companion_format: Convert a VAR(p) model to VAR(1) companion format

In nielsaka/zeitreihe: Simulate, Estimate, Select, and Forecast Multiple Time Series Processes

Description

Convert matrices containig observations, coefficients, residuals and covariances to their VAR(1) companion format.

Usage

companion_format(Y, nu, A, U)

big_A(A)

big_Y(Y, p)

big_nu(nu, p)

big_U(U, p)

big_SIGMA(SIGMA, p)

Arguments

Y	A (K x N+p) matrix carrying the data for estimation. There are N observations for each of the K variables with p pre-sample values.
nu	A (K x 1) matrix of intercepts.
A	A (K x Kp) matrix, providing the coeffcients for lag 1 to p with the first row containing the coefficents of the first equation. Parameter p is the maximum lag length and K the number of variables.
U	A (K x N) matrix of residuals.
p	An integer scalar. The lag length of the VAR(p) system.
SIGMA	A (K x K) matrix of covariances. The covariance matrix of the residuals U.

Details

Convert objects such as

• A - a matrix of slope parameters

• nu - a column vector of intercepts

• Y - a matrix of observations

• U - a matrix of residuals

• SIGMA - a covariance matrix

such that they correspond to objects taken from a VAR(1) representation.

Value

• companion_format
  A list with five elements. Four elements Y, nu, A and U correspond to the output of big_Y, big_nu, big_A, and big_U. The fifth element is the (Kp x N) regressor matrix Z populated with lags of Y.

• big_A
  A (Kp x Kp) matrix with the slope parameters on the first K rows. The remaining rows carry block-diagonal unit matrices and the remainder are zeros.

• big_Y
  A (Kp x N) matrix. The p-1 lags of Y are pasted as rows below Y. The p pre-sample observations are deleted.

• big_nu
  A (Kp x 1) colum matrix. The first K elements correspond to nu. The remaining elements are zero.

• big_U
  A (Kp x N) matrix. The first K rows contain the residuals, the last K(p-1) rows consist of zeros.

• big_SIGMA
  A (Kp x Kp) matrix. The upper left (K x K) block will contain the original covariance matrix. The remaining elements will be zero.

Note

The difference between big_Y() and Y2Z() is that observations in matrix Z are shifted back by one time period in comparison to Y.

Examples

set.seed(8191)

K <- 3
N <- 5
p <- 2

nu <- matrix(1:K, ncol = 1)
A  <- matrix(0.1, K, K * p); diag(A) <- 1:K / 10
U  <- matrix(rnorm(K * N), K, N)
Y0 <- matrix(0, K, p)
Y  <- create_varp_data(A, Y0, U)

cf <- companion_format(Y, nu, A, U)

cf$U
cf$Y - cf$A %*% cf$Z

## Not run: 
# input has to be in matrix form
 companion_format(Y[1, ], nu[1, ], A[1, ], U[1,])

## End(Not run)

big_A(A)

Y <- matrix(seq_len(K * N), K, N)
big_Y(Y, p)

big_nu(nu, p)

## Not run: 

big_nu(c(nu), p)

## End(Not run)

U <- matrix(seq_len(K * N), K, N)
big_U(U, p)

SIGMA <- matrix(0.5, K, K)
big_SIGMA(SIGMA, p = p)

nielsaka/zeitreihe documentation built on March 17, 2020, 8:38 p.m.