vars (version 1.1-9)

VARselect: 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.

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.

encoding

latin1

concept

• VAR
• Vector autoregressive model
• Information criteria
• Final Prediction Error
• Akaike
• Hannan-Quinn
• Schwarz

Details

Estimates a VAR by OLS per equation. The model is of the following form: $$\bold{y}_t = A_1 \bold{y}_{t-1} + \ldots + A_p \bold{y}_{t-p} + CD_t + \bold{u}_t$$ 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 $A_1, \ldots, A_p$ 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: $$AIC(n) = \ln \det(\tilde{\Sigma}_u(n)) + \frac{2}{T}n K^2 \quad,$$ $$HQ(n) = \ln \det(\tilde{\Sigma}_u(n)) + \frac{2 \ln(\ln(T))}{T}n K^2 \quad,$$ $$SC(n) = \ln \det(\tilde{\Sigma}_u(n)) + \frac{\ln(T)}{T}n K^2 \quad,$$ $$FPE(n) = \left ( \frac{T + n^*}{T - n^*} \right )^K \det(\tilde{\Sigma}_u(n)) \quad ,$$ with $\tilde{\Sigma}_u (n) = T^{-1} \sum_{t=1}^T \bold{\hat{u}}_t \bold{\hat{u}}_t'$ and $n^*$ is the total number of the parameters in each equation and $n$ assigns the lag order.

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. Cs�ki (eds.), 2nd International Symposium on Information Theory, Acad�mia Kiad�, 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. L�tkepohl, 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")