timsac (version 1.3.0)

mulmar: Multivariate Case of Minimum AIC Method of AR Model Fitting

Description

Fit a multivariate autoregressive model by the minimum AIC procedure. Only the possibilities of zero coefficients at the beginning and end of the model are considered. The least squares estimates of the parameters are obtained by the householder transformation.

Usage

mulmar(y, max.order=NULL, plot=FALSE)

Arguments

y

a multivariate time series.

max.order

upper limit of the order of AR model. Default is $2 \sqrt{n}$, where $n$ is the length of the time series y.

plot

logical. If TRUE daic[[1]],...,daic[[d]] are plotted, where $d$ is the dimension of the multivariate time series.

Value

• mean
  mean.
• var
  variance.
• v
  innovation variance.
• aic
  AIC.
• aicmin
  minimum AIC.
• daic
  AIC-aicmin.
• order.maice
  order of minimum AIC.
• v.maice
  MAICE innovation variance.
• np
  number of parameters.
• jnd
  specification of $i$-th regressor.
• subregcoef
  subset regression coefficients.
• rvar
  residual variance.
• aicf
  final estimate of AIC ($=n\log$(rvar)$+2$np).
• respns
  instantaneous response.
• matv
  innovation variance matrix.
• morder
  order of the MAICE model.
• arcoef
  AR coefficients. arcoef[i,j,k] shows the value of $i$-th row, $j$-th column, $k-$th order.
• aicsum
  the sum of aicf.

Details

Multivariate autoregressive model is defined by $$y(t) = A(1)y(t-1) + A(2)y(t-2) +\ldots+ A(p)y(t-p) + u(t),$$ where $p$ is order of the model and $u(t)$ is Gaussian white noise with mean $0$ and variance matrix matv. AIC is defined by $$AIC = n \log(det(v)) + 2k,$$ where $n$ is the number of data, $v$ is the estimate of innovation variance matrix, $det$ is the determinant and $k$ is the number of free parameters.

References

G.Kitagawa and H.Akaike (1978) A Procedure for The Modeling of Non-stationary Time Series. Ann. Inst. Statist. Math., 30, B, 351--363. H.Akaike, G.Kitagawa, E.Arahata and F.Tada (1979) Computer Science Monograph, No.11, Timsac78. The Institute of Statistical Mathematics.

Examples

# Example 1
  data(Powerplant)
  z <- mulmar(Powerplant, max.order=10)
  z$arcoef

  # Example 2
  ar <- array(0,dim=c(3,3,2))
  ar[,,1] <- matrix(c(0.4,  0,   0.3,
                      0.2, -0.1, -0.5,
                      0.3,  0.1, 0),3,3,byrow=TRUE)
  ar[,,2] <- matrix(c(0,  -0.3,  0.5,
                      0.7, -0.4,  1,
                      0,   -0.5,  0.3),3,3,byrow=TRUE)
  x <- matrix(rnorm(200*3),200,3)
  y <- mfilter(x,ar,"recursive")
  z <- mulmar(y, max.order=10)
  z$arcoef