ECONOMETRIC METHODS II TA session 1 MATLAB Intro: Simulation of VAR(p) processes

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1 ECONOMETRIC METHODS II TA session 1 MATLAB Intro: Simulation of VAR(p) processes Fernando Pérez Forero April 19th, Introduction In this rst session we will cover the simulation of Vector Autoregressive (VAR) processes of order p. Besides, we will cover how to compute Impulse Response Functions (IRF) and the Forecast Error Variance Decomposition (FEVD). It is assumed that students are familiar with general concepts of Time Series Econometrics and Matrix Algebra, i.e. we will not put many de nitions explicitly, since we assume that they were learnt by heart in the past. This session will be particularly useful for those students who are not familiar with MATLAB 1. It is important to emphasize that when we produce codes, they need to be as general as possible, so that it is easy to consider di erent parametrizations in the future. However there is no free lunch, i.e. this comes at the cost of investing a large amount of time in debugging the code. On the other hand, MATLAB is easy to use relative to other languages such as C++ or Phyton. In particular, since we will not perform object-oriented programming, it will not be necessary to declare variables before using them. Most of the material covered in this session can be found in Lütkepohl (2005) (L) and Hamilton (1994) (H). In order to provide a better understanding of each step, I will indicate the number of equation of each reference as follows: (Book, equation). PhD student. Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25 27, Barcelona, Spain ( fernandojose.perez@upf.edu) 1 A very nice MATLAB primer by Winistorfer and Canova (2008) can be also found in 1

2 2 Preliminaries 2.1 VAR(p) process Consider a VAR(p) process (L,2.1.1), y t = v + A 1 y t A p y t p + u t ; t = 0; 1; 2; : : : ; (2.1) where y t ; u t and v are (K 1) vectors and A i are (K K) matrices for each i = 1; : : : ; p. In addition, the error term u t is a white noise random vector such that E (u t ) = 0, E (u t u 0 t) = u and E (u t u 0 s) = 0 for s 6= t, where u is a (K K) positive de nite matrix. It is useful to re-write (2:1) in its companion form (L,2.1.8), Y t = v + AY t 1 + U t (2.2) where Y t = y 0 t; y 0 t 1; : : : ; y 0 t p+1 0, v = [v 0 ; 0 0 ; : : : ; 0 0 ] 0 and U t = [u 0 t; 0 0 ; : : : ; 0 0 ] 0 are (Kp 1) vectors and 2 A = 6 4 A 1 A 2 A p 1 A p I K I K I K 0 is a (Kp Kp) matrix. The model (2:2) is said to be stable i max (j A j) < 1, where A denotes the vector of eigenvalues of A. The stability condition implies that the model can be re-written as a Moving-Average (MA) representation (L,2.1.13): X y t = JY t = J + J A i U t i (2.3) h where =E (Y t ) = (I Kp A) 1 v is the unconditional expectation and J = I K 0 0 is a (K Kp) selection matrix. Notice also that U t = J 0 JU t and u t = JU t, thus i=0 i y t = JY t = J+ 1X i=0 JA i J 0 JU t i where i = JA i J 0 denotes the i th matrix lag on the MA representation (L,2.1.17) y t = JY t = J+ 1X i=0 JA i J 0 u t i (2.4) Fernando Pérez Forero 2

3 2.2 Impulse Response Function (IRF) In multivariate systems one could be interested in explore the dynamic propagation of innovations across variables. For that purpose, there exists the impulse response function. Basically, one could study the e ect of a one-time innovation in a particular variable n on another variable m over time. The duration of that typoe of innovation will dep exclusively on the structure of the system. In this section we compute the Impulse Response function for the model (2:1). We proceed as follows: If the matrix u is positive de nite, it can be decomposed such that u = P P 0, where P is a lower triangular nonsingular matrix with positive diagonal elements. Moreover, we can re-write (2:4) as (L,2.3.33): y t = + 1X i! t i (2.5) i=0 where i = JA i J 0 P is a (K K) matrix and! t = P 1 u t is a white noise process with covariance matrix equal to the identity matrix, i.e.! = I K. Thus, the impulse response function t = i ; i = 0; 1; 2; : : : (2.6) where i = i jk is such that i jk denotes the response of variable j to a shock in variable k after i periods. 2.3 Forecast Error Variance Decomposition (FEVD) We have shown how to compute the responses of variables after innovations in the system (2:5). It is also possible to perform a forecasting exercise using the latter system. Denote the forecast error h periods ahead as (L,2.3.34): Xh 1 e t+h = y t+h y t (h) = i! t+h i (2.7) Denote the contribution of innovation on variable k to the forecast error h periods ahead of variable j (L,2.3.37): where! jk;h = MSE [y j;t (h)] = i=0 P h 1 i=0 e 0 j i e k 2 MSE [y j;t (h)] Xh 1 i=0 KX k=1 (2.8) i jk 2 (2.9) Fernando Pérez Forero 3

4 3 Introducing the model in MATLAB In this section we simulate data using the system (2:1). We set matrices A i, i = 1; : : : ; p such that the VAR(p) is stationary. 3.1 Data simulation We set K = 4, p = 2 and the number of observations T = 100. p=4; % Lag-order T=100; % Observations K=4; % Variables; Then, we set the matrix u such that it is positive de nite: scale=2*rand(k,1)+1e-30; Sigma_U=0.5*rand(K,K)+diag(scale); Sigma_U=Sigma_U *Sigma_U; if all(eig((sigma_u+sigma_u )/2)) >= 0 disp( Sigma_U is positive definite ) else error( Sigma_U must be positive definite ) And we set lag matrices A i (K K) for each i = 1; : : : ; p in a 3-dimensional vector: Alag=zeros(K,K,p); lambda=2.5; for k=1:p Alag(:,:,k)=(lambda^(-k))*rand(K,K)-((2*lambda)^(-k))*ones(K,K); With all these ingredients, we procced to simulate T = 100 observations of the process (2:1) assuming u t N (0; u ) and p initial conditions Y p+1 ; : : : ; Y 1 ; Y 0 that come from the same distribution. Besides, the intercept v is a vector of ones: Ylag=zeros(K,p); for k=1:p Ylag(:,k)=mvnrnd(zeros(K,1),Sigma_U) ; Y=Ylag; v=ones(k,1); for i=1:t Fernando Pérez Forero 4

5 temp=v+mvnrnd(zeros(k,1),sigma_u) ; for k=1:p temp=temp+alag(:,:,k)*ylag(:,k); Y=[Y,temp]; Ylag(:,p)=temp; for k=1:p-1 Ylag(:,k)=Ylag(:,k+1); clear Ylag temp A_coeff=v; for k=1:p A_coeff=[A_coeff,Alag(:,:,k)]; Y=Y(:,p+1:); We then plot the simulated data in Figure 3.1: FS=15; LW=2; gr_size2=ceil(k/3); figure(1) set(0, DefaultAxesColorOrder,[0 0 1],... DefaultAxesLineStyleOrder, -j-j- ) set(gcf, Color,[1 1 1]) set(gcf, defaultaxesfontsize,fs) for k=1:k subplot(gr_size2,2,k) plot(y(k,:), b, Linewidth,LW) title(sprintf( Y_%d,k)) Fernando Pérez Forero 5

6 Figure 3.1: Simulated Data Next step is to generate the companion form (2:2) and check for stationarity given the parameter values: F1=[A_coeff(:,2:)]; Q1=Sigma_U; if p>1 % Matrix F - [H, ] F2=cell(1,p-1); [F2{:}]=deal(sparse(eye(K))); F2=blkdiag(F2{:}); F2=full(F2); F3=zeros(K*(p-1),K); F=[F1;F2,F3]; % Var_cov Q - [H, ] Q2=cell(1,p-1); [Q2{:}]=deal(sparse(zeros(K,K))); Q2=blkdiag(Q2{:}); Fernando Pérez Forero 6

7 Q2=full(Q2); Q=blkdiag(Q1,Q2); clear F1 F2 F3 else F=F1; Q=Q1; clear F1 Q1 if max(abs(eig(f)))<1 % [H, ] disp( VAR is stationary ) else error( VAR is not stationary! ) The companion form vector Y t has a covariance matrix Y. Using the fact that the model is stationary, we take variances on (2:2), (H, ) Y = A Y A 0 + Q (3.1) where Q =E (U t U 0 t). (3:1) is a discrete Lyapunov equation that can be solved in two ways: i) using a while loop and iterating equation this equation or ii) using a built-in MATLAB function dlyap.m. We use both methods and compare our results: % Companion form s Variance-Covariance matrix % [H, ] Cap_Sigma0=eye(size(F)); Cap_Sigma1=F*Cap_Sigma0*F +Q; diff=max(max(abs(cap_sigma1-cap_sigma0))); diff1=[]; epsilon=1e-10; iter=0; figure(1) set(gcf, Color,[1 1 1]) tic disp( Iterating covariance matrix... ) while diff>epsilon iter=iter+1; Cap_Sigma0=Cap_Sigma1; Cap_Sigma1=F*Cap_Sigma0*F +Q; Fernando Pérez Forero 7

8 diff=max(max(abs(cap_sigma1-cap_sigma0))); diff1=[diff1,diff]; figure(2) plot(diff1) title( Evolution of convergence, FontSize,14) sprintf( Convergence achieved after %d iterations,iter) % Alternatively, solve the discrete Lyapunov equation Cap_Sigma1l=dlyap(F,Q); % Compare results diff2=max(max(abs(cap_sigma1l-cap_sigma1))); toc Figure 3.2 shows that convergence is roughly achieved after iterating this equation 7 times. However, since we impose a convergence criteria of " = , the loop converges after 20 iterations. Moreover, in this application di 2 is less than , which shows that the loop produces a good approximation. We now proceed to the next section for a more interesting analysis. Figure 3.2: Evolution of convergence of Y Fernando Pérez Forero 8

9 3.2 Impulse Response Function (IRF) Once we have generated the companion form components in (2:2), computing Impulse Responses is extremely simple. First, we set a nite horizon h and we generate matrices P and J. h=24; % Horizon P=chol(Sigma_U); % Cholesky decomposition P=P ; % Following the notation of Lutkepohl - Section 3.7 : capj=zeros(k,k*p); capj(1:k,1:k)=eye(k); U=P*E Then we construct matrices i(kk) according to equation (2:6) and store all of them in a 3-dimensional object: Iresp=zeros(K,K,h); Iresp(:,:,1)=P; for j=1:h-1 temp=(f^j); Iresp(:,:,j+1)=capJ*temp*capJ *P; clear temp Finally, we plot the Impulse Responses in Figure 3.3: FS=15; LW=2; figure(3) set(0, DefaultAxesColorOrder,[0 0 1],... DefaultAxesLineStyleOrder, -j-j- ) set(gcf, Color,[1 1 1]) set(gcf, defaultaxesfontsize,fs-5) for i=1:k for j=1:k subplot(k,k,(j-1)*k+i) plot(squeeze(iresp(i,j,:)), Linewidth,LW) hold on plot(zeros(1,h), k ) title(sprintf( Y_{%d} to U_{%d},i,j)) xlim([1 h]) set(gca, XTick,0:ceil(h/4):h) Fernando Pérez Forero 9

10 Figure 3.3: Impulse responses 3.3 Forecast Error Variance Decomposition (FEVD) Once we have computed the Impulse Responses, we can now proceed to compute the contribution of each variable to the Forecast Error variance j = 1; : : : ; h periods ahead. We rst compute the numerator and denominator of equation (2:8) separately, i.e. the absolute contribution and the Mean-Squared-Error (MSE). MSE=zeros(K,h); CONTR=zeros(K,K,h); for j=1:h % Compute MSE temp2=eye(k); for i=1:k if j==1 Fernando Pérez Forero 10

11 MSE(i,j)=temp2(:,i) *Iresp(:,:,j)*Sigma_U*Iresp(:,:,j) *temp2(:,i); for k=1:k CONTR(i,k,j)=(temp2(:,i) *Iresp(:,:,j)*P*temp2(:,k))^2; else MSE(i,j)=MSE(i,j-1)+temp2(:,i) *Iresp(:,:,j)*Sigma_U*Iresp(:,:,j) *temp2(:,i); for k=1:k CONTR(i,k,j)=CONTR(i,k,j-1)+(temp2(:,i) *Iresp(:,:,j)*P*temp2(:,k))^2; % (L,2.3.36) Then we compute the fraction (2:8) and plot the results in Figure 3.4: VD=zeros(K,h,K); for k=1:k VD(:,:,k)=squeeze(CONTR(:,k,:))./MSE; % (L,2.3.37) clear temp2 CONTR MSE figure(4) set(0, DefaultAxesColorOrder,[0 0 1],... DefaultAxesLineStyleOrder, -j-j- ) set(gcf, Color,[1 1 1]) set(gcf, defaultaxesfontsize,fs-5) for i=1:k for j=1:k subplot(k,k,(j-1)*k+i) plot(squeeze(vd(i,:,j)), Linewidth,LW) hold on plot(zeros(1,h), k ) title(sprintf( U{%d} to Var(e_{%d,t+h}).,j,i)) xlim([1 h]) set(gca, XTick,0:ceil(h/4):h) ylim([0 1]) set(gca, YTick,0:0.25:1) Fernando Pérez Forero 11

12 Figure 3.4: Forecast Error Variance decomposition Fernando Pérez Forero 12

13 A List of new commands Type help followed by the command name for more details. clear: Clear variables and functions from memory clc: Clear command window rand(m,n): Produces a m by n matrix with uniformly distributed pseudorandom numbers. diag: Produces diagonal matrices and captures the main diagonal of a matrix. eig: Eigenvalues and eigenvectors all: True (=1) if all elements of a vector are nonzero. disp: displays the array, without printing the array name. error: Display message and abort function. zeros(m,n): Produces a m by n matrix of zeros. ones(m,n): Produces a m by n matrix of ones. eye(m): Produces an identity matrix of order m. for: Repeat statements a speci c number of times. while: Repeat statements an inde nite number of times. figure: Create gure window set: Set object properties. mvnrnd(mu,sigma): Random vectors from the multivariate normal distribution (use also randn(m,n)). ceil(x): rounds the elements of X to the nearest integers towards in nity. plot: Linear plot. See also subplot and options (e.g., title, xlim, etc.) cell(m,n): Create a m by n cell array of empty matrices. sparse(x): converts a sparse or full matrix to sparse form by squeezing out any zero elements. deal: matches up the input and output lists. Fernando Pérez Forero 13

14 blkdiag: Block diagonal concatenation of matrix input arguments. full: Convert sparse matrix to full matrix. abs(x): Produces the absolute value of X. The number X can be either real or complex. max(x) / min(x): Captures the largest / smallest component. chol(x): Cholesky factorization. Uses only the diagonal and upper triangle of X. squeeze(x): returns an array with the same elements as X but with all the singleton dimensions removed. sprintf: Write formatted data to string. B MATLAB tips B.1 Growing arrays Sometimes we do not know the nal size of a vector/matrix in advance. In such cases it is useful to create an empty vector and make it grow at each iteration. However, if you know the exact size of the vector in advance, you should generate (e.g., x=zeros(1,k)) to avoid out-of-memory outcomes. x=[]; A0=rand; A1=rand; diff=abs(a1-a0); while diff>1e-2 x=[x,diff]; A0=rand; A1=rand; diff=abs(a1-a0); B.2 Cumulative sums In MATLAB we can always assign a new value of an object using the previous one. This is useful for cumulative sums: a=0; Fernando Pérez Forero 14

15 for i=1:10; a=a+1; B.3 Matrix partitions It is possible to extract a partition of an existing matrix. As a matter of fact, it is also possible to select and reorder rows and columns: A=magic(4); B=A(2:3,1:2); C=A([2 4 1],[1 3]); References Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press. Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer. Fernando Pérez Forero 15