# Time-Varying Volatility and Conditional Likelihood

Engle's comment in his seminal paper "Risk and Volatility: Econometric models and Financial Practice" mentions that

I had recently worked extensively with the Kalman Filter and knew that a likelihood function could be decomposed into the sum of its predictive or conditional densities.

What is the significance of this statement? Could someone elaborate what this means and how this can be used?

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Arch and Garch parameters are typically estimated via maximum likelihood. In MLE, parameters are estimated by $$\theta \equiv argmax\left\{ \sum_{t=1}^{T}ln\left(f\left(x_{t}|\theta\right)\right)\right\}$$ where $\theta$ are some parameters and $f(x)$ is the probability density function. Note that this applies the same $\theta$ to each of the different $x$'s. A Kalman filter is often used for modelling time-varying coefficients. In that case, the distribution is different in every period. So it's like adjusting the above to $$\theta \equiv argmax\left\{ \sum_{t=1}^{T}ln\left(f\left(x_{t}|\mu_{t}, \sigma \right)\right)\right\}$$ where the $\mu_{t}$ represents the mean of a distribution as it changes in time based on the Kalman filter. The logic can be extended to the Arch/Garch case. However, instead of focusing on a time-varying mean, they are focusing on a time-varying variance. So it might instead look something like $$\theta \equiv argmax\left\{ \sum_{t=1}^{T}ln\left(f\left(x_{t}|\mu, \sigma_{t} \right)\right)\right\}$$ where $\sigma_{t}$ is conditional on the parameters $\theta$ that would be estimated as part of the Garch process.