mulbar {timsac}

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/2d where n is the length and d is the dimension of the time series y. Default is min(2 \sqrt{n}, n/2d).

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 \times \frac{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,\ldots,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) + \ldots + W(lag).

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

ek = \{ D(1)^2 + \ldots + D(lag)^2 \} id + \frac{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