A Heteroskedasticity-Robust F-Test Statistic for Individual Effects

Figures & data

TABLE 1 Rejection frequencies of the asymptotic and wild-bootstrap modified F-tests and modified random effects tests under homoskedastic errors (HET0)

H 0: α i  = 0									H 1: var(α i ) = 0.1, α i correlated with regressors
	Asymptotic tests				Bootstrap tests				Asymptotic tests				Bootstrap tests
ω	1				1				1				1
	SN								SN
N, T	F ω								F ω
20, 5	5.8	2.8	4.7	5.9	6.2	5.9	5.9	6.1	29.0	18.0	25.4	29.4	30.4	29.9	29.8	30.4
50, 5	5.2	2.8	4.5	5.2	5.7	5.8	5.8	5.7	46.8	38.6	44.4	47.1	47.9	48.8	48.3	47.8
100, 5	4.7	3.1	4.1	4.8	5.3	5.2	5.3	5.3	72.1	66.6	70.8	72.3	73.0	74.3	73.6	73.0
50, 10	4.3	1.9	3.1	4.3	4.5	4.7	4.4	4.5	94.2	87.9	92.6	94.2	94.4	94.0	94.4	94.4
50, 20	4.7	1.6	3.4	4.8	5.1	5.2	5.0	5.1	100.0	100.0	100.0	100.0	100.0	100.0	100.0	100.0
	R ω								R ω
20, 5	5.1	1.5	3.7	5.2	6.1	5.9	6.1	6.1	23.9	10.8	18.8	24.4	27.0	25.4	26.1	26.9
50, 5	4.6	2.3	3.9	4.5	5.6	5.6	5.6	5.5	32.5	20.4	29.3	32.7	35.9	35.1	35.3	35.8
100, 5	4.4	2.8	3.8	4.6	5.3	5.2	5.1	5.3	55.8	44.7	52.6	55.9	57.6	57.1	57.5	57.7
50, 10	4.1	1.7	2.9	4.0	4.5	4.8	4.5	4.5	89.7	77.7	87.2	89.8	90.5	89.6	90.3	90.5
50, 20	4.6	1.6	3.4	4.6	5.2	5.3	5.0	5.2	100.0	99.9	100.0	100.0	100.0	100.0	100.0	100.0
	t 5								t 5
N, T	F ω								F ω
20, 5	4.8	2.4	3.9	5.3	5.5	5.4	5.5	5.4	30.1	20.7	27.6	31.5	31.9	31.7	31.4	31.7
50, 5	4.6	2.7	4.0	4.9	5.3	5.0	5.3	5.2	47.9	40.3	46.4	49.0	49.2	50.5	50.0	49.1
100, 5	5.3	3.5	4.5	5.3	5.9	5.9	5.9	5.8	72.6	68.2	71.4	72.7	73.3	74.7	74.0	73.3
50, 10	5.2	2.1	4.0	5.3	5.7	5.4	5.6	5.7	93.6	87.1	92.0	93.6	94.0	93.3	93.5	93.9
50, 20	4.8	1.5	3.4	4.8	5.1	5.2	5.1	5.1	100.0	99.9	100.0	100.0	100.0	100.0	100.0	100.0
	R ω								R ω
20, 5	4.4	1.5	3.2	4.6	5.7	5.6	5.4	5.4	24.2	12.7	21.2	25.8	28.6	26.7	27.7	28.6
50, 5	4.1	2.3	3.6	4.5	5.5	5.1	5.3	5.4	32.9	21.8	30.2	33.9	36.4	35.1	36.1	36.5
100, 5	5.0	3.3	4.3	5.0	5.8	6.1	5.8	5.9	56.6	47.1	55.1	57.7	58.8	58.7	59.2	58.8
50, 10	5.0	2.0	3.8	5.0	5.9	5.4	5.6	5.8	89.3	77.6	87.1	89.7	90.1	88.3	89.9	90.1
50, 20	4.6	1.5	3.4	4.5	5.2	5.2	5.1	5.2	100.0	99.8	100.0	100.0	100.0	100.0	100.0	100.0
N, T	F ω								F ω
20, 5	4.5	2.3	3.6	4.4	4.7	5.1	4.7	4.5	30.1	19.7	27.4	31.0	31.5	31.6	32.2	31.4
50, 5	5.1	2.3	3.6	4.8	4.9	5.0	5.0	4.9	46.3	38.4	44.2	46.6	47.2	49.1	47.6	47.3
100, 5	4.9	3.0	4.0	4.8	5.0	5.3	5.2	5.0	72.8	67.8	72.3	73.6	74.4	75.5	74.9	74.3
50, 10	4.5	1.9	3.3	4.4	5.1	5.4	5.0	5.1	93.0	86.9	91.5	93.3	93.3	92.7	93.4	93.4
50, 20	5.2	1.8	3.6	4.9	5.1	5.5	5.4	5.1	100.0	99.9	100.0	100.0	100.0	100.0	100.0	100.0
	R ω								R ω
20, 5	4.3	1.4	2.9	3.9	4.7	5.4	4.6	4.4	24.3	11.4	20.3	25.2	27.8	26.8	27.5	27.6
50, 5	4.6	1.8	3.1	4.1	4.9	4.9	4.9	4.8	31.5	20.5	27.8	31.6	34.2	33.3	34.1	34.2
100, 5	4.8	2.8	3.9	4.6	5.1	5.4	5.4	4.9	57.0	45.9	53.9	57.5	59.2	58.2	58.6	59.3
50, 10	4.4	1.7	3.2	4.3	5.1	5.3	5.0	5.1	89.1	76.5	86.7	89.3	89.8	88.1	89.5	89.9
50, 20	5.1	1.7	3.6	4.8	5.2	5.5	5.4	5.2	100.0	99.9	100.0	100.0	100.0	100.0	100.0	100.0

TABLE 2 Rejection frequencies of the asymptotic and wild-bootstrap modified F-tests and modified random effects tests under cross-sectional one-break-in-volatility heteroskedastic scheme (HET1)

H 0: α i  = 0									H 1: var(α i ) = 0.1, α i correlated with regressors
	Asymptotic tests				Bootstrap tests				Asymptotic tests				Bootstrap tests
ω	1				1				1				1
	SN								SN
N, T	F ω								F ω
20, 5	9.4	2.7	4.9	6.6	6.1	5.8	5.9	5.9	26.9	14.2	18.4	21.0	20.5	22.8	20.5	20.0
50, 5	9.2	2.8	4.7	5.6	5.8	5.5	5.8	5.7	37.6	21.2	26.3	28.8	28.0	31.6	29.1	27.6
100, 5	9.1	3.2	4.7	5.4	5.5	5.4	5.5	5.5	55.2	40.4	43.3	44.7	44.2	49.5	46.2	44.1
50, 10	9.1	1.8	3.8	5.2	5.2	4.9	5.1	5.2	82.7	68.1	73.3	74.5	74.1	81.5	78.0	74.1
50, 20	8.7	1.2	2.9	4.5	4.8	4.8	4.9	4.8	99.8	98.5	99.5	99.4	99.5	99.7	99.7	99.5
	R ω								R ω
20, 5	8.6	1.6	3.7	5.6	6.1	5.6	5.8	5.9	22.5	8.3	13.8	16.6	18.0	19.8	18.2	17.7
50, 5	8.8	2.0	4.1	5.2	5.8	5.6	5.7	5.7	27.9	10.3	16.3	19.0	20.9	21.4	21.0	20.7
100, 5	8.7	2.8	4.3	5.1	5.5	5.3	5.4	5.4	42.7	24.9	29.7	31.9	32.9	36.3	34.0	32.6
50, 10	8.7	1.6	3.5	5.0	5.2	4.9	5.2	5.2	75.5	53.9	63.3	65.7	66.1	73.4	69.3	66.0
50, 20	8.6	1.2	3.0	4.6	4.8	4.7	4.9	4.8	99.7	97.6	99.0	99.1	99.0	99.5	99.5	99.0
	t 5								t 5
N, T	F ω								F ω
20, 5	8.5	2.7	4.6	6.0	5.8	5.6	5.4	5.6	26.6	16.0	20.0	21.9	21.7	23.5	22.4	21.1
50, 5	8.6	2.9	4.3	5.5	5.2	5.3	5.4	5.1	39.3	24.4	28.6	30.9	30.7	33.1	31.5	30.3
100, 5	10.4	3.3	4.8	6.2	5.8	6.0	6.1	5.8	56.9	43.2	46.4	47.4	47.3	52.3	49.5	47.1
50, 10	9.2	1.8	3.9	5.2	5.6	5.2	5.5	5.5	82.1	68.0	73.7	74.6	73.6	80.4	77.1	73.5
50, 20	8.8	1.2	3.0	4.7	5.1	5.3	4.7	5.1	99.7	98.2	99.0	99.3	99.2	99.4	99.4	99.2
	R ω								R ω
20, 5	7.6	1.8	3.6	5.1	5.9	5.5	5.6	5.6	22.4	9.3	14.6	17.6	19.2	21.0	19.4	18.4
50, 5	8.1	2.1	3.8	4.9	5.3	5.2	5.2	5.2	28.6	11.7	17.3	20.5	21.9	22.7	22.0	21.6
100, 5	9.9	2.8	4.4	5.7	6.0	6.0	6.0	5.9	44.5	27.2	32.4	35.1	35.6	38.8	37.1	35.5
50, 10	8.8	1.7	3.7	5.1	5.5	5.1	5.6	5.6	75.3	56.3	64.6	66.5	67.1	72.7	70.2	67.0
50, 20	8.7	1.3	3.0	4.7	5.1	5.3	4.8	5.0	99.5	96.9	98.5	99.0	98.9	99.0	99.1	98.9
N, T	F ω								F ω
20, 5	8.4	2.6	4.5	5.5	5.1	5.4	5.0	4.9	26.4	15.1	18.8	20.8	19.9	23.0	21.1	19.3
50, 5	8.4	2.2	3.7	4.5	4.5	4.9	4.7	4.4	36.6	21.2	25.7	27.6	27.5	30.2	28.4	27.1
100, 5	9.5	3.0	4.4	5.2	5.5	5.5	5.3	5.4	57.3	41.0	44.5	45.9	45.7	50.6	47.6	45.6
50, 10	9.1	1.7	3.4	4.7	4.8	5.1	5.0	4.8	81.4	67.6	72.7	74.0	73.9	80.0	76.7	73.9
50, 20	8.5	1.6	3.4	5.1	4.7	5.1	4.7	4.7	99.7	98.4	99.4	99.5	99.4	99.5	99.6	99.4
	R ω								R ω
20, 5	7.6	1.4	3.2	4.7	5.3	5.2	5.2	5.0	22.2	8.5	14.1	16.2	17.8	19.7	18.5	17.3
50, 5	7.9	1.7	3.1	4.0	4.5	4.9	4.7	4.4	26.7	10.6	15.5	18.3	19.8	20.2	20.1	19.5
100, 5	9.1	2.5	4.1	4.9	5.4	5.4	5.4	5.4	43.9	25.9	30.2	32.9	33.2	37.1	34.8	33.0
50, 10	8.9	1.6	3.1	4.4	4.9	5.0	5.0	4.8	74.6	53.7	63.2	65.7	66.1	71.5	69.5	66.0
50, 20	8.4	1.6	3.4	5.1	4.8	5.2	4.7	4.8	99.6	97.3	99.0	99.1	99.1	99.3	99.4	99.0

TABLE 3 Rejection frequencies of the asymptotic and wild-bootstrap modified F-tests and modified random effects tests under time-series one-break-in-volatility heteroskedastic scheme (HET2)

H 0: α i  = 0									H 1: var(α i ) = 0.1, α i correlated with regressors
	Asymptotic tests				Bootstrap tests				Asymptotic tests				Bootstrap tests
ω	1				1				1				1
	SN								SN
N, T	F ω								F ω
20, 5	3.1	3.1	4.3	5.2	5.5	5.6	5.7	5.6	27.2	20.4	25.8	32.7	33.3	29.5	29.7	33.5
50, 5	3.2	3.6	4.5	5.1	5.5	5.9	5.6	5.5	44.9	41.5	46.0	51.5	52.2	49.8	49.2	52.2
100, 5	2.9	3.7	4.5	4.8	5.5	5.5	5.5	5.5	70.7	68.6	71.0	75.7	76.2	74.3	73.2	76.3
50, 10	3.8	1.8	3.1	4.5	4.7	4.7	4.8	4.7	85.8	75.2	80.5	87.0	86.8	85.7	83.9	86.9
50, 20	4.2	1.6	3.3	4.5	5.1	5.2	5.4	5.1	99.9	99.5	99.7	99.9	99.9	99.8	99.9	99.9
	R ω								R ω
20, 5	2.6	2.0	3.6	4.5	5.7	5.4	5.7	5.7	21.3	12.1	19.1	26.5	30.5	25.2	26.1	30.6
50, 5	2.7	2.7	3.8	4.5	5.6	5.7	5.7	5.7	29.8	23.3	29.7	36.0	39.6	35.2	35.6	39.5
100, 5	2.7	3.3	4.1	4.4	5.6	5.5	5.4	5.6	53.3	48.6	53.4	59.8	62.3	57.5	57.7	62.3
50, 10	3.5	1.6	3.0	4.1	4.7	4.7	4.8	4.7	78.9	62.1	71.7	80.2	81.5	78.0	76.9	81.5
50, 20	4.1	1.6	3.2	4.4	5.1	5.2	5.3	5.1	99.8	99.0	99.5	99.8	99.9	99.7	99.7	99.9
	t 5								t 5
N, T	F ω								F ω
20, 5	3.0	3.4	4.0	4.9	5.4	5.4	5.4	5.5	29.1	23.3	29.6	35.8	36.2	32.2	33.2	36.3
50, 5	2.9	3.4	4.0	4.6	5.2	5.2	5.3	5.1	46.5	44.4	48.4	53.8	54.6	51.5	51.1	54.5
100, 5	3.3	4.4	5.0	5.5	6.0	6.0	6.1	6.0	71.9	70.7	72.9	76.9	77.7	75.1	75.0	77.6
50, 10	4.4	2.2	4.2	5.0	5.7	5.3	5.7	5.7	85.3	75.9	80.7	86.8	86.9	84.8	83.9	86.9
50, 20	4.0	1.7	3.2	4.5	5.1	5.2	4.9	5.1	99.9	99.1	99.5	99.9	99.9	99.7	99.6	99.9
	R ω								R ω
20, 5	2.3	1.9	2.8	3.9	5.4	5.4	5.3	5.3	22.9	13.9	21.8	29.3	32.9	27.8	29.4	33.1
50, 5	2.3	2.7	3.4	4.0	5.3	5.3	5.4	5.3	31.0	25.7	32.4	38.4	41.8	37.1	38.2	41.7
100, 5	3.0	3.9	4.5	5.0	5.8	6.1	6.1	5.8	55.5	52.6	56.7	63.2	64.8	60.5	61.0	64.9
50, 10	4.1	2.0	3.9	4.7	5.7	5.4	5.8	5.8	79.2	63.6	72.9	80.7	81.9	78.0	77.5	81.8
50, 20	3.9	1.6	3.2	4.3	5.0	5.2	5.0	5.0	99.9	98.7	99.3	99.9	99.9	99.5	99.5	99.8
N, T	F ω								F ω
20, 5	3.2	2.8	3.8	4.3	4.6	5.2	4.7	4.6	28.8	22.7	27.9	35.1	35.9	31.1	32.0	35.9
50, 5	3.4	3.0	3.8	4.2	4.8	4.6	4.7	4.7	44.2	41.4	45.2	51.0	52.2	49.4	49.0	52.1
100, 5	3.3	3.7	4.5	4.6	5.1	5.0	5.2	5.0	71.8	70.7	72.9	77.4	77.6	75.6	74.6	77.6
50, 10	4.4	2.3	3.7	4.9	5.3	5.5	5.4	5.3	84.1	74.7	79.8	85.5	85.8	83.8	82.8	85.8
50, 20	4.8	2.0	3.5	4.8	4.8	5.2	4.8	4.8	99.9	99.2	99.7	99.9	99.9	99.9	99.9	99.9
	R ω								R ω
20, 5	2.5	1.9	2.7	3.4	4.5	5.1	4.6	4.4	22.1	14.3	21.1	28.1	32.1	26.8	27.8	32.2
50, 5	2.9	2.5	3.2	3.7	4.7	4.9	4.6	4.6	27.9	22.2	28.1	35.0	38.3	33.9	34.4	38.5
100, 5	3.1	3.4	4.1	4.4	5.2	5.2	5.2	5.1	54.7	50.2	54.8	62.0	64.1	59.0	59.1	64.2
50, 10	4.2	1.9	3.4	4.6	5.1	5.5	5.4	5.1	77.8	61.4	71.5	79.7	81.0	76.9	77.0	81.0
50, 20	4.7	2.0	3.5	4.7	4.7	5.0	4.8	4.7	99.9	98.7	99.4	99.9	99.9	99.7	99.8	99.9

Table

Approximate significance levels of F N
	T = 5			N = 50
	N = 20	N = 50	N = 100	T = 10	T = 20
HET1:	8.8%	9.2%	9.4%	9.2%	9.2%
HET2:	3.5%	3.4%	3.3%	4.8%	4.8%

TABLE 4 Rejection frequencies of the asymptotic and wild-bootstrap modified F-tests and modified random effects tests under conditional heteroskedasticity depending on a regressor (HET3)

H 0: α i  = 0									H 1: var(α i ) = 0.1, α i correlated with regressors
	Asymptotic tests				Bootstrap tests				Asymptotic tests				Bootstrap tests
ω	1				1				1				1
	SN								SN
N, T	F ω								F ω
20, 5	2.1	3.9	4.6	5.3	5.7	5.5	5.8	5.4	9.8	13.1	18.5	20.1	20.8	17.5	20.6	20.5
50, 5	5.7	3.1	4.1	4.8	5.4	5.1	5.1	5.1	18.5	13.9	16.9	18.9	19.8	19.0	19.6	19.4
100, 5	5.6	3.9	4.9	5.3	6.0	6.3	5.9	5.8	30.0	25.9	29.7	31.1	32.7	32.7	32.5	32.4
50, 10	5.5	2.6	4.0	5.1	5.5	5.5	5.6	5.5	47.5	31.4	40.5	45.2	46.3	43.5	45.3	46.1
50, 20	5.4	2.2	3.9	5.1	5.5	5.4	5.5	5.5	80.6	64.2	75.6	79.6	80.0	77.7	79.7	80.0
	R ω								R ω
20, 5	1.7	2.5	3.4	4.4	5.9	6.0	5.9	6.0	7.2	7.8	13.4	16.0	18.9	15.8	18.6	18.6
50, 5	5.1	2.5	3.3	4.1	5.6	5.3	5.2	5.3	13.1	7.6	10.8	12.6	15.4	13.8	15.1	14.8
100, 5	5.3	3.5	4.4	4.9	6.0	6.3	6.0	5.8	21.7	16.1	20.7	22.7	25.2	23.9	24.9	24.9
50, 10	5.3	2.5	3.7	4.8	5.5	5.4	5.5	5.4	41.0	22.6	33.0	38.5	40.5	36.5	38.8	40.2
50, 20	5.3	2.2	3.7	5.0	5.4	5.4	5.4	5.4	76.9	57.5	71.6	75.9	77.0	73.8	76.8	76.9
	t 5								t 5
N, T	F ω								F ω
20, 5	1.6	4.1	4.7	5.2	5.4	5.4	5.3	5.4	10.4	15.5	20.8	22.5	23.5	19.8	22.8	23.1
50, 5	5.6	3.7	4.6	5.3	5.7	5.7	5.4	5.6	18.7	16.6	20.0	21.5	23.1	22.2	23.3	22.9
100, 5	4.4	3.4	4.2	4.5	5.4	5.4	5.4	5.3	30.9	28.8	33.0	35.1	36.7	35.5	36.4	36.3
50, 10	6.1	2.9	4.2	5.5	5.9	5.8	5.9	5.8	49.7	35.6	44.8	48.5	49.4	46.8	48.7	49.4
50, 20	4.6	1.8	3.2	4.6	5.1	5.5	5.0	5.1	79.4	65.1	76.2	79.6	79.8	77.5	79.5	79.7
	R ω								R ω
20, 5	1.3	2.4	3.1	3.8	5.6	5.7	5.3	5.3	7.7	9.2	15.2	17.9	21.2	17.4	20.4	20.8
50, 5	5.1	2.7	3.7	4.3	5.7	5.6	5.4	5.5	13.3	9.1	12.4	14.9	17.6	15.4	17.0	17.3
100, 5	4.2	3.1	3.8	4.3	5.3	5.2	5.3	5.2	21.9	18.0	23.3	25.6	28.0	26.4	27.8	27.6
50, 10	5.7	2.4	3.8	5.2	5.9	5.8	5.9	5.8	43.3	26.7	38.0	42.8	44.4	40.0	43.0	44.2
50, 20	4.6	1.7	3.2	4.4	5.1	5.5	5.0	5.1	76.3	59.8	72.8	76.5	77.4	73.7	76.8	77.3
N, T	F ω								F ω
20, 5	1.8	4.4	4.9	5.3	5.7	6.0	5.5	5.5	10.6	13.2	17.9	19.3	20.7	17.6	20.3	20.2
50, 5	5.4	3.4	4.3	4.8	5.4	5.0	5.0	5.0	16.9	13.3	15.8	17.2	18.5	18.1	18.6	18.0
100, 5	5.4	3.7	4.2	4.7	5.6	5.5	5.5	5.4	27.8	25.1	28.1	29.9	31.4	31.1	31.0	31.0
50, 10	6.0	2.5	3.7	4.6	5.0	5.5	5.5	4.9	47.9	33.7	41.5	46.1	47.3	46.6	46.7	47.2
50, 20	4.5	1.5	3.1	3.9	4.2	4.5	4.7	4.2	80.0	66.7	77.0	80.5	80.9	79.4	80.9	80.8
	R ω								R ω
20, 5	1.4	2.7	3.2	4.0	5.7	5.9	5.6	5.6	8.0	8.2	13.1	14.8	18.4	15.3	17.9	18.0
50, 5	4.8	2.8	3.6	4.2	5.3	5.1	5.1	4.9	12.3	6.9	9.7	11.4	14.4	12.6	13.8	14.1
100, 5	5.2	3.4	3.9	4.4	5.6	5.4	5.4	5.3	20.5	14.7	19.0	21.2	23.5	21.8	22.8	23.1
50, 10	5.7	2.3	3.5	4.4	5.1	5.5	5.4	5.0	40.9	23.7	33.2	38.6	40.6	38.6	39.7	40.3
50, 20	4.4	1.4	3.0	3.8	4.2	4.4	4.7	4.2	76.5	60.5	72.9	77.0	78.0	75.8	77.9	77.9

TABLE 5 Rejection frequencies of the asymptotic and wild-bootstrap modified F-tests and modified random effects tests under conditional heteroskedasticity, GARCH(1,1) (HET4)

H 0: α i  = 0									H 1: var(α i ) = 0.1, α i correlated with regressors
	Asymptotic tests				Bootstrap tests				Asymptotic tests				Bootstrap tests
ω	1				1				1				1
	SN								SN
N, T	F ω								F ω
20, 5	7.2	2.3	4.5	6.0	6.0	5.8	5.8	5.8	30.5	17.6	24.3	27.5	27.5	28.8	28.1	27.1
50, 5	7.9	2.3	4.4	5.5	5.8	5.6	5.4	5.7	47.2	34.9	39.1	41.2	40.9	46.2	42.8	40.7
100, 5	8.8	2.9	5.1	6.3	6.0	5.8	6.0	6.0	71.1	60.8	62.9	64.4	63.4	70.4	66.2	63.3
50, 10	6.8	1.8	3.9	5.2	5.7	5.9	5.5	5.7	92.7	85.3	89.9	90.9	90.8	92.6	91.9	90.8
50, 20	5.6	1.6	3.6	4.9	5.3	5.6	5.3	5.3	100.0	99.9	100.0	100.0	100.0	100.0	100.0	100.0
	R ω								R ω
20, 5	6.5	1.3	3.5	5.3	6.0	5.7	5.8	5.8	25.7	10.1	17.9	22.5	24.2	24.6	24.5	23.8
50, 5	7.4	1.9	3.7	5.1	5.8	5.4	5.5	5.7	33.7	18.4	24.9	28.6	29.2	32.3	30.4	29.0
100, 5	8.6	2.6	4.5	5.9	6.0	5.8	5.9	6.0	55.6	40.3	45.9	48.1	49.0	53.2	50.5	49.0
50, 10	6.5	1.6	3.7	5.1	5.6	5.8	5.5	5.4	88.6	74.0	83.2	86.0	85.7	87.4	87.0	85.7
50, 20	5.5	1.6	3.6	4.8	5.3	5.6	5.3	5.3	100.0	99.8	100.0	99.9	100.0	100.0	100.0	100.0
	t 5								t 5
N, T	F ω								F ω
20, 5	7.9	1.9	4.2	5.7	5.3	5.1	5.1	5.1	32.7	20.8	27.0	29.7	29.0	31.4	29.8	28.5
50, 5	9.2	2.6	4.4	5.8	5.4	5.3	5.1	5.3	49.7	36.5	41.1	42.7	41.4	47.1	43.6	41.1
100, 5	11.5	3.5	5.6	6.5	6.3	6.2	6.4	6.3	70.8	59.0	59.9	60.3	59.3	67.3	62.2	59.2
50, 10	8.2	1.9	4.0	5.5	5.6	5.5	5.3	5.5	91.9	82.8	86.9	87.8	86.8	90.6	88.8	86.8
50, 20	6.9	1.5	3.8	5.3	5.4	5.5	5.5	5.3	99.9	99.3	99.6	99.4	99.3	99.7	99.7	99.3
	R ω								R ω
20, 5	7.4	1.3	3.2	5.1	5.4	5.1	5.2	5.2	27.7	12.0	19.8	24.0	25.7	26.3	26.2	25.2
50, 5	8.7	2.0	3.7	5.3	5.4	5.1	5.2	5.3	36.4	20.0	26.3	29.2	30.5	33.3	31.4	30.2
100, 5	11.1	3.0	5.0	6.2	6.3	6.3	6.4	6.3	56.9	39.5	44.6	46.3	46.6	52.4	48.8	46.4
50, 10	8.0	1.8	3.6	5.3	5.6	5.4	5.3	5.6	87.7	72.1	80.2	82.3	81.9	85.4	84.4	81.9
50, 20	6.8	1.5	3.7	5.2	5.4	5.5	5.5	5.3	99.9	99.0	99.4	99.4	99.2	99.7	99.6	99.1
N, T	F ω								F ω
20, 5	6.9	1.9	3.3	4.4	3.7	4.2	3.7	3.4	29.8	17.2	23.2	26.2	25.1	28.0	25.9	24.6
50, 5	8.1	2.0	3.3	4.7	4.3	4.2	4.1	4.2	46.2	33.1	36.9	38.9	37.6	44.6	40.3	37.3
100, 5	9.3	1.8	3.0	4.6	3.8	4.4	3.5	3.7	68.3	56.2	57.3	58.7	57.3	66.5	60.3	57.0
50, 10	7.4	1.2	2.9	4.5	4.3	4.7	4.1	4.3	92.4	84.4	87.6	89.2	88.4	92.3	89.9	88.3
50, 20	6.4	1.1	2.7	4.5	4.8	4.8	4.4	4.8	100.0	99.8	99.9	99.9	99.8	100.0	99.9	99.8
	R ω								R ω
20, 5	6.1	1.2	2.2	3.9	3.7	4.1	3.9	3.5	24.5	9.7	16.5	20.5	22.3	23.6	22.7	21.7
50, 5	7.6	1.5	2.5	4.0	4.2	4.4	4.0	4.1	33.6	17.1	22.8	26.1	27.0	30.2	27.7	26.5
100, 5	9.0	1.5	2.7	4.2	3.8	4.4	3.6	3.8	53.1	35.4	40.1	42.6	43.0	48.8	44.6	42.9
50, 10	7.2	1.0	2.6	4.4	4.3	4.8	4.2	4.3	87.8	72.4	81.1	84.0	83.3	86.3	85.1	83.2
50, 20	6.3	1.1	2.7	4.4	4.8	4.8	4.4	4.8	100.0	99.6	99.8	99.9	99.8	100.0	99.9	99.8

TABLE 6 Rejection frequencies of the asymptotic and wild-bootstrap modified F-tests and modified random effects tests under conditional heteroskedasticity, GJR-GARCH(1,1) (HET5)

H 0: α i  = 0									H 1: var(α i ) = 0.1, α i correlated with regressors
	Asymptotic tests				Bootstrap tests				Asymptotic tests				Bootstrap tests
ω	1				1				1				1
	SN								SN
N, T	F ω								F ω
20, 5	7.7	2.4	4.8	6.4	6.2	5.6	5.9	6.1	29.3	16.7	23.0	26.2	26.1	27.9	26.6	25.7
50, 5	8.4	2.5	4.3	5.6	5.6	5.5	5.4	5.5	45.1	32.1	36.8	38.5	37.6	43.7	39.6	37.2
100, 5	9.9	3.0	4.9	6.4	6.2	6.1	5.8	6.0	68.3	56.8	59.2	60.4	59.1	66.4	61.4	59.1
50, 10	7.6	1.4	3.8	5.8	5.7	5.9	5.8	5.7	90.9	80.9	86.0	87.7	86.8	90.3	88.6	86.8
50, 20	6.5	1.3	3.5	5.2	5.4	5.2	5.2	5.4	100.0	99.8	99.9	100.0	99.9	100.0	99.9	99.9
	R ω								R ω
20, 5	7.0	1.4	3.7	5.5	6.3	5.6	6.1	6.1	24.7	9.5	17.0	21.2	23.0	23.8	23.1	22.5
50, 5	7.8	1.9	3.7	5.2	5.6	5.3	5.3	5.4	32.5	16.3	22.3	26.2	26.8	30.0	27.6	26.6
100, 5	9.6	2.4	4.5	6.1	6.1	6.0	5.7	6.1	53.3	35.9	41.7	44.8	45.5	49.8	46.7	45.3
50, 10	7.3	1.3	3.6	5.6	5.7	5.8	5.9	5.7	85.5	69.6	78.5	81.6	81.3	84.5	82.8	81.1
50, 20	6.4	1.3	3.5	5.2	5.5	5.2	5.2	5.5	100.0	99.6	99.9	99.9	99.9	99.9	99.9	99.9
	t 5								t 5
N, T	F ω								F ω
20, 5	8.2	1.9	4.3	6.2	5.6	5.0	5.4	5.4	31.5	20.1	26.1	28.0	27.4	30.2	28.7	26.9
50, 5	10.0	2.6	4.6	6.3	5.7	5.5	5.3	5.5	48.0	35.1	38.2	40.0	38.9	44.9	40.8	38.5
100, 5	12.6	3.6	5.4	6.8	6.5	6.1	6.3	6.4	67.8	54.9	56.2	56.8	55.3	63.5	58.3	55.0
50, 10	9.3	1.6	3.6	5.6	5.5	5.5	5.1	5.5	90.2	79.2	82.4	83.8	82.2	88.4	84.9	82.1
50, 20	8.0	1.5	3.3	5.5	5.4	5.1	5.3	5.4	99.8	98.9	99.0	99.1	98.7	99.5	99.2	98.7
	R ω								R ω
20, 5	7.6	1.2	3.3	5.4	5.7	5.1	5.3	5.4	26.5	11.6	19.3	23.1	24.5	25.8	25.2	24.0
50, 5	9.6	1.9	3.7	5.8	5.7	5.2	5.2	5.5	35.8	18.0	25.1	27.7	28.9	32.3	29.9	28.6
100, 5	12.0	2.9	4.8	6.4	6.5	6.1	6.2	6.4	54.9	36.0	41.1	43.8	42.8	49.1	45.3	42.6
50, 10	9.0	1.5	3.3	5.4	5.5	5.5	5.0	5.5	85.0	67.9	75.4	77.3	76.5	82.3	79.4	76.4
50, 20	7.9	1.4	3.3	5.5	5.4	5.1	5.2	5.4	99.7	98.3	98.9	98.9	98.5	99.4	99.0	98.4
N, T	F ω								F ω
20, 5	6.5	2.2	3.4	4.7	4.2	4.8	4.4	4.1	32.1	19.6	26.3	29.0	28.5	31.5	29.7	28.2
50, 5	7.2	2.6	3.8	4.8	4.7	5.0	4.7	4.6	49.5	38.0	42.9	44.3	43.9	49.6	46.2	43.5
100, 5	7.6	2.3	3.4	4.4	4.1	5.1	4.2	4.1	74.0	64.3	67.0	68.0	67.4	73.1	69.9	67.3
50, 10	6.6	1.6	3.3	4.5	4.7	5.4	4.7	4.7	94.6	88.0	91.7	92.8	92.4	94.1	93.4	92.3
50, 20	5.7	1.3	3.1	4.5	4.7	5.0	4.6	4.7	100.0	99.9	100.0	100.0	100.0	100.0	100.0	100.0
	R ω								R ω
20, 5	5.9	1.3	2.5	4.1	4.3	5.0	4.6	4.1	26.3	10.9	19.6	23.4	25.2	26.1	25.7	24.8
50, 5	6.7	2.0	3.2	4.3	4.7	4.9	4.5	4.6	35.1	20.1	26.6	30.2	31.1	33.9	31.9	30.8
100, 5	7.5	2.0	3.1	4.1	4.1	5.1	4.2	4.0	57.6	42.8	48.6	50.3	51.5	56.0	53.0	51.3
50, 10	6.3	1.4	3.1	4.3	4.8	5.5	4.7	4.7	90.7	77.6	86.1	88.4	88.0	89.6	89.4	87.9
50, 20	5.6	1.2	3.1	4.4	4.7	5.0	4.7	4.7	100.0	99.9	100.0	100.0	100.0	100.0	100.0	100.0