vectorizing for-loops for autoregressive or other recursive processes

[Ginters Bušs]

Dear all,

I am inspired by the recent Allan's comment that I can vectorize for-loops.


Say, I have a for-loop for AR(2) process
y(t+1)=f1*y(t)+f2*y(t-1)+epsilon(t+1) with a degenerate disturbance term:

f1=1.2; f2=-0.5;
n=100000;
c=1:n;
c(2)=f1*c(1)+c(2);
for i=3:n;
    c(i)=c(i)+f1*c(i-1)+f2*c(i-2);
end

Allan proposed trying

f1=1.2; f2=-0.5;
n=100000;
b=1:n;
b(2)=f1*b(1)+b(2);
b(3:n) = b(3:n)+f1*b(2:n-1)+f2*b(1:n-2);


but the result is different.  Then, I thought I can try to use ode; say, for
AR(1) process y(t+1) = a*y(t)+ epsilon(t+1) it would look like

deff("yp=a_function(k,y)","yp=a*y+sigma*u(k)")
y0=0;
a=.9;
n=100000;
u=1:n;
y=ode("discrete",y0,1,1:n,a_function);

but it turns out that for-loop is a bit faster than this ode code.

Another idea would be using Wold representation of (only stationary) AR
process by rewriting the above AR(2) process as

y(t+1) = ((1-f1*L - f2*L^2)^(-1))*epsilon(t+1)

where (1-f1*L - f2*L^2) is a lag polynomial, L defined as Ly(t)=y(t-1)

and trying to get the series representation of  (1-f1*L - f2*L^2)^(-1) but
I'm stuck here, and this would work only for a stationary process.

Do you have any ideas/experience with rewriting for-loops for AR or other
recursive processes more efficiently than the above first code?
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