Recitation 2: Time Series in Matlab

Time Series in Matlab 1 14.384 Time Series Analysis, Fall 2007 Recitation by Paul Schrimpf Supplementary to lectures given by Anna Mikusheva September 11, 2008 Recitation 2: Time Series in Matlab Time Series in Matlab In problem set 1, you need to estimate spectral densities and apply common filters. You can use any software you would like, but we recommend using Matlab. It may be easier to do simple things using more statistics oriented programs like Stata or RATs, since these programs include pre-packaged commands for many common tasks, but you will learn more by writing the code yourself. Also, it is generally easier to write programs for new estimators in a full-featured programming language like Matlab’s than in the language of statistics oriented programs. Finally, I recommend using Matlab because I happen to use Matlab, and I will be more likely to be able to provide help if you need it. These notes cover some slightly obscure Matlab commands that can be useful for time series. For a more general overview, see http://web.mit.edu/~paul_s/www/14.170/matlab.html Disclaimer: I wrote these notes last year, and I am not entirely sure that they are com­ pletely correct. Many of the commands covered are from Matlab’s signal processing toolbox, and they have different names and may do slightly different things than what an econometri­ cian would expect. Always read Matlab’s help and documentation before using a command. When in doubt, double check that the command does what you think. Simulating an ARMA Model a(L)yt = b(L)et 1 2 3 4 5 6 clear; a = [1 0.5]; % AR coeffs b = [1 0.4 0.3]; % MA coeffs T = 1000; e = randn(T,1); % generate gaussian white noise y = filter(b,a,e); % generate y The filter function can be used to generate data from an ARMA model, or apply a filter to a series. Impulse-Response 2 Impulse-Response To graph the impulse response of an ARMA, use fvtool 1 2 % create an impulse response fvtool(b,a,'impulse'); Sample Covariances 1 2 3 4 [c lags]=xcov(y,'biased'); figure; plot(lags,c); title('Sample Covariances'); The option, 'biased', says to use �ˆk = �T −k 1 �ˆk = T −k t=1 (yt − ȳ)(yt−k − ȳ) 1 T �T −k t=1 (yt − y)(y ¯ t−k − ȳ); 'unbiased' would use Spectral Analysis Population spectral density for an ARMA: 1 2 3 4 % population density w = 0:0.1:pi; % frequencies to compute density at h = freqz(b,a,w); % returns frequency response = b(eˆ{iw})/a(eˆ{iw}) sd = abs(h).ˆ2./sqrt(2*pi); % make into density Estimating the Spectral Density Parametric methods These estimate an AR(p) and use it to compute the spectrum. 1 [sdc wc] = pcov(y,8); % estimate spectral density by fitting AR(8) 2 3 4 5 [sdy wy] = pyulear(y,8); % estimate spectral density by fitting AR(8) % using % the Yule−walker equations Non-parametric methods Definition 1. The sample periodogram of {xt }Tt=1 is Ŝ(�) = T1 | �T t=1 e−i�t xt |2 Filtering 3 Remark 2. The sample periodogram is equal to the Fourier transform of the sample autoco­ variances T T −1 � 1 � −i�t 2 ˆ S(�) = | e xt | = �ˆk e−i�k T t=1 k=−T −1 1 [sdp wp] = periodogram(y,[],'onesided'); % estimate using sample Definition 3. A smoothed periodogram estimate of the spectral density is � � T 1 � −i�t 2 hT (� − �) | e xt | d� Ŝ(�) = T t=1 −� where hT () is some kernel weighting function. A smoothed periodogram is a weighting moving average of the sample periodogram. The�following code estimates a smoothed periodogram using a Parzen kernel with band­ width T . 1 2 rT = round(sqrt(T)); [sdw ww] = pwelch(y,parzenwin(rT),rT−1,[],'onesided'); % smoothed periodogram Definition 4. A weighted covariance estimate of the spectrum is: Ŝ(�) = ST � �ˆk gT (k)e−i�k k=−ST where gT (k) is some kernel. 1 2 3 4 5 6 7 8 9 10 % bartlett weighted covariances wb = 0:0.1:pi; rT = sqrt(T); [c t]=xcov(y,'biased'); weight = 1−abs(t)/rT; weight(abs(t)>rT) = 0; for j=1:length(wb) sdb(j) = sum(c.*weight.*exp(−i*wb(j)*t)); end sdb = sdb / sqrt(2*pi); Filtering See: http://ideas.repec.org/c/wpa/wuwppr/0507001.html for code for common filters. Example: Simulating an ARMA and estimating the spectrum Example: Simulating an ARMA and estimating the spectrum 1 2 clear; close all; % closes all open figure windows 3 4 5 6 7 8 9 % a b T e y model: y t = 0.9 y {t−1} + b(L) e t = [1 −0.7]; % AR coeffs = [1 0.3 2]; % MA coeffs = 200; = randn(T,1); % generate gaussian white noise = filter(b,a,e); % generate y 10 11 12 13 14 15 16 % plot y figure; plot(y); xlabel('t'); ylabel('y'); title('ARMA(1,2)'); 17 18 19 % create an impulse response fvtool(b,a,'impulse'); 20 21 22 23 24 25 % calculate and plot sample auto−covariances [c lags]=xcov(y,'biased'); figure; plot(lags,c); title('Sample Covariances'); 26 27 % estimate spectral density 28 29 30 31 32 33 % parametric [sdc wc] = pcov(y,8); % estimate spectral density by fitting AR(8) [sdy wy] = pyulear(y,8); % estimate spectral density by fitting AR(8) % using % the Yule−walker equations 34 35 36 37 38 39 40 % nonparametric [sdp wp] = periodogram(y,[],'onesided'); % estimate using unsmoothed % periodogram rT = round(sqrt(T))*3; [sdw ww] = pwelch(y,parzenwin(rT),rT−1,[],'onesided'); % smoothed periodogram 41 42 43 44 45 46 47 48 % bartlett weighted covariances [c lags]=xcov(y,'biased'); t = −(T−1):(T−1); weight = 1−abs(t')/rT; weight(abs(t')>rT) = 0; wb = ww; for j=1:length(wb) 4 Example: Simulating an ARMA and estimating the spectrum 5 ARMA(1,2) 15 10 y 5 0 −5 −10 −15 49 50 51 0 50 100 t 150 200 sdb(j) = sum(c.*weight.*exp(−i*wb(j)*(−(T−1):(T−1))')); end sdb = sdb / sqrt(2*pi); 52 53 54 55 56 % population density w = wb; h = freqz(b,a,w); sd = abs(h).ˆ2./sqrt(2*pi); 57 58 59 60 61 figure; plot(wp,sdp,wc,sdc,ww,sdw,wb,sdb,wy,sdy,w,sd); legend('raw periodogram','parametric AR','smoothed periodogram', ... 'bartlett weighted cov','Yule−Walker','population density'); Example: Simulating an ARMA and estimating the spectrum 6 Impulse Response 2.5 Amplitude 2 1.5 1 0.5 0 0 5 10 15 20 25 Samples Sample Covariances 16 14 12 10 8 6 4 2 0 −2 −4 −200 −150 −100 −50 0 50 100 150 200 Example: Simulating an ARMA and estimating the spectrum 7 70 raw periodogram parametric AR smoothed periodogram bartlett weighted cov Yule−Walker population density 60 50 40 30 20 10 0 0 0.5 1 1.5 2 2.5 3 3.5