## Copyright (C) 1995, 1996, 1997 Kurt Hornik ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 2, or (at your option) ## any later version. ## ## This program is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this file. If not, write to the Free Software Foundation, ## 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. ## usage: [pval, lm] = arch_test (y, X, p) ## [pval, lm] = arch_test (y, k, p) ## ## arch_test (y, X, p) performs a Lagrange Multiplier (LM) test of the ## null hypothesis of no conditional heteroscedascity in the linear ## regression model y = X * b + e against the alternative of CH(p). ## I.e., the model is ## y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t), ## where given y up to t-1 and x up to t, e(t) is N(0, h(t)) with ## h(t) = v + a(1) * e(t-1)^2 + ... + a(p) * e(t-p)^2, ## and the null is a(1) == ... == a(p) == 0. ## ## arch_test (y, k, p) does the same in a linear autoregression model of ## order k, i.e., with [1, y(t-1), ..., y(t-k)] as the t-th row of X. ## ## Under the null, lm approximately has a chisquare distribution with p ## degrees of freedom. pval is the p-value (1 minus the CDF of this ## distribution at lm) of the test. ## ## If no output argument is given, the p-value is displayed. ## Author: KH <Kurt.Hornik@ci.tuwien.ac.at> ## Description: Test for conditional heteroscedascity function [pval, lm] = arch_test (y, X, p) if (nargin != 3) error ("arch_test needs 3 input arguments"); endif if !(is_vector (y)) error ("arch_test: y must be a vector"); endif T = length (y); y = reshape (y, T, 1); [rx, cx] = size (X); if ((rx == 1) && (cx == 1)) X = autoreg_matrix (y, X); elseif !(rx == T) error (["arch_test: ", ... "either rows(X) == length(y), or X is a scalar"]); endif if !(is_scalar(p) && (rem(p, 1) == 0) && (p > 0)) error ("arch_test: p must be a positive integer."); endif [b, v_b, e] = ols (y, X); Z = autoreg_matrix (e.^2, p); f = e.^2 / v_b - ones (T, 1); f = Z' * f; lm = f' * inv (Z'*Z) * f / 2; pval = 1 - chisquare_cdf (lm, p); endfunction

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