recserar#

Purpose#

Computes a vector of autoregressive recursive series.

Format#

y = recserar(x, y0, rho)#

Parameters:

x (NxK matrix) – If simulating an AR process, this would contain the error term and constant if included in the model.
y0 (PxK matrix) – The starting values for the series.
rho (PxK matrix) – The AR parameters.

Returns:

y (NxK matrix) – Contains the series.

Examples#

AR(1) without constant#

// Starting value for the time series
y0 = 0;

// AR(1) parameter
rho = 0.6;

// Innovations
eps = rndn(10, 1);

// Simulate AR(1) model
y = recserar(eps, y0, rho);

AR(2) with constant#

// Starting value for the time series
y0 = { 0, 0 };

// AR(2) parameters
rho = { 0.6, -0.3 };

// Constant term
const = 1.3;

// Innovations
eps = rndn(10, 1);

// Simulate AR(2) model with constant
y = recserar(eps + const, y0, rho);

Example 3#

n = 10;

sig = { 1 -.3, -.3 1 };
mu = { 0, 0 };
e = rndMVn(n, mu, sig);

x = ones(n, 1)~rndn(n, 3);

b = 1|2|3|4;

rho = { 0.5, 0.3 };
y0 = zeros(1, 2);
y = recserar(x*b+e, y0, rho);

In this example, two autoregressive series are formed using simulated data. The general form of the series can be written:

y[1,t] = rho[1,1]*y[1,t-1] + x[t,.]*b + e[1,t];
y[2,t] = rho[2,1]*y[2,t-1] + x[t,.]*b + e[2,t];

The error terms (\(e[1,t]\) and \(e[2,t]\)) are not individually serially correlated, but they are contemporaneously correlated with each other. The variance-covariance matrix is \(\sigma\).

Remarks#

recserar() is particularly useful in dealing with time series.

Typically, the result would be thought of as \(K\) vectors of length \(N\).

y0 contains the first \(P\) values of each of these vectors (thus, these are prespecified). The remaining elements are constructed by computing a \(P^{th}\) order “autoregressive” recursion, with weights given by a, and then by adding the result to the corresponding elements of x. That is, the \(t^{th}\) row of y is given by:

y[t,.] = x[t,.] + a[1,.] * y[t-1,.] +...+ a[P,.] * y[t-p,.], t = P + 1,...N

and

y[t,.] = y0[t,.], t = 1,...,P

Note that the first \(P\) rows of x are not used.