ECONOMETRIC METHODS II TA session 1 MATLAB Intro ...

ECONOMETRIC METHODS II 

TA session 1 

MATLAB Intro: Simulation of VAR(p) processes 

1 Introduction 

Fernando Pérez Forero 

April 19th, 2012 

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, 

08005 Barcelona, Spain (email: fernandojose.perez@upf.edu) 

1 A very nice MATLAB primer by Winistorfer and Canova (2008) can be also found in 

http://www.crei.cat/people/canova/Matlab-intro%281%29.pdf 

1

ECONOMETRIC METHODS II TA Session 1 

2 Preliminaries 

2.1 VAR(p) process 

Consider a VAR(p) process (L,2.1.1), 

yt = v + A1yt 1 + + Apyt p + ut; t = 0; 1; 2; : : : ; (2.1) 

where yt; ut and v are (K 1) vectors and Ai are (K K) matrices for each i = 1; : : : ; p. 

In addition, the error term ut is a white noise random vector such that E (ut) = 0, 

E (utu 0 t) = u and E (utu 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), 

where Yt = y 0 t; y 0 t 1; : : : ; y 0 t p+1 

vectors and 

6 

A = 6 

4 

Yt = v + AYt 1 + Ut (2.2) 

0 , v = [v 0 ; 0 0 ; : : : ; 0 0 ] 0 and Ut = [u 0 t; 0 0 ; : : : ; 0 0 ] 0 are (Kp 1) 

2 

A1 A2 Ap 1 Ap 

IK 0 0 0 

0 IK 0 0 

.. . . . . 

0 0 IK 0 

is a (Kp Kp) matrix. The model (2:2) is said to be stable i¤ max (j Aj) < 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): 

yt = JYt = J + J 

3 

7 

5 

1X 

A i Ut i (2.3) 

where =E (Yt) = (IKp A) 1 h 

i 

v is the unconditional expectation and J = IK 0 0 

is a (K Kp) selection matrix. Notice also that Ut = J 0 JUt and ut = JUt, thus 

yt = JYt = J + 

1X 

i=0 

i=0 

JA i J 0 JUt i 

where i = JA i J 0 denotes the i th matrix lag on the MA representation (L,2.1.17) 

yt = JYt = J + 

1X 

i=0 

JA i J 0 ut i (2.4) 

Fernando Pérez Forero 2


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 depend 

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): 

yt = + 

1X 

i=0 

i!t i (2.5) 

where i = JA i J 0 P is a (K K) matrix and !t = P 1 ut is a white noise process with 

covariance matrix equal to the identity matrix, i.e. ! = IK. 

Thus, the impulse response function is 

@yt+i 

@!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): 

et+h = yt+h yt (h) = 

Xh 

1 

i=0 

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 [yj;t (h)] = 

P h 1 

i=0 e0 j iek 

MSE [yj;t (h)] 

Xh 

1 

i=0 

KX 

k=1 

2 

i 

jk 

2 

(2.8) 

(2.9) 



3 Introducing the model in MATLAB 

In this section we simulate data using the system (2:1). We set matrices Ai, 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’) 

end 

And we set lag matrices Ai (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); 

end 

With all these ingredients, we procced to simulate T = 100 observations of the process 

(2:1) assuming ut N (0; u) and p initial conditions Y p+1; : : : ; Y 1; Y0 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)’; 

end 

Y=Ylag; 

v=ones(K,1); 

for i=1:T 



temp=v+mvnrnd(zeros(K,1),Sigma_U)’; 

for k=1:p 

temp=temp+Alag(:,:,k)*Ylag(:,k); 

end 

Y=[Y,temp]; 

Ylag(:,p)=temp; 

for k=1:p-1 

Ylag(:,k)=Ylag(:,k+1); 

end 

end 

clear Ylag temp 

A_coeff=v; 

for k=1:p 

A_coeff=[A_coeff,Alag(:,:,k)]; 

end 

Y=Y(:,p+1:end); 

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)) 

end 



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:end)]; 

Q1=Sigma_U; 

if p>1 

% Matrix F - [H,10.1.10] 

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,10.1.11] 

Q2=cell(1,p-1); 

[Q2{:}]=deal(sparse(zeros(K,K))); 

Q2=blkdiag(Q2{:}); 



Q2=full(Q2); 

Q=blkdiag(Q1,Q2); 

clear F1 F2 F3 

else 

F=F1; 

Q=Q1; 

clear F1 Q1 

end 

if max(abs(eig(F)))epsilon 

iter=iter+1; 

Cap_Sigma0=Cap_Sigma1; 

Cap_Sigma1=F*Cap_Sigma0*F’+Q; 



diff=max(max(abs(Cap_Sigma1-Cap_Sigma0))); 

diff1=[diff1,diff]; 

figure(2) 

plot(diff1) 

title(’Evolution of convergence’,’FontSize’,14) 

end 

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 " = 1 10 10 , the loop 

converges after 20 iterations. Moreover, in this application di¤2 is less than 1 10 10 , 

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 



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 : U=P*E 

capJ=zeros(K,K*p); 

capJ(1:K,1:K)=eye(K); 

Then we construct matrices i(K K) 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; 

end 

clear temp 

Finally, we plot the Impulse Responses in Figure 3.3: 

FS=15; 

LW=2; 

figure(3) 




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) 



end 

end 

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 



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; 

end 

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) 

end 

end 

end 

end 

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) 

end 

clear temp2 CONTR MSE 

figure(4) 




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) 

end 

end 



Figure 3.4: Forecast Error Variance decomposition 



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. 



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); 

end 

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; 



for i=1:10; 

a=a+1; 

end 

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.

ECONOMETRIC METHODS II TA session 1 MATLAB Intro ...

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