A stochastic recurrence equations approach for score driven correlation models

Figures & data

Figure 1 Stationarity and ergodicity sufficiency regions for the normal distribution and different scaling choices S(f) = (ℐ_t(f))^−a for a ∈ {0, 1/2, 1}. The two panels contain different regions obtained by parameterizing the matrix roots h(f) with ψ(ρ) = k_ψarcsin (ρ). Panel (a) contains the results for the symmetric matrix root (k_ψ = 1/2), and panel (b) corresponds to the Cholesky decomposition (k_ψ = 1).

Figure 2 Empirical results. Panel (a) shows 60-day rolling correlations and the filtered correlations between the FTSE 100 (U.K.) and Athex Composite (Greece) equity index returns. The right panel puts the empirical estimates obtained by unconstrained estimation into the zoomed stationarity and ergodicity region perspective.

Table 1 Estimation results.

	EWMA	EWMA	EWMA	t(∞)-GAS	t(∞)-GAS	t(∞)-GAS	t(5)-GAS	t(5)-GAS	t(5)-GAS	t(λ)-GAS
				(a = 0)	(a = 1/2)	(a = 1)	(a = 0)	(a = 1/2)	(a = 1)	(a = 1)
λ	5	5	8.9321	∞	∞	∞	5	5	5	9.1411
			(0.0801)							(0.9287)
ω		0.0003	0.0012	0.0117	0.0115	0.0112	0.0089	0.0089	0.0088	0.0114
		(0.0003)	(0.0003)	(0.0016)	(0.0014)	(0.0013)	(0.0011)	(0.0011)	(0.0010)	(0.0013)
α				0.0254	0.0283	0.0314	0.0335	0.0316	0.0297	0.0360
				(0.0040)	(0.0045)	(0.0050)	(0.0048)	(0.0046)	(0.0043)	(0.0050)
β	0.9757	0.9756	0.9739	0.9752	0.9757	0.9763	0.9797	0.9798	0.9800	0.9762
	(0.0085)	(0.0063)	(0.0030)	(0.0021)	(0.0016)	(0.0012)	(0.0013)	(0.0011)	(0.0009)	(0.0012)
Log-likelihood	−7, 847	−7, 847	−7, 812	−7, 891	−7, 890	−7, 890	−7, 845	−7, 845	−7, 845	-7,809
AIC	15, 697	15, 699	15, 631	15, 778	15, 787	15, 787	15, 696	15, 696	15, 696	15,626
BIC	15, 703	15, 711	15,649	15, 814	15, 805	15, 805	15, 714	15, 714	15, 714	15,649
# parameters	1	2	3	3	3	3	3	3	3	4
H: ρt ≡ 0	reject	reject					reject	reject	reject
[3pt] H:		reject	reject	reject	reject	reject	reject	reject	reject
Inside SE region?
Cholesky (kψ = 1)				Yes	Yes	No	No	No	No	No
Symmetric ()				Yes	Yes	Yes	Yes	Yes	Yes	Yes

Heteroskedasticity and autocorrelation consistent (HAC) standard errors in parentheses. Best log-likelihood, AIC, and BIC values across models are printed in boldface. H: ρ_t ≡ 0 and H: indicate whether the Nyblom test among residuals rejects constant zero or estimated correlation, respectively. HAC standard errors are computed using Newey–West weights with min(⌊1.2 × T^1/3⌋, T) lags.