ON THE CAUSALITY BETWEEN MULTIPLE LOCALLY STATIONARY PROCESSES

In discussion of the relations between time series, concepts of dependence and causality are frequently invoked. Geweke (1982) and Hosoya (1991) have proposed measures of dependence and causality for multiple stationary processes. They have also showed that these measures can be additively decomposed into frequencywise. However, it seems to be restrictive that these measures are constants all the time. Priestley (1981) has developed the extensions of prediction and filtering theory to nonstationary processes which have evolutionary spectra. Here, we generalize measures of dependence and causality to multiple locally stationary processes.

In discussion of the relations between time series, concepts of dependence and causality are frequently invoked.Geweke (1982) and Hosoya (1991) have proposed measures of dependence and causality for multiple stationary processes.They have also showed that these measures can be additively decomposed into frequencywise.However, it seems to be restrictive that these measures are constants all the time.Priestley (1981) has developed the extensions of prediction and filtering theory to nonstationary processes which have evolutionary spectra.Here, we generalize measures of dependence and causality to multiple locally stationary processes.
For the d (Z) -dimensional locally stationary process {Z t,T }, we introduce H, the Hilbert space spanned by Z (j) t,T , j = 1, . . ., d (Z) , t = 0, ±1, . . .and call H (Z t,T ) the closed subspace spanned by Z (j) s,T , j = 1, . . ., d (Z) , s ≤ t.We obtain the best one-step linear predictor of Z t+1,T by projecting the components of the vector onto H (Z t,T ), so here projection implies componentwise projection.We call the error of prediction ξ t+1,T .Then, for locally stationary process we have where δ s,t is the Kronecker delta function.Note that ξ t,T 's are uncorrelated but do not have identical covariance matrices, namely G t,T are time-dependent.Now, we impose the following assumption on G t,T .

Assumption 1.
The covariance matrices of errors G t,T are non-singular for all t and T . Define as a one-sided linear process and where coefficient matrices are Note that each H t,T (j)ξ t−j,T , j = 0, 1, . . . is projection of Z t,T onto the closed subspace spanned by ξ t−j,T .Now, we have the following Wold decomposition for locally stationary processes.

Lemma 1 (Wold Decomposition).
If {Z t,T } is a locally stationary vector process of d (Z) components, then Z t,T = u t,T + v t,T , where u t,T is given by ( 1), ( 2) and (4), v t,T is deterministic and If only u t,T occurs we say that Z t,T is purely nondeterministic.
Assumption 2. Z t,T is purely nondeterministic.
In view of Lemma 1, we can see that under Assumptions 1 and 2, Z t,T becomes a one-side linear process given by (2).For locally stationary process, if we choose an orthonormal basis ε (j) t , j = 1, . . ., d (Z) , in the closed subspace spanned by ξ t,T , then {ε t } will be an uncorrelated stationary process.We call {ε t } a fundamental process of {Z t,T } and C t,T (j), j = 0, 1, . . .denote the corresponding coefficients, i.e., Let f t,T (λ) be the time varying spectral density matrix of Z t,T .A process is said to have the maximal rank if it has non-degenerate spectral density matrix a.e.

Assumption 3.
The locally stationary process {Z t,T } has the maximal rank for all t and T .In particular π −π log |f t,T (λ)| dλ > −∞, for all t and T, (6) where |D| denotes the determinant of the matrix D.
We will say that a function φ(z), analytic in the unit disc, belongs to the class Under Assumptions 1-3, it follows that {Z t,T } has a time varying spectral density f t,T (λ) which has rank d (Z) for almost all λ, and is representable in the form where D * denotes the complex conjugate of matrix D, and Φ t,T (e iλ ) is the boundary value of a in the unit disc, and it holds that Φ t,T (0 Now, we introduce measures of linear dependence, linear causality and instantaneous linear feedback at time t.
-dimensional locally stationary process, which has time varying spectral density matrix We shall find the partitions ξ t,T = ξ t,T are the residuals of the projections of X t,T and Y t,T onto H (X t−1,T ) and H (Y t−1,T ), respectively.
We define the measures of linear dependence, linear causality from {Y t,T } to {X t,T }, from {X t,T } to {Y t,T } and instantaneous linear feedback, at time t as and respectively, then we have Next, we decompose measures of linear causality into frequencywise.To define frequencywise measures of causality, we introduce the following analytic facts.

Assumption 4 (Kolmogorov's formula).
Under Assumptions 1-3, an analytic matrix Φ t,T (z) satisfying the boundary condition (8), will be maximal if and only if Now we define the process {η t,T } as t,T is the residuals of the projection of X t,T onto H (X t−1,T , Y t,T ), whereas η (2) t,T is the residuals of the projection of Y t,T onto H (X t,T , Y t−1,T ).Furthermore, we have Cov ξ so we can see that η (2) we have ξ we have the following lemma.
Lemma 3. Φ t,T (z) is a analytic function in the unit disc with Φ t,T (0) Φ t,T (0) * = G t,T and thus maximal, such that the time varying spectral density f t,T (λ) has a factorization From this lemma, it is seen that time varying spectral density is decomposed into two parts where Γ (1,1) The former part is related to the process ξ (1) t,T whereas the latter part is related to the process η (2) t,T , which is orthogonal to ξ (1) t,T .This relation suggests that frequencywise measure of causality, from {Y t,T } to {X t,T } at time t Similarly, we propose = log (2,2) t,T (z).Now, we introduce the following assumption.

Assumption 5.
The roots of Γ t,T (z) all lie outside the unit circle.
The relation of frequencywise measure to overall measure is addressed in the following result.
In this section we discuss the testing problem for linear dependence.The average measure of linear dependence is given by the following integral functional of time varying spectral density lim Next, we introduce a measure of goodness of our test.Consider a sequence of alternative spectral density matrices Let E gT (•) and V f (•) denote the expectation under g T (u, λ) and the variance under f (u, λ), respectively.It is natural to define an efficacy of L T by eff (L T ) = lim in line with the usual definition for a sequence of "parametric alternatives".Then we see that For another test L * T we can define an asymptotic relative efficiency (ARE) of L T relative to L * T by If we take the test statistic based on stationary assumption as another test L * T , we can measure the effect of nonstationarity when the process concerned is locally stationary process.
Finally, we discuss a testing problem of linear dependence for stock prices of Tokyo Stock Exchange.The data are daily log-returns of 7 companies; 1:HI-TACHI 2:MATSUSHITA 3:SHARP 4:SONY 5:HONDA 6:NISSAN 7:TOYOTA.The individual time series are 1174 data points since December 28, 1999 until October 1, 2004.We compute L T in (39) for each two companies.The selected parameters are T = 1000, N = 175, and M = 8, where N is the length of segment which the localized periodogram is taken over and M is the bandwidth of the weight function.
The results are listed in Table 1.It shows that all values for each two companies are large.Since under null hypothesis the limit distribution of L T is standard normal, we can conclude hypothesis is rejected.Namely, the linear dependencies exist at each two companies.In particular, the values both among electric appliance companies and among automobile companies are significantly large.Therefore, we can see that the companies in the same business have strong dependence.
Table 1 is about here.
In Figures 1 and 2, the daily linear dependence between HONDA and TOY-OTA and between HITACHI and SHARP are plotted.They show that the daily dependencies are not constant and change in time.So, it seems to be reasonable that we use the test statistic based on nonstationary assumption.
denote the covariance matrices of the onestep-ahead errors ξ (X) t,T and ξ (Y ) t,T when X t,T and Y t,T are forecasts from their own pasts alone, namely, ξ (X) t,T and ξ (Y )

Figures
Figures 1 and 2 are about here.

Figure 1 :
Figure 1: The daily linear dependence between HONDA and TOYOTA.

Figure 2 :
Figure 2: The daily linear dependence between HITACHI and SHARP.

Table 1 :
1 and 2 are about here.L T in (39) for each two companies.