function tsensdemo % % % This program calculates the ASYMPMTOTIC decision (Rule of Thumb) of the t-statistic % ( linear restrictions case) being sensitive to nuisance parameter in the variance covariance matrix. % Cut and paste the function in your own file % Reference: Banerjee, A.N. and J.R. Magnus (2000), On the sensitivity of the usual % t- and F-tests to covariance misspecification, Journal of Econometrics, % Vol 95(10), pp 157-176. X = [ones(100,1) (1:100)' rand(100,1)]; % generated data C1 = [1 0 0]; % restriction on the intercept C2 = [0 0 1]; % restriction on the \beta_3 n = length(X); A = Derivative(n); % This is the derivative of the Omega matrix [decision1, RHO,ASY_MEAN,ASY_VARIANCE] = t_sens(X,C1,A) [decision2, RHO,ASY_MEAN,ASY_VARIANCE] = t_sens(X,C2,A) function [decision, RHO,ASY_MEAN,ASY_VARIANCE] =t_sens(X,C,T) % % Rule of Thumb: Banerjee, A.N. and J.R. Magnus (2000) % Testing for restriction C \beta = c0 % The t-statistic is sensitive (at the 50% level) % to covariance misspecification if and only if % ???/c>0.40 % input: X = the matrix of independent data (n x k) % C = the restriction k x 1 % T = the Derivative of the covariance matrix at \theta = 0 % In this demo the derivative matrix A is the derivative of % variance covariance matrix of AR(1)(same as MA(1))) at the \theta =0 % where \theta is the AR(1) parameter. % % Output : decision = 1 then t-statistic is sensitive to % nuisance parameter \theta % = 0 otherwise % RHO = ???/c % % ASY_MEAN,ASY_VARIANCE = asymptotic mean and variance of RHO % ------------------------------------------------------------------------ % Reference: Banerjee, A.N. and J.R. Magnus (2000), On the sensitivity of the usual % t- and F-tests to covariance misspecification, Journal of Econometrics, % Vol 95(10), pp 157-176. % ------------------------------------------------------------------------ % % /* ........... PROCEDURES FOR SENSITIVITY ............................ % % Anurag N Banerjee % Durham University, % UK % anurag_banerjee@hotmail.com Date 17/11/2009 % .............................................................. % This program is in the public domain. While the author disclaims % any responsibility for the performance of this software, he % would appreciate receiving any comments. % % This written by Anurag N Banerjee and may be distributed as freeware % for public non-commercial use. Please provide appropriate % acknowledgment if this supports supports published work. % .................................................................. [n k] = size(X); XtX_1 = inv(X'*X); P = X*XtX_1*X'; M = eye(n) - P; B = X*XtX_1*C'/sqrt(C*XtX_1*C'); BTMTB = B'*T*M*T*B; BDB = B'*T*B ; CONST = sqrt(BDB^2+4*BTMTB); % Calculating constant (c) RHO = BDB/CONST; % Calculating r of the product normal distribution */ decision = (abs(RHO) > 0.4); % the asymptotic mean and variance from the product normal approximation */ ASY_MEAN = CONST*RHO; ASY_VARIANCE = CONST*sqrt(1+RHO^2); % @ -------------------- % Derivative of Co-Variance Matrix of AR1 process % ---------------------- @ function DV = Derivative(n); DV = zeros(n,n); for i =1:n ; for j=1:n; if (abs(i-j) == 1) ; DV(i,j) = -1; else; DV(i,j) = 0; end end end