Maximum Likelihood estimation for time series models

In this post … $a=5$ asdf

\[L(\boldsymbol{\theta})=f_{Y_{T},Y_{T-1},...,Y_{1}}(y_{T},y_{T-1},...,y_{1};\boldsymbol{\theta})\]

This approach requires specifying a partiular distribution for the noise proccess. For example the Gaussian white noise:

\[\epsilon_{t}\sim i.i.d.\,N(0,\sigma^{2})\]

…

This has to be maximised with respect to $\boldsymbol{\theta}.$ It is often conditioned on the first $p$ observation (Conditional Maximum Likelihood Estimation), therefore $y_{p},…,y_{1}$ are often substituted with their observed or expected values. Thus the loglikelihood can be simplified to

Posted on March 13th, 2021 by You!