Table 6 Three candidate processes for the study population via time series models per year

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