R: Multivariate Bayesian Method of AR Model Fitting

mulbar {timsac} | R Documentation

## Multivariate Bayesian Method of AR Model Fitting

### Description

Determine multivariate autoregressive models by a Bayesian procedure. The basic least squares estimates of the parameters are obtained by the householder transformation.

### Usage

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

### Arguments

`y` | a multivariate time series.
`max.order` | upper limit of the order of AR model, less than or equal to n/(2*d) where n is the length and d is the dimension of the time series `y`. Default is min(2*sqrt(n), n/(2*d)).
`plot` | logical. If TRUE `daic` is plotted.

### Details

The statistic 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.

Bayesian weight of the m-th order model is defined by

W(n) = const * C(m) / (m+1),

where const is the normalizing constant and C(m)=exp(-0.5 AIC(m)). The Bayesian estimates of partial autoregression coefficient matrices of forward and backward models are obtained by (m = 1,...,lag)

G(m) = G(m) D(m),

H(m) = H(m) D(m),

where the original G(m) and H(m) are the (conditional) maximum likelihood estimates of the highest order coefficient matrices of forward and backward AR models of order m and D(m) is defined by

D(m) = W(m) + ... + W(lag).

The equivalent number of parameters for the Bayesian model is defined by

ek = (D(1)^2 + ... + D(lag)^2) id + id (id+1)/2

where id denotes dimension of the process.

### 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.
`bweight` | Bayesian weights.
`integra.bweight` | integrated Bayesian Weights.
`arcoef.for` | AR coefficients (forward model). `arcoef.for[i,j,k]` shows the value of i-th row, j-th column, k-th order.
`arcoef.back` | AR coefficients (backward model). `arcoef.back[i,j,k]` shows the value of i-th row, j-th column, k-th order.
`pacoef.for` | partial autoregression coefficients (forward model).
`pacoef.back` | partial autoregression coefficients (backward model).
`v.bay` | innovation variance of the Bayesian model.
`aic.bay` | equivalent AIC of the Bayesian (forward) model.

### References

H.Akaike (1978) A Bayesian Extension of The Minimum AIC Procedure of Autoregressive Model Fitting. Research Memo. NO.126, The Institute of Statistical Mathematics.

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

      data(Powerplant)
      z <- mulbar(Powerplant, max.order=10)
      z$pacoef.for
      z$pacoef.back

[Package timsac version 1.3.5 Index]