Time Series Analysis and Examples: Multivariate Time Series Analysis

Time Series Analysis and Examples

Multivariate Time Series Analysis

The subroutines TSMULMAR, TSMLOMAR, and TSPRED analyze multivariate time series. The periodic AR model, TSPEARS, can also be estimated by using a vector AR procedure, since the periodic AR series can be represented as the covariance-stationary vector autoregressive model.

The stationary vector AR model is estimated and the order of the model (or each variable) is automatically determined by the minimum AIC procedure. The stationary vector AR model is written

${y}_t & = & {a}_0 + {a}_1 {y}_{t-1} + ... + {a}_p {y}_{t-p} + \epsilon_t \ \epsilon_t & \sim & n(0, \sigma)$

Using the ${l}{d}{l}^'$ factorization method, the error covariance is decomposed as

$\sigma = {l}{d}{l}^'$

where

is a unit lower triangular matrix and

is a diagonal matrix. Then the instantaneous response model is defined as

${c}{y}_t = {a}_0^* + {a}_1^*{y}_{t-1} + ... + {a}_p^*{y}_{t-p} + \epsilon_t^*$

where ${c}= {l}^{-1}$ , ${a}_i^* = {l}^{-1}{a}_i$ for

, and $\epsilon_t^* = {l}^{-1}\epsilon_t$ . Each equation of the instantaneous response model can be estimated independently, since its error covariance matrix has a diagonal covariance matrix

. Maximum likelihood estimates are obtained through the least squares method when the disturbances are normally distributed and the presample values are fixed.

The TSMULMAR subroutine estimates the instantaneous response model. The VAR coefficients are computed by using the relationship between the VAR and instantaneous models.

The general VARMA model can be transformed as an infinite-order MA process under certain conditions.

${y}_t = \mu + \epsilon_t + \sum_{m=1}^{\infty} \psi_m \epsilon_{t-m}$

In the context of the VAR(

) model, the coefficient $\psi_m$ can be interpreted as the

-lagged response of a unit increase in the disturbances at time

$\psi_m = \frac{\partial {y}_{t+m}}{\partial \epsilon^'_t}$

The lagged response on the one-unit increase in the orthogonalized disturbances $\epsilon_t^*$ is denoted

$\frac{\partial {y}_{t+m}}{\partial \epsilon_{jt}^*} = \frac{\partial{\rm e}({y}_{t+m}| y_{jt},y_{j-1,t}, ... ,{x}_t)} {\partial y_{jt}} = \psi_m {l}_j$

where ${l}_j$ is the

th column of the unit triangular matrix

and ${x}_t=[{y}_{t-1}, ... ,{y}_{t-p}]$ . When you estimate the VAR model by using the TSMULMAR call, it is easy to compute this impulse response function.

The MSE of the -step prediction is computed as

${\rm e}({y}_{t+m}-{y}_{t+m| t})({y}_{t+m}-{y}_{t+m| t})^' = \sigma + \psi_1 \sigma \psi^'_1 + ... + \psi_{m-1} \sigma \psi^'_{m-1}$

Note that $\epsilon_t = {l}\epsilon_t^*$ . Then the covariance matrix of $\epsilon_t$ is decomposed

$\sigma = \sum_{i=1}^n {l}_i {l}^'_i d_{ii}$

where $d_{ii}$ is the

th diagonal element of the matrix

and

is the number of variables. The MSE matrix can be written

$\sum_{i=1}^n d_{ii} [ {l}_i {l}^'_i + \psi_1 {l}_i {l}^'_i \psi^'_1 + ... + \psi_{m-1} {l}_i {l}^'_i \psi^'_{m-1} ]$

Therefore, the contribution of the

th orthogonalized innovation to the MSE is

${v}_i = d_{ii} [ {l}_i {l}^'_i + \psi_1 {l}_i {l}^'_i \psi^'_1 + ... + \psi_{m-1} {l}_i {l}^'_i \psi^'_{m-1} ]$

The

th forecast error variance decomposition is obtained from diagonal elements of the matrix ${v}_i$ .

The nonstationary multivariate series can be analyzed by the TSMLOMAR subroutine. The estimation and model identification procedure is analogous to the univariate nonstationary procedure, which is explained in the section "Nonstationary Time Series".

A time series is periodically correlated with period if ${\rm e}y_t = {\rm e}y_{t+d}$ and ${\rm e}y_s y_t = {\rm e}y_{s+d}y_{t+d}$ . Let be autoregressive of period with AR orders - that is,

$y_t = \sum_{j=1}^{p_t} \alpha_{jt}y_{t-j} + \epsilon_t$

where $\epsilon_t$ is uncorrelated with mean zero and ${\rm e}\epsilon_t^2 = \sigma_t^2$ , $p_t = p_{t+d}$ , $\sigma_t^2 = \sigma_{t+d}^2$ , and $\alpha_{jt} = \alpha_{j,t+d} (j=1, ... ,p_t)$ . Define the new variable such that $x_{jt} = y_{j+d(t-1)}$ . The vector series, ${x}_t = (x_{1t}, ... ,x_{dt})^'$ , is autoregressive of order

, where $p = \max_j{\rm int}((p_j - j)/d) + 1$ . The TSPEARS subroutine estimates the periodic autoregressive model by using minimum AIC vector AR modeling.

The TSPRED subroutine computes the one-step or multistep forecast of the multivariate ARMA model if the ARMA parameter estimates are provided. In addition, the subroutine TSPRED produces the (intermediate and permanent) impulse response function and performs forecast error variance decomposition for the vector AR model.

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