ECON734: Spatial Econometrics – Lab 3 Term I, 2018-2019 Yang Zhenlin zlyang@smu.edu.sg http://www.mysmu.edu/faculty/zlyang/

Monte Carlo Simulation We introduce basic idea for Monte Carlo Simulation. The purpose of Monte Carlo simulation is for assessing the finite sample performance of the estimators, test statistics. In the context of spatial linear regression (SLR) model, it is known that QMLEs are downward biased. How biased can they be? To do so, we need a model, or a data generating process (DGP) to (i) generate data, (ii) calculate the QMLEs, (iii) repeat this for many times, and (iv) calculate Monte Carlo (empirical) means, standard deviation (sd), and root-mean-squared error (rmse). Consider the following DGP – an SLD model: Yn = WnYn + 01n+ X11 + X22 + en where  = (0, 1, 2) = (5, 1, 1),  = 1,  = {.5, .25, 0, -.25, -.5}.

Monte Carlo – QMLE of SLD Model Regressors’ values are generated according to either: 𝑥 1𝑖 𝑖𝑖𝑑 𝑁 0,1 / 2 , 𝑥 2𝑖 𝑖𝑖𝑑 𝑁 0,1 / 2 , and 𝑥 1𝑖 and 𝑥 2𝑖 are independent. The spatial weight matrix Wn is generated according to (a) Rook contiguity, (b) Queen contiguity, and (c) group interaction. The errors {eni} are iid and are generated from (a) standard normal, (b) normal mixture, (c) lognormal, where in (b) and (c), random variates are standardized to have mean 0 and variance 1. See Yang (2015, JOE: A general method …) for details on these; also for details on 2nd- and 3rd-order bias-corrected QMLEs.

Monte Carlo – QMLE of SLD Model Matlab files: SldQmle1117.m: for QMLE of 𝜆 𝑛 SldQmleBC1117.m: for 𝜆 𝑛 , 2nd- and 3rd-order bias corrected 𝜆 𝑛 FnSarN.m: returns the QMLE 𝜆 𝑛 queen.m: returns Wn under Queen contiguity rook.m: returns Wn under Rook contiguity group.m: returns Wn under group interaction