Model-Free Tests for Series Correlation in Multivariate Linear Regression

1. Introduction

Linear regression is an important topic in statistics and has been found to be useful in almost all aspects of data science, especially in business and economics statistics and biostatistics. Consider the following multivariate linear regression model

(1.1)

Y = X^{'} β + ε,

where $Y$

is the response variable,

X = (x_{1}, x_{2}, \dots, x_{p})^{'}

is a

p

-dimensional vector of regressors,

β = {(β_{1}, β_{2}, \dots, β_{p})}_{1}^{'}

is a

p

-dimensional regression coefficient vector and

ε

is random errors with zero mean. Suppose we obtain

n

samples from this model, that is,

Y = (y_{1}, y_{2}, \dots, y_{n})^{'}

with design matrix

X = (x_{1}, x_{2}, \dots, x_{n})^{'}

, where for

i = 1, \dots, n

x_{i} = {(x_{i 1}, x_{i 2}, \dots, x_{i p})}_{i 1}^{'}

. The first task in a regression problem is to make statistical inference about the regression coefficient vector. By applying the ordinary least squares (OLS) method, we obtain the estimate

^β = {(X^{'} X)}^{- 1} X^{'} Y

for coefficient vector

β

. In most applications of linear regression models, we need the assumption that the random errors

{ε_{i}}_{i = 1}^{n}

are uncorrelated and homoscedastic. That is to say, we assume

C o v (ε_{i}, ε_{j}) = {\begin{matrix} σ^{2} & for i = j, 0 & for i \neq j, \end{matrix}

where $σ^{2}$ are unknown. With this assumption, the Gauss-Markov theorem states that the ordinary least squares estimate (OLSE) $^β$

is the best linear unbiased estimator (BLUE). When this assumption does not hold, we suffer from a loss of efficiency and, even worse, make wrong inferences in using OLS. For example, positive serial correlation in the regression error terms will typically lead to artificially small standard errors for the regression coefficient when we apply the classic linear regression method, which will cause the estimated t-statistic to be inflated, indicating significance even when there is in fact none. Therefore, tests for heteroscedasticity and series correlation are important when applying linear regression.

For detecting heteroscedasticity, in one of the most cited papers in econometrics, White [White(1980)] proposed a test based on comparing the Huber-White covariance estimator to the usual covariance estimator under homoscedasticity. Many other researchers have considered this problem, for example, Breusch and Pagan [Breusch and Pagan(1979)], Dette and Munk [Dette and Munk(1998)], Glejser [Glejser(1969)], Harrison and McCabe [Harrison and McCabe(1979)], Cook and Weisberg [Cook and Weisberg(1983)], and Azzalini and Bowman[Azzalini and Bowman(1993)]. Recently, Li and Yao [Li and Yao(2015)] and Bai, Pan and Yin [Bai et al.(2018)Bai, Pan, and Yin] proposed tests for heteroscedasticity that are valid in both low- and high-dimensional regressions. Their tests were shown by simulations to perform better than some classic tests.

The most famous test for series correlation, the Durbin-Watson test, was proposed in [Durbin and Watson(1950), Durbin and Watson(1951), Durbin and Watson(1971)]. The Durbin-Watson test statistic is based on the residuals $e_{1}, e_{2}, \dots, e_{n}$ from linear regression. The researchers considered the statistic

d = \frac{\sum_{i = 2}^{n} {(e_{i} - e_{i - 1})}_{i}^{2}}{\sum_{i = 1}^{n} e_{i}^{2}},

whose small-sample distribution was derived by John von Neumann. In the original papers, Durbin and Watson investigated the distribution of this statistic under the classic independent framework, described the test procedures and provided tables of the bounds of significance. However, the asymptotic results were derived under the normality assumption on the error term, and as noted by Nerlove and Wallis [Nerlove and Wallis(1966)]

, although the Durbin-Watson test appeared to work well in an independent observations framework, it may be asymptotically biased and lead to inadequate conclusions for linear regression models containing lagged dependent random variables. New alternative test procedures, for instance, Durbin’s h-test and t-test

[Durbin(1970)], were proposed to address this problem; see also Inder [Inder(1986)], King and Wu [King and Wu(1991)], Stocker [Stocker(2007)], Bercu and Proïa [Bercu and Proia(2013)], Gençay and Signori [Gençay and Signori(2015)] and Li and Gençay [Li and Gençay(2017)]

and references therein. However, all these tests were proposed under some model assumptions on the regressors and/or the response variable. Moreover, Durbin’s h-test requires a Gaussian distribution of the error term. Thus, some common models are excluded. In fact, since it is difficult to assess whether the regressors and/or the response are lag dependent, model-free tests for the regressors and response variable appear to be appropriate.

The present paper proposes a simple test procedure without assumptions on the response variable and regressors that is valid in both low- and high-dimensional multivariate linear regression. The main idea, which is simple but proves to be useful, is to express the mean and variance of the test statistic by making use of the residual maker matrix. In addition to a general joint central limit theorem for several quadratic forms, which is proved in this paper and may have its own interest, we consider a Box-Pierce-type test for series correlation. Monte Carlo simulations show that our test procedures perform well in situations where some classic test procedures are inapplicable.

2. Test for series correlation in linear regression model

2.1. Notation

Let $X$ be the design matrix, and let $R = (r_{i, j}) = I_{n} - X (X^{'} X)^{- 1} X^{'}$ be the residual maker matrix, where $H = (h_{i, j}) = X (X^{'} X)^{- 1} X^{'}$ is the hat matrix (also known as the projection matrix). We assume that the noise vector $ε = Σ^{1 / 2} ϵ,$ where $ϵ$ is an $n$ -dimensional random vector whose entries $ϵ_{1}, \dots, ϵ_{n}$

are independent with zero means, unit variances and the same finite fourth-order moments

M_{4}

, and

Σ^{1 / 2}

is an n-dimensional nonnegative definite nonrandom matrix with bounded spectral norm. Then, the OLS residuals are

e = (e_{1}, \dots, e_{n})^{'} = R Σ^{1 / 2} ϵ

. We note that we will use

\circ

to indicate the Hadamard product of two matrices in the rest of this paper.

2.2. Test for a given order series correlation

To test for a given order series correlation, for any number $q \leq n$ , denote

where $P_{τ} = {(p_{i, j}^{(τ)})}_{n \times n}$ with $p_{i, j}^{(τ)} = {\begin{matrix} 1 & for i - j = τ, 0 & other . \end{matrix}$

First, for $0 \leq τ \leq q,$ we have

(2.1)

E γ_{τ} = E t r ϵ^{'} Σ^{1 / 2} R P_{τ} R Σ^{1 / 2} ϵ = t r Σ^{1 / 2} R P_{τ} R Σ^{1 / 2} = t r R P_{τ} R Σ .

Denote $Σ^{1 / 2} R P_{τ_{1}} R Σ^{1 / 2} = (a_{i, j})$ and $Σ^{1 / 2} R P_{τ_{2}} R Σ^{1 / 2} = (b_{i, j})$ , and set $ν_{4} = M_{4} - 3;$ we then have, for $τ_{1}, τ_{2} = 0, \dots, q,$

(2.2)			$C o v (γ_{τ_{1}}, γ_{τ_{2}})$
	$=$	$E (ϵ^{'} Σ^{1 / 2} R P_{τ_{1}} R Σ^{1 / 2} ϵ - t r R P_{τ_{1}} R Σ) (ϵ^{'} Σ^{1 / 2} R P_{τ_{2}} R Σ^{1 / 2} ϵ - t r R P_{τ_{2}} R Σ)$
	$=$	$E ⎛ ⎝ \sum i a_{i, i} (ϵ_{i}^{2} - 1) + \sum i \neq j a_{i, j} ϵ_{i} ϵ_{j} ⎞ ⎠ ⎛ ⎝ \sum i b_{i, i} (ϵ_{i}^{2} - 1) + \sum i \neq j b_{i, j} ϵ_{i} ϵ_{j} ⎞ ⎠$
	$=$
		$+ t r (Σ^{1 / 2} R P_{τ_{1}} R Σ^{1 / 2}) {(Σ^{1 / 2} R P_{τ_{2}} R Σ^{1 / 2})}_{τ_{2}}^{'}$
	$=$

Note that $C o v (ε, ε) = Σ .$ We want to test the hypothesis for $1 \leq τ \leq q,$

H_{0} : Σ = σ^{2} I_{n}, where 0 < σ^{2} < \infty,

against

H_{1, τ} : C o v (ε_{i}, ε_{i - τ}) = ρ \neq 0.

Under the null hypothesis, due to (

2.1) and (2.2), we obtain

(2.3)

E γ_{τ} = σ^{2} t r P_{τ} R,

and

(2.4)

Specifically, we have $E γ_{τ_{0}} = σ^{2} t r R = σ^{2} (n - p)$ and

(2.5)

The validity of our test procedure requires the following mild assumptions.

(1): Assumption on $p$ and $n$ :: The number of regressors $p$ and the sample size $n$ satisfy that $p / n \to c \in [0, 1)$ as $n \to \infty$ .
(2): Assumption on errors:: The fourth-order cumulant of the error distribution $ν_{4} \neq - 2$ .

Assumption $(2)$ excludes the rare case where the random errors are drawn from a two-point distribution with the same masses $1 / 2$ at $- - 1$ and $1$ . However, if this situation occurs, our test remains valid if the design matrix satisfies the mild condition that

limsup n \to \infty \frac{t r R \circ R}{n - p} = limsup n \to \infty \frac{\sum_{i = 1}^{n} r_{i, i}^{2}}{\sum_{i = 1}^{n} r_{i, i}} < 1.

These assumptions ensure that $V a r (γ_{τ_{0}})$ has the same order as $n$ as $n \to \infty$ , thus satisfying the condition assumed in Theorem 4.1.

Define

m_{τ} = \frac{E γ_{τ}}{σ^{2}} = t r P_{τ} R, v_{τ_{1} τ_{2}} = \frac{C o v (γ_{τ_{1}}, γ_{τ_{2}})}{n σ^{4}} .

By applying Theorem 4.1 presented in Section 4, we obtain that for $1 \leq τ \leq q,$

\frac{1}{\sqrt{n}} ((\begin{matrix} γ_{τ} γ_{0} \end{matrix}) - (\begin{matrix} m_{τ} m_{0} \end{matrix})) \sim N (0, (\begin{matrix} v_{τ τ} & v_{τ 0} v_{0 τ} & v_{00} \end{matrix})) .

Then, by the delta method, we obtain, as $n \to \infty$ ,

T_{τ} = \frac{\sqrt{n} (\frac{γ_{τ}}{γ_{0}} - μ_{τ 0})}{σ_{τ 0}} \sim N (0, 1),

where $μ_{τ 0} = \frac{m_{τ}}{m_{0}}$ and

(2.6)		$σ_{τ 0}^{2} =$	$(\begin{matrix} \frac{1}{m_{0} / n} & \frac{m_{1} / n}{{(m_{0} / n)}_{0}^{2}} \end{matrix}) (\begin{matrix} v_{11} & v_{10} v_{01} & v_{00} \end{matrix}) ⎛ ⎜ ⎝ \begin{matrix} \frac{1}{m_{0} / n} \frac{m_{1} / n}{{(m_{0} / n)}_{0}^{2}} \end{matrix} ⎞ ⎟ ⎠$
	$=$	$n^{2} (\begin{matrix} \frac{1}{n - p} & \frac{m_{1}}{(n - p)^{2}} \end{matrix}) (\begin{matrix} v_{11} & v_{10} v_{01} & v_{00} \end{matrix}) ⎛ ⎜ ⎝ \begin{matrix} \frac{1}{n - p} \frac{m_{1}}{(n - p)^{2}} \end{matrix} ⎞ ⎟ ⎠ .$

We reject $H_{0}$ in favor of $H_{1, τ}$ if a large $| T_{τ} |$ is observed.

2.3. A portmanteau test for series correlation

In time series analysis, the Box-Pierce test proposed in [Box and Pierce(1970)] and the Ljung-Box statistic proposed in [Ljung and Box(1978)] are two portmanteau tests of whether any of a group of autocorrelations of a time series are different from zero. For a linear regression model, consider the following hypothesis

H_{0} : Σ = σ^{2} I_{n},

against

H1:there\ exist 1≤τ≤q such\ that Cov(εi,εi−τ)=ρ≠0.

Applying Theorem 4.1

and the delta method, we shall now consider the following asymptotically standard normally distributed statistic

T (q) = \frac{\sqrt{n} (\sum_{τ = 1}^{q} {(2 - \frac{γ_{τ}}{γ_{0}})}^{2} - \sum_{τ = 1}^{q} (2 - μ_{τ 0})^{2})}{σ_{T}} \sim N (0, 1),

as $n \to \infty,$ where $μ_{τ 0} = \frac{m_{τ}}{m_{0}}$ and $σ_{T} = \sqrt{\nabla^{'} Σ_{T} \nabla}$ with

\nabla = {⎛ ⎜ ⎝ \frac{2 n \sum_{τ = 1}^{q} m_{τ} (2 - \frac{m_{τ}}{n - p})}{(n - p)^{2}}, \frac{- 2 n (2 - \frac{m_{1}}{n - p})}{(n - p)}, \frac{- 2 n (2 - \frac{m_{2}}{n - p})}{(n - p)}, \dots, \frac{- 2 n (2 - \frac{m_{q}}{n - p})}{(n - p)} ⎞ ⎟ ⎠}^{'},

and

Σ_{T} = {(v_{i, j})}_{(q + 1) \times (q + 1)}, i, j = 0, \dots, q .

Then, we reject $H_{0}$ in favor of $H_{1}$ if $| T (q) |$ is large.

2.4. Discussion of the statistics

In the present subsection, we discuss the asymptotic parameters of the two proposed statistics.

If the entries in design matrix $X$ are assumed to be i.i.d. standard normal, then we know that as $n \to \infty$ , the diagonal entries in the symmetric and idempotent matrices $H$ and $R = I_{n} - H$ are of constant order while the off-diagonal entries are of order $n^{- 1 / 2}$ . Then, the order of $m_{τ} = t r P_{τ} R$ for a given $τ > 0$ is at most $n^{1 / 2}$ since it is exactly the summation of the $n - τ$ off-diagonal entries of $R$ . Thus, elementary analysis shows that $σ_{τ 0}^{2} = \frac{n^{2} v_{11}}{(n - p)^{2}} + o (1)$ .

For a fixed design or a more general random design, it become almost impossible to study matrices $H$ and $R = I_{n} - H$ , except for some of the elementary properties. Thus, for the purpose of obtaining an accurate statistical inference, we suggest the use of the original parameters since we have little information on the distribution of the regressors in a fixed design, and the calculation of those parameters is not excessively complex.

3. Simulation studies

In this section, Monte Carlo simulations are conducted to investigate the performance of our proposed tests.

3.1. Performance of test for first-order series correlation

First, we consider the test for first-order series correlation of the error terms in multivariate linear regression model (1.1). Note that although our theory results were derived by treating the design matrix as a constant matrix, we also need to obtain a design matrix under a certain random model in the simulations. We thus consider the situation where the regressors $x_{1}, x_{2}, \dots, x_{f}$ are lagged dependent. Formally, for a given $f$ , we set

x_{t, j} = r x_{t - 1, j} + u_{t}, j = 1, \dots, f, t = 1, \dots, n,

where $r = 0.2$ and ${u_{t}}$ are independently drawn from N(0,1). While ${x_{i, j}}, f + 1 \leq j \leq p, 1 \leq i \leq n$

are independently chosen from a Student’s t-distribution with 5 degrees of freedom. The random errors

ε

obey (1) the normal distribution N(0,1) and (2) the uniform distribution U(-1,1). The significant level is set to

α = 0.05.

Table 1 and Table 2 show the empirical size of our test (denoted as

‘ ‘ F D W T "

) for different

p, n, f

under the two error distributions. To investigate the power of our test, we randomly choose a

ε_{0}

and consider the following AR(1) model:

ε_{t} = ρ ε_{t - 1} + ϑ_{t}, t = 1, \dots, n

where ${ϑ_{t}}$ are independently drawn from (1) N(0,1) and (2) U(-1,1). Tables 3 and 4 show the empirical power of our proposed test for different $p, n, f, ρ$ under the two error distributions.

These simulation results show that our test always has good size and power when $n - p$ is large and is thus applicable under the framework that $p / n \to [0, 1)$ as $n \to \infty$ .

$p, n$	f	FDWT	$p, n$	f	FDWT
2,32	1	0.0486	8,32	2	0.0428
8,32	4	0.0410	8,32	8	0.0434
16,64	4	0.0446	16,64	12	0.0463
32,64	12	0.0420	32,64	24	0.0414
32,128	12	0.0470	32,128	24	0.0478
64,128	12	0.0479	64,128	36	0.0430
128,256	12	0.0509	128,256	24	0.0486
128,256	64	0.0504	128,256	128	0.0422
128,512	24	0.0519	128,512	64	0.0496
128,512	96	0.0487	128,512	128	0.0497
256,512	64	0.0469	256,512	96	0.0492
256,512	144	0.0472	256,512	256	0.0486
256,1028	64	0.0457	256,1028	96	0.0498
256,1028	144	0.0473	256,1028	256	0.0487
512,1028	12	0.0463	512,1028	96	0.0506
512,1028	144	0.0520	512,1028	256	0.0478
512,1028	288	0.0460	512,1028	314	0.0442
512,1028	440	0.0438	512,1028	512	0.0443

Table 1. Empirical size under Gaussian error assumption

$p, n$	f	FDWT	$p, n$	f	FDWT
2,32	1	0.0410	2,32	2	0.0421
8,32	4	0.0414	8,32	8	0.0468
16,64	4	0.0467	16,64	12	0.0450
32,64	12	0.0450	32,64	24	0.0419
32,128	12	0.0456	32,128	24	0.0458
64,128	12	0.0479	64,128	36	0.0460
128,256	12	0.0509	128,256	24	0.0476
128,256	64	0.0461	128,256	128	0.0412
128,512	24	0.0497	128,512	64	0.0505
128,512	96	0.0508	128,512	128	0.0501
256,512	64	0.0525	256,512	96	0.0455
256,512	144	0.0443	256,512	256	0.0461
256,1028	64	0.0509	256,1028	96	0.0455
256,1028	144	0.0482	256,1028	256	0.0465
512,1028	12	0.0491	512,1028	96	0.0461
512,1028	144	0.0483	512,1028	256	0.0480
512,1028	288	0.0447	512,1028	314	0.0468
512,1028	440	0.0453	512,1028	512	0.0459

Table 2. Empirical size under uniform distribution U(-1,1) error assumption

$p, n$	f	$ρ = 0.2$	$ρ = - 0.3$	$ρ = 0.5$	$p, n$	f	$ρ = 0.2$	$ρ = - 0.3$	$ρ = 0.5$
2,32	1	0.1363	0.2550	0.5409	8,32	2	0.1056	0.1705	0.3224
8,32	4	0.0906	0.1724	0.3841	8,32	8	0.1093	0.1831	0.3597
16,64	4	0.1888	0.3672	0.6783	16,64	12	0.1987	0.3764	0.7199
32,64	12	0.1055	0.1584	0.3542	32,64	24	0.1030	0.1739	0.3791
32,128	12	0.3673	0.6637	0.9552	32,128	24	0.3706	0.6655	0.9556
64,128	12	0.1754	0.3335	0.6345	64,128	36	0.1897	0.3519	0.6639
128,256	12	0.3255	0.6104	0.9160	128,256	24	0.3324	0.6037	0.9225
128,256	64	0.3362	0.6200	0.9345	128,256	128	0.3362	0.6515	0.9438
128,512	24	0.9064	0.9981	1.0000	128,512	64	0.9151	0.9976	1.0000
128,512	96	0.9167	0.9981	1.0000	128,512	128	0.9196	0.9981	1.0000
256,512	64	0.5880	0.8951	0.9975	256,512	96	0.6041	0.9029	0.9980
256,512	144	0.6019	0.8963	0.9990	256,512	256	0.6117	0.9103	0.9987
256,1028	64	0.9970	1.0000	1.0000	256,1028	96	0.9973	1.0000	1.0000
256,1028	144	0.9971	1.0000	1.0000	256,1028	256	0.9976	1.0000	1.0000
512,1028	12	0.8766	0.9957	1.0000	512,1028	96	0.8829	0.9958	1.0000
512,1028	144	0.9201	0.9979	1.0000	512,1028	256	0.8967	0.9954	1.0000
512,1028	288	0.9125	0.9986	1.0000	512,1028	314	0.8946	0.9969	1.0000
512,1028	440	0.8942	0.9975	1.0000	512,1028	512	0.8937	0.9979	1.0000

Table 3. Empirical power under Gaussian error assumption

$p, n$	f	$ρ = 0.2$	$ρ = - 0.3$	$ρ = 0.5$	$p, n$	f	$ρ = 0.2$	$ρ = - 0.3$	$ρ = 0.5$
2,32	1	0.1457	0.2521	0.5548	8,32	2	0.1245	0.1721	0.3478
8,32	4	0.1245	0.1754	0.3548	8,32	8	0.1254	0.1845	0.3547
16,64	4	0.1987	0.3789	0.6567	16,64	12	0.1879	0.3478	0.7456
32,64	12	0.1145	0.1544	0.3582	32,64	24	0.1125	0.1555	0.3548
32,128	12	0.3825	0.6647	0.9845	32,128	24	0.3845	0.6789	0.9677
64,128	12	0.1863	0.3765	0.6748	64,128	36	0.1758	0.3877	0.6478
128,256	12	0.3358	0.5978	0.9185	128,256	24	0.3495	0.6657	0.9244
128,256	64	0.3378	0.5899	0.9578	128,256	128	0.3392	0.6788	0.9584
128,512	24	0.9114	0.9945	1.0000	128,512	64	0.9121	0.9944	1.0000
128,512	96	0.9102	0.9977	1.0000	128,512	128	0.9157	0.9945	0.9999
256,512	64	0.6053	0.8979	0.9969	256,512	96	0.6020	0.9456	0.9978
256,512	144	0.6151	0.8966	1.0000	256,512	256	0.6135	0.9678	1.0000
256,1028	64	0.9975	1.0000	1.0000	256,1028	96	0.9972	1.0000	1.0000
256,1028	144	0.9921	1.0000	1.0000	256,1028	256	0.9982	1.0000	1.0000
512,1028	12	0.8787	0.9944	1.0000	512,1028	96	0.8800	0.9976	1.0000
512,1028	144	0.9201	0.9913	1.0000	512,1028	256	0.8881	0.9964	1.0000
512,1028	288	0.9165	0.9959	1.0000	512,1028	314	0.8957	0.9967	1.0000
512,1028	440	0.8978	0.9944	1.0000	512,1028	512	0.8959	0.9947	1.0000

Table 4. Empirical power under uniform distribution U(-1,1) error assumption

3.2. Performance of the Box-Pierce type test

This subsection investigates the performance of our proposed Box-Pierce type test statistic $T (q)$ in subsection 2.3. The design matrix $X$ is obtained in the same way as in the last subsection, with $f = p / 2$

, and the random error terms are assumed to obey a (1) normal distribution N(0,1) and a (2) gamma distribution with parameters 4 and 1/2. Table

5 and Table 6 show the empirical size of our test with different

n, p, q

under the two error distributions. We consider the following AR(2) model to assess the power:

ε_{t} = ρ_{1} ε_{t - 1} + ρ_{2} ε_{t - 2} + ϑ_{t}, t = 1, \dots, n

where ${ϑ_{t}}$ are independently drawn from (1) N(0,1) and (2) Gamma(4,1/2). The design matrix $X$ is obtained in the same way as before, with $f = p / 2$ . Tables 7 and 8 show the empirical power of our proposed test for different $p, n, ρ$ under the two error distributions.

As shown by these simulation results, the empirical size and empirical power of the portmanteau test improve as $n - p$ tends to infinity.

$p, n$	$n - p$	$q = 3$	$q = 5$	$p, n$	$n - p$	$q = 3$	$q = 5$
2,32	30	0.0389	0.0402	8,32	24	0.0351	0.0350
16,32	16	0.0299	0.0349	24,32	8	0.0208	0.0132
2,64	62	0.0443	0.0505	32,64	32	0.0391	0.0420
32,128	96	0.0436	0.0501	64,128	64	0.0402	0.0427
32,256	224	0.0489	0.0470	64,256	192	0.0475	0.0485
128,256	128	0.0452	0.0477	16,512	496	0.0499	0.0494
64,512	448	0.0490	0.0486	128,512	384	0.0502	0.0513
256,512	256	0.0473	0.0438	64,1028	964	0.0461	0.0494
128,1028	900	0.0480	0.0485	256,1028	772	0.0492	0.0501

Table 5. Empirical size under Gaussian error assumption

$p, n$	$n - p$	$q = 3$	$q = 5$	$p, n$	$n - p$	$q = 3$	$q = 5$
2,32	30	0.0359	0.0374	8,32	24	0.0390	0.0383
16,32	16	0.0265	0.0281	24,32	8	0.0129	0.0087
2,64	62	0.0444	0.0426	32,64	32	0.0385	0.0365
32,128	96	0.0430	0.0448	64,128	64	0.0439	0.0417
32,256	224	0.0497	0.0437	64,256	192	0.0509	0.0514
128,256	128	0.0487	0.0465	16,512	496	0.0504	0.0498
64,512	448	0.0479	0.0511	128,512	384	0.0498	0.0458
256,512	256	0.0518	0.0523	64,1028	964	0.0500	0.0489
128,1028	900	0.0490	0.0513	256,1028	772	0.0439	0.0503

Table 6. Empirical size under Gamma(4,1/2) error assumption

		$ρ_{1} = 0.2$	$ρ_{1} = 0$			$ρ_{1} = 0.2$	$ρ_{1} = 0$
$p, n$	$n - p$	$ρ_{2} = - 0.3$	$ρ_{2} = 0.3$	$p, n$	$n - p$	$ρ_{2} = - 0.3$	$ρ_{2} = 0.3$
2,32	30	0.2630	0.1960	8,32	24	0.1699	0.1265
16,32	16	0.0890	0.0694	24,32	8	0.0760	0.0205
2,64	62	0.5698	0.4064	32,64	32	0.1708	0.1210
32,128	96	0.6660	0.4775	64,128	64	0.2764	0.2232
32,256	224	0.9849	0.9278	64,256	192	0.9369	0.8167
128,256	128	0.6147	0.4335	16,512	496	1.0000	1.0000
64,512	448	1.0000	1.0000	128,512	384	0.9991	0.9897
256,512	256	0.9155	0.7551	64,1028	964	1.0000	1.0000
128,1028	900	1.0000	1.0000	256,1028	772	1.0000	1.0000

Table 7. Empirical power under Gaussian error assumption

		$ρ_{1} = 0.2$	$ρ_{1} = 0$			$ρ_{1} = 0.2$	$ρ_{1} = 0$
$p, n$	$n - p$	$ρ_{2} = - 0.3$	$ρ_{2} = 0.3$	$p, n$	$n - p$	$ρ_{2} = - 0.3$	$ρ_{2} = 0.3$
2,32	30	0.2657	0.1892	8,32	24	0.1202	0.1822
16,32	16	0.0519	0.0281	24,32	8	0.0202	0.0198
2,64	62	0.5721	0.3981	32,64	32	0.1190	0.1998
32,128	96	0.6738	0.5285	64,128	64	0.2853	0.1757
32,256	224	0.9291	0.8898	64,256	192	0.9034	0.7370
128,256	128	0.6320	0.4225	16,512	496	1.0000	0.9998
64,512	448	1.0000	0.9989	128,512	384	0.9989	0.9893
256,512	256	0.9137	0.7530	64,1028	964	1.0000	1.0000
128,1028	900	1.0000	1.0000	256,1028	772	1.0000	1.0000

Table 8. Empirical power under Gamma(4,1/2) error assumption

3.3. Parameter estimation under the null hypothesis

In practice, if the error terms are not Gaussian, we need to estimate the fourth-order cumulant to perform the test. We now give a suggested estimate under the additional assumption that the error terms are independent under the null hypothesis. Note that an unbiased estimate of variance $σ^{2}$ under the null hypothesis is

{^σ}_{n}^{2} = \frac{γ_{0}}{n - p},

and

E n \sum i = 1 e_{i}^{4}

= 3 σ^{4} n \sum i = 1 \sum j_{1}, j_{2} h_{i j_{1}}^{2} h_{i j_{2}}^{2} + ν_{4} σ^{4} n \sum i = 1 n \sum j = 1 h_{i j}^{4} = 3 σ^{4} t r (R \circ R) + ν_{4} σ^{4} t r (R \circ R)^{2} .

Then, $ν_{4}$ can be estimated by a consistent estimator

{^ν}_{4} = \frac{\sum_{i = 1}^{n} e_{i}^{4} - 3 {^σ}_{n}^{4} t r (R \circ R)}{{^σ}_{n}^{4} t r (R \circ R)^{2}} .

4. A general joint CLT for several general quadratic forms

In this section, we establish a general joint CLT for several general quadratic forms, which helps us to find the asymptotic distribution of the statistics for testing the series correlations. We believe that the result presented below may have its own interest.

4.1. A brief review of random quadratic forms

Quadratic forms play an important role not only in mathematical statistics but also in many other branches of mathematics, such as number theory, differential geometry, linear algebra and differential topology. Suppose $ε = {(ε_{1}, ε_{2}, \dots, ε_{n})}_{1}^{'}$ , where ${ε_{i}}_{i = 1}^{n}$ is a sample of size $n$ drawn from a certain standardized population. Let $A = (a_{i j})_{n \times n}$ be a matrix. Then, $ε^{'} A ε = \sum_{i, j} a_{i j} ε_{i} ε_{j}$ is called a random quadratic form in $ε$ . The random quadratic forms of normal variables, especially when $A$ is symmetric, have been considered by many authors, who have achieved fruitful results. We refer the reader to [Bartlett et al.(1960)Bartlett, Gower, and Leslie, Darroch(1961), Gart(1970), Hsu et al.(1999)Hsu, Prentice, Zhao, and Fan, Forchini(2002), Dik and De Gunst(2010), Al-Naffouri et al.(2016)Al-Naffouri, Moinuddin, Ajeeb, Hassibi, and Moustakas]. Furthermore, many authors have considered the more general situation, where $ε$ follow a non-Gaussian distribution. For the properties of those types of random quadratic forms, we refer the reader to [Fox and Taqqu(1985), Cambanis et al.(1985)Cambanis, Rosinski, and Woyczynski, de Jong(1987), Gregory and Hughes(1995), Gotze and Tikhomirov(1999), Liu et al.(2009)Liu, Tang, and Zhang, Deya and Nourdin(2014), Oliveira(2016)] and the references therein.

However, few studies have considered the joint distribution of several quadratic forms. Thus, in this paper, we want to establish a general joint CLT for several random quadratic forms with general distributions.

4.2. Assumptions and results

To this end, suppose

Ξ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} ε_{1}^{(1)} & ε_{2}^{(1)} & \dots & ε_{n}^{(1)} ε_{1}^{(2)} & ε_{2}^{(2)} & \dots & ε_{n}^{(2)} ⋮ & ⋱ & ⋮ ε_{1}^{(q)} & ε_{2}^{(q)} & \dots & ε_{n}^{(q)} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

is a $q \times n$ random matrix. Let ${A_{l} = a_{i, j}^{(l)}}, 1 \leq i, j \leq n, 1 \leq l \leq q$ be $q$ nonrandom $n$ -dimensional matrices. Define $Q_{l} = \sum_{i, j = 1}^{n} a_{i, j}^{(l)} ε_{i}^{(l)} ε_{j}^{(l)}$ for $1 \leq l \leq q .$ We are interested in the asymptotic distribution, as $n \to \infty$ , of the random vector $(Q_{1}, Q_{2}, \dots, Q_{q})$ , which consists of $q$ random quadratic forms. Now, we make the following assumptions.

${ε_{j}^{(i)}}_{1 \leq j \leq n, 1 \leq i \leq q}$ are standard random variables (mean zero and variance one) with uniformly bounded fourth-order moments $M_{4, j}^{(i)}$ .
The columns of $Ξ$ are independent.
The spectral norms of the $q$ $n \times n$ square matrices ${A_{l}}_{1 \leq l \leq q}$ are uniformly bounded in $n$ .

Clearly, for $1 \leq l \leq q$ , we have $E Q_{l} = t r A_{l}$ , and for $1 \leq l_{1}, l_{2} \leq q$ , we obtain

(4.1)			$C o v (Q_{l_{1}}, Q_{l_{2}}) = E (Q_{l_{1}} - E Q_{l_{1}}) (Q_{l_{2}} - E Q_{l_{2}})$
	$=$	$E ⎛ ⎝ n \sum i = 1 a_{i, i}^{(l_{1})} ({(ε_{i}^{(l_{1})})}^{2} - 1) + \sum i \neq j a_{i, j}^{(l_{1})} ε_{i}^{(l_{1})} ε_{j}^{(l_{1})} ⎞ ⎠ ⎛ ⎝ n \sum i = 1 a_{i, i}^{(l_{2})} ({(ε_{i}^{(l_{2})})}^{2} - 1) + \sum i \neq j a_{i, j}^{(l_{2})} ε_{i}^{(l_{2})} ε_{j}^{(l_{2})} ⎞ ⎠$
	$=$	$n \sum i = 1 a_{i, i}^{(l_{1})} a_{i, i}^{(l_{2})} E ({(ε_{i}^{(l_{1})})}^{2} - 1) ({(ε_{i}^{(l_{2})})}^{2} - 1) + n \sum i \neq j a_{i, j}^{(l_{1})} a_{i, j}^{(l_{2})} E ε_{i}^{(l_{1})} ε_{i}^{(l_{2})} E ε_{j}^{(l_{1})} ε_{j}^{(l_{2})}$
		$+ n \sum i \neq j a_{i, j}^{(l_{1})} a_{j, i}^{(l_{2})} E ε_{i}^{(l_{1})} ε_{i}^{(l_{2})} E ε_{j}^{(l_{1})} ε_{j}^{(l_{2})} .$

Let $l_{1} = l_{2} = l$ ; then, we have

V a r (Q_{l}) = n \sum i = 1 (M_{4, i}^{(l)} - 3) {(a_{i, i}^{(l)})}^{2} + t r A_{l} A_{l}^{'} + t r A_{l}^{2} .

Thus, according to assumptions $(a) - (c)$ , for any $1 \leq l \leq q,$ $V a r (Q_{l})$ at most has the same order as $n$ . This result also holds for any $C o v (Q_{l_{1}}, Q_{l_{2}})$ by applying the Cauchy-Schwartz inequality. We then have the following theorem.

Theorem 4.1.

In addition to assumptions (a)-(c), suppose that there exists an $i$ such that $V a r (Q_{i})$ has the same order as $n$ when $n \to \infty$ . Then, the distribution of the random vector $n^{- 1 / 2} (Q_{1}, Q_{2}, \dots, Q_{q})$ is asymptotically $q$ -dimensional normal.

4.3. Proof of Theorem 4.1

We are now in position to present the proof of the joint CLT via the method of moments. The procedure of the proof is similar to that in [Bai et al.(2018)Bai, Pan, and Yin] but is more complex since we need to establish the CLT for a $q$ -dimensional, rather than 2-dimensional, random vector. Moreover, we do not assume the underlying distribution to be symmetric and identically distributed. The proof is separated into three steps.

4.3.1. Step 1: Truncation

Noting that ${sup}_{i, j} E (ε_{j}^{(i)})^{4} < \infty$ , $j = 1, \dots, n, i = 1, \dots, q$ , for any $δ > 0$ , we have ${sup}_{i, j} δ^{- 4} n P (| (ε_{j}^{(i)}) | \geq δ n^{1 / 4}) \to 0.$ Thus, we may select a sequence $δ_{n} \to 0$ such that ${sup}_{i, j} δ_{n}^{- 4} n P (| (ε_{j}^{(i)}) | > δ_{n} n^{1 / 4}) \to 0$ . The convergence rate of $δ_{n}$ to 0 can be made arbitrarily slow. Define $({˜ Q}_{1}, {˜ Q}_{2}, \dots, {˜ Q}_{q})$ to be the analogue of $(Q_{1}, Q_{2}, \dots, Q_{q})$ with $ε_{j}^{(i)}$ replaced by ${~ ε}_{j}^{(i)}$ , where ${~ ε}_{j}^{(i)} = ε_{j}^{(i)} I (| (ε_{j}^{(i)}) | < δ_{n} n^{1 / 4})$ . Then,

P (({˜ Q}_{1}, {˜ Q}_{2}, \dots, {˜ Q}_{q}) \neq (Q_{1}, Q_{2}, \dots, Q_{q})) \leq q \sum i = 1 n \sum j = 1 P (ε_{j}^{(i)} \neq {~ ε}_{j}^{(i)}) \leq q n P (| (ε_{j}^{(i)}) | \geq δ_{n} n^{1 / 4}) \to 0.

Therefore, we need only to investigate the limiting distribution of the vector $({˜ Q}_{1}, {˜ Q}_{2}, \dots, {˜ Q}_{q})$ .

4.3.2. Step 2: Centralization and Rescaling

Define $({˘ Q}_{1}, {˘ Q}_{2}, \dots, {˘ Q}_{q})$ to be the analogue of $({˜ Q}_{1}, {˜ Q}_{2}, \dots, {˜ Q}_{q})$ with ${˜ ε}_{j}^{(i)}$ replaced by ${˘ ε}_{j}^{(i)} = \frac{{~ ε}_{j}^{(i)} - E {~ ε}_{j}^{(i)}}{\sqrt{V a r {~ ε}_{j}^{(i)}}}$ . Denote by $d (X, Y) = \sqrt{E | X - Y |^{2}}$ the distance between two random variables $X$ and $Y$ . Additionally, denote ${˜ ε}^{(i)} = {({˜ ε}_{1}^{(i)}, {˜ ε}_{2}^{(i)}, \dots, {˜ ε}_{n}^{(i)})}^{'}$ , ${˘ ε}^{(i)} = {({˘ ε}_{1}^{(i)}, {˘ ε}_{2}^{(i)}, \dots, {˘ ε}_{n}^{(i)})}^{'}$ and $Ψ^{(i)} = D i a g (\sqrt{V a r {~ ε}_{1}^{(i)}}, \sqrt{V a r {~ ε}_{2}^{(i)}}, \dots, \sqrt{V a r {~ ε}_{n}^{(i)}}) .$ We obtain that for any $l = 1, \dots, q,$

(4.2)	$d^{2} ({˜ Q}_{l}, {˘ Q}_{l})$	$= E \| ({˘ ε}^{' (l)} A_{l} {˘ ε}^{(l)} - {˜ ε}^{' (l)} A_{l} {˜ ε}^{(l)}) \|^{2}$
	$= E \| ({˘ ε}^{' (l)} A_{l} {˘ ε}^{(l)} - {˘ ε}^{' (l)} Ψ^{(l)} A_{l} Ψ^{(l)} {˘ ε}^{(l)} - {E ~ ε}^{' (l)} Ψ^{(l)} A_{l} Ψ^{(l)} E {~ ε}^{(l)}) \|^{2}$
	$\leq 2 (E \| {˘ ε}^{' (l)} A_{l} {˘ ε}^{(l)} - {˘ ε}^{' (l)} Ψ^{(l)} A_{l} Ψ^{(l)} {˘ ε}^{(l)} \|^{2} + \| {E ~ ε}^{' (l)} Ψ^{(l)} A_{l} Ψ^{(l)} E {~ ε}^{(l)} \|^{2})$
	$≜ 2 (Υ_{l, 1} + Υ_{l, 2}) .$

Noting that ${˘ ε}_{j}^{(i)}$ ’s are independent random variables with 0 means and unit variances, it follows that

	$Υ_{l, 1}$	$= E {({˘ ε}^{' (l)} A_{l} {˘ ε}^{(l)} - {˘ ε}^{' (l)} Ψ^{(l)} A_{l} Ψ^{(l)} {˘ ε}^{(l)})}_{l}^{2} = E {({˘ ε}^{' (l)} (A_{l} - Ψ^{(l)} A_{l} Ψ^{(l)}) {˘ ε}^{(l)})}^{2}$
		$= ν_{4} t r ((A_{l} - Ψ^{(l)} A_{l} Ψ^{(l)}) \circ (A_{l} - Ψ^{(l)} A_{l} Ψ^{(l)})) + t r {(A_{l} - Ψ^{(l)} A_{l} Ψ^{(l)})}_{l}^{2}$
		$+ t r ((A_{l} - Ψ^{(l)} A_{l} Ψ^{(l)}) {(A_{l} - Ψ^{(l)} A_{l} Ψ^{(l)})}_{l}^{'}) .$

Since $E {~ ε}_{j}^{(l)} = E ε_{j}^{(l)} I (| ε_{j}^{(l)} | \geq δ_{n} n^{1 / 4}) \leq C δ_{n}^{- 3} n^{- 3 / 4},$ and

1 - E ({~ ε}_{j}^{(l)})^{2} = E (ε_{j}^{(l)})^{2} I (| ε_{j}^{(l)} | \geq δ_{n} n^{1 / 4}) \leq C δ_{n}^{- 2} n^{- 1 / 2},

we know that

(4.3)

∥ (I - Ψ^{(l)}) ∥ = max j = 1, \dots, n | 1 - \sqrt{V a r {~ ε}_{j}^{(l)}} | \leq max j = 1, \dots, n \sqrt{| 1 - V a r {~ ε}_{j}^{(l)}} | \leq C δ_{n}^{- 1} n^{- 1 / 4} .

Then, we have

∥ (A_{l} - Ψ^{(l)} A_{l} Ψ^{(l)}) ∥ \leq ∥ A_{l} ∥ ∥ (I - Ψ^{(l)}) ∥ + ∥ (I - Ψ^{(l)}) ∥ ∥ A_{l} ∥ ∥ Ψ^{(l)} ∥ = O (δ_{n}^{- 1} n^{- 1 / 4}) .

It follows that $Υ_{l, 1} = O (δ_{n}^{- 2} n^{1 / 2})$ and $Υ_{l, 2} \leq ∥ Ψ^{(l)} ∥^{2} ∥ A_{l} ∥ \sum_{j = 1}^{n} {(E {~ ε}_{j}^{(l)})}^{2} = O (δ_{n}^{- 6} n^{- 1 / 2}) .$ By combining the above estimates, we obtain that $d ({˜ Q}_{l}, {˘ Q}_{l}) = O (δ_{n}^{- 1} n^{1 / 4})$ for $l = 1, \dots, q .$

Noting that the entries in the covariance matrix of the random vector $(Q_{1}, Q_{2}, \dots, Q_{q})^{'}$ have at most the same order as $n$ , we conclude that $n^{- 1 /}$

Model-Free Tests for Series Correlation in Multivariate Linear Regression

Authors

Correlation Estimation System Minimization Compared to Least Squares Minimization in Simple Linear Regression

Nonparametric Tests in Linear Model with Autoregressive Errors

Max-sum tests for cross-sectional dependence of high-demensional panel data

Smoothing-based tests with directional random variables

Linear regression and its inference on noisy network-linked data

A non-inferiority test for R-squared with random regressors

A test for partial correlation between repeatedly observed nonstationary nonlinear timeseries

1. Introduction

2. Test for series correlation in linear regression model

2.1. Notation

2.2. Test for a given order series correlation

2.3. A portmanteau test for series correlation

2.4. Discussion of the statistics

3. Simulation studies

3.1. Performance of test for first-order series correlation

3.2. Performance of the Box-Pierce type test

3.3. Parameter estimation under the null hypothesis

4. A general joint CLT for several general quadratic forms

4.1. A brief review of random quadratic forms

4.2. Assumptions and results

Theorem 4.1.

4.3. Proof of Theorem 4.1

4.3.1. Step 1: Truncation

4.3.2. Step 2: Centralization and Rescaling

Model-Free Tests for Series Correlation in Multivariate Linear Regression

Authors

Correlation Estimation System Minimization Compared to Least Squares Minimization in Simple Linear Regression

Nonparametric Tests in Linear Model with Autoregressive Errors

Max-sum tests for cross-sectional dependence of high-demensional panel data

Smoothing-based tests with directional random variables

Linear regression and its inference on noisy network-linked data

A non-inferiority test for R-squared with random regressors

A test for partial correlation between repeatedly observed nonstationary nonlinear timeseries

This week in AI

1. Introduction

2. Test for series correlation in linear regression model

2.1. Notation

2.2. Test for a given order series correlation

2.3. A portmanteau test for series correlation

2.4. Discussion of the statistics

3. Simulation studies

3.1. Performance of test for first-order series correlation

3.2. Performance of the Box-Pierce type test

3.3. Parameter estimation under the null hypothesis

4. A general joint CLT for several general quadratic forms

4.1. A brief review of random quadratic forms

4.2. Assumptions and results

Theorem 4.1.

4.3. Proof of Theorem 4.1

4.3.1. Step 1: Truncation

4.3.2. Step 2: Centralization and Rescaling