Parameter Estimation from a Time Series Characteristic Function

Summary

The document poses a parameter estimation problem for a time series modeled by a process with a known-form characteristic function and unknown parameters. The author has considered recovering a probability density through Fourier inversion and then applying maximum likelihood, but says the characteristic function is too complicated for that route. They also consider fitting an empirical characteristic function by least squares, but are unsure how to form it when observations are ordered in time and may depend on one another.

This is a research question rather than a worked solution: it provides no estimator, assumptions about stationarity or dependence, or empirical results. Its useful point is the distinction between treating observations as independent draws and estimating from a stochastic process. Any practical method would need to account for temporal dependence and specify which distribution or time-indexed increment is being modeled; the document leaves those choices open.

Key ideas

• The target is a vector of unknown parameters in a process characteristic function.
• Fourier inversion followed by maximum likelihood is considered impractical for the stated model.
• An empirical characteristic function and least squares are proposed as an alternative direction.
• Temporal dependence makes it necessary to define the sampling unit and account for the process structure.
• The document asks the estimation question but provides no solution or assumptions.

Tags

# Time series (stochastic process) estimating parameters using characteristic function


# Time series (stochastic process) estimating parameters using characteristic function












I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector of unknown parameters. I need to esimate vector of unknown parameters $\theta$ of characteristic function.

I tried to find a PDF using the inverse Fourier transform to use the maximum likelihood method, but the characteristic function is too complicated for that. I also thought about building the empirical characteristic function using the time series of assets and to estimate parameters using the least square method, but I do not know how to build the empirical characteristic function, because time series is not just a sample of random variables, it is a random process that depends on time.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.