Implementation of Multivariate Continuous-Time ARMA Models

Abstract

The multivariate continuous-time ARMA model is a tool to capture the relationship between multivariate time series. In this chapter, a particular computational implementation of a stationary normal multivariate CARMA model is illustrated. A review of a parametric setup is shown. Data are assumed to be observed at irregular non-synchronous discrete time points. The computational approach for calculating the likelihood is based on a state-space form and the Kalman filter. Interpretation of the CARMA models is discussed. The computational algorithms have been implemented in R packages. Examples of a simulated and real data are shown.

Appendices

Appendix 1: Implementation of Kalman Filter

The likelihood function is calculated by use of the Kalman-filter recursions. The dynamics are defined by Eqs. (15.15) and (15.16). The sample selection matrix C contains 1 at coordinate ii when coordinate i is observed at time point t _i and is zero otherwise. Given information at time t _i−1,the optimal estimate of the state vector is X(t _i−1|t _i−1) and the corresponding variance matrix P _X(t _i−1|t _i−1). The prediction step is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{X}(t_i|t_{i-1})&\displaystyle =&\displaystyle \boldsymbol{e}^{A(t_i-t_{i-1})}\boldsymbol{X}(t_{i-1}|t_{i-1}),\\ P(t_i|t_{i-1})&\displaystyle =&\displaystyle \boldsymbol{e}^{A(t_i-t_{i-1})}P(t_{i-1}|t_{i-1})\boldsymbol{e}^{A'(t_i-t_{i-1})}\\ &\displaystyle &\displaystyle +\Gamma_{\boldsymbol{X}}(0)-\boldsymbol{e}^{A(t_i-t_{i-1})} \Gamma_{\boldsymbol{X}}(0) \boldsymbol{e}^{A'(t_i-t_{i-1})},\\ \boldsymbol{F}(t_i|t_{i-1})&\displaystyle =&\displaystyle \boldsymbol{C}(t_i)P(t_i|t_{i-1})\boldsymbol{C}'(t_i),\\ \hat{\boldsymbol{y}}(t_i|t_{i-1})&\displaystyle =&\displaystyle \boldsymbol{C}(t_i)\boldsymbol{X}(t_i|t_{i-1}). \end{array} \end{aligned} $$

The corresponding updating equations:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{X}(t_i|t_i)&\displaystyle =&\displaystyle \boldsymbol{X}(t_i|t_{i-1})+P(t_i|t_{i-1})\boldsymbol{C}(t_i)\boldsymbol{F}^*(t_i|t_{i-1})(\boldsymbol{y}(t_i) -\hat{\boldsymbol{y}}(t_i|t_{i-1})),\\ P(t_i|t_i)&\displaystyle =&\displaystyle P(t_i|t_{i-1})-P(t_i|t_{i-1})\boldsymbol{C}(t_i)\boldsymbol{F}^*(t_i|t_{i-1})\boldsymbol{C}'(t_i)P(t_i|t_{i-1}). \end{array} \end{aligned} $$

Here F ^∗(t _i|t _i−1) denotes a generalised inverse of F(t _i|t _i−1). The normal likelihood is easily calculated in the non-synchronous case, because at each time point, observed data are essentially univariate. The observed value is compared to its predicted value. The corresponding variance is the non-zero element of element of F(t _i|t _i−1)).

Appendix 2: A Brief Description of the R Packages

The R packages are based on objects. In the ctarmaRcpp package, a ctarma object contains the observed series, y=y(t ₁), …, y(t _n); the time points of observations, tt=t ₁, …, t _n; the AR parameters, a=(α ₁, …, α _p); the MA parameters b=(β ₁, …, β _q); and the standard deviation of the innovations, sigma=σ.

 names(ctarma1) [1] "y"     "tt"    "a"     "b"     "sigma" >ctarma1$y[1:5] [1] -1.330619311  0.215559470  0.218091305 -0.004512067 -0.124261787 >ctarma1$tt[1:5] [1] 0.08360339 0.51537932 0.65898476 0.77687963 0.86278019 >> ctarma1$a [1]  2 40 > ctarma1$b [1] 1.00 0.15 >ctarma1$sigma [1] 8

The package ctarmaRcpp contains functions on ctarma objects for, e.g., calculating and maximising the log-likelihood. Spectral densities can also be calculated and plotted.

In the multivariate mctarmaRcpp, a second compatible (same number of AR and MA coefficients) ctarma object ctarma2 can be merged with ctarma1 into a bivariate mctarma object.

> cc=cbind(as.list(ctarma0),as.list(ctarma2)) > mctobj=mctobjnonsync(cc) > mctobj$A     [,1] [,2] [,3] [,4] [1,]   -2    0    1    0 [2,]    0   -2    0    1 [3,]  -40    0    0    0 [4,]    0  -40    0    0 > mctobj$R      [,1] [,2] [1,] 0.15 0.00 [2,] 0.00 0.15 [3,] 1.00 0.00 [4,] 0.00 1.00 > mctobj$sigma      [,1] [,2] [1,]   64    0 [2,]    0   64

That is, the multivariate object mctobj now contains the matrix of AR parameters A, the matrix of MA parameters R and the variance matrix of the innovations Σ. The object also contains a vector measurements y, an ordered vector of time points of measurements tt and a vector z denoting which coordinate of the y vector was observed.

cbind(mctobj$y,mctobj$tt,mctobj$z)[1:5,]            [,1]       [,2] [,3] [1,] -0.6597594 0.03740642    2 [2,] -1.3306193 0.08360339    1 [3,]  0.9946577 0.27623398    2 [4,]  1.1272836 0.31624703    2 [5,]  1.2388762 0.32568241    2

The observations are non-synchronous, i.e. only one coordinate of the y vector is observed at each time point. Observations 1 and 3–5 are of coordinate 2; the second is of coordinate 1.