Conditional Likelihood for Time-Varying Volatility Models
Summary
The answer explains conditional likelihood as a way to estimate models in which the distribution of observations changes over time. In a standard maximum likelihood setup, the same parameter values govern each observation. With time-varying parameters, each observation instead has a density conditional on information available at that time, and estimation sums the log of those conditional densities.
The discussion connects this idea to the Kalman filter, which can model a changing mean or coefficient, and to ARCH or GARCH models, where conditional variance changes over time. The likelihood is then built from period-specific volatility estimates linked to model parameters. This is a conceptual sketch rather than a complete derivation: it does not specify a particular distribution, filtering procedure, or estimation algorithm, and its notation simplifies the role of conditioning information.
Key ideas
- Maximum likelihood can sum log densities that differ across time periods.
- Time-varying models condition each observation’s distribution on information available at that time.
- Kalman filters can represent changing means or coefficients.
- ARCH and GARCH models apply the conditional likelihood idea to changing variance.
Tags
Full text
# Time-Varying Volatility and Conditional Likelihood
# 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?
## Answer by John (score 4, accepted)
https://quant.stackexchange.com/a/10375
Basically he's just saying that you don't have to estimate parameters assuming they're the same in every period.
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.