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.
VARselect(y, lag.max = 10, type = c("const", "trend", "both", "none"),
season = NULL, exogen = NULL)A list with the following elements:
Vector with the optimal lag number according to each criterium.
A matrix containing the values of the criteria up to
lag.max.
Data item containing the endogenous variables
Integer for the highest lag order (default is
lag.max = 10).
Type of deterministic regressors to include.
Inlusion of centered seasonal dummy variables (integer value of frequency).
Inlusion of exogenous variables.
Bernhard Pfaff
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.
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.
VAR
data(Canada)
VARselect(Canada, lag.max = 5, type="const")
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