Table 6.

Three candidate processes for the study population via time series models per year

Model
ARMA (1, 1)	$\widehat{b}$	S.E. ($\widehat{b}$)	p value	RMSE
Constant	5.90	1.67	0.001*	8.68
AR (1)	0.66	0.13	< 0.001*
MA (1)	−0.37	0.16	0.025*
Modified Box-Pierce test	Lag	Chi-square	df	p value
	12	24.0	9	0.004*
	24	33.9	21	0.037*
ARMA(1, 2)	$\widehat{b}$	S.E. ($\hat{b}$)	p value	RMSE
Constant	0.72	0.07	< 0.001*	10.06
AR (1)	0.96	0.25	< 0.001*
MA(1)	0.25	0.29	0.384
MA (2)	0.71	0.20	0.001*
Modified Box-Pierce test	Lag	Chi-square	df	p value
	12	40.7	8	< 0.001*
	24	80.7	20	< 0.001*
ARMA (1, 1) × (0, 1)12	$\hat{b}$	S.E. ($\widehat{b}$)	p value	RMSE
Constant	6.42	2.21	0.006*	8.09
AR (1)	0.63	0.14	< 0.001*
MA (1)	−0.38	0.16	0.026*
SMA (1)	−0.41	0.14	0.005*
Modified Box-Pierce test	Lag	Chi-square	df	p value
	12	13.2	8	0.104
	24	19.9	20	0.466

ARMA auto-regressive moving average, ARMA (p, q) × (P, Q)_h mixed seasonal ARMA, $\widehat{\mathit{b}}$ coefficient, df degree of freedom, S.E. ( $\widehat{\mathit{b}}$ ) standard error of coefficient, RMSE root mean square error

*p value <0.05 is significant