. 2022 Dec 8;11(1):30. doi: 10.1186/s40008-022-00291-7

Table 3.

Quantile-mean covariance (QC) test of normality

		ε = 0.001	ε = 0.01	ε = 0.05	ε = 0.10	ε = 0.15	ε = 0.20
hr	$T_{1 n}$	9.008***	9.008***	9.008***	9.008***	1.834***	0.751***
	$T_{2 n}$	81.160***	81.160***	81.160***	81.160***	3.365***	0.564***
	$T_{3 n}$	5.966***	5.884***	4.931***	1.887***	0.266***	0.180***
fr	$T_{1 n}$	11.375***	11.375***	11.375***	11.375***	9.968***	2.088***
	$T_{2 n}$	129.405***	129.405***	129.405***	129.405***	99.371***	4.363***
	$T_{3 n}$	12.976***	12.829***	11.672***	8.546***	2.901***	1.322***
ovx	$T_{1 n}$	12.981***	12.981***	12.981***	12.981***	12.981***	12.386***
	$T_{2 n}$	168.508***	168.508***	168.508***	168.508***	168.508***	153.429***
	$T_{3 n}$	47.495***	47.237***	46.272***	43.403***	37.200***	28.295***
oilcv	$T_{1 n}$	16.418***	16.418***	16.418***	16.418***	16.418***	16.418***
	$T_{2 n}$	269.559***	269.559***	269.559***	269.559***	269.559***	269.559***
	$T_{3 n}$	34.647***	34.415***	33.317***	30.398***	24.442***	15.123***
oil	$T_{1 n}$	4.627***	4.627***	4.627***	4.627***	3.194***	1.719***
	$T_{2 n}$	21.409***	21.409***	21.409***	21.409***	10.206***	2.957***
	$T_{3 n}$	2.636***	2.586***	2.101***	1.712***	0.685***	0.284***
bnd	$T_{1 n}$	4.653***	4.653***	3.933***	3.933***	2.837***	2.837***
	$T_{2 n}$	21.658***	21.658***	15.473***	15.473***	8.052***	8.052***
	$T_{3 n}$	3.306***	3.291***	2.645***	2.233***	1.698***	1.605***

^*, **, *** denote significance at 10%, 5% and 1% level, respectively. T1, T2, T3 refer to Bera et al. (2016) statistics: $T_{1 n} : = \begin{matrix} sup \\ τ E T \end{matrix} |\overset{\land}{C_{n}} (τ)|, T_{2 n} : = \begin{matrix} sup \\ τ E T \end{matrix} \overset{\land}{C_{n}} {(τ)}^{2}, T_{3 n} : = \int_{τ E T} \overset{\land}{C_{n}} {(τ)}^{2} d r,$ where $\overset{\land}{C_{n}} (τ)$ is the quantile-mean covariance (QC) function, which is the asymptotic covariance between the sample quantiles and the sample mean