The role of the financial sector in the UK economy: evidence from a seasonal cointegration analysis

Figures & data

Table 1. Seasonal unit roots test (1965Q1–2011Q4).

Variables	Deterministic Components	Auxiliary Regression
		t(Π1)	C.V.	t(Π2)	C.V.	t(Π3)	C.V.	t(Π4)	C.V.	F(Π3∩Π4)	C.V.
LCAPt	C, SD, T	−8.31	−3.51	−9.07	−2.92	−9.66	−3.43	−5.31	−1.93	74.55	6.60
LBCt	C, SD, T	−5.58	−3.51	−5.68	−2.92	−5.95	−3.43	−2.24	−1.93	89.70	6.60
LMCt	C, SD, T	−7.13	−3.51	−7.29	−2.92	−6.34	−3.43	−3.84	−1.93	85.01	6.60
LVt	C, SD, T	−6.76	−3.51	−7.26	−2.92	−6.72	−3.43	−4.02	−1.93	71.65	6.60
LKRt	C, SD, T	−5.26	−3.51	−6.16	−2.92	−5.21	−3.43	−3.32	−1.93	73.51	6.60
LHRt	C, SD, T	−5.43	−3.51	−5.76	−2.92	−6.42	−3.43	−3.91	−1.93	79.14	6.60
LNGDt	C, SD, T	−6.03	−3.51	−6.23	−2.92	−6.44	−3.43	−3.02	−1.93	82.65	6.60

Notes: The reported critical values at 5% significance level are obtained from Hylleberg et al. (1990). It is worth noting that the critical values for 172 numbers of observations do not exist in the relevant table that is tabulated by Hylleberg et al. (1990). Hence, we have calculated the reported critical values approximately for 172 numbers of observations by using the relevant table. Dependent variables in the table are in the seasonal differences, C intercept; SD seasonal dummies; and T trend. Source: Authors' analysis.

Table 2. Cointegration test at frequency 0: the long run.

Cointegration Regression			Auxiliary Regression		Calculated ADF Residuals
Equation	R^2	Deterministic Components	Deterministic Part	Augmentation	ADF, t(Φ)
LCAP1t = f (LKR1t, LHR1t, LNGD1t LBC1t, LMC1t, LV1t)	0.96	C,T	–	1, 2	−4.69

Notes: The tests are based upon the ordinary ADF regression where u_t is the residual from the cointegration regression. The critical value is –4.28 at 5% significance level obtained from Engle and Yoo (1987), Engle and Granger (1987) and Mackinnon (1991). It is worth noting that deterministic components (i.e. trend and intercept) are included for the equation in the table above. Source: Authors' analysis.

Table 3. Cointegration test at frequency ½ : biannual.

Cointegration Regression			Auxiliary Regression		Calculated ADF Residuals
Equation	R^2	Deterministic Components	Deterministic Part	Augmentation	ADF, t(Φ)
LCAP2t = f (LKR2t, LHR2t, LNGD2t,LBC2t, LMC2t, LV2t)	0.97	C, T	–	1, 2	−7.52

Notes: The tests are based upon the auxiliary regression where V_t is the residual from the cointegration regression. The critical value is –4.28 at 5% significance level obtained from Engle and Yoo (1987), Engle and Granger (1987) and Mackinnon (1991). It is worth noting that deterministic components (i.e. trend and intercept) are included for the equation in the table above. Source: Authors' analysis.

Table 4. Cointegration test at frequencies ¼ and ¾: annual.

Cointegration Regression			Auxiliary Regression		HEGY Test
Equation	R^2	Deterministic Components	Deterministic Part	Augmentation	tλ1	tλ2	Fλ1∩λ2
LCAP3t = f (LKR3t, LHR3t, LNGD3t,LBC3t, LMC3t, LV3t)	0.94	C, T	–	1, 2	−5.73	−3.25	19.34

Notes: The tests are based upon the auxiliary regression,

where w_t is the residual from the cointegration regression. The critical values are at 5% significance level t: λ_1=-3.22, t: λ₂ = -2.13, and F: λ₁∩λ₂ = 7.01, obtained from Engle et al. (1993). Note that deterministic components (i.e. trend and intercept) are included for the relevant equation. Source: Authors' analysis.

Table 5. Seasonal error correction models.

Explanatory Variables	Dependent Variable
ECT	Δ4LCAP
C	0.04***	0.05***	0.03***
	(3.21)	(3.32)	(4.17)
ECT1t (-1)	−0.25***	-
	(−5.14)
ECT2t (-1)	–	−0.21***
		(−4.31)
ECT3t (-1)	–	–	−0.19***
			(−2.59)
Δ4LCAP(-1)	0.54***	0.61***	0.49***
	(7.03)	(4.58)	(3.83)
Δ4LBC(-1)	0.04^	0.06^	0.08^
	(1.54)	(1.59)	(1.61)
Δ4LMC(-1)	0.03*	0.14*	–
	(1.71)	(1.77)
Δ4LMC(-2)	–	–	0.14***
			(2.81)
Δ4LV(-1)	–	−0.11*	–
		(−1.93)
Δ4LV(-2)	−0.19**	−0.21*	−0.15*
	(−2.27)	(−1.68)	(−1.72)
Δ4LHR(-1)	0.35***	0.37**	−0.13**
	(3.09)	(2.24)	(2.19)
Δ4LKR(-1)	0.43***	0.41**	0.44***
	(3.15)	(2.07)	(2.94)
Δ4LNGD(-2)	−0.17*	−0.18*	−0.21*
	(−1.64)	(−1.66)	(−1.65)

Notes: ECT(-1) is the residual series from the OLS cointegrating regression in tables 2, 3 and 4. Numbers in the parentheses are t values. ***, **, * indicate 1%, 5%, and 10% level of significance respectively and ^ shows that they are not insignificant at the conventional levels. All diagnostic passes at 5% level of significance. Source: Authors' analysis.

Table 6. Johansen Maximum Likelihood (ML) procedure cointegration likelihood ratio (LR) test to determine the number of cointegration vectors (r) based upon Maximal Eigen value of the stochastic matrix, and trace of the stochastic matrix from a seasonal point of view suggested by Lee (1992).

Cointegration Regression	Null Hypothesis	Alternative Hypothesis	λmax	Critical Value at 5%	λtrace	Critical Value at 5%
Long run: (0)	r = 0	r = 1	76.25	43.61	122.99	115.85
	r <= 1	r = 2	35.26	37.86	86.31	87.17
	r <= 2	r = 3	23.55	31.79	56.47	63.00
LCAP1t = f (LKR1t, LHR1t, LNGD1t,LBC1t, LMC1t, LV1t)	r <= 3	r = 4	21.72	25.42	38.17	42.34
	r <= 4	r = 5	10.35	19.22	18.21	25.77
	r <= 5	r = 6	4.82	12.39	4.82	12.39
Biannual: ½	r = 0	r = 1	72.88	43.61	135.88	115.85
	r <= 1	r = 2	30.43	37.86	87.00	87.17
	r <= 2	r = 3	28.17	31.79	61.57	63.00
LCAP2t = f (LKR2t, LHR2t, LNGD2t,LBC2t, LMC2t, LV2t)	r <= 3	r = 4	23.46	25.42	35.67	42.34
	r <= 4	r = 5	11.51	19.22	23.02	25.77
	r <= 5	r = 6	9.42	12.39	9.42	12.39
Annual: 1/3 (or ¾)	r = 0	r = 1	82.43	43.61	136.59	115.85
	r <= 1	r = 2	36.29	37.86	83.85	87.17
	r <= 2	r = 3	26.79	31.79	57.06	63.00
LCAP3t = f (LKR3t, LHR3t, LNGD3t,LBC3t, LMC3t, LV3t)	r <= 3	r = 4	21.15	25.42	38.76	42.34
	r <= 4	r = 5	18.39	19.22	22.48	25.77
	r <= 5	r = 6	9.08	12.39	9.08	12.39

Notes: r indicates the number of cointegrating vectors. λ_max is the maximum eigenvalue statistics, λ_trace is the trace statistics. VAR 4 based on a common practise in which four lags are used due to quarterly data (see Bahmani-Oskooee and Shabsigh (1996). The unrestricted constant and restricted trends are not rejected in all cases. The critical values are obtained from Osterwald-Lenum (1992). Source: Authors' analysis.

Table 7. Testing for weak exogeneity using Johansen approach from a seasonal point of view.

Relation	Variables	Test for Weak Exogeneity
		Test Statistics	Conclusion/Loading factor/p-value
	LBC1t	χ^2(1) = 3.73 (.04)	Accept /0.08/0.05
Long Run	LMC1t	χ^2(1) = 0.42 (.54)	Accept/0.10/0.04
Or	LV1t	χ^2(1) = 2.39 (.12)	Accept/-0.12/0.05
Zero Frequency	LKR1t	χ^2(1) = 2.73 (.10)	Accept/0.25/0.02
Normalised on LCAP1t	LHR1t	χ^2(1) = 1.32 (.25)	Accept/0.21/0.01
	LNGD1t	χ^2(1) = 0.45 (.51)	Accept/-0.16/0.10
	LBC2t	χ^2(1) = 0.06 (.80)	Accept/0.11/0.06
Biannual	LMC2t	χ^2(1) = 1.33 (.26)	Accept/0.13/0.03
Or	LV2t	χ^2(1) = 1.99 (.44)	Accept/-0.12/0.07
½ Frequency	LKR2t	χ^2(1) = 0.06 (.80)	Accept/0.29/0.01
Normalised on LCAP2t	LHR3t	χ^2(1) = 2.57 (.11)	Accept/0.22/0.03
	LNGD1t	χ^2(1) = 3.82(.05)	Accept/-0.09/0.08
	LBC3t	χ^2(1) = 0.10 (.75)	Accept/0.09/0.05
Annual	LMC3t	χ^2(1) = 0.05 (.90)	Accept/0.14/0.03
Or	LV3t	χ^2(1) = 3.81 (.05)	Accept/-0.11/0.07
¼ (and ¾) Frequency	LKR3t	χ^2(1) = 0.78 (.37)	Accept/0.31/0.02
Normalised on LCAP3t	LHR3t	χ^2(1) = 3.80 (.04)	Accept/0.19/0.03
	LNGD3t	χ^2(1) = 2.75 (.15)	Accept/-0.11/0.09

Notes: Table 7 indicates the results that the hypothesis of weak exogeneity cannot be rejected at 5% level. The tabulated test statistics of χ²(1) is 3.84. Source: Authors' analysis.

Table 8. Selection of lag lengths using fFPE.

Dependent Variable	Independent Variable	m^*	n^*	FPE (m^*)	FPE (m^*, n^*)
D4LCAP	D4LBC	5	1	0.151 x 10^−3	0.163 x 10^−3
D4LBC	D4LCAP	5	1	3.175 x 10^−3	3.331 x 10^−3
D4LCAP	D4LMC	5	1	0.1619 x 10^−3	0.1612 x 10^−3
D4LMC	D4LCAP	5	1	18.37 x 10^−3	18.42 x 10^−3
D4LCAP	D4LV	5	1	0.2119 x 10^−3	0.2114 x 10^−3
D4LV	D4LCAP	5	1	19.54 x 10^−3	19.65 x 10^−3
D4LCAP	D4LHR	5	1	0.1726 x 10^−3	0.1719 x 10^−3
D4LHR	D4LCAP	5	1	2.31x 10^−3	2.39 x 10^−3
D4LCAP	D4LKR	5	1	0.563 x 10^−3	0.561 x 10^−3
D4LKR	D4LCAP	5	1	4.21 x 10^−3	4.48 x 10^−3

Notes: If FPE (m^*, n^*) < FPE (m^*), Y Granger-causes X.

m^* denotes maximum lag on dependent variable.

n^* denotes minimum lag on independent variable. Source: Authors' analysis.

Table 9. Vector autoregression models; GC and HH tests.

Dependent Variable	Independent Variable	Degrees Of freedom^a	Wald Test	Sim’s LR Test	m^*	n^*	HH Multiple-rankF-test	Causal Inference
D4LCAP	D4LBC	1	0.029	0.031	5	1	0.168	NC
							(1.128)^b
D4LBC	D4LCAP	1	0.056	0.059	5	1	0.006	NC
							(1.128)
D4LCAP	D4LMC	1	11.21^*	11.89^*	5	1	5.06^*	MC → CAP
							(1.128)
D4LMC	D4LCAP	1	3.31	3.34	5	1	1.25	NC
							(1.128)
D4LCAP	D4LV	1	7.24^*	7.43^*	5	1	4.52^**	V → CAP
							(1.128)
D4LV	D4LCAP	1	2.31	2.47	5	1	3.27	NC
							(1.128)
D4LCAP	D4LHR	1	6.84^*	6.89^*	5	1	3.86^**	HR → CAP
							(1.128)
D4LHR	D4LCAP	1	3.77	3.81	5	1	3.45	NC
							(1.128)
D4LCAP	D4LKR	1	5.55^*	5.86^*	5	1	4.18^**	KR → CAP
							(1.128)
D4LKR	D4LCAP	1	3.82	3.83	5	1	3.29	NC
							(1.128)

Notes:

^* indicates significance at the 1% level.

^**indicates significance at the 5% level.

^aχ² degrees of freedom for both Wald and Sims’ LR tests.

^bdegrees of freedom for HH multiple-rank F-test.

NC no causality. Source: Authors' analysis.

Table 10. Summary of causality results χ²-Test.

Seasonal Differences
FPE	χ^2-Test	HH Multiple-Rank F-Test
CAP – BC(NC)	CAP – BC(NC)	CAP – BC(NC)
BC – CAP(NC)	CAP – BC(NC)	CAP – BC(NC)
CAP – MC(NC)	CAP – MC(NC)	CAP – MC(NC)
MC → CAP	MC → CAP	MC → CAP
CAP – V(NC)	CAP – V(NC)	CAP – V(NC)
V → CAP	V → CAP	V → CAP
CAP – HR(NC)	CAP – HR(NC)	CAP – HR(NC)
HR → CAP	HR → CAP	HR → CAP
CAP – KR(NC)	CAP – KR(NC)	CAP – KR(NC)
KR → CAP	KR → CAP	KR → CAP

Notes:

CAP; Real GDP per worker, BC; Domestic bank credit, MC; Stock market capitalisation, V; stock market volatility, HR; human capital or tertiary enrolment rate, KR; physical capital or fixed capital formation and NC; No causality. Source: Authors' analysis.

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