timsac (version 1.3.8-4)

mulbar: 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)

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.

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.

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