NAG Toolbox for Matlab nag_rand_times_garch_exp (g05pg)

Transcription

1 1 Purpose NAG Toolbox for Matlab nag_rand_times_garch_exp () nag_rand_times_garch_exp () generates a given number of terms of an exponential GARCHðp; qþ process (see Engle and Ng (1993)). 2 Syntax [ht, et, r, state, ifail] = nag_rand_times_garch_exp(dist, num, ip, iq, theta, df, fcall, r, state, lr, lr) [ht, et, r, state, ifail] = (dist, num, ip, iq, theta, df, fcall, r, state, lr, lr) 3 Description An exponential GARCHðp; qþ process is represented by: lnðh t Þ ¼ 0 þ Xq i z t i þ Xq i ðj z t i j E½jz t i jš Þþ Xp j¼1 j ln h t j, t ¼ 1; 2;...;T; where z t ¼ p t ffiffiffiffi, E½jz t i jš denotes the expected value of jz t i j, and t j t 1 ¼ Nð0;h t Þ or h t t j t 1 ¼ S t ðdf ;h t Þ. Here S t is a standardized Student s t-distribution with df degrees of freedom and variance h t, T is the number of observations in the sequence, t is the observed value of the GARCHðp; qþ process at time t, h t is the conditional variance at time t, and t the set of all information up to time t. One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_times_garch_exp (). 4 References Bollerslev T (1986) Generalised autoregressive conditional heteroskedasticity Journal of Econometrics Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation Econometrica Engle R and Ng V (1993) Measuring and testing the impact of news on volatility Journal of Finance Glosten L, Jagannathan R and Runkle D (1993) Relationship between the expected value and the volatility of nominal excess return on stocks Journal of Finance Hamilton J (1994) Time Series Analysis Princeton University Press 5 Parameters 5.1 Compulsory Input Parameters 1: dist string The type of distribution to use for t. dist ¼ N A Normal distribution is used..1

2 NAG Toolbox for Matlab Manual dist ¼ T A Student s t-distribution is used. Constraint: dist ¼ N or T. 2: num int32 scalar T, the number of terms in the sequence. Constraint: num 0. 3: ip int32 scalar The number of coefficients, i, for i ¼ 1; 2;...;p. Constraint: ip 0. 4: iq int32 scalar The number of coefficients, i, for i ¼ 1; 2;...;q. Constraint: iq 1. 5: thetað2 iq þ ip þ 1Þ double array The initial parameter estimates for the vector. The first element must contain the coefficient o and the next iq elements must contain the autoregressive coefficients i, for i ¼ 1; 2;...;q. The next iq elements must contain the coefficients i, for i ¼ 1; 2;...;q. The next ip elements must contain the moving average coefficients j, for j ¼ 1; 2;...;p. Constraints: X p i 6¼ 1:0; 0 X p 1 logðx02amfþ. i 6: df int32 scalar The number of degrees of freedom for the Student s t-distribution. If dist ¼ N, df is not referenced. Constraint: ifdist ¼ T, df > 2. 7: fcall logical scalar If fcall ¼ true, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in r. 8: rðlrþ double array lr, the dimension of the array, must satisfy the constraint lr 2 ðip þ 2 iq þ 2Þ. The array contains information required to continue a sequence if fcall ¼ false. 9: stateð:þ int32 array Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg). Contains information on the selected base generator and its current state..2

3 5.2 Optional Input Parameters 1: lr int32 scalar Default: The dimension of the array r. Constraint: lr 2 ðip þ 2 iq þ 2Þ. 5.3 Input Parameters Omitted from the MATLAB Interface None. 5.4 Output Parameters 1: htðnumþ double array The conditional variances h t, for t ¼ 1; 2;...;T, for the GARCHðp; qþ sequence. 2: etðnumþ double array The observations t, for t ¼ 1; 2;...;T, for the GARCHðp; qþ sequence. 3: rðlrþ double array Contains information that can be used in a subsequent call of nag_rand_times_garch_exp (), with fcall ¼ false. 4: stateð:þ int32 array Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg). Contains updated information on the state of the generator. 5: ifail int32 scalar ifail ¼ 0 unless the function detects an error (see [Error Indicators and Warnings]). 6 Error Indicators and Warnings Errors or warnings detected by the function: ifail ¼ 1 On entry, dist 6¼ N or T. ifail ¼ 2 On entry, num < 0. ifail ¼ 3 On entry, ip < 0. ifail ¼ 4 On entry, iq < 1. ifail ¼ 6 On entry, dist ¼ T and df 2. ifail ¼ 10 The value of ip or iq is not the same as when r was set up in a previous call..3

4 NAG Toolbox for Matlab Manual ifail ¼ 11 On entry, lr < 2 ðip þ iq þ 2Þ. ifail ¼ 12 On entry, state vector was not initialized or has been corrupted. ifail ¼ 20 Invalid sequence generated, use different parameters. 7 Accuracy Not applicable. 8 Further Comments None. 9 Example function nag_rand_times_garch_exp_example % Initialize the seed seed = [int32( )]; % genid and subid identify the base generator genid = int32(1); subid = int32(1); dist = N ; num = 10; ip = 1; iq = 1; theta = [0.1; -0.3; 0.1; 0.9]; df = int32(0); fcall = true; r = zeros(2*(2*ip+iq+2),1); % Initialize the generator to a repeatable sequence [state, ifail] = nag_rand_init_repeat(genid, subid, seed); % Generate the first realisation [ht, et, r, state, ifail] = nag_rand_times_garch_exp(dist, int32(num), int32(ip), int32(iq),... theta, df, fcall, r, state); % Display the results if ifail == 0 fprintf( \n Realisation Number 1\n ); fprintf( I HT(I) ET(I)\n ); fprintf( \n ); for i=1:num fprintf( %2d %16.4f %16.4f\n, i, ht(i), et(i)); else fprintf( \n nag_rand_times_garch_exp exited with ifail = %d \n, ifail); % Generate a second realisation fcall = false; [ht, et, r, state, ifail] = nag_rand_times_garch_exp(dist, int32(num), int32(ip), int32(iq),... theta, df, fcall, r, state); % Display the results if ifail == 0 fprintf( \n Realisation Number 2\n ); fprintf( I HT(I) ET(I)\n ); fprintf( \n ); for i=1:num.4

5 fprintf( %2d %16.4f %16.4f\n, i, ht(i), et(i)); else fprintf( \n nag_rand_times_garch_exp exited with ifail = %d \n, ifail); Realisation Number 1 I HT(I) ET(I) Realisation Number 2 I HT(I) ET(I) (last)