**Open Journal of Statistics** Vol.1 No.3(2011), Article ID:8066,6 pages DOI:10.4236/ojs.2011.13017

Testing for Deterministic Components in Vector Seasonal Time Series

Departamento de Economía, Universidad de Cantabria, Santander, Spain

E-mail: jose.gallego@unican.es, carlos.diazv@unican.es

Received April 29, 2011; revised May 30, 2011; accepted June 10, 2011

**Keywords:** Vector Time Series, Deterministic Components, Parametric Stability, Non-Invertibility, Unit Roots

Abstract

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.

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(d_{0}) = I_{T}. [17] found that the Locally Best Invariant (LBI) test statistic of the null hypothesis H_{0}: d = d_{0} against the one-sided alternative H_{1}: d > d_{0} 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 = d_{0}. 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 H_{0}: d = d_{0} against the two-sided alternative H_{1}: d ≠ d_{0} is given by (2) but with K being the second derivative d^{2}W(d)/dd^{2 }evaluated at d = d_{0}.

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 B_{1} + … + 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 Statistics

3.1. 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 = [y_{1},…, y_{T}]¢, A = [a_{1},…, a_{T}]¢, U = [u_{1},…, u_{T}]¢ and V = [v_{1}, , v_{T}]¢, (4) can be written in matrix form as

, (5)

where A_{0} = [a_{–k+1}, …, a_{0}]¢ 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 A_{0} is assumed to be fixed, it follows that (4) is a special case of (1) with

, ,

where, i_{n} = C_{n}d_{n} and X is a T × k matrix of seasonal dummy variables.

The LBI test statistic for testing the null hypothesis of deterministic seasonality (H_{0}: r = 0) against the alternative of seasonal random walk (H_{1}: 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 B_{0} = [β_{–k+1}, …, β_{0}]¢ is a k × m fixed matrix of initial conditions for If A_{0} is fixed, (7) is a special case of (1) with X = [i_{n} Ä I_{k}, t_{n} Ä I_{k}], P = [A_{0}, B_{0}] 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 (H_{0}: r = 0) against vector seasonal drifted random walk (H_{1}: 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 = [A_{0}, b_{0}] and W(r) as in (7). By the same token, the LBI statistic for testing H_{0}: r = 0 against H_{1}: 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)_{k} 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^{k}y_{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 H_{0}: r = 0 versus H_{1}: r > 0 is equivalent to the two-sided one H_{0}: Θ = 1 versus H_{1}: 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 H_{0}: Θ = 1 versus H_{1}: 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 d_{1} + … + 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 = [x_{1}, …, x_{k}]', P = [β_{1,0}, …, β_{k,0}]' and W(r') = I_{T} + r'(d_{}A_{1} + … + d_{k}A_{k}), where x_{j }= (x_{j1}, …, x_{jT})', A_{j} = x_{j}_{°}C_{T}C'_{T}_{°}x_{j}, and the operator denotes the Hadamard product. The LBI test statistic for testing the null hypothesis of deterministic seasonality (H_{0}: d_{1} + … + d_{k} = k) against the alternative hypothesis of mixed deterministic-stochastic seasonality (H_{1}: d_{1} + … + d_{k} = r < k), say SD_{m,k,n} (r), is given by (2) with K = (d_{}A_{1} + … + d_{k}A_{k})/k, which coincides with RW_{m,k,n} when r = k. Noting that MA_{j}MA_{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_{j}S), where r_{j} = a_{j}r/k^{2} 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_{jt}cos(πjt/k) + β_{j+1,t}sin(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 (H_{0}: d_{1} + … + d_{k} = k) against the alternative of mixed deterministic-stochastic seasonality (H_{1}: d_{1} + … + d_{k} = 2) is given by (2) with K = a_{j} (A_{j} + A_{j+1})/k^{2} 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, H_{0}: r_{} = 0 versus H_{1}: 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 = (a_{2}A_{2} + … + a_{k}A_{k})/k^{2}. We find that the eigenvalues of MK have mean and mean-square given by

and

4. Approximate Distributions and Accuracy

4.1. Asymptotic Samples

Following [11] and [21], it can be shown that the limiting distribution of RW_{m,k,n}/mn under testing H_{0}: 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 μ^{3}/λ= r/45, we obtain that the fitted IG(r/6, 45r^{2}/6^{3}) distribution has third and fourth cumulants given by κ_{3} = 8r/900 and κ_{4} = 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/6^{3}. We can see that the IG approximation provides a very good fit on both tails

Figure 1. Accuracy of the IG approximation.

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_{n}Ñ¢_{n})^{–1}Ñ_{n}] Ä I_{k}. Hence, MK = (Ñ_{n}Ñ¢_{n})^{–1} Ä I_{k} and its eigenvalues are the reciprocals of those of the tridiagonal matrix Ñ_{n}Ñ¢_{n}, λ_{t} = [4sin^{2}(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.

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