Estimating Student-t Degrees of Freedom from Tail Behavior
Summary
The document asks how to estimate Student-t distribution parameters for Value-at-Risk analysis of assets whose returns are not normally distributed. One answer connects the Student-t degrees of freedom to the tail shape exponent and suggests estimating that exponent with the Hill method, which uses observations from the distribution's extremes. It emphasizes that the estimator depends on selecting a suitable tail cutoff: observations should be far enough into the tail, and double-log plots can help identify a region consistent with Pareto-like behavior.
Other replies point to fitting a Student-t distribution by maximum likelihood in statistical software and describe a modified Hill approach that compares the sample estimate with the estimate from a normal distribution at the same cutoff. These are alternative suggestions, not a head-to-head evaluation. The source gives no dataset or performance evidence, and warns that Hill-based methods are criticized; estimates can be sensitive to tail selection and the assumed tail model.
Key ideas
- For Student-t distributions, the tail exponent is linked to the degrees of freedom parameter.
- The Hill method estimates tail shape using observations selected from the distribution's extremes.
- Double-log plots can help assess where observations begin to show Pareto-like tail behavior.
- Likelihood-based distribution fitting and modified Hill estimators are presented as other approaches.
- Hill estimates can be sensitive to the cutoff and are subject to methodological criticism.
Tags
Full text
# Portfolio Optimization using Parametric Portfolio Policy Method in MATLAB # Portfolio Optimization using Parametric Portfolio Policy Method in MATLAB I want to contruct an optimized stock portfolio with the restriction of a zero-investment strategy. The portfolio weight in each stock needs to be modeled as a function of state variables (factors that have an effect on stock performance). This method for portfolio optimization is colled "parametric portfolio policy" and developed by Brandt et al (2009),so I need to find the weight for each stock in the optimized portfolio and maximize the investor’s expected utility, like following; Here is the expression of the weight of the stock i `w (i,t)`: `Teta`is a vector of coefficients of state variables and `x` presents the state variables. So, the optimization problem becomes; How to program the problem using Matlab? I am not good in modeling and optimization..I would be grateful for any help ## Answer by phdstudent (score 2) https://quant.stackexchange.com/a/24762 Find below a link for a dummy implementation in excel using VBA, which I did couple of years ago. http://www.speedyshare.com/xJZJ8/PPPs-Copy.xlsm On this implementation I am allocating between Dax returns and a German bond using as state variables the dividend yield, the inflation and the ECB interest rate. The implementation is out of sample, and simply uses the solver algorithm to maximize the utility. In matlab you can do exactly the same using the function `fminsearch` or something similar. However, if you are looking to replicate the paper, you should do the analysis in sample and use GMM to maximize equation (11) using equation (12) as the optimal weighting matrix. There are several resources online on how to implement GMM on matlab. Hope this helps.
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.