R: Information criteria and FPE for different VAR(p)

VARselect {vars} | R Documentation

## Information criteria and FPE for different VAR(p)

### Description

The function returns infomation criteria and final prediction error for sequential increasing the lag order up to a VAR(p)-proccess. which are based on the same sample size.

### Usage

    VARselect(y, lag.max = 10, type = c("const", "trend", "both", "none"),
    season = NULL, exogen = NULL)

### Arguments

`y` | Data item containing the endogenous variables
`lag.max` | Integer for the highest lag order (default is `lag.max = 10`).
`type` | Type of deterministic regressors to include.
`season` | Inlusion of centered seasonal dummy variables (integer value of frequency).
`exogen` | Inlusion of exogenous variables.

### Details

Estimates a VAR by OLS per equation. The model is of the following form:

\bold{y}_t = A_1 \bold{y}_{t-1} + … + A_p \bold{y}_{t-p} + CD_t + \bold{u}_t

where \bold{y}_t is a K \times 1 vector of endogenous variables and u_t assigns a spherical disturbance term of the same dimension. The coefficient matrices A_1, …, A_p are of dimension K \times K. In addition, either a constant and/or a trend can be included as deterministic regressors as well as centered seasonal dummy variables and/or exogenous variables (term CD_T, by setting the `type` argument to the corresponding value and/or setting `season` to the desired frequency (integer) and/or providing a matrix object for `exogen`, respectively. The default for `type` is `const` and for `season` and `exogen` the default is set to `NULL`.
Based on the same sample size the following information criteria and the final prediction error are computed:

AIC(n) = \ln \det(\tilde{Σ}_u(n)) + \frac{2}{T}n K^2 \quad,

HQ(n) = \ln \det(\tilde{Σ}_u(n)) + \frac{2 \ln(\ln(T))}{T}n K^2 \quad,

SC(n) = \ln \det(\tilde{Σ}_u(n)) + \frac{\ln(T)}{T}n K^2 \quad,

FPE(n) = ≤ft ( \frac{T + n^*}{T - n^*} \right )^K \det(\tilde{Σ}_u(n)) \quad ,

with \tilde{Σ}_u (n) = T^{-1} ∑_{t=1}^T \bold{\hat{u}}_t \bold{\hat{u}}_t' and n^* is the total number of the parameters in each equation and n assigns the lag order.

### Value

A list with the following elements:

`selection` | Vector with the optimal lag number according to each criterium.
`criteria` | A matrix containing the values of the criteria up to `lag.max`.

### Author(s)

Bernhard Pfaff

### References

Akaike, H. (1969), Fitting autoregressive models for prediction, Annals of the Institute of Statistical Mathematics, 21: 243-247.

Akaike, H. (1971), Autoregressive model fitting for control, Annals of the Institute of Statistical Mathematics, 23: 163-180.

Akaike, H. (1973), Information theory and an extension of the maximum likelihood principle, in B. N. Petrov and F. Csáki (eds.), 2nd International Symposium on Information Theory, Académia Kiadó, Budapest, pp. 267-281.

Akaike, H. (1974), A new look at the statistical model identification, IEEE Transactions on Automatic Control, AC-19: 716-723.

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Hannan, E. J. and B. G. Quinn (1979), The determination of the order of an autoregression, Journal of the Royal Statistical Society, B41: 190-195.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

Quinn, B. (1980), Order determination for a multivariate autoregression, Journal of the Royal Statistical Society, B42: 182-185.

Schwarz, G. (1978), Estimating the dimension of a model, Annals of Statistics, 6: 461-464.

### See Also

`VAR`

### Examples

    data(Canada)
    VARselect(Canada, lag.max = 5, type="const")

[Package vars version 1.5-3 Index]