as instrument as it is by construction correlated with the fixed effect µ_do in the levels equation. Given the large number of exogenous variables, we try to avoid an overfitting bias resulting from using too many instruments (see Zil-iak, 1997; Arellano, 2003) by (i) only using the first three available instruments

8Ln MST_dotis predetermined as it is defined as the migrant stock at the beginning of the period.

for the predetermined variables (ln M_dot−1 and ln MST_dot) and the contempora-neous values for the exogenous variables and (ii) stacking the instrument matrix as suggested by Roodman (2009). We report both one-step and two-step GMM estimates.

Simulation results

The simulation results are presented in Tables 2.1 and 2.2. For each estimator, we report mean bias, standard deviation (Std) and root mean squared error (Rmse) in estimating θ1and θ2.

First looking at the performance in estimating θ₁, we observe the following re-sults for both types of models. As expected, the FE estimator is biased downward because of the correlation between the transformed lagged dependent variable and the transformed error term. Correcting for the dynamic panel bias by performing BCFE significantly reduces the bias while maintaining the low dispersion associ-ated with the uncorrected FE. The bias of the GMMd1 and GMMd2 estimators is of the same order as in BCFE, but they have a much larger dispersion and rmse.

The GMMs estimators have a sizable bias in all cases. This suggests that the extra moment conditions imposed in the level part of the system, from a restriction on the initial conditions process generating ln M_do1, is violated.

Second, regarding the relative performance in the estimation of θ₂, the GMMd estimators have the smallest bias, followed by the FE and BCFE estimators. How-ever, the standard deviation of the GMMd estimators is always bigger compared to the FE and BCFE estimators. This results in (i) the lowest rmse for the BCFE estimator using the advanced dataset and the partial model developing dataset and (ii) a fairly similar rmse for the BCFE and GMMd estimates for the complete model developing dataset. The GMMs estimators again have a sizable bias in most cases.

Table 2.1: Monte Carlo results based on database with advanced origins (T = 9, N = 247)

Bias θ1 Std θ1 Rmse θ1 Bias θ2 Std θ2 Rmse θ2

Full model, θ1= 0.61 and θ2= 0.46

FE -0.192 0.021 0.193 0.035 0.047 0.059

BCFE -0.011 0.025 0.028 -0.026 0.045 0.052

GMMd1 -0.014 0.138 0.138 0.003 0.064 0.064

GMMd2 -0.014 0.141 0.141 0.002 0.065 0.065

GMMs1 0.226 0.026 0.228 -0.249 0.031 0.251

GMMs2 0.200 0.031 0.202 -0.204 0.042 0.208

Partial model with θ₃= θ4= 0, θ1= 0.64 and θ2= 0.49

FE -0.185 0.021 0.186 0.079 0.044 0.091

BCFE -0.011 0.025 0.028 -0.024 0.043 0.049

GMMd1 -0.006 0.076 0.076 0.002 0.066 0.066

GMMd2 -0.005 0.076 0.076 0.001 0.066 0.066

GMMs1 -0.112 0.070 0.132 -0.090 0.062 0.109

GMMs2 -0.099 0.071 0.122 -0.103 0.065 0.122

Notes: θ1 and θ2denote the coefficients for ln Mdot−1and ln MSTdot, respectively. θ3and θ4represent the coefficients of the strictly exogenous variables. For the GMM estimators, ‘1’ refers to one-step estimates and ‘2’ refers to two-step estimates.

In conclusion, due to its small bias combined with a relatively small standard deviation, the BCFE estimator is shown to outperform the alternative estimators in terms of rmse given the specificities of our model and sample data. As such, we take it as our preferred estimator in the next section.