Generalized True Random Effects Model
Next we are interested in estimating the Generalized True Random
Effects Model (GTRE) model of Filippini Greene (2016) using simulated
maximum likelihood. We will begin by using a simulated data set. We use
the “data_gen_p” call to create a simulated data set. The psfm call runs
the likelihood through three successive optimizer rountines, only
updating the parameter values if there is an improvement in the
likelihood. We have opted for 100 iterations of the “bobyqa” procedure,
8 for “psoptim”, and 8 for “optim”. “rand” signifies the seed, using
set.seed() for replication. Other variables are for the sigmas and betas
in the model, with cons for beta0/constant.
data_trial <- data_gen_p(t=10,N=100,rand = 16, sig_u = 0.3,sig_v = 0.1, sig_r = 0.1, sig_h = 0.3, cons = 0.5, beta1 = 0.5, beta2 = 0.5)
p.gtre_sml <- psfm(formula = y_gtre ~ x1 + x2,
model_name = "GTRE",
data = data_trial,
individual = "name",
PSopt = TRUE,
optHessian = TRUE,
maxit.bobyqa = 150,
maxit.psoptim= 10,
maxit.optim = 10)
#> Warning in commonArgs(par, fn, control, environment()): maxfun < 10 *
#> length(par)^2 is not recommended.
summary(p.gtre_sml)
#> --- SFA Regression Model Summary ---
#> Formula: y_gtre ~ x1 + x2
#> Total time: 18.55371
#> Model Output:
#> par st_err t-val
#> lambda 3.32985362 0.457061209 7.28535600
#> sigma 0.32713857 0.011443273 28.58784909
#> sigr 0.00280835 0.070474823 0.03984898
#> sigh 0.39948213 0.027595726 14.47623172
#> (Intercept) 0.59193150 0.031720137 18.66106392
#> x1 0.50385689 0.006790335 74.20206907
#> x2 0.50032519 0.007462036 67.04942137
mean(p.gtre_sml$U)
#> [1] 0.7919856
mean(p.gtre_sml$H)
#> [1] 0.7376397
GTRE Results
The results give the model parameter estimates in the classic \(\lambda\)-\(\sigma\) framework as well as the mean
efficiency scores. We see that most parameters are estimated quite well,
with large t-values. For example, \(\hat
\lambda = 3.14\) while the true \(\lambda = 0.3/0.1=3\). The optimization
struggled a bit with \(\sigma_r\)=sig_r, which is reflected in the
less accurate parameter and lower t-value. Increasing the number of
iterations in the optimization procedure typically improves accuracy,
but will increase the run time. We also see that the mean persistent
technical efficiency is 0.75 and the mean transient technical efficiency
is 0.80.
We may be interested in plotting the densities of these efficiency
scores:
plot(density(p.gtre_sml$U),main="Density of Transient TE")
plot(density(p.gtre_sml$H),main="Density of Persistent TE")
To get the total technical efficiency, we would simply multiply the
TE’s of U and H in the following way:
total_te <- rep(p.gtre_sml$H, each=10) * p.gtre_sml$U
plot(density(total_te),main="Density of Total TE")
NB: Using the constant 10 in rep(p.gtre_sml$H, each=10) only works
for a balanced panel with t=10 for each individual. For an unbalanced
panel, the each argument in rep would not work. Instead, use the times
argument and a vector of length N (number of individuals), with each of
the time period lengths, e.g. rep(c(2,5),times=c(5,2)).