Estimation of ARMA: state space vs. alternatives I am interested in estimation of ARMA models. I understand that a popular approach is to write the model down in the state space form and then maximize the likelihood of the model using some optimization routine. 
Question: Why rewrite the model into its state space representation and maximize the corresponding likelihood -- instead of maximizing the "naive" or "direct" likelihood?
(I could imagine that a different parameterization can make the optimization easier -- is that the case here?)
Related questions:


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*"Comparison of estimation techniques for ARIMA model"

*"What state-space representation of VARMA is commonly used for fitting"

*"What is the requirement for Kalman filters"
I am also aware of some general advantages and disadvantages of the state space representation as mentioned in "What are disadvantages of state-space models and Kalman Filter for time-series modelling?".
 A: Rewriting an ARMA model into its state space form and maximizing the corresponding likelihood, rather than maximizing the "naive" or "direct" likelihood, is a popular approach because it has several advantages:

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*Flexibility: The state space representation allows for a greater degree of flexibility in modeling the system. For example, it is easy to include exogenous variables in the model, to model non-stationary systems, or to estimate time-varying parameters. In general it is easier to estimate an ARMA model in its state space form than via the direct likelihood. The direct likelihood for an ARMA model can be very complicated, involving sums of products of polynomials and powers of lagged variables. On the other hand, the state space representation of an ARMA model is relatively simple and can be written as a set of linear, time-invariant difference equations. This makes it much easier to write down the likelihood function and to compute the gradients required for optimization.


*Computational efficiency: The state space representation is computationally efficient, particularly when the model is estimated using Kalman filtering algorithms. These algorithms are able to efficiently compute the likelihood and gradients required for optimization even when the number of observations is very large. Some reference that discuss the estimation of ARMA models using state-space approach are "Time series analysis with applications in R" by Jonathan Cryer and Kung-Sik Chan, "Forecasting with Univariate Box-Jenkins Models: Concepts and Cases" by George E. P. Box and Gwilym M. Jenkins.


*Robustness: The state space representation is more robust to model misspecification than the "naive" likelihood. When the model is written in the state space form, the parameters of the model can be estimated independently of the initial conditions, which makes the estimates more robust to model misspecification. Initial conditions can be dropped from the likelihood by using conditional likelihood, however the state space representation is more general, it handles both cases (conditional or unconditional) and can be used for estimating models with both time-invariant and time-varying parameters. Model misspecification in this context refers to when the assumptions of the model do not match the true underlying process. For example, if the true process is non-stationary but the model assumes stationarity, or if the true process has different order of the ARMA model assumed.


*Estimation of missing values: In state space representation, it's possible to incorporate missing data in the model and use methods like "Kalman filter" or "Kalman smoother" to estimate missing values.
It's worth noting that the state-space representation of ARMA process is equivalent to the direct likelihood for Gaussian noise and stationary process, However, for non-stationary process and non-gaussian noise, the state-space representation is more useful. Also, the ARMA model can be represented as a special case of the state-space model, so the estimation of ARMA model using the state space representation doesn't change the interpretation or the estimation of the parameters.
More references:

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*"Kalman Filtering: Theory and Practice Using MATLAB" by Mohinder S. Grewal and Angus P. Andrews.

*"Modern Statistical Methods for Time Series Analysis" by Friedrich Leisch

