*3.1. Simulation Study*

A simulation study was conducted, by running the model-fitting process on batches of randomly generated AR or TAR models of observation length 500 and batch size 50 for all the results in this section (i.e., 50 simulated observations of length 500 were considered in each simulation experiment). The completion of each simulation took around half a day on a laptop. The following tables give the results for the bias and mean square error in the simulation study. Tables 1 and 2 give the results for the threshold models and Table 3 gives the result for AR models. The correct estimation of AR order and lag meant that the estimation of both the AR order p and the lag *d* were in line with the true parameters. The bias and MSE were calculated with the results in the simulations which gave the correct estimation of AR order and lag. The parameter for the true TAR model was selected such that the TAR structure was reasonably demonstrated (i.e., there were not too few observations in any regime).

**True Model** *α***1** *β***1** *α***2** *β***2** *ϕ***1,1** *ϕ***1,2** *ϕ***2,1** *ϕ***2,2 T** 5 2 5 2 0.5 0.3 0.3 0.2 30 AIC Proportion of correct estimation of Autoregressive (AR) order and Lag: 44/50 AIC Bias 0.032 0.023 0.341 −0.022 0.013 −0.007 0.003 −0.004 0.001 AIC MSE 1.245 0.083 2.759 0.123 0.002 0.003 0.002 0.002 0.001 BIC Proportion of correct estimation of AR order and Lag: 50/50 BIC Bias 0.015 0.022 0.384 −0.015 0.012 −0.006 0.004 −0.005 0.001 BIC MSE 1.198 0.08 3.887 0.143 0.002 0.003 0.002 0.002 0.001

**Table 1.** Simulation results for the threshold Autoregressive (TAR) (2) model with *d* = 2.


**Table 2.** Simulation results for the TAR (1) model with *d* = 1.

**Table 3.** Simulation results for the AR (2) model.


The estimates of the AR order and lag were generally good, except the AIC criterion for the AR model, as the AIC tends to pick a more complicated model. In fact, the AIC estimated the AR order correctly in 36 out of 50 cases; yet, in most of these cases, it preferred a threshold structure.

The simulation results show that the model could identify the correct AR order p and the lag *d* with good accuracy in general, the estimate for the threshold T was very consistent; and the results for the AR parameters *ϕ* were rather accurate when p and *d* were estimated correctly. It should be noted that the accuracy here is defined as the probability of identifying the correct AR model order and correct threshold, given that the underlying model was indeed an AR/TAR model.
