Independence Test For High Dimensional Random Vectors

Transcription

1 Independence Test For High Dimensional Random Vectors Guangming Pan, Jiti Gao, Yanrong Yang, Meihui Guo June 6, 2 Abstract This paper proposes a new test statistic for the hypothesis testing of cross sectional independence. This test is based on the empirical spectral distribution of the sample covariance matrix. We study the limiting distributions of this test under both null and alternative hypotheses, as dimensionality and the sample size of the data go to infinity together, with their ratio converging to a finite nonzero limit. Finite sample properties are investigated through simulations on some classical time series. As an empirical study, the proposed test is applied to test the random walk hypothesis for closed stock prices of 62 companies in New York Stock Exchange(NYSE) market from the date to the date , to detect whether the observed closed stock prices of each company follow a random walk. The results illustrate that most stocks in NYSE do not follow random walks, in agreement with the conclusions of some modern literature research works. Keywords: Independence test, empirical spectral distribution, asymptotic distribution, characteristic function, Marcenko-Pastur Law. Introduction Suppose that X ij are real valued random variables. For j p, let x j = (X j,, X jn ) T denote the j th time series and X = ( x T,, ) xt p be a panel of p time series, where n usually denotes the sample size in each of the time series data. In both theory and practice, it is not uncommon to assume that each of the time series (X j, X j2,, X jp ) is statistically independent, but it may be unrealistic to assume that x, x 2,, x p are independent and even uncorrelated. This is because there is no natural ordering for cross sectional indices. There are such cases in various disciplines. In economics and finance, for example, it is not unreasonable to expect that there is significant evidence of cross sectional dependence in output innovations across p countries and regions in the World. In the field of climatology, there is also some evidence to show that climatic variables in different stations may be cross sectionally dependent and the level of cross sectional dependence may be determined by some kind of physical distance. Moreover, one would expect that climatic variables, such as temperature and rainfall variables, in a station in Australia have higher level dependence with the same type of climatic variables in a station in New Zealand than those in the United States. In such situations, it may be necessary to test whether x, x 2,, x p are independent before a statistical model is used to model such data. In the econometrics and statistics literature, several papers have considered testing for cross sectional independence for the residuals involved in some specific regression models. Such studies include (38) for the parametric linear model case, (28) for the parametric nonlinear case, and (23) for the nonparametric nonlinear case.

2 As the main motivation of this paper, we will propose using an empirical spectral distribution function based test statistic for cross sectional independence of x, x 2,, x p. Our aim is to test H : x,, x p are independent; against H : x,, x p are not independent. () In this paper a new test for hypothesis () is proposed allowing p proportional to n. Our approach relies on large random matrix theory. The reminder of the paper is organized as follows. Section 2 introduces the test statistic and some related random matrix results. Section 3 shows the central limit theorem for the proposed test statistic. Section 4 reports simulation results, including the sizes and powers of the proposed test for normal and gamma distributed data respectively, and the powers for testing some classical time series, such as ARCH() process, MA() process and nonlinear MA process. The goal is to check the efficiency of our test in application for time series possessing different kinds of dependence. A specific random matrix called Vandermonde matrix is also investigated, which has important applications in engineering. Section 5 provides an empirical study, the proposed test is applied to the random walk hypothesis for the closed stock prices of 62 companies from New York Stock Exchange. Section 6 contains the conclusions. Proofs and some lemmas are deferred to the Appendix. 2 Methodology We first make the following assumption: Assumption. For each j, X j,, X jn are i.i.d random variables with mean zero and variance one. Denote the sample covariance matrix by A n = n XH X where X = (x,..., x p ) = (y,..., y n ) T and the superscript H stands for the complex conjugate transpose. The empirical spectral distribution (ESD) of the sample covariance matrix A n is defined as F An (x) = p I(λ i x), (2) p i= where λ λ 2... λ p are eigenvalues of A n. It is well-known that if x,, x p are independent and p/n c (, ) then F A p (x) converges with probability one to the Marcenko- Pastur Law (M-P Law) F c (x) ( see Marcenko-Pastur (967) and (6)) whose density has an explicit expression { p c (x) = 2πxc (b x)(x a), a x b; (3), otherwise; and a point mass /c at the origin if c >, where a = ( c) 2 and b = ( + c) 2. When there is some correlation among x,, x p, denote by T p the covariance matrix of the first row, y T, of X. Then, under some structure of each x j, when F T p D (x) H(x), F An converges with probability one to a non random distribution function F c,h whose Stieltjes transform satisfies (see (42)) m = dh(x). (4) x( c czm) z 2

3 Here the Stieltjes transformation m G for any c.d.f G is defined as m G (z) = dg(λ), Imz > (5) λ z and G can be recovered by the inversion formula G{[x, x 2 ]} = π lim ε + x2 x Im(m G (x + iε))dx where x and x 2 are continuity points of G. Moreover, the equation (4) takes a simpler form when F c,h is replaced by F c,h = ( c)i [, ] + cf c,h, (6) which is the limiting ESD of A p = n XXH. Its Stieltjes transform has an inverse m(z) = c z z = z(m) = m + c + cm(z) (7) x dh(x). (8) + xm Our test statistics relies on the following observation: the limit of the ESD of the sample covariance matrix A n is the M-P law by (3) when x,, x p are independent, while, the limit of the ESD is determined from (4) when T p is different from the identity matrix; Moreover, preliminary investigations indicate that when x,, x p are only uncorrelated (without any further assumptions) the limit of the ESD of A n is not the M-P law either (see (4)). So, these motivate us to use the ESD of A n, F An (x), as a test statistic. However, there is no central limit theorem for (F A n (x) F c,h (x)), as argued by (5). Therefore, instead, we consider the characteristic function of F A n (x). The characteristic function of F A n (x) is s n (t) Our test statistic is then proposed as follows M n = e itx df A n (x) = p e itλ i. i= s n (t) s(t) 2 du(t), (9) where s(t) is the characteristic function of F c n (x), obtained from the M-P law F c (x) with c replaced by c n = p/n, and U(t) is a distribution function with its support on a compact interval. To develop the asymptotic distribution of M n the following assumptions are needed. Assumption 2. let T p be a p p random Hermitian nonnegative definite matrix and (T /2 ) 2 = T. let y T j = ut j T/2 with u j being i.i.d random vectors. The empirical spectral distribution F Tp of T p converges weakly to a distribution H on [, ) as n ; the jth entry of u i is denoted as y ij, for each p, y ij, i n, j p are i.i.d., i.d. for all p, i, j, Ey =, E y 2 = and E y 4 < ; Assumption 3. p = p(n) with p/n c (, ). 3

4 3 Main results Write G n (x) = p[f An (x) F cn (x)]. () Then p(s n (t) s(t)) can be decomposed as the sum of random part and non-random part as follows: p(s n (t) s(t)) = e itx dg n (x) = e itx dp(f A n (x) F c n,h n (x)) + e itx dp(f c n,h n (x) F c n (x)), () where F c n,h n is obtained from F c,h with c and H replaced by c n = p/n and H n = F T n. (5) established the remarkable central limit theorem for functional of eigenvalues of A n under the additional assumption that E X 4 = 3 while (37) provided a supplement to this theorem by eliminating the condition to some extent. From Theorem 4 of (37) and () we can directly obtain the following results. Theorem. In addition to Assumption?? and 2, suppose that n e i T /2 n (m(z )T n + I) T /2 n e i e i T /2 n (m(z 2 )T n + I) T /2 n e i h (z, z 2 ) (2) n and i= n n i= e i T /2 n (m(z)t n + I) T /2 n e i e i T /2 n (m(z)t n + I) 2 T /2 n e i h 2 (z), (3) where e i is the n-dimensional row vector with the i-th element being and others. Then, for any positive integer k, ( ) cos(t x)dg n (x),..., cos(t k x)dg n (x), sin(t x)dg n (x),..., sin(t k x)dg n (x) converges in distribution to Gaussian vectors (V + δ (t ),..., V k + δ (t k ), Z + δ 2 (t ),..., Z k + δ 2 (t k )), where δ (t) = lim cos(tx)dp(f c n,h n (x) F c n (x)), δ 2 (t) = lim sin(tx)dp(f c n,h n (x) F c n (x)). n n The means covariances of V j and Z j are specified as follows: If X is real, then for any j =,..., k, EV j = c m 3 (z)τ 2 ( + τm(z)) 3 dh(τ) cos(t j z) 2πi ( c m 2 (z)τ 2 ( + τm(z)) 2 dh(τ)) 2 dz EX4 3 cm 3 (z)h 2 (z) cos(t j z) 2πi c m 2 (z)τ 2 ( + τm(z)) 2 dh(τ) dz. Replacing cos(t j z) in EV j by sin(t j z) yields the expression of EZ j. For any j, h =,..., k, Cov(V j, Z h ) = cos(tj z )sin(t h z 2 ) d 2π 2 (m(z ) m(z 2 )) 2 m(z 2 ) d m(z )dz dz 2 dz 2 dz c(ex4 3) 4π 2 d 2 (4) cos(t j z )sin(t h z 2 ) [m(z )m(z 2 )h (z, z 2 )]dz dz 2. dz dz 2 4

5 The covariances cov(v j, V h ) and cov(z j, Z h ) are similar except replacing cos(t j z) and (or) sin(t j z) by sin(t j z) and (or) sin(t j z) correspondingly. If X is complex with EX 2 =, then EV j = E X 4 2 cm 3 (z)h 2 (z) cos(t j z) 2πi c m 2 (z)τ 2 dh(τ)/( + τm(z)) 2 dz, and the covariance Cov(V j, Z h ) = cos(tj z )sin(t h z 2 ) d 4π 2 (m(z ) m(z 2 )) 2 m(z 2 ) d m(z )dz dz 2 dz 2 dz c(e X 4 2) d 2 4π 2 cos(t j z )sin(t h z 2 ) [m(z )m(z 2 )h (z, z 2 )]dz dz 2. dz dz 2 Remark. When T n = I, (5) calculated the asymptotic expectations and covariances for power functions f(x) = x r, r Z + for real X when EX 4 = 3 or complex X when E X 4 = 2. (37) eliminated this fourth moment condition and provided the expectation and covariance for f(x) = x r. These are helpful in simulations because the asymptotic expectations and covariances for f (x) = sintx and f 2 (x) = costx can be derived by Taylor series of sintx and costx. Remark 2. Assumption?? assumes that all the entries of the time series y are identical distributed. This applies to stationary time series. For a non-stationary time series, we can normalize it so that it has zero mean and unit variance. But the fourth moments of all the entries of the time series are not necessarily identical after normalization. In this case we only need to impose the additional assumptions that sup EYij 4 < and for any η >, i,j η 4 np E( Y jk 4 I Yjk η n). (5) jk A careful checking on the arguments of Theorem. of (5) and Theorem.4 of (37) indicates that Theorem still holds. For the expressions of the expectation and covariance of asymptotic distribution, we substitute the fourth moment E Y 4 with the average of all the fourth moments p j= E Y ji 4 of all the entries, i.e. p for any i. Therefore the proposed test can be applied to detect the independence for non-stationary time series with finite fourth moment. Based on Theorem, we can derive the central limit theorem for the proposed test statistic S n. Theorem 2. With the same assumptions as in Theorem, the proposed test statistic M n converges in distribution to a random variable R, where R = t2 t ( V (t) + δ (t) 2 + Z(t) + δ 2 (t) 2 )du(t), (6) (V (t), Z(t)) is the Gaussian limit of ( costxdg n (x), sintxdg n (x)) as n, p and δ (t), δ 2 (t) are defined in the above theorem. Particularly, under the null hypothesis, the statistic M n converges in distribution to + ( V (t) 2 + Z(t) 2 )du(t). The proof is delegated to Appendix. 5

6 Remark 3. There exists a problem how to choose the distribution U(t) appearing in the proposed test statistic S n. A practical consideration is that the choice of U(t) guarantees that the expression of the proposed statistic S n is easy to calculate in practice. In our simulation, we choose U(t) as the uniform distribution. Another consideration is how the values of t influence the values of the asymptotic random variables appearing in the central limit theorem of S n. From (5), the expectation and covariance of the asymptotic random variable for ( f (x)dp(f A n (x) F c n,h n (x)), f 2 (x)dp(f A n (x) F c n,h n (x))) are and EX f = 2π Cov(X f, X f2 ) = 2π 2 f (x)arg( c t 2 m 2 (x) dh(t))dx, (7) ( + tm(x)) 2 f (x)f 2(y)ln( m + 4 i (x)m i (y) )dxdy. (8) m(x) m i (y) 2 If f (x) = costx and f 2 (x) = sintx, then the limiting expectations and variances include t and t 2 respectively. From this point, large t will cause large covariance while small t that approximates to zero will incur small covariances. In practice, this will incur a larger empirical size than the significant level. But for the proposed test statistic, t will not have large influences on the the value of the test statistic S n, since t only appears in the function sintx or costx. So we should not choose large t or small t. Therefore, in our simulation, we choose the interval t [.2,.7]. 4 Simulations This section reports the simulation results about the finite sample properties of the proposed test. Simulations are used to find the size and power of the proposed independence test and hence evaluate the performance of this test. Comparison with LRT test conducted by (2) is provided for normal distributed time series. Some classical time series such as MA() process, uncorrelated nonlinear MA process and ARCH() process are tested by the proposed test. A specific matrix called Vandermonde matrix is also detected. 4. Empirical size and empirical power First we introduce the calculation of empirical size and empirical power. Let z α be the ( α)% quantile of the asymptotic null distribution of the test statistic M n. With K replications of the data set simulated under the null hypothesis, we calculate the empirical size as ˆα = { of M n H z α }, (9) K where Mn H represents the values of the test statistic M n based on the data simulated under the null hypothesis. In our simulation, we choose K = as the number of repeated simulations. The significance level is α =.5. Since the asymptotic null distribution of the test statistic is not a classical distribution, we need to estimate the quantile z α. Naturally, we do as follows: generate K replications of the asymptotic distributed random variable and then select the (Kα)-th largest value as the estimated ( α) quantile of the asymptotic distributed random variable. To compute the empirical power, we use the estimated critical point ẑ α under the null hypothesis. ẑ α is the Kα-th largest value of the test statistic by K repeated simulations under 6

7 the null hypothesis. Then the empirical power is ˆβ = { of M n A ẑ α }, (2) K where Mn A represents the values of the test statistic M n based on the data simulated under the alternative hypothesis. For the proposed independence test, we generate the number n of p-dimensional independent and identical distributed random vectors {x k } n k=, each with means p and covariance matrix Σ = I p. Under Assumption?? and the null hypothesis, {x k } n k= are generated in two scenarios:. Each y k is p-dimensional normal random vector with mean p and covariance matrix Σ = I p ; T p = I p ; 2. Each y k consists of i.i.d. random variables with standardized Gamma(4,2) distribution, so they have zero mean and unit variance; T p = I p. Under the alternative hypothesis, we consider the case: T /2 p = (.95I p,.5 p ), where p is p-dimensional vector with as entries. In this case, the population covariance matrix of x is Σ =.95I p +.5 p p, which is called compound symmetric covariance matrix. In our simulation, we choose p = 2 : 2 : 2 and n = 2 : 2 : 2. The significant level is.5. In each case, we run simulations. For normal distributed data, the fourth moment E X 4 = 3; for standardized Gamma distributional data, E X 4 = 4.5. This is calculated by a formula as follows: the k-th moment of X which is Gamma(α, λ) is E X k = (α + k ) (α + )α λ k. (2) Table and Table 2 show the empirical sizes and empirical powers of our proposed test and LRT((2)) test for normal distributed random vectors respectively. From Table and Table 2, we can see that LRT test does not work when p is large while the proposed test possesses good performance when p and n are both large. The LRT test is only applicable to the case that p is fixed and n goes to infinity. When the difference between p and n are large, the sizes and the powers of the proposed test become worse. This is the reason that our test is proposed under the case that p and n are in the same order when they approximate to infinity. The proposed test also works well for gamma random vectors while LRT test is not applicable to gamma case, since, in theory, LRT test is given under normal random vectors. Table 3 provides the empirical sizes and empirical powers of the proposed test for the gamma case. The empirical powers are all under the alternative hypothesis that Σ =.5I p +.95 p p. Reports for empirical powers show that the proposed test can check independence for both normal and gamma vectors well. Moreover, the empirical powers converge to as n, p. Since independence test is important in time series, we apply the proposed test to several time series, such as MA() model, uncorrelated nonlinear MA model and ARCH() model. Furthermore, a special kind matrix called Vandermonde matrix is also tested. 7

8 4.2 Some time series and Vandermonde matrix 4.2. MA() model and nonlinear MA model Moving average of order (MA()) model with length p is a p-dimensional random vector which is defined as v t = z t +.5z t, t =,..., p; (22) where z t W N(, ), i.e. white noise process with mean and variance σ 2. The correlation matrix is ρ() ρ() ρ() ρ() , (23) ρ() ρ() ρ() ρ() where ρ() = cov(vt,v t ). var(vt ) Since the entries of v = (v,..., v p ) are correlated and satisfy Assumption??. Hence, from Theorem 2, our test can capture the dependence of MA() process as n and p go to infinity in the same order. The simulation results in Table 4 show that the proposed test can indeed test the dependence of MA() process. Moreover, the proposed test can also distinguish the dependence for non-martingale difference sequences with zero autocorrelations. Consider, for example, the following uncorrelated nonlinear MA process: r t = z t z t 2 (z t 2 + z t + ), t =,..., p; (24) where z t W N(, ), i.e. white noise process with mean and variance σ 2. Its correlation matrix is a diagonal matrix. This model is provided by (32) which tests the martingale difference hypothesis. Our proposed independence test can be applied to this nonlinear MA model, and the powers in Table 5 show that this test performs well for this model. This result also states that the empirical spectral distribution of nonlinear MA model is not M-P law since the proposed test statistic is established on the characteristic function of M-P law ARCH() model This section considers autoregressive conditional heteroscedastic(arch()) model: w t = z t α + α wt 2, t =,..., p; (25) where z t W N(, ), i.e. white noise process with mean and variance. ARCH() model is a martingale difference sequence and it is a strictly stationary stochastic process. There exists no theoritical results stating that the LSD of sample covariance matrix for ARCH() model is M-P Law, but from Figure, we can see that the LSD of the sample covariance matrix for ARCH() model is M-P Law. For ARCH() model, the proposed test can not capture its dependence directly, but we can use our test to test the dependence of {w 2,..., w2 p}. Since this test can tell us that {w 2,..., w2 p} is not an independent sequence, 8

9 naturally it can be concluded that ARCH() is not an independent sequence. This idea is similar to (27) who use the second derivative of empirical spectral density function to capture the pairwise dependence for ARCH() model. Here we take α =.9 and α =.. Table 6 shows the powers of our test for testing dependence of ARCH() model. Figure : Empirical Spectral Distribution of ARCH() and I.I.D Normal Vector n=,p=2 n=45,p= Empirical Spectral Distribution ARCH() i.i.d Normal Empirical Spectral Distribution ARCH() i.i.d Normal x x n=2,p= n=5,p= Empirical Spectral Distribution ARCH() i.i.d Normal Empirical Spectral Distribution ARCH() i.i.d Normal x x Vandermonde matrix Consider n p vandermonde matrix V of the form V = e iω e iω p n..... e i(n )ω e i(n )ω p (26) where ω i, i =,..., p are called phased distributions and are assumed i.i.d on [, 2π). Then the entries of V lie on the unit circle. Obviously, all the elements of each column of V are not independent while the elements from different columns are independent. Denote the sample covariance matrix of V as D = V H V. Vandermonde matrices play an important role in signal processing and wireless applications such as direction of arrival estimation, precoding or sparse sampling theory, etc.. (4) established that as n, p go to, the limiting spectral distribution of D = V H V is not the Marcenko-Pastur Law. We use the proposed test statistic S to test the dependence of every column of the matrix V. Since for any k =,..., n and j =,..., p, E(e ikω j ) 2 = and E e ikω j 4 =, Theorem 9

10 2 can be applied directly without normalize process. The empirical powers in table 7 show that the proposed test works well for Vandermonde matrix. 5 Empirical study: random walk hypothesis The random walk hypothesis is defined by x t = µ + x t + ε t, t =,..., p; (27) where x t is the stock price at time t, µ is a drift parameter and ε t is a random disturbance term. All ε t, t =,..., p are assumed to be independent. To investigate whether the stock prices {x t } p t= follow a random walk is equal to testing independence of the first difference {ε t = x t x t } p t=. We apply our method to test random walk hypothesis for financial data in stock markets: daily closed stock prices of 62 companies from New York Stock Exchange(NYSE) from the date to the date For each company, there are 28 closed stock prices x t, t =,..., 28. Then we can make difference and get the random disturbance series {ε t } with length 28, i.e. t =,..., 28. In order to capture the independence of the time series {ε t, t =,..., 28}, we reshape the time series as a 4 7 matrix W and apply the proposed test to the matrix Q = n W W T. Fig.4 provides the histograms of the p-values of the proposed test for 62 stocks. The significance level is.5. The figure shows that there are 445 stocks whose p-values are below.5 and others are above.5. These results show that 76 stocks are accepted to follow random walk while 445 stocks are rejected. To be more clear, we plot autocorrelation functions(acf) of the random disturbance time series for 8 companies, of which 4 time series are accepted and 4 time series are rejected by our test. The ACF graphs of Figure 2 and Figure 3 illustrate that the proposed test can relatively accurately distinguish dependence from independence, since we accept the time series with relatively smaller sample ACFs and reject those with larger sample ACFs. Our test states that 445 stocks of the total 62 stocks are rejected as random walk hypothesis, which constitute a high proportion of the total investigated stocks. This result is in conformity with many research works on random walk hypothesis for various stock markets, e.g.(5), (2), (36). Some stock markets follow random walk hypothesis well while some not. From our proposed test, we can conclude that closed stock prices in New York Stock Exchange do not follow random walk hypothesis. Moreover, more and more professional traders and researchers find that random walk hypothesis is bogus. Many complex and more accurate models are proposed to describe stock markets or other financial markets. 6 Conclusions This paper provides an independence test based on the characteristic function of the empirical spectral distribution of the sample covariance matrix of a time series. The proposed test can test independence of all the entries of time series with length p, where p goes to infinity. Finite sample simulations show that our test also performs well when applying to time series with relatively small length. We do not assume any specific distribution on the time series and moreover, the proposed test can be applied to non-stationary time series with finite fourth moment condition, while most literatures impose stationary condition on the tested time series, such as (2). The proposed test is consistent for a class of random vectors with the type X = TY, where Y is a

11 Sample Autocorrelation Figure 2: ACF Graphs HONEYWELL INTERNATIONAL INC ARCHER DANIELS MIDLAND CO..8.6 Sample Autocorrelation Lag Lag. COCA COLA CO. EATON CORP Sample Autocorrelation Sample Autocorrelation Lag Lag *Autocorrelation function graphs of random disturbance series of stock prices from four companies in NYSE, whose stock prices are accepted as random walks by the proposed test. The time series are all length of 28, from the date 99.. to the date All the closed stock prices are from WRDS database. random vector with independent entries and T is a nonnegative definite matrix. Many literatures investigate independence by autocorrelation functions, e.g.(), (26),etc.. This method can not capture the dependence of some time series with uncorrelated entries. (27) offered a generalized spectral density approach and his test can apply to many uncorrelated time series. But Hong s method is based on testing mutually independence of any two entries of a time series and can not test for a time series with length of more than 2. Moreover, his test is also restricted on stationary time series. For uncorrelated time series, such as nonlinear MA model investigated in the paper, which is not a martingale difference sequence, simulation results tell that our test can capture the dependence between its entries. For ARCH() model, which is a martingale difference sequence, we propose a modification to the proposed test and the modified version performs well on ARCH() model. This paper is contributed to study independence test for large-dimensional random vectors, which case is missed by classical test methods, such as LRT test come up by (2). LRT test includes the sample covariance matrix as the estimator of the population covariance matrix. When the dimensionality p is larger than the sample size n, sample covariance matrix is singular. Hence larger dimensionality causes lesser efficiency in classical statistics. Nowadays, the theory of random matrix is applied to deal with large dimensional data((4)). This paper contributes to this work.

12 Figure 3: ACF Graphs. EXXON MOBIL CORP. IDACORP INC Sample Autocorrelation Sample Autocorrelation Lag Lag. SCHAWK INC. AMERON INTERNATIONAL CORP DEL Sample Autocorrelation Sample Autocorrelation Lag Lag *Autocorrelation function graphs of random disturbance series of stock prices from four companies in NYSE, whose stock prices are rejected to follow random walks by the proposed test. The time series are all length of 28, from the date 99.. to the date All the closed stock prices are from WRDS database. 7 Appendix Lemma (Theorem 8. of (9)). Let P n, P be probability measures on (C, ϕ). If the finite dimensional distributions of P n converge weakly to those of P, and if {P n } is tight, then P n P. Lemma 2 (Theorem 2.3 of (9)). The sequence {X n } is tight if it satisfies these two conditions (I) The sequence {X n ()} is tight. (II) There exist constants γ, α >, and a nondecreasing, continuous function F on [, ] such that holds for all t, t 2, and n. E{ X n (t 2 ) X n (t ) γ } F (t 2 ) F (t ) α (28) Lemma 3 (Continuous Theorem). Let X n and X be random elements defined on a metric space S. Suppose g : S S has a set of discontinuous points D g such that P (X D g ) =. Then X n d X g(x n ) d g(x). (29) Lemma 4 (Lemma 2.4 of (24)). Let Ω be an open convex set in C. If f : Ω C is an analytic function and a, b are distinct points in Ω, then there exist points u, v on L(a, b) such that Re( f(a) f(b) ) = Re(f f(a) f(b) (u)), Im( ) = Im(f (v)), (3) a b a b where Re(z) and Im(z) are the real and imaginary parts of z respectively; and L(a, b) {a + t(b a) : t (, )}. 2

13 Figure 4: P-value of the proposed test P values for the proposed test number of stocks p value *These are p-values for 62 stocks from NYSE from the date 99.. to the date All the closed stock prices are from WRDS database. Proof of Theorem 2. Let φ n (t) = costxdg n (x) and ψ n (t) = sintxdg n (x). To finish Theorem 2, by Lemma 3 it is enough to prove that (φ n, ψ n ) D (V + δ, Z + δ 2 ), where φ n = {φ n (t)}, ψ n = {ψ n (t)}, V = {V (t)}, Z = {Z(t)} and δ = {δ (t)}, δ 2 = {δ 2 (t)}. To this end, in view of Lemma and Theorem, we need to prove the tightness of (φ n (t)}, ψ n (t)) for t [t, t 2 ]. Repeating the same truncation and centralization steps as those in (5), we may assume that X ij < δ n n, EXij =, E X ij 2 =, E X ij 4 <. (3) Set M n (z) = n[m F An (z) m F cn,hn (z)]. By the Cauchy theorem f(x)dg(x) = f(z)m G (z)dz, (32) 2πi where G is a c.d.f and f is an analytic function on an open set containing the support of G. Therefore, with probability one, for all n large, φ n (t) can be written as φ n (t) = cos(tz)m n (z)dz. (33) 2πi The contour C involved in the above integral is specified as follows. Let C u = {x + iv : x [x l, x r ]}, (34) where v >, x r is any number greater than lim sup T n ( + c) 2, x l is any negative number if n c and otherwise choose x l (, lim sup T n ( c) 2 ). Then the contour C is defined by n C = {x l + iv : v [, v ]} C u {x r + iv : v [, v ]}. (35) We consider ˆM n (z), a truncated version of M n (z), as in (5). Let C l = { {xl + iv : v [n ε n, v ]}, x l >, {x l + iv : v [, v ]}, x l <, (36) 3

14 and C r = {x r + iv : v [n ε n, v ]} (37) where ε n is a decreasing sequence to zero satisfying for some α (, ), ε n n α. (38) Set C n = C l C u C r. The process ˆM n (z) is defined as M n (z), z C n, ˆM n (z) = M n (x r + in ε n ), x = x r, v [, n ε] and if x l >, M n (x l + in ε n ), x = x l, v [, n ε]. (39) Then Write φ () n (t) = 2πi M n (z) = M n (z) ˆM n (z) + ˆM n (z). (4) cos(tz)[m n (z) ˆM n (z)]dz, φ (2) n (t) = 2πi (costz) ˆM n (z)dz. Then φ n (t) can be decomposed as φ n (t) = φ () n (t) + φ (2) n (t). (4) There is a similar decomposition for ψ n (t). For concreteness, we only prove the tightness for the part {φ n (t)}. For ( ˆM n (z) M n (z)), the argument in page 563 of (5) ensures that cos(tz)(m n (z) ˆM n (z))dz a.s., uniformly in t [t, t 2 ]. Obviously, this implies for any ε >, lim lim sup P [ sup (cost z cost 2 z)[m n (z) ˆM n (z)]dz ε] =. (42) δ n t t 2 <δ By Theorem 7.3 of (9), {φ () n (t)} is tight. For ˆM n (z), write ˆM () n (z) = ˆM n (z) + (2) ˆM n (z), where and Based on these we may further write ˆM () n (z) = n[ ˆM F An (z) E ˆM F An (z)] (43) ˆM (2) n (z) = n[ ˆM EF An (z) ˆM F cn,hn (z)]. (44) φ (2) n (t) = φ 2, n (t) + φ 2,2 n (t), (45) where φn 2, (t) = 2πi cos(tz) () ˆM n (z)dz, φ 2,2 n (t) = 2πi cos(tz) (2) ˆM n (z)dz. 4

15 In order to prove the tightness of {φ 2, n (t)}, we use a tightness criteria, i.e. Theorem 2.3 of (8). It follows from Lemma 4 that sup E φ2, n (t ) φ 2, n (t 2 ) 2 t,t 2,n t t 2 ( = sup t,t 2,n 2π E C Re[zsin(t 3 z)] + i Im[zsin(t 4 z)]) ˆM n (z)dz 2, where t 3 and t 4 belong to L[t, t 2 ]. Since the integral contour C is bounded, by Cauchy s inequality, E (Re[zsin(t 3 z)] + i Im[zsin(t 4 z)]) ˆM [ n(z)dz 2 K E C C ] () ˆM n (z) 2 dz, where K is a constant number which may be different from line to line. From the argument in Page 569, (2.5) and Section 3 in (5) we know that on the contour C E ˆM () n (z) 2 K. This implies that {φ 2, n (t)} is tight in t. For the non-random part {φn 2,2 (t)}, we only need to prove that it is bounded and equicontinuous with respect to t. Since (5) has proved that ˆM n(z) 2 is uniformly bounded with respect to z and the integral contour is also bounded, we obtain φ 2,2 n (t) K. (46) By Lemma 4 and the fact that ˆM n(z) 2 is uniformly bounded, {φ 2,2 n (t)} is equicontinuous. Hence the tightness of {φn 2,2 (t)} is also completed. Therefore {φ n (t)} is tight. References [] W.L. Andrew, A.C. Mackinlay, stock market prices do not follow random walks: evidence from a simple specification test, Rev. Financ. Stud., ()(988), [2] T.W. Anderson, An introduction to multivariate statistical analysis, New York: Wiley, 2th edtion(984). [4] Z.D. Bai, H. Saranadasa, Effect of high dimension comparison of significance tests for a high dimensional two sample problem, Statist. Sinica, 6(996), [5] Z.D. Bai, J.W. Silverstein, CLT for linear spectral statistics of large-dimensional sample covariance matrices, Ann. Probab., 32(A)(24), [6] Z.D. Bai, J.W. Silverstein, Spectral analysis of large dimensional random matrices, Sringer, second edition(29). [7] Z.D. Bai, D.D. Jiang, J.F. Yao, S.R. Zheng, Corrections to LRT on large-dimensional covariance matrix by RMT, Ann. Statist., 37(6B)(29), [8] P. Billingsley, Convergence of Probability Measures, Wiley, New York first edition (968). [9] P. Billingsley, Convergence of probability measures, Wiley-Interscience Publication, second edition(999). 5

16 [] G. Box and D. Pierce, Distribution of residual autocorrelations in autoregressive integrated moving average time series models, J. Am. Statist. Ass., 65(97), [] N.H. Chan, L.T. Tran, Nonparametric tests for serial dependence, J. Time. Ser. Anal., 3(992), 2-3. [2] J.Y. Campbell, W.L. Andrew and A.C. MacKinlay, The econometrics of financial markets, Princeton University Press,(997). [3] D.L. Donoho, High-dimensional data analysis: The curses and blessings of dimensionality, American Math. Society Lecture, Math. Challenges of the 2st Century, (2). [4] T.W. EPPS, A test for normality based on the empirical characteristic function, Biometrika, 7(3)(983), [5] E.F. Fama, The Behavior of Stock Market Prices, J. Bus., 38(965), [6] E.F. Fama, Efficient capital markets: a review of theory and empirical work, J. Financ., 25(97), [7] E.F. Fama, Efficient capital markets: II, J. Financ., 46(99), [2] M.D. Godfrey, C.W.J. Granger, O. Morgenstern, The random-walk hypothesis of stock market behavior, Kyklos, 7()(964), -3. [22] C. Granger, T. Terasvirta, Modelling nonlinear economic relationships, New York: Oxford University Press, (993). [23] Chen, J., Gao, J. and Li, D. (29). A new diagnostic test for cross sectional uncorrelatedness in nonparametric panel data models. Working paper available at [24] C.H. Guo, N.J. Higham, A Schur-Newton method for the matrix p-th root and its inverse, SIAM Journal on Matrix Analysis and Applications, 28(3)(26). [25] R.E. Hall, Stochastic implications of the life cycle-permanent income hypothesis: theory and evidence, J. Polit. Econ., 86(978), [26] Y.M. Hong, Testing for pairwise serial independence via the empirical distribution function, J. R. Statist. Soc. B, 6(2)(998), [27] Y.M. Hong, Hypothesis testing in time series via the empirical characteristic function: a generalized spectral density approach, J. Am. Statist. Ass., 94(448)(999), [28] Hsiao, C., Pesaran, M. H. and Pick, A. (27). Diagnostic tests of cross section independence for nonlinear panel data models. IZA discussion paper No. 27 [29] S. John, Some optimal multivariate tests, Biometrika, 58(97), [3] I. Johnstone, on the distribution of the largest principal component, Ann. Statist., 29(2), [3] I. Johnstone, D.M. Titterington, Statistical challenges of high-dimensional data, Phil. Trans. R. Soc. A, 367(29), [32] C.M. Kuan and W.M. Lee, A new test for the martingale difference hypothesis, Studies in Nonlinear Dynamics and Econometrics, 8(4)(24),

17 [33] G.M. Ljung, G. Box, on a measure of lack of fit in time series models, Biometrika, 65(978), [34] V.A. Marcenko, L.A. Pastur, Distribution for some sets of random matrices, Math. USSR-Sb., (967), [35] H. Nagao, On some test criteria for covariance matrix, Ann. Statist., (973), [36] T. Nakamura, M. Small, Tests for the random walk hypothesis for financial data, Phys. A, 377(27), [37] G.M. Pan, W. Zhou, Central limit theorem for signal-to-interference ratio of reduced rank linear receiver, Ann. Appl. Probab., 8(3)(28), [38] Pesaran, M. H. (24). General diagnostic tests for cross section dependence in panels. Cambridge Working Paper in Economics No [39] V. Plerou, P. Gopikrishnan and B. Rosenow, A random matrix approach to cross-correlations in financial data, Phys. Rev. E: Stat., 65(6)(22), 6626(-8). [4] P.M. Robinson, Consistent nonparametric entropy-based testing, Rev. Econ. Stud., 58(99), [4]. Ryan, M. Debbah, Asymptotic behaviour of random vandermonde matrices with entries on the unit circle, IEEE Trans. on Information Theory, 55(7)(29), [42] J.W. Silverstein, Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices, J. Multivariate Anal. 55 (995), [43] H.J. Skaug, D. Tjøstheim, Nonparametric test of serial independence based on the empirical distribution function, Biometrika, 8(993), [44] H. Tong, Nonlinear time series: A dynamic system approach, Oxford, U.K.: Clarendon Press, (99). Table : Empirical sizes of the proposed test and the LRT test at significant level.5 for standardized normal distributed random vectors. p n The proposed test The LRT test

18 Table 2: Empirical powers of the proposed test and the LRT test at significant level.5 for standardized normal distributed random vectors. p n The proposed test The LRT test *The powers are under the alternative hypothesis that the population covariance matrix is Σ =.5I p +.95 p p. Table 3: Empirical sizes and powers of the proposed test at.5 significance level for standardized gamma distributed random vectors. p n Empirical sizes Empirical powers *The powers are under the alternative hypothesis that the population covariance matrix is Σ =.5I p +.95 p p. 8

19 Table 4: Empirical powers of the proposed test at.5 significance level for MA() model. p n Table 5: Empirical powers of the proposed test at.5 significance level for nonlinear MA model. p n Table 6: Empirical powers of the proposed test at.5 significance level for ARCH() model. p n Table 7: Empirical powers of the proposed test at.5 significance level for Vandermonde Matrix. p n