Mixed Portmanteau Test for Diagnostic Checking of Time Series Models

Table 4

Empirical power by fitting an AR(1) process against an ARMA(1,1) process: 𝑦_𝑡=𝜙𝑦_𝑡−1+𝜀_𝑡+0.8𝜀_𝑡−1 using 1000 Monte Carlo runs.

𝑚	𝑇	𝜙
		0.1	0.4	0.7	0.8	0.9
5	𝑄^∗𝑚(𝑟)̂	0.955	0.991	1	1	1
	𝑄^∗𝑚(𝜔)	0.992	0.997	1	1	1
	𝑄^∗𝑚(𝑟,̂𝜔)	0.997	0.999	1	1	1
10	𝑄^∗𝑚(𝑟)̂	0.819	0.941	0.983	0.986	0.995
	𝑄^∗𝑚(𝜔)	0.980	1	1	1	1
	𝑄^∗𝑚(𝑟,̂𝜔)	0.985	1	1	1	1
15	𝑄^∗𝑚(𝑟)̂	0.748	0.871	0.940	0.957	0.976
	𝑄^∗𝑚(𝜔)	0.958	0.990	1	1	1
	𝑄^∗𝑚(𝑟,̂𝜔)	0.953	0.970	0.995	0.997	1
20	𝑄^∗𝑚(𝑟)̂	0.680	0.828	0.900	0.940	0.948
	𝑄^∗𝑚(𝜔)	0.926	0.989	0.995	0.999	1
	𝑄^∗𝑚(𝑟,̂𝜔)	0.943	0.957	0.987	0.998	0.999
25	𝑄^∗𝑚(𝑟)̂	0.614	0.810	0.872	0.910	0.934
	𝑄^∗𝑚(𝜔)	0.864	0.974	0.990	0.999	1
	𝑄^∗𝑚(𝑟,̂𝜔)	0.927	0.940	0.975	0.995	0.998