ols_cholesky: Estimate Contemporaneous Structural Effects

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

Description

Estimate structural matrices By and Be by least squares or maximum likelihood.

Usage

ols_cholesky(Y, p)

mle_svar(Y, p, By, Be)

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.
p	An integer scalar. The lag length of the VAR(p) system.
By	A (K x K) matrix. Determines the contemporaneous relation between the endogeneous variables y_t. See examples and details below.
Be	A (K x K) matrix. Determines the contemporaneous impact of a structural shock e_t on the endogeneous variables y_t. See examples and details below.

Details

The two matrices By and Be refer to the instantaneous effects matrices in a vector autoregressive system

By * y_t = A_s(L) y_t + Be * e_t.

Thus, By describes the contemporaneous effects between the elements of y. Be describes the contemporaneous impact a structural error vector e_t has on y_t.

• ols_cholesky
  Apply OLS and a Cholesky decomposition for recovering Be from a simple recursive system. As usual, the ordering of the variables will matter for a standard Cholesky decomposition. The effect matrix By is assumed to be an identity matrix.

• mle_svar
  Estimate By and Be by maximising the concentrated log-likelihood. The reduced-form parameters are first estimated by OLS while the remaining structural parameters are estimated by MLE in a second step. The function covers recursive and non-recursive, exactly identified and over-identified systems. The necessary restrictions on matrices By and Be are imposed by assigning numeric values to the restricted elements. Unrestricted elements are left to be estimated by setting them to NA.

Value

The results may not necessarily be unique. The column sign, for example, could be indeterminate.

• ols_cholesky
  A (K x K) numeric lower triangular matrix.

• mle_svar
  A list with two elements By and Be. Both are (K x K) numeric matrices. The elements which previously held NAs were replaced with estimates.

Examples

set.seed(8191)

# number of variables, observations and lag length
K <- 3
N <- 1E6
p <- 2

# prepare input
A <- cbind(matrix(0.1, K, K), matrix(-0.05, K, K)); diag(A) <- 0.4
Be <- matrix(0.4, K, K); Be[upper.tri(Be)] <- 0
Y0 <-matrix(0, nrow = K, ncol = p)
W <- matrix(rnorm(N * K), nrow = K, ncol = N)

# draw data and estimate Be
Y <- create_svar_data(A, Be, Y0, W)
Be_hat <- ols_cholesky(Y, p)
Be_hat
set.seed(8191)

K <- 3
N <- 1E6
p <- 2

A <- cbind(matrix(0.1, K, K), matrix(-0.05, K, K)); diag(A) <- 0.4
Be <- matrix(0.4, K, K); Be[upper.tri(Be)] <- 0
Y0 <-matrix(0, nrow = K, ncol = p)
W <- matrix(rnorm(N * K), nrow = K, ncol = N)

Y <- create_svar_data(A, Be, Y0, W)

By_init <- diag(K)
Be_init <- matrix(0, K, K)
Be_init[lower.tri(Be_init, diag = TRUE)] <- NA

mle_svar(Y, p, By_init, Be_init)

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