Monte Carlo results

This section describes the Monte Carlo experiment. It shows how the power of the DLR tests for functional form and spatial error dependence in finite samples. The model considered in the simulations is:

ε γ λ β λ λ = − + − + − W y I W X I W Z I ) r ( ) r ( ) ( ( ) ( ) _,

where the number of crimes is represented by (r)

y and household value and income byX(r).

The only constant term is in Z. The Monte Carlo experiment follows the work done by Anselin and Ray (1991), Anselin et al. (1996), and Baltagi and Li (2001). The spatial weight matrix is computed by using 49 observations in the Anselin (1988a) data set. The number of replications is 1000. The regressors included in X are generated from a uniform (0, 10) distribution with its coefficient β set to be 1, while the constant term Z has a coefficient γ assumed to be 4. Error terms, ε , are randomly generated from a standard normal distribution. The DLR tests are evaluated at their asymptotic critical value at the 5% level. This essay follows the Monte Carlo setups in Baltagi and Li (2001). The power of the DLR tests is reported in Figures 1.2 through 1.10.

Figure 1. 2: Power of DLR joint test under hypothesis H₀1:r =0and λ=0

Figures 1.2 and 1.3 show the power of the DLR joint tests under the null hypotheses

1 0

H and H , respectively. The power increases when r and₀2 λ depart from their

hypothesized values. More specifically, under the hypothesis 1 0

H , the power dramatically

increases when r moves toward 1 and λ moves toward either 1 or 1− . Moreover, the power quickly converges to 100% of rejection when r moves away from 0. This result implies that the power of the DLR joint test under hypothesis 1

0

H would be 100% if the true model is

linear. The power of the DLR joint test under the hypothesis 2 0

H also increases when r

moves toward 0 and λ moves toward either 1 or 1− . The power of this test quickly reaches 100% of rejection when r departs from 1, and it is 100% if the true model is log-linear.

Figure 1. 4: Power of DLR one-direction test under hypothesis H₀3:λ =0 assuming r=0

Figure 1.4 presents the power of the DLR one-direction test in 1000 replications under the hypothesis 3

one-direction test under the hypothesis increases when the spatial coefficient moves away from its hypothesized value. The power of this test is also sensitive to a movement of the functional form coefficient.

Figure 1. 5: Power of DLR one-direction test under hypothesis H₀4 :λ =0 assuming r=1

Figure 1.5 shows the power of the DLR one-direction test under the hypothesis 4 0 H : no

spatial error dependence assuming a linear model. For r=1, the power of this test increases when λ departs from its hypothesized value. In particular, it increases when the spatial error coefficient moves from 0 to either 1 or 1− . However, the power of this test is also sensitive to departures of the functional form coefficient from 1 to 0.

Figure 1.6 shows the power of the DLR one-direction test under the hypothesis 5 0 H .

moves toward 1, regardless of the true spatial error dependence. It quickly converges to 100% of rejection when the log-linear functional form moves toward linearity.

Figure 1. 6: Power of DLR one-direction test under hypothesis H₀5:r=0 assuming λ=0

Figure 1.7 on the next page provides the power of the DLR one-direction test under the hypothesis 6

0

H : linearity assuming no spatial error dependence. When λ =0, the power of

this DLR test increases as the functional coefficient moves far away from its null hypothesized value. Under the hypothesis, however, the frequency of rejections quickly converges to 100% regardless of the true spatial error dependence coefficients. It is likely that the power of the DLR test under the hypothesis 6

0

H is not sensitive to movements of the

Figure 1. 7: Power of DLR one-direction test under hypothesis H₀6:r=1 assuming λ=0

Figure 1.8 on the preceding page plots the power of the DLR conditional test for spatial error dependence conditional on a general Box-Cox model. The power of this test depends on both the functional form and the spatial error dependence. It increases when the spatial coefficient departs from its hypothesized value of zero.

Figure 1.9 shows the power of the DLR conditional test under the hypothesis 8 0 H . The

power of this test increases when the functional form coefficient moves away from its hypothesized value of zero. The power of this DLR conditional test converges to 100% of rejection as the functional form moves from log-linearity to linearity.

Figure 1. 10: Power of the DLR conditional test under hypothesis H₀9:r=1|unkownλ

Figure 1.10 plots the power of the DLR conditional test under the hypothesis 9 0 H . The

figure shows that the power of the DLR test increases when the functional form coefficient departs from its null hypothesized value of 1. It also converges to 100% of rejection when the functional form closes to log-linearity.

The Monte Carlo experiment results have shown that the power of the DLR tests increases when the functional coefficient and/or the spatial coefficient deviates from their hypothesized value(s). The tests perform reasonably well. They are more sensitive to the functional coefficient than to the spatial error coefficient. The DLR tests perform similarly to their LM counterparts in Baltagi and Li (2001) in the Monte Carlo simulations. Our graphs show almost identical patterns as those shown in Baltagi and Li (2001).