Having decided the orders (p,q,d) of an ARIMA model - why do we use Maximum Likelihood Estimation instead of least squares to determine the coefficients?
An ARIMA model is after all very similar to a multivariable regression.
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Sign up to join this communityHaving decided the orders (p,q,d) of an ARIMA model - why do we use Maximum Likelihood Estimation instead of least squares to determine the coefficients?
An ARIMA model is after all very similar to a multivariable regression.
OLS only works when all regressors are observable, because it employs the design matrix $X$ in estimating the model coefficients ($\beta=(X'X)^{-1}X'y$). Meanwhile, the ARIMA model specification contains unobservable variables in the moving average part (the lagged errors); hence, the OLS estimator is infeasible when the moving-average order $q>0$. On the other hand, an AR model can be estimated with OLS, and this is in fact quite a common approach. For normally distributed errors, it is also very close to the Maximum Likelihood (ML) solution. (There are subtle differences on how to treat the initial values in the ML estimation, but other than that it coincides with OLS for AR models.)