Table 2. Results of linearity tests...

Table 2.

Results of linearity tests

McLeod and Li Model: $y_{t} = β_{0} + Σ_{i = 1}^{p} δ_{i} y_{t - p} + Σ_{j = 1}^{q} δ_{j} ε_{t - q} + ε_{t}$
Keenan Model 1: $y_{t} = β_{0} + Σ_{i = 1}^{p} δ_{i} y_{t - p} + ε_{t}$
Keenan Model 2: $y_{t}^{2} = β_{0} + Σ_{i = 2}^{p} δ_{i} y_{t - i} + V_{t}$
Tsay Model 1: $y_{t} = β_{0} + Σ_{i = 1}^{p} δ_{i} y_{t - p} + ε_{t}$
Tsay Model 2: $y_{t} = β_{0} + Σ_{i = 1}^{p} δ_{i} y_{t - p} + Σ_{l = 1}^{p} θ_{l} y_{t - l}^{2} + Σ_{j = 2}^{p} ϑ_{j} y_{t - 1} y_{t - j} + Σ_{j = 3}^{p} \emptyset_{j} y_{t - 2} y_{t - j} + \dots + Σ_{j = p}^{p} φ_{j} y_{t - p - 1} y_{t - p} + V_{t}$
Harvey and Leybourne Model: $y_{t} = β_{0} + β_{1} y_{t - 1} + β_{2} y_{t - 1}^{2} + β_{3} y_{t - 1}^{3} + β_{4} Δ y_{t - 1} + β_{5} {(Δ y_{t - 1})}^{2} + β_{6} {(Δ y_{t - 1})}^{3} + ε_{t}$
Harvey et al. Model ^I(0): $y_{t} = β_{0} + β_{1} y_{t - 1} + β_{2} y_{t - 2}^{2} + β_{3} y_{t - 3}^{3} + Σ_{i = 1}^{k} δ_{i} Δ y_{t - i} + ε_{t}$
Harvey et al. Model ^I(1): $y_{t} = λ_{1} Δ y_{t - 1} + λ_{2} {(Δ y_{t - 1})}^{2} + λ_{3} {(Δ y_{t - 1})}^{3} + Σ_{i = 1}^{k} δ_{i} Δ y_{t - i} + ε_{t}$
Tests	McLeod and Li	k	Keenan	Tsay	Harvey and Leybourne			Harvey et al.
Bulgaria	(0.00)^***	3	0.467	0.713	9.87	9.75^**	9.68	5.41^*
Croatia	(0.00)^***	6	2.409	2.018^***	15.98^***	15.79	15.68	13.44^***
CzechR	(0.00)^***	4	0.026	1.676^*	1.32	1.28	1.25	0.26
Estonia	(0.00)^***	10	8.537^***	1.619^**	7.48	7.42	7.38	3.30
Hungary	(0.00)^***	3	0.036	0.532	2.33	2.30	2.28	0.73
Kazakhstan	(0.00)^***	3	1.478	7.186^***	49.52^***	49.00	48.71	58.96^***
KyrgyzR	(0.00)^***	1	0.402	0.132	10.43	10.25^**	10.15	0.58
Latvia	(0.00)^***	9	9.594^***	0.874	7.35	7.29	7.25	3.74
Lithuania	(0.00)^***	7	5.275^**	1.686^**	7.72	7.64	7.59	3.90
Poland	(0.00)^***	2	0.003	0.967	3.44	3.31	3.23	3.29
Romania	(0.00)^***	2	0.047	1.929	5.71	5.48	5.35	4.80^*
SlovakR	(0.00)^***	2	0.145	4.712^***	5.36	5.13	5.00	2.92
Slovenia	(0.00)^***	7	1.745	1.107	2.40	2.37	2.35	0.43

McLeod and Li Model: $y_{t} = β_{0} + Σ_{i = 1}^{p} δ_{i} y_{t - p} + Σ_{j = 1}^{q} δ_{j} ε_{t - q} + ε_{t}$
Bulgaria	(0.00)^***	3	0.467	0.713	9.87	9.75^**	9.68	5.41^*
Croatia	(0.00)^***	6	2.409	2.018^***	15.98^***	15.79	15.68	13.44^***
CzechR	(0.00)^***	4	0.026	1.676^*	1.32	1.28	1.25	0.26
Estonia	(0.00)^***	10	8.537^***	1.619^**	7.48	7.42	7.38	3.30
Hungary	(0.00)^***	3	0.036	0.532	2.33	2.30	2.28	0.73
Kazakhstan	(0.00)^***	3	1.478	7.186^***	49.52^***	49.00	48.71	58.96^***
KyrgyzR	(0.00)^***	1	0.402	0.132	10.43	10.25^**	10.15	0.58
Latvia	(0.00)^***	9	9.594^***	0.874	7.35	7.29	7.25	3.74
Lithuania	(0.00)^***	7	5.275^**	1.686^**	7.72	7.64	7.59	3.90
Poland	(0.00)^***	2	0.003	0.967	3.44	3.31	3.23	3.29
Romania	(0.00)^***	2	0.047	1.929	5.71	5.48	5.35	4.80^*
SlovakR	(0.00)^***	2	0.145	4.712^***	5.36	5.13	5.00	2.92
Slovenia	(0.00)^***	7	1.745	1.107	2.40	2.37	2.35	0.43

Notes: McLeod and Li (1983) test statistics are obtained using the squares of residuals obtained from the ARMA model in the Ljung and Box (1978) statistic; in the Tsay (1986) test, residuals obtained from Models 1 and 2 equations are regressed and F test statistics are obtained from this equation; in the Keenan (1985) test, residuals obtained from Models 1 and 2 equations are regressed and F test statistics are obtained from two equations; $χ_{4}^{2}$ Table critical values of and the $χ_{2}^{2}$ table critical values of Harvey et al. (2008) test are 9.21, 5.99, 4.60, respectively;

***,

** and

*simply rejection of the linearity at the 1%, 5% and 10% levels, respectively; values in parentheses indicate prob. values

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