Fitting t distribution to data

A Student t distribution is generally a much better fit to daily, weekly, and monthly returns than a normal distribution.

You can create one by using the fit.st() function in the QRM package. The resulting fitted model has a parameter estimates component par.ests which can be assigned to a list tpars in order to store its values of nu, mu, and sigma for later use:

> tfit <- fit.st(ftse)
> tpars <- tfit$par.ests
> tpars
          nu           mu        sigma
2.949514e+00 4.429863e-05 1.216422e-02

In this exercise, you will fit a Student t distribution to the daily log-returns of the Dow Jones index from 2008-2011 contained in djx. Then, you will plot a histogram of the data and superimpose a red line to the plot showing the fitted t density. The djx data and QRM package have been loaded for you.

This exercise is part of the course

Quantitative Risk Management in R

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Exercise instructions

Use fit.st() to fit a Student t distribution to the data in djx and assign the results to tfit.
Assign the par.ests component of the fitted model to tpars and the elements of tpars to nu, mu, and sigma, respectively.
Fill in hist() to plot a histogram of djx.
Fill in dt() to compute the fitted t density at the values djx and assign to yvals. Refer to the video for this equation.
Fill in lines() to add a red line to the histogram of djx showing the fitted t density.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Fit a Student t distribution to djx
tfit <- ___(___)

# Define tpars, nu, mu, and sigma
tpars <- ___
nu <- ___
mu <- ___
sigma <- ___

# Plot a histogram of djx
hist(___, nclass = 20, probability = TRUE, ylim = range(0, 40))

# Compute the fitted t density at the values djx
yvals <- dt((___ - ___)/___, df = ___)/___

# Superimpose a red line to show the fitted t density
lines(___, yvals, col = "red")

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