Abstract

OJS

Open Journal of Statistics

2161-718X

Scientific Research Publishing

10.4236/ojs.2011.13017

OJS-8066

Articles

Physics&Mathematics

Testing for Deterministic Components in Vector Seasonal Time Series

osé

Luis Gallego

^*Carlos

Díaz

* E-mail:jose.gallego@unican.es(OLG);carlos.diazv@unican.es(CD);

20102011

0103145150April 29, 2011May 30, 2011 June 10, 2011

2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Certain locally optimal tests for deterministic components in vector time series have associated sampling distributions determined by a linear combination of Beta variates. Such distributions are nonstandard and must be tabulated by Monte Carlo simulation. In this paper, we provide closed form expressions for the mean and variance of several multivariate test statistics, moments that can be used to approximate unknown distributions. In particular, we find that the two-moment Inverse Gaussian approximation provides a simple and fast method to compute accurate quantiles and p-values in small and asymptotic samples. To illustrate the scope of this approximation we review some standard tests for deterministic trends and/or seasonal patterns in VARIMA and structural time series models.

Vector Time Series Deterministic Components Parametric Stability Non-Invertibility Unit Roots

Abstract

1. Introduction

A wide class of test statistics for detecting the presence of deterministic components in univariate linear time series models can be derived following the [1] approach to the problem of testing for a scalar error covariance matrix in linear regression models with non-spherical disturbances. Members of this class are either locally best invariant (LBI) tests or LBI unbiased (LBIU) tests depending on whether the null hypothesis of a particular deterministic component is confronted with a one-sided or two-sided alternative, respectively. Some examples are the LBI test statistics for a null variance ratio or parametric stability proposed by [2-8], as well as the LBIU test statistics for non-invertibility or moving average (MA) unit roots derived by [9-12]. All these LBI and LBIU test statistics can be formulated as ratios of quadratic forms in normal variables, whose distribution functions are usually computed by numerical inversion of the corresponding characteristic functions using the [13] or [14] procedures. Moreover, their limiting distributions are related to that of the Cramèr-von Mises test statistics for goodness-of-fit derived and tabulated by [15]. Several non-parametric and parametric corrections have been proposed to cope with serially correlated errors so that the modified statistics follow the same limiting distributions.

Multivariate versions of these tests have been only derived in the framework of structural time series models by [5] and [16] based on the multivariate generalization of the [1] approach given by [17]. As in the univariate case, the limiting distributions of the multivariate test statistics are also related to the Cramèr-von Mises distribution. However, the small sample distributions are unknown and must be evaluated by Monte Carlo simulation. A similar problem arises in the analysis of cointegrated VAR models with Dickey-Fuller type tests, where procedures for easily computing p-values and quantiles has been proposed among others by [18], using a response surface approach, and [19], fitting a Gamma distribution with moments estimated from a response surface regression. Approximating the unknown distribution of the tests for deterministic components has been also suggested by [17], who found that such distributions can be expressed as linear combinations of Beta variates and gave closed form expressions for their first two moments, which involve the computation of the eigenvalues of a matrix whose order depends on the sample size. However, a tentative family has not been proposed yet.

In this paper, we use the results of [17] to derive closed form expressions for the mean and variance of several test statistics for deterministic components that depends on the sample size and avoid the computation of eigenvalues. Besides, we propose a two-moment Inverse Gaussian (IG) approximation to the distribution of a linear combination of Beta variates. To illustrate some applications of this approximation we provide seasonal extensions of the [5] tests that can be used to test for non-invertibility in vector seasonal ARIMA models, as well as to derive easily the [16] tests for deterministic seasonality at specific frequencies.

The paper is organized as follows. In Section 2, we summarize the main results of the [17] approach. In Section 3, we review some relevant test statistics for deterministic components in vector time series and give exact expressions for the first two moments. In Section 4, we describe the two-moment IG approximation and assess its accuracy. Finally, in Section 5, we conclude with some extensions.

2. Invariant Tests for Covariance Structures

Consider the multivariate linear regression model

, (1)

where Y and E are T × m random matrices, X is a T × k fixed design matrix, P is a k × m matrix of parameters, W (d) and S are T × T and m × m positive definite matrices, respectively, and d is the parameter of interest determining whether or not the columns of E are i.i.d. Tdimensional errors, i.e., W(d₀) = I_T. [17] found that the Locally Best Invariant (LBI) test statistic of the null hypothesis H₀: d = d₀ against the one-sided alternative H₁: d > d₀ has the following general expression

(2)

where tr is the trace operator, Ê = MY is the residual matrix in the ordinary least squares regression of Y on X, M = I_T – X(X¢X)^–¹ X¢ and K is the first derivative d W (d)/dd evaluated at d = d₀. Invariance is defined against the group of transformations Y ® YP + XA for an arbitrary k × m matrix A and a positive definite m × m matrix P. Thus, without loss of generality, it can be assumed that S = I_T. From [17], and following [1], it can also be proved that the LBIU test statistic of H₀: d = d₀ against the two-sided alternative H₁: d ≠ d₀ is given by (2) but with K being the second derivative d²W(d)/dd²evaluated at d = d₀.

The null distribution of L can be characterized rewriting it as

(3)

where l_t are the non-null eigenvalues of the product matrix MK, e_t ~ N(0, I_m), B_t ~ Beta (m/2, (T – k – m)/2) and B₁ + … + B_T–k = m, see, e.g., [20] p. 540. [17] found that the first two moments of L are given by

and

with c = 2m(T – k – m)/[(T – k – 1)(T – k + 2)] and

In the next section, we give formulae to compute tr (MK) and tr(MK)² for some useful test statistics.

3. Test Statistics3.1. Multivariate Seasonal Random Walk Plus Noise Model

[21] derived the LBI test statistic for a vector deterministic level in the multivariate local level model. We consider here the seasonal extension of this multivariate model given by

, , , (4)

where the vector time series y_t = (y_1t, …, y_mt)¢ is decomposed into the sum of a vector seasonal random walk a_t = (a_1t, …, a_mt)¢ plus a vector Gaussian white noise u_t = (u_1t, …, u_mt)¢ ~ N(0, S), the vector Gaussian white noise v_t = (v_1t, …, v_mt)¢ ~ N(0, rS) is assumed to be independent of u_t, and the parameter r > 0 quantifies the degree of stochasticity of a_t. Without loss of generality we assume that the seasonal period k is even and that the dataset is balanced, T = nk.

Defining the T × m matrices Y = [y₁,…, y_T]¢, A = [a₁,…, a_T]¢, U = [u₁,…, u_T]¢ and V = [v₁, , v_T]¢, (4) can be written in matrix form as

, (5)

where A₀ = [a_–k+1, …, a₀]¢ is a k × m matrix of initial conditions, Ä denotes the Kronecker or tensor product, D_n is an n × n lower bidiagonal matrix with 1s on the main diagonal and –1s on the first sub-diagonal, which can be horizontally partitioned as D_n = [d_n|Ñ_n]¢, being d_n = (1, 0, …, 0)¢ and Ñ_n the (n – 1) × n first-order differencing matrix. If A₀ is assumed to be fixed, it follows that (4) is a special case of (1) with

, ,

where, i_n = C_nd_n and X is a T × k matrix of seasonal dummy variables.

The LBI test statistic for testing the null hypothesis of deterministic seasonality (H₀: r = 0) against the alternative of seasonal random walk (H₁: r > 0), denoted by RW_m,k,n, is given by (2) with

(6)

and Ê being the residual matrix in the multivariate regression of Y on the full set of k seasonal dummies. To compute the two first moments of, we find that the mean and mean-square of the eigenvalues of MK are given by

and

which suggest to correct the RW_m,k,n test statistic by a factor depending of the sample size so that it converges to a non-degenerate limiting distribution. Some candidates are n, mn or m (n – k). It should be noted that RW_m,k,n/mn has asymptotic mean 1/6 and variance 1/45 mk.

[21] also derived the LBI test statistic for a vector deterministic linear trend. We obtain here the LBI test statistic for a vector deterministic seasonal linear trend by including a vector seasonal drift β_t in (4)

, , (7)

whose matrix form is given by

where B₀ = [β_–k+1, …, β₀]¢ is a k × m fixed matrix of initial conditions for If A₀ is fixed, (7) is a special case of (1) with X = [i_n Ä I_k, t_n Ä I_k], P = [A₀, B₀] and W(r) = I_T + r(C_n Ä I_k), where t_n = (1, 2, …, n)'. It is now clear that the inclusion of the vector seasonal drift only affects the mean vector of the sampling distribution of Y, , but not its covariance matrix. Therefore, the LBI statistic for testing the null hypothesis of vector deterministic seasonal linear trend (H₀: r = 0) against vector seasonal drifted random walk (H₁: r > 0) in (7), say DRW_m,k,n, is computed as RW_m,k,n, being now Ê the residual matrix in the multivariate regression of Y on k seasonal dummies and k seasonal lineal trends. We find for DRW_m,k,n that the mean and mean-square of the eigenvalues of MK are given by

and

and so DRW_m,k,n has asymptotic mean 1/15 and variance 11/6300 mk.

We also obtain another relevant modification of the RW_m,k,n test statistic by including the time index t as a regressor in (4)

, (8)

or in matrix form,

which is a special case of (1) with X = [i_n Ä I_k, t_T], P = [A₀, b₀] and W(r) as in (7). By the same token, the LBI statistic for testing H₀: r = 0 against H₁: r > 0 in (8), say TRW_m,k,n, is computed as RW_m,k,n, being now Ê the residual matrix in the multivariate regression of Y on k seasonal dummies and a regular lineal trend. We find for TRW_m,k,n that the mean and mean-square of the eigenvalues of MK are given by

and

It should be noted that DRW_1,1,n= TRW_1,1,n is the [3] test statistic for a univariate deterministic linear trend, and that the asymptotic mean and variance of TRW_1,1,n/n are 1/15 and 11/6300, which agree with those obtained by [3] in a rather complicated proof.

3.2. Vector Seasonal IMA(1,1)<sub>k</sub> Model

Multivariate structural model (4) can be written as a vector seasonal IMA(1,1)_k process

, (9)

where B is the backshift operator such that B^ky_t = y_t–k, a_t= (a_1t, …, a_mt)', W_a is a m × m positive definite matrix, and the parameter is restricted to be positive so that ρ = (1 – Θ)²/Θ > 0. Process (9) is said to be invertible when Θ < 1 and strictly non-invertible when Θ = 1. In the last case, the cancellation of the matrix polynomials on both sides of the equation reveals the presence of deterministic seasonality. Noting that ρ(Θ) = ρ(1/Θ), the one-sided testing problem H₀: r = 0 versus H₁: r > 0 is equivalent to the two-sided one H₀: Θ = 1 versus H₁: Θ ≠ 1. Hence, the LBI test statistic RW_m,k,n for a null variance ratio in (4) is the LBIU test statistic for strict noninvertibility in (9). Note that RW_1,k,n is the [11] test statistic for a seasonal MA unit root.

Analogously, it can be proved that DRW_m,k,n and TRW_m,k,n are the LBIU test statistics of H₀: Θ = 1 versus H₁: Θ ≠ 1 in the reduced form of (7),

and (8)

respectively. Note that DWR_1,k,n is closely related to the [22] test statistic, while TRW_1,k,n is the [12] test statistic for non-invertibility in the seasonal IMA(1,1)_k model.

3.3. Dynamic Seasonal Linear Models

[6] considered testing the null hypothesis of deterministic seasonality against the alternative of mixed deterministic and stochastic seasonality. A related multivariate LBI test statistic can be derived in the seasonal linear regression

(10)

where x_jt (j = 1, …, k) are a full set of seasonal dummy variables, β_jt is either a time-varying parameter if d_j = 1 or a nuisance constant parameter if d_j = 0, u_t ~ N(0, S), v_jt ~ N(0, (r/k)S) are mutually and serially uncorrelated vector errors, and is here divided by k for comparison purposes given that (10) reduces to (4) when d₁ + … + d_K = k. Assuming that the initial conditions β_1,0, …, β_k,0 are fixed, (10) is a special case of (1) with X = [x₁, …, x_k]', P = [β_1,0, …, β_k,0]' and W(r') = I_T + r'(d_A₁ + … + d_kA_k), where x_j= (x_j1, …, x_jT)', A_j = x_j_°C_TC'_T_°x_j, and the operator denotes the Hadamard product. The LBI test statistic for testing the null hypothesis of deterministic seasonality (H₀: d₁ + … + d_k = k) against the alternative hypothesis of mixed deterministic-stochastic seasonality (H₁: d₁ + … + d_k = r < k), say SD_m,k,n (r), is given by (2) with K = (d_A₁ + … + d_kA_k)/k, which coincides with RW_m,k,n when r = k. Noting that MA_jMA_i = 0 for j ≠ i, we find that the eigenvalues of MK have mean and mean-square are given by

and

Analogously, when the explanatory variables in (10) are trigonometric seasonal variables (x_1t = 1, x_jt = cos(jtπ/k) for j even, and x_jt = sin[(j – 1)tπ/k] for j odd and j > 1), it is convenient to assume that v_jt ~ N(0, r_jS), where r_j = a_jr/k² with a_j = 1 (j = 1, k) and a_j = 2 (j = 2, …, k – 1). Now, as before, (10) reduces to (4) when r = k. Here, we can focus our attention on testing the deterministic or stochastic nature of the local level β_1t, the (j/2)-th harmonic β_jtcos(πjt/k) + β_j+1,tsin(πjt/k) (j = 2, 4, …, k) or any combination of these k/2 harmonics. Taking as illustration the (j/2)-th harmonic, the LBI test statistic for testing the null hypothesis of deterministic seasonality (H₀: d₁ + … + d_k = k) against the alternative of mixed deterministic-stochastic seasonality (H₁: d₁ + … + d_k = 2) is given by (2) with K = a_j (A_j + A_j+1)/k² with A_j as defined before. We find that the eigenvalues of MK have mean and mean-square given by

and

where b_j = 0 (j = 1, k) and b_j = 1 (j = 2, …, k – 1). Note that these expressions are also valid to compute the mean and variance of the LBI test statistic for a deterministic level in presence of deterministic seasonality, H₀: r_ = 0 versus H₁: r_ > 0, which is closely related to the KPSS test with seasonal dummies proposed by [23]. Furthermore, [7,8] derived the LBI test statistic, say TV_1,k,n, for the dual testing problem of deterministic seasonality in presence of a deterministic level, which was generalized to the multivariate case by [16]. This test statistic is given by (2) with K = (a₂A₂ + … + a_kA_k)/k². We find that the eigenvalues of MK have mean and mean-square given by

and

4. Approximate Distributions and Accuracy4.1. Asymptotic Samples

Following [11] and [21], it can be shown that the limiting distribution of RW_m,k,n/mn under testing H₀: r = 0 is

(11)

where ξ_j ~ iid. [2] noted that RW_1,1,n/n follows the same limiting distribution as the Cramèr-von Mises goodness-of-fit test statistic. Hence, (11) is the average of mk copies of the Cramèr-von Mises distribution, denoted by CvM(mk)/mk. [24] found that the first four cumulants of a CvM(r) distribution are given by , , , which reveal that the distribution is strongly rightskewed and leptokurtic. We observe that the two parameter Inverse Gaussian distribution, IG(µ, λ), can be fitted to possess similar characteristics. The first four cumulants of this distribution are , , , and matching the first two cumulants, μ = r/6 and μ³/λ= r/45, we obtain that the fitted IG(r/6, 45r²/6³) distribution has third and fourth cumulants given by κ₃ = 8r/900 and κ₄ = 8r/1350, which seem to be quite close to those of the CvM (r) distribution. The accuracy of the IG approximation is illustrated in Figure 1, which shows the limit pdf of RW_1,1,n/n evaluated by the [13] procedure (solid line), along with the pdf of the fitted IG(µ, λ) distribution (dashed line) given by

with x > 0, μ = 1/6 and λ = 45m/6³. We can see that the IG approximation provides a very good fit on both tails

of the distribution. Therefore, accurate asymptotic pvalues for RW_m,k,n/mn can be computed from the cdf of the IG(µ, λ) distribution.

4.2. Finite Samples

The goodness-of-fit in the asymptotic case take us to ask if the approximation will be also good in finite samples. To evaluate the exact null distribution of RW_m,k,n/mn from (3)-(6), we must determine the eigenvalues of the matrix MK. To this end, it is convenient to note that the projection matrix M can be alternatively written as M = [Ñ¢_n(Ñ_nÑ¢_n)^–1Ñ_n] Ä I_k. Hence, MK = (Ñ_nÑ¢_n)^–1 Ä I_k and its eigenvalues are the reciprocals of those of the tridiagonal matrix Ñ_nÑ¢_n, λ_t = [4sin²(tπ/2n)]^–1 (t = 1, 2, …, n – 1), each one with multiplicity k. In the case m = 1, (3)-(6) can be expressed as a ratio of quadratic forms in normal variables whose distribution was tabulated by [11] using the [13] procedure (they used n – 1 as correction factor instead of n). However, when m > 1, similar tables can be obtained by Monte Carlo simulation from (2)-(6).

To assess the accuracy of the IG approximation in small samples we simply compare the approximate pvalues with the nominal sizes for n = 10, 20, 30, 50 10, k = 1, 2, 3, and m = 1, 2, 3, 4, 5. In general, the approximate p-values agree closely with the nominal sizes even in small samples, being the mean absolute errors less than 0.003. Such discrepancies seem not to be relevant in practical applications. Similar results have been found for DRW_m,k,n and TV_m,k,n. The results of this simulation study are not presented here due to space restrictions but are available from authors at request.

5. Conclusions

We have presented seasonal extensions of the [21] test for a deterministic level in multivariate models that can be used to detect different forms of non-invertibility in VARIMA models. The two-moment IG approximation to the null distribution of these test statistics, along with the closed forms expressions for the first two moments here derived, provide a simple and fast way to compute accurate critical values and p-values in practical applications. The proposed approximation could be also useful when modifying the test statistics to deal with intervention variables and serially correlated errors. Finally, the testing procedures described have been implemented in a computer program that can be freely obtained from authors at request.

6. References

[1] M. L. King and G. H. Hillier, “Locally Best Invariant Tests of the Error Covariance Matrix of the Linear Regression Model,” Journal of the Royal Statistical Society, Series B (Methodological), Vol. 47, No. 1, 1985, pp. 98- 102.

[2] J. Nyblom and T. Mäkeläinen, “Comparisons of Tests for the Presence of Random Walk Coefficients in a Simple Linear Model,” Journal of the American Statistical Association, Vol. 78, No. 384, 1983, pp. 856-864. doi:10.2307/2288196

[3] J. Nyblom, “Testing for Deterministic Linear Trend in Time Series,” Journal of the American Statistical Association, Vol. 81, No. 394, 1986, pp. 545-549. doi:10.2307/2289247

[4] D. Kwiatkowski, P. C. B. Phillips, P. Schmidt and Y. Shin, “Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root,” Journal of Econometrics, Vol. 54, No. 1-3, 1992, pp. 159-178. doi:10.1016/0304-4076(92)90104-Y

[5] J. Nyblom and A. Harvey. “Testing against Smooth Stochastic Trends,” Journal of Applied Econometrics, Vol. 16, 2001, pp. 415-429. doi:10.1002/jae.604

[6] F. Canova and B. E. Hansen, “Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability,” Journal of Business and Economic Statistics, Vol. 13, No. 3, 1995, pp. 237-252. doi:10.2307/1392184

[7] M. Caner, “A Locally Optimal Seasonal Unit-Root Test,” Journal of Business and Economic Statistics, Vol. 16, No. 3, 1998, pp. 349-356. doi:10.2307/1392511

[8] F. Busetti and A. Harvey, “Seasonality Tests,” Journal of Business and Economic Statistics, Vol. 21, No. 3, 2003, pp. 420-436. doi:10.1198/073500103288619061

[9] K. Tanaka, “Testing for a Moving Average Unit Root,” Econometric Theory, Vol. 6, No. 4, 1990, pp. 433-444. doi:10.1017/S0266466600005442

[10] P. Saikkonen and R. Luukkonen, “Testing for a Moving Average Unit Root in Autoregressive Integrated Moving Average Models,” Journal of the American Statistical Association, Vol. 88, No. 422, 1993, pp. 596-601. doi:10.2307/2290341

[11] W. Tam and G. C. Reinsel, “Tests for Seasonal Moving Average Unit Root in ARIMA Models,” Journal of the American Statistical Association, Vol. 92, No. 438, 1997, pp. 725-738. doi:10.2307/2965721

[12] W. Tam and G. C. Reinsel, “Seasonal Moving-Average Unit Root tests in the Presence of a Linear Trend,” Journal of Time Series Analysis, Vol. 19, No. 5, 1998, pp. 609-625. doi:10.1111/1467-9892.00112

[13] J. P. Imhof, “Computing the Distribution of Quadratic Forms in Normal Variables,” Biometrika, Vol. 48, No. 3-4, 1961, pp. 419-426. doi:10.1093/biomet/48.3-4.419

[14] R. B. Davies, “Numerical Inversion of a Characteristic Function,” Biometrika, Vol. 60, No. 2, 1973, pp. 415-417. doi:10.1093/biomet/60.2.415

[15] T. W. Anderson and D. A. Darling, “Asymptotic Theory of Certain ‘Goodness of Fit’ Criteria Based on Stochastic Processes,” The Annals of Mathematical Statistics, Vol. 23, No. 2, 1952, pp. 193-212. doi:10.1214/aoms/1177729437

[16] F. Busetti, “Tests of Seasonal Integration and Cointegration in Multivariate Unobserved Component Models,” Journal of Applied Econometrics, Vol. 21, 2006, pp. 419- 438. doi:10.1002/jae.852

[17] J. Nyblom, “Invariant Tests for Covariance Structures in Multivariate Linear Model,” Journal of Multivariate Analysis, Vol. 76, 2001, pp. 294-315. doi:10.1006/jmva.2000.1918

[18] J. MacKinnon, “Approximate Asymptotic Distribution Functions for Unit-Roots and Cointegration Tests,” Journal of Business and Economic Statistics, Vol. 12, 1994, pp. 167-176. doi:10.2307/1391481

[19] J. A. Doornik, “Approximation to the Asymptotic Distributions of Cointegration Tests,” Journal of Economic Surveys, Vol. 12, 1998, pp. 573-593. doi:10.1111/1467-6419.00068

[20] C. R. Rao, “Linear Statistical Inference and its Applications,” 2nd Edition, Wiley, New York, 1973. doi:10.1002/9780470316436

[21] J. Nyblom and A. Harvey, “Tests of Common Stochastic Trends,” Econometric Theory, Vol. 16, 2000, pp. 176-199. doi:10.1017/S0266466600162024

[22] A. M. R. Taylor, “Locally Optimal Tests against Unit Roots in Seasonal Time Series Processes,” Journal of Time Series Analysis, Vol. 24, No. 5, 2003, pp. 591-612. doi:10.1111/1467-9892.00324

[23] P. C. B. Phillips and S. Jin, “The KPSS Test with Seasonal Dummies,” Economics Letters, Vol. 77, 2002, pp. 239-243. doi:10.1016/S0165-1765(02)00127-1

[24] B. M. Brown, “Cramèr-Von Mises Distributions and Permutation Tests,” Biometrika, Vol. 69, No. 3, 1982, pp. 619-624.

References1

M. L. King and G. H. Hillier, “Locally Best Invariant Tests of the Error Covariance Matrix of the Linear Regression Model,” Journal of the Royal Statistical Society, Series B (Methodological), Vol. 47, No. 1, 1985, pp. 98- 102.

J. Nyblom and T. M?kel?inen, “Comparisons of Tests for the Presence of Random Walk Coefficients in a Simple Linear Model,” Journal of the American Statistical Association, Vol. 78, No. 384, 1983, pp. 856-864. doi:10.2307/2288196

J. Nyblom, “Testing for Deterministic Linear Trend in Time Series,” Journal of the American Statistical Association, Vol. 81, No. 394, 1986, pp. 545-549. doi:10.2307/2289247

D. Kwiatkowski, P. C. B. Phillips, P. Schmidt and Y. Shin, “Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root,” Journal of Econometrics, Vol. 54, No. 1-3, 1992, pp. 159-178. doi:10.1016/0304-4076(92)90104-Y

J. Nyblom and A. Harvey. “Testing against Smooth Stochastic Trends,” Journal of Applied Econometrics, Vol. 16, 2001, pp. 415-429. doi:10.1002/jae.604

F. Canova and B. E. Hansen, “Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability,” Journal of Business and Economic Statistics, Vol. 13, No. 3, 1995, pp. 237-252. doi:10.2307/1392184

M. Caner, “A Locally Optimal Seasonal Unit-Root Test,” Journal of Business and Economic Statistics, Vol. 16, No. 3, 1998, pp. 349-356. doi:10.2307/1392511

F. Busetti and A. Harvey, “Seasonality Tests,” Journal of Business and Economic Statistics, Vol. 21, No. 3, 2003, pp. 420-436. doi:10.1198/073500103288619061

K. Tanaka, “Testing for a Moving Average Unit Root,” Econometric Theory, Vol. 6, No. 4, 1990, pp. 433-444. doi:10.1017/S0266466600005442

P. Saikkonen and R. Luukkonen, “Testing for a Moving Average Unit Root in Autoregressive Integrated Moving Average Models,” Journal of the American Statistical Association, Vol. 88, No. 422, 1993, pp. 596-601. doi:10.2307/2290341

W. Tam and G. C. Reinsel, “Tests for Seasonal Moving Average Unit Root in ARIMA Models,” Journal of the American Statistical Association, Vol. 92, No. 438, 1997, pp. 725-738. doi:10.2307/2965721

W. Tam and G. C. Reinsel, “Seasonal Moving-Average Unit Root tests in the Presence of a Linear Trend,” Journal of Time Series Analysis, Vol. 19, No. 5, 1998, pp. 609-625. doi:10.1111/1467-9892.00112

J. P. Imhof, “Computing the Distribution of Quadratic Forms in Normal Variables,” Biometrika, Vol. 48, No. 3-4, 1961, pp. 419-426. doi:10.1093/biomet/48.3-4.419

R. B. Davies, “Numerical Inversion of a Characteristic Function,” Biometrika, Vol. 60, No. 2, 1973, pp. 415-417. doi:10.1093/biomet/60.2.415

T. W. Anderson and D. A. Darling, “Asymptotic Theory of Certain ‘Goodness of Fit’ Criteria Based on Stochastic Processes,” The Annals of Mathematical Statistics, Vol. 23, No. 2, 1952, pp. 193-212. doi:10.1214/aoms/1177729437

F. Busetti, “Tests of Seasonal Integration and Cointegration in Multivariate Unobserved Component Models,” Journal of Applied Econometrics, Vol. 21, 2006, pp. 419- 438. doi:10.1002/jae.852

J. Nyblom, “Invariant Tests for Covariance Structures in Multivariate Linear Model,” Journal of Multivariate Ana- lysis, Vol. 76, 2001, pp. 294-315. doi:10.1006/jmva.2000.1918

J. MacKinnon, “Approximate Asymptotic Distribution Functions for Unit-Roots and Cointegration Tests,” Journal of Business and Economic Statistics, Vol. 12, 1994, pp. 167-176. doi:10.2307/1391481

J. A. Doornik, “Approximation to the Asymptotic Distributions of Cointegration Tests,” Journal of Economic Surveys, Vol. 12, 1998, pp. 573-593. doi:10.1111/1467-6419.00068

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