Multivariate Time Series

Transcription

1 Multivariate Time Series Notation: I do not use boldface (or anything else) to distinguish vectors from scalars. Tsay (and many other writers) do. I denote a multivariate stochastic process in the form {X t }, where, for any t, X t is a vector of the same order. We denote the individual elements with two subscripts: X it denotes the i th element of the multivariate time series at time t. Warning: I m not promising I ll do this! This is Tsay s notation, and one of the two obvious notations. One common introductory-graduate-level textbook on time series is by Shumway and Stoffer, and another is by Brockwell and Davis. Shumway and Stoffer use X ti. (I like this.) Brockwell and Davis use X it. One further thing on notation: you do not need to use the transpose (as Tsay and many others do) on X t = (X 1t,..., X kt ). It s a column vector, but you re not drawing a picture! 1

2 Marginal Model Components In a multivariate time series {X t : t..., 1,0,1,...}, each univariate series {X it : t..., 1,0,1,...} is called a marginal time series. The model for {X it : t..., 1,0,1,...} is called a marginal model component of the model for {X t : t..., 1,0,1,...}. This illustrates one of the disadvantages of not having a special notation for vectors! 2

3 Stationary Multivariate Time Series First of all, let s establish that in time series, stationary means weakly stationary. ( Strictly stationary means strictly stationary or strongly.) If a time series is stationary, then its first and second moments are time invariant. (This is not the definition.) For a stationary time series {X t }, we will generally denote the first moment as the vector µ: µ = E(X t ) and the variance-covariance as Γ 0 : Γ 0 = V(X t ) = E ( (X t µ)(x t µ) T). (Note on notation: Γ, with or without serifs, is used to denote the gamma function; it is a reserved symbol. Γ is used to denote various things.) 3

4 Stationary Multivariate Time Series If a time series is stationary, then its first and second moments are time invariant. Tsay (page 390) states the converse of this. That is true, if second moments means second auto-moments at fixed lags (including the 0 lag). I looked back to see how clear Tsay has been on that point. He has not been clear. On page 39 (in the middle), he makes an argument that is based on the time invariance of the first and second moments and the finiteness of the autocovariance imply stationarity. In other places, he indicates that time invariance of the autocorrelations is also required, but because he does not write as a mathematician, it is sometimes not clear. 4

5 Stationary Multivariate Time Series So let s be very clear. Analogously to the autocovariance γ s,t, let s define the cross-autocovariance matrix Γ s,t : ( Γ s,t = E (X s µ s )(X t µ) T). ( cross-autocovariance matrix is quite a mouthful, so we ll just call it the cross-covariance matrix.) Now, suppose Γ s,t is constant if s t is constant. (Notice I did not say if s t is constant.) Under that additional condition (together with the time-invariance of the first two moments), we say that the multivariate time series is stationary. 5

6 Stationary Multivariate Time Series In the case of stationarity, we can use the notation Γ h, which is consistent with the notation Γ 0 used before. We refer to the h = 0 case as concurrent or contemporaneous. The matrix Γ 0 is the concurrent cross-covariance matrix. We now introduce another notational form so that we can easily refer to the elements of the matrices. We equate Γ(h) with Γ h. Now, we can denote the ij th element of Γ(h) as Γ ij (h). (There is an alternative meaning of Γ p. We have used it (I think) to refer to the p p symmetric matrix of autocovariances of orders 0 through p. It is often seen in the Yule-Walker equations.) The meaning is made clear in the context. 6

7 Stationary Multivariate Time Series Notice that stationarity of the multivariate time series implies stationarity of the individual univariate time series. The univariate autocovariance functions are the diagonal elements of Γ h. We sometimes use the phrase jointly stationary to refer to a stationary multivariate time series. (This excludes the case of a multivariate time series each of whose components is stationary, but the cross-covariances are not constant at constant lags.) 7

8 Cross-Correlation Matrix The standard deviation of the i th component of a multivariate time series in the standard notation is Γ ii (0). ( For a k-variate time series, the matrix D = diag Γ11 (0),..., ) Γ kk (0) is very convenient. All variances are assumed to be positive, so D 1 exists, and D 1 Γ h D 1 is the cross-correlation matrix of lag h. (If h = 0, of course, it is the concurrent cross-correlation matrix.) Tsay denotes it as ρ 0 or ρ l. I like to use uppercase rho, R 0 or R h. Of course, either way, we may use an alternative notation, ρ ij (l) or R ij (h). Furthermore, note that in my notation, I may use ρ ij (h) in place of R ij (h). 8

9 Properties of the Cross-Covariance and Cross-Correlation Matrices Notice that Γ 0 is symmetric; it s an ordinary variance-covariance matrix. On the other hand, Γ h is not necessarily symmetric; in fact, Γ T h = Γ h. The elements of the matrix Γ(h) have directional meanings, and these have simple interpretations. First of all, we need to be clear about what kind of relationships that covariance or correlation relates to. Covariance or correlation relates to linear relationships. For X centered on 0, the correlation between X and Y = X 2 is 0. 9

10 Properties of the Cross-Covariance and Cross-Correlation Matrices Consider a given i and j representing the i th and j th marginal component time series. The direction of time is very important in characterizing the relationships between the i th and j th series. In the following, which is consistent with the nontechnical language in time series analysis, we will use the term depend loosely. (We need the linear qualifier.) If Γ ij (h) = 0 for all h, then the series X it does not depend on the past values of the series X jt. If Γ ij (h) 0 for some h, then the series X it does depend on the past values of the series X jt. In this case we say X jt leads X it, or X jt is a leading indicator of X jt. If Γ ij (h) = Γ ji (h) = 0 for all h, then neither depends on the past values of the other series, and we say the series are uncoupled. 10

11 The Cross-Covariance Matrix in a Strictly Stationary Process Strict stationarity is a restrictive property. Notice, first of all, that it requires more than just time-invariance of the first two moments; it requires time-invariance of the whole distribution. It also requires stronger time-invariance of auto-properties. An iid process is obviously strictly stationary, and such a process is often our choice for examples. The following process, however, is also strictly stationary:..., X, X, X, X,... 11

12 The Cross-Covariance Matrix in a Strictly Stationary Process and in a Serially Uncorrelated Process The cross-covariance matrix alone does not tell us whether the process is stationary; we also need time-invariance of the first two moments. Given stationarity, it is not possible to tell from the cross-covariance matrix whether or not a process is strictly stationary. In a serially uncorrelated process, Γ h is a hollow matrix for all h 0. 12

13 Sample Cross-Covariance and Cross-Correlation Matrices Given a realization of a multivariate time series {x t : t = 1,...,n}, where each x t is a k-vector, the sample cross-covariance and cross-correlation matrices are formed from in the obvious ways. We use the hat notation to indicate that these are sample quantities, and also because they are often used as estimators of the population quantities. We also use the notation x to denote x t /n, and ˆσ i to denote t(x it x i ) 2 /n. (Note the divisor. It s not really important, of course.) 13

14 Sample Cross-Covariance and Cross-Correlation Matrices The sample cross-covariance matrix is (Note the divisor.) Γ h = 1 n n i=1+h (x t x)(x t h x) T. Letting D = diag(ˆσ 1,...,ˆσ k ), the sample cross-correlation matrix is R h = D 1 Γ h D 1. 14

15 Sample Cross-Correlation Matrices There is a nice simplified notation that Tiao and Box introduced to indicate leading and lagging indicators as measured by a sample cross-covariance matrix. Each cell has an indicator of significant positive, significant negative, insignificant sample correlation. Here, significant is defined with respect to twice the asymptotic 5% critical value of a sample correlation coefficient under the assumption that the true correlation coefficient is 0. The comparison value is 2/ n, and the indicators are +,, and. ; thus, at a specific lag, a correlation matrix for three component time series may be represented as

16 Multivariate Portmanteau Tests Recall the test statistic for the portmanteau test of Ljung and Box (first, recall what the portmanteau test tests): Q(m) = n(n + 2) m h=1 1 n hˆρ2 h. The multivariate version for a k-variate time series is Q k (m) = n 2 m h=1 1 n h tr ( Γ T h Note the similarities and the differences. Γ 1 0 Γ h ) 1 Γ 0. Under the null hypothesis and some regularity conditions, this has an asymptotic distribution of chi-squared with k 2 m degrees of freedom. 16

17 VAR Models In time series, VAR means vector autoregressive. In finance generally, VaR means value at risk. A VAR(1) model is X t = φ 0 + ΦX t 1 + A t, where X t, φ 0, and X t 1 are k-vectors, Φ is a k k matrix, and {A t } is a sequence of serially uncorrelated k-vectors with 0 mean and constant positive definite variance-covariance matrix Σ. Note that the systematic term may be bigger than it looks. There are k linear terms. The key is that they only go back in time one step. This form is called the reduced-form of a VAR model. 17

18 Structural Equations of VAR Models The relationships among the component time series arise from the off-diagonal elements of Σ. To exhibit the concurrent relationships among the component time series, we do a diagonal decomposition of Σ, writing it as Σ = LGL T, where L is a lower triangular matrix whose diagonal entries are all 1 (which means that it is of full rank), and G is a diagonal matrix with positive entries. (Such a decomposition exists for any positive definite matrix.) Note that G = L 1 Σ(L 1 ) T. 18

19 Structural Equations of VAR Models Now we transform the reduced-form equation by premultiplying each term by L 1 : L 1 X t = L 1 φ 0 + L 1 ΦX t 1 + L 1 A t = φ 0 + Φ X t 1 + B t. This is one of the most fundamental transformations in statistics. The important result is that the variance-covariance that ties the component series together concurrently, that is, V(A t ), has been replaced by V(B t ), which is diagonal. Because of the special structure of L, we can see concurrent linear dependencies of the k th series on the others. And by rearranging the terms in the series, we can make any component series the last one. 19

20 Structural Equations of VAR Models The last row of L 1 has a 1 in the last position. Call the other elements w k1, w k2, etc. Then the last equation in the matrix equation can be written as X kt + k 1 i=1 L 1 X t = φ 0 + Φ X t 1 + B t. w ik X it = φ k,0 + k i=1 φ k,i X it 1 + B kt. This shows the concurrent relationship of X kt on the other series. 20

21 Properties of a VAR(1) Process There are several important properties we can see easily. Because the {A } are serially uncorrelated, Cov(A t, X t h ) = 0 for all h > 0. Also, we see Cov(A t, X t ) = V(A t ) = Σ. Also, we see the X t depends on the j th previous X (and A by Φ j. The process would be explosive (i.e., the variance would go to infinity) unless Φ j 0 as j. This will be guaranteed if all eigenvalues of Φ are less than 1 in modulus. (Remember this?) Also, just as in the univariate case, we have the recursion from which we get Γ h = ΦΓ h 1 Γ h = Φ h Γ 0. 21

22 VAR(p) Processes and Models A VAR(p) model, for p > 0 is X t = φ 0 + Φ 1 X t Φ p X t p + A t, where X t, φ 0, and X t i are k-vectors, Φ 1,..., Φ p are k k matrices, with Φ p 0, and {A t } is a sequence of serially uncorrelated k-vectors with 0 mean and constant positive definite variancecovariance matrix Σ. We also can write this using the back-shift operator as or (I Φ 1 B Φ p B p )X t = φ 0 + A t, Φ(B)X t = φ 0 + A t, We also can work out the autocovariance for a VAR(p) process: Γ h = Φ 1 Γ h Φ p Γ h p. This is the multivariate Yule-Walker equations. 22

23 The Yule-Walker Equations Let s just consider an AR(p) model. We have worked out the autocovariance function of an AR(p) model. It is and γ(h) = φ 1 γ(h 1) + + φ p γ(h p) σ 2 A = γ(0) φ 1γ(1) φ p γ(p). 23

24 The Yule-Walker Equations The equations involving the autocovariance function are called the Yule-Walker equations. There are p such equations, for h = 1,...,p. For an AR(p) process that yields the two sets of equations on the previous slide, we can write them in matrix notation as and Γ p φ = γ p σ 2 A = γ(0) φt γ p 24

25 Yule-Walker Estimation of the AR Parameters After we compute the sample autocovariance function for a given set of observations, we merely solve the Yule-Walker equations to get our estimators: and ˆφ = Γ 1 p ˆγ p ˆσ 2 A = ˆγ(0) ˆφ Tˆγ p Instead of using the sample autocovariance function, we usually use the sample ACF: ˆφ = R 1 p ˆρ p. 25

26 Large Sample Properties of the Yule-Walker Estimators A result that can be shown for the Yule-Walker estimator ˆφ is n(ˆφ φ) d N p (0,σ 2 w Γ 1 p ) and ˆσ 2 A p σ 2 A. 26

27 Yule-Walker Equation for a VAR(p) Model The multivariate Yule-Walker equations, can also be used in estimation. Γ h = Φ 1 Γ h Φ p Γ h p They are often expressed in the correlation from where Υ i = D 1/2 Φ i D 1/2. R h = Υ 1 R h Υ p R h p, 27

28 Companion Matrix We can sometimes get a better understanding of a k-dimensional VAR(p) process by writing it as a kp VAR(1). It is Y t = Φ X t 1 + B t where 0 I I I 0 0 Φ p Φ p 1 Φ p 2 Φ 1. This is sometimes called the companion matrix. 28