Table 1 Log-likelihood values of the...

Table 1

Log-likelihood values of the different models and specification tests

Countries	Types of models			First step		Second step
	Model designation	Parameters θ₁ and θ₂		Negativity conditions not imposed			Negativity conditions imposed
	Model designation	Parameters θ₁ and θ₂		Log L	LR^a	LRC^b,c	Log L	BIC^d
Germany	General	θ₁ = 0.858^***	θ₂ = 0.153	379.184
	Rotterdam	θ₁ = 0.000	θ₂ = 0.000	374.349	9.670	6.080
	CBS^d	θ₁ = 1.000	θ₂ = 0.000	378.908	0.552	0.347	367.192	−307.554
	AID	θ₁ = 1.000	θ₂ = 1.000	373.076	12.216	7.681
	NBR	θ₁ = 0.000	θ₂ = 1.000	370.918	16.532	10.395
Italy	General	θ₁ = 0.542	θ₂ = 0.642	298.003
	Rotterdam	θ₁ = 0.000	θ₂ = 0.000	297.153	1.700	1.113	293.105	−248.028
	CBS	θ₁ = 1.000	θ₂ = 0.000	296.025	3.956	2.589	295.964	−250.887
	AID	θ₁ = 1.000	θ₂ = 1.000	296.565	2.876	1.882	283.870	−250.656
	NBR	θ₁ = 0.000	θ₂ = 1.000	297.695	0.616	0.403	284.899	−251.685
United Kingdom	General	θ₁ = 0.470^**	θ₂ = 0.590^*	292.742
	Rotterdam	θ₁ = 0.000	θ₂ = 0.000	290.220	5.044	3.172	288.180	−223.572
	CBS	θ₁ = 1.000	θ₂ = 0.000	283.314	18.856	11.856
	AID	θ₁ = 1.000	θ₂ = 1.000	290.462	4.560	2.867	280.045	−227.862
	NBR	θ₁ = 0.000	θ₂ = 1.000	288.487	8.510	5.351	274.559	−221.976

Countries	Types of models			First step		Second step
	Model designation	Parameters θ₁ and θ₂		Negativity conditions not imposed			Negativity conditions imposed
	Model designation	Parameters θ₁ and θ₂		Log L	LR^a	LRC^b,c	Log L	BIC^d
Germany	General	θ₁ = 0.858^***	θ₂ = 0.153	379.184
	Rotterdam	θ₁ = 0.000	θ₂ = 0.000	374.349	9.670	6.080
	CBS^d	θ₁ = 1.000	θ₂ = 0.000	378.908	0.552	0.347	367.192	−307.554
	AID	θ₁ = 1.000	θ₂ = 1.000	373.076	12.216	7.681
	NBR	θ₁ = 0.000	θ₂ = 1.000	370.918	16.532	10.395
Italy	General	θ₁ = 0.542	θ₂ = 0.642	298.003
	Rotterdam	θ₁ = 0.000	θ₂ = 0.000	297.153	1.700	1.113	293.105	−248.028
	CBS	θ₁ = 1.000	θ₂ = 0.000	296.025	3.956	2.589	295.964	−250.887
	AID	θ₁ = 1.000	θ₂ = 1.000	296.565	2.876	1.882	283.870	−250.656
	NBR	θ₁ = 0.000	θ₂ = 1.000	297.695	0.616	0.403	284.899	−251.685
United Kingdom	General	θ₁ = 0.470^**	θ₂ = 0.590^*	292.742
	Rotterdam	θ₁ = 0.000	θ₂ = 0.000	290.220	5.044	3.172	288.180	−223.572
	CBS	θ₁ = 1.000	θ₂ = 0.000	283.314	18.856	11.856
	AID	θ₁ = 1.000	θ₂ = 1.000	290.462	4.560	2.867	280.045	−227.862
	NBR	θ₁ = 0.000	θ₂ = 1.000	288.487	8.510	5.351	274.559	−221.976

Note(s): Log L = Log likelihood, LR = likelihood ratio statistic, LRC = Corrected likelihood ratio statistic, BIC=Bayesian information criterion

^aLR = −2 × (LogL_restricted-Log L_general)

^b $L R C = L R [\frac{M T - 0.5 (N_{U} + N_{R}) - 0.5 M (M + 1)}{M T}]$ where M is the number of equations, T is the number of time series observations, N_U is the number of parameters in the unrestricted model specification (i.e. Barten's general model), N_R is the number of parameters in the restricted model specification

^cThe null hypothesis of selecting one of the four restricted model specifications is not rejected if the LRC statistic is smaller than a critical χ² value with two degrees of freedom. The adopted critical χ² value to perform the above model specification tests is 5.99 for a 5% level of significance

BIC = −2 × Log L + Log(MT) × N where N is the number of parameters

^dHeadings and figures marked in bold characters are the selected model specifications

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