  When using R package tmbstan for Bayesian inference, the built-in feature
Laplace approximation to the marginal likelihood with random effects integrated
out can be switched on and off. There exists no guideline on whether Laplace
approximation should be used to achieve better efficiency especially when the
statistical model for estimating selection is complicated. To answer this
question, we conducted simulation studies under different scenarios with a
state-space model employing a VAR(1) state equation. We found that turning on
Laplace approximation in tmbstan would probably lower the computational
efficiency, and only when there is a good amount of data, both tmbstan with and
without Laplace approximation are worth trying since in this case, Laplace
approximation is more likely to be accurate and may also lead to slightly
higher computational efficiency. The transition parameters and scale parameters
in a VAR(1) process are hard to be estimated accurately and increasing the
sample size at each time point do not help in the estimation, only more time
points in the data contain more information on these parameters and make the
likelihood dominate the posterior likelihood, thus lead to accurate estimates
for them.
