# Time Varying Coefficients vs Rolling Estimation

What are the practical differences for forecasting from fitting a model with time varying coefficients vs. estimating a model with fixed parameters over rolling windows? Intuitively it seems that these approaches both attempt to capture time variation in parameters, but the former model does this explicitly whereas the latter model handles this implicitly. I am trying to understand the tradeoffs with the two approaches, in particular

• Besides various cross validation methods, are there other approaches to determine which model would be suitable for a given use case? Things like AIC and BIC come to mind but in the rolling case, since we are actually refitting many models I'm not sure how these would be applicable?
• Are there certain situations where one class of model is generally preferred? e.g. Is quantifying things like forecast uncertainty more tractable via time varying parameter models?

Within the context of statespace models, the models I am referencing have the following form. Quoting from Time Series Analysis by State Space Methods by J. Durbin and S.J. Koopman Section 3.6,

$$y_t = X_t \beta + \epsilon_t \hspace{10mm} \epsilon_t \sim N(0, > H_t) \hspace{10mm} (3.30)$$ for $$t = 1,...,n$$ where $$X_t$$ is the $$1 \times k$$ regressor vector with exogenous variables, $$\beta$$ is the $$k \times 1$$ vector of regression coefficients and $$H_t$$ is the known variance that possibly varies with $$t$$. This model can be represented in the state space form (3.1) with $$Z_t = X_t, T_t = I_k$$ and $$R_t = Q_t = 0$$, so that $$\alpha_t = \alpha_1 = \beta$$
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Suppose that in the linear regression model (3.30) we wish the coefficient vector $$\beta$$ to vary over time. A suitable model for this is to replace $$\beta$$ in (3.30) by $$\alpha_t$$ and to permit each coefficient $$\alpha_{it}$$ to vary according to a random walk $$\alpha_{i,t+1} = \alpha_{it} + \eta_{it}$$. This gives a state equation for the vector $$\alpha_t$$ in the form $$\alpha_{t+1} = \alpha_t + \eta_t$$.

The referenced form in (3.1) is given by

\begin{align} y_t &= Z_t \alpha_t + \epsilon_t, \hspace{10mm} \epsilon_t \sim N(0, H_t), \\ \alpha_{t+1} &= T_t \alpha_t + R_t\eta_t, \hspace{5mm} \eta_t \sim N(0, Q_t), \hspace{8mm} t=1,...n, \end{align}