Yule-Walker Estimation for the Moving-Average Model

Table 2

Approximate bias, variance, and mean squared error of the method of moments and the maximum likelihood estimators (based on 𝑁=𝑛² observations) for the two-dimensional MA parameters 𝛽 and 𝛾 from 1000 replications.

	𝛽=0.5			𝛾=0.45
𝑛=7	Bias	Var	MSE	Bias	Var	MSE
MME	0.02842	0.00010879	0.00091649	−0.07858	0.00001209	0.0061869
MLE	0.0006	0.0002023	0.00020266	−0.00604	0.00006244	0.00009892
𝑛=30	Bias	Var	MSE	Bias	Var	MSE
MME	0.0149	0.00037173	0.00059374	−0.00746	0.0002023	0.00025795
MLE	−0.00184	0.0003022	0.00030558	0.00066	0.00016793	0.00016837
𝑛=100	Bias	Var	MSE	Bias	Var	MSE
MME	0.00576	0.00025984	0.00029302	0.0065	0.00023986	0.00028211
MLE	−0.00044	0.00022068	0.00022087	−0.00024	0.0002023	0.00020236
	𝛽=0.15			𝛾=−0.3
𝑛=30	Bias	Var	MSE	Bias	Var	MSE
MME	0.02638	0.00004406	0.00073996	0.03274	0.00001688	0.00108879
MLE	0.00012	0.00002248	0.00002249	0.00004	0.00010879	0.00010879
𝑛=100	Bias	Var	MSE	Bias	Var	MSE
MME	−0.0422653	0.00021173	0.00199808	0.0123275	0.00097912	0.00113109
MLE	−0.00058	0.00002248	0.00002281	0.00026	0.00000402	0.00008408