A mixture autoregressive model based on Student’s t–distribution

Figures & data

Figure 1. Left panel: Daily ${\text{RK}}_t$ (lower solid) and $\text{log}\hspace{0.17em}({\text{RK}}_t)$ (upper solid), and mixing weights based on the estimates of the StMAR(4,2) model in Table 1 (dot-dash) for the $\text{log}\hspace{0.17em}({\text{RK}}_t)$ series. The mixing weights ${\widehat{\alpha }}_{1,t}$ are scaled from $(0, 1)$ to $(\min \hspace{0.17em}\text{log}\hspace{0.17em}({\text{RK}}_t), \max \hspace{0.17em}\text{log}\hspace{0.17em}({\text{RK}}_t)).$ Right panel: A kernel density estimate of the $\text{log}\hspace{0.17em}({\text{RK}}_t)$ observations (solid), and the mixture density (dashes) implied by the same StMAR model as in the left panel.

Table 1. Parameter estimates for three selected StMAR models and the $\text{log}\hspace{0.17em}({\text{RK}}_t)$ data over the period 3 January 2000–3 June 2014.

	StMAR(4, 1)		StMAR(4, 2)		StMAR(4, 3)
${\phi }_{1,0}$	–0.746	(0.089)	–0.851	(0.112)	–0.859	(0.116)
${\phi }_{1,1}$	0.428	(0.017)	0.432	(0.024)	0.407	(0.024)
${\phi }_{1,2}$	0.224	(0.019)	0.221	(0.025)	0.216	(0.025)
${\phi }_{1,3}$	0.121	(0.019)	0.122	(0.025)	0.123	(0.025)
${\phi }_{1,4}$	0.150	(0.017)	0.134	(0.024)	0.162	(0.024)
${\sigma }_1^2$	0.298	(0.011)	0.285	(0.015)	0.283	(0.015)
ν1	11.999	(1.173)	10.510	(1.628)	10.695	(2.070)
${\phi }_{2,0}$			–5.381	(1.036)	–5.745	(1.050)
${\phi }_{2,1}$			0.289	(0.046)	0.300	(0.049)
${\phi }_{2,2}$			0.129	(0.049)	0.121	(0.051)
${\phi }_{2,3}$			0.023	(0.047)	0.015	(0.049)
${\phi }_{2,4}$			0.047	(0.053)	0.019	(0.056)
${\sigma }_2^2$			0.287	(0.022)	0.290	(0.023)
ν2			29.031	(1.957)	32.143	(4.503)
${\phi }_{3,0}$					–5.459	(1.068)
${\phi }_{3,1}$					0.479	(0.102)
${\phi }_{3,2}$					0.334	(0.125)
${\phi }_{3,3}$					0.206	(0.111)
${\phi }_{3,4}$					–0.728	(0.122)
${\sigma }_3^2$					0.088	(0.032)
ν3					35438.182	(0.137)
α1			0.724	(0.065)	0.721	(0.060)
α2					0.256	(0.062)
$T{L}_T^{(c)}({\widehat{\mathit{\theta }}}_T)$	–2854.153		–2832.665		–2814.469
aic	5722.306		5695.330		5674.937
hqc	5737.741		5728.406		5725.653
bic	5765.613		5788.131		5817.233

Numbers in parentheses are standard errors based on a numerical Hessian.

Table 2. The percentage shares of cumulative realized kernel observations that belong to the 99%, 95% and 90% one-sided upper prediction intervals based on the distribution of 1,000,000 simulated conditional sample paths.

	Daily			Weekly
	99%	95%	90%	99%	95%	90%
AR(11)	98.99	95.97	90.52	96.54	91.26	86.18
HAR	98.59	94.76	90.32	96.14	91.06	86.99
StMAR(4,1)	98.99	95.97	92.14	98.17	95.12	90.24
StMAR(4,2)	99.19	95.97	92.54	97.97	94.92	90.65
StMAR(4,3)	99.19	96.17	92.54	97.76	94.72	90.45
	Biweekly			Monthly
	99%	95%	90%	99%	95%	90%
AR(11)	94.05	89.53	85.63	94.11	88.63	85.47
HAR	93.63	88.71	85.01	91.79	87.37	84.00
StMAR(4,1)	97.33	93.22	90.76	97.89	93.89	91.79
StMAR(4,2)	97.33	93.22	90.76	97.47	94.11	91.16
StMAR(4,3)	97.33	93.22	90.55	97.68	93.89	91.37