How to find parameters of a time series by MLE

Hi, I have a set of data (a time series) Zt (t=1:400) I need to find MLE of parameters a,b,c and d by following equation which is AR(2) model for my time series link of the image for equation: http://cl.ly/image/1i1z0q412T1s

How can I find these parameters? I guess I can use fmincon function but I don't know how can I use it for this equation which has two part and is dependent on the previous values of the time series Thanks

Answer 1

[Ahmed A. Selman]

Hi.

The equation listed in the figure appears so uneasy to handle. That is true since it has cyclic-like behavior with three levels.

However, I've tried simple math, and came up with another, equivalent, yet much clearer formula.

first let

aZ(t-1) = Z1,  

bZ(t-2) = Z2,

1./((2Pi)^0.5*d) = A -- The amplitude of et.

2*d^2 = B

and let go a little further and write:

Z1 + Z2 + c = E 

Zt-E = K

So, your equation will reduce to (after eliminating et from both equations):

Log(K) + K/B = Log(A/B)

with K being the main parameter. This, I think, can be handled easier than the original formula, and it is valid only if it DOES NOT alter the behavior of Z(t-1) and Z(t-2), and that I don't know much about.