Likelihood Construction for Independent CIR Factors

Summary

The document asks how to construct a maximum likelihood objective when modeling a process with multiple independent Cox–Ingersoll–Ross factors. It contrasts adding the factors’ conditional densities at each observation and taking the log with summing the log likelihoods of the individual factors across observations. The distinction is whether the observed variable is modeled as a sum of factor processes or whether each factor’s observations are modeled separately; those are different statistical models and require likelihoods that match the observed data and its distribution.

The post does not provide a resolution, derivation, or empirical evidence, so it leaves the central question open. In particular, adding densities is not generally equivalent to multiplying independent likelihoods: a sum of independent random variables has a convolution density. The correct objective depends on what is observed and how the factors enter that observation.

Key ideas

• The likelihood must correspond to the distribution of the observed data.
• Independent factor likelihoods multiply when the individual factors are observed jointly.
• The density of a sum of independent factors is generally obtained by convolution.
• The post poses alternative objectives but does not resolve which model applies.

Tags

# MLE for two independent factor CIR


# MLE for two independent factor CIR












Following the maximun likelihood estimation as done in Klavidko I would like to generalize this to more independent factors . In first istance I would use the transition function at time t as a sum of the non chi squared conditional distributions for each factor: \begin{equation} p_t = p_{1 t} +.. + p_{N t} \end{equation} then take the log of the product of all t times used in the data and optimize: \begin{equation} L = \sum_{t}log(p_t) \end{equation} On the other hand in the paper multi they just optimize the sum of each log product \begin{equation} L = \sum_{t,i}log(p_{i t}) \end{equation} Which one is correct? Thanks

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