Assignment Problem: This exercise will help you to understand how to implement different models used to forecast volatility. It will also help you to assess the performance of each model. Follow these steps to set up the data needed for this exercise.
Section I:
Questions:
Now starting as early as possible, compatibly with the specific model used, forecast the variance for the next day according to:
a) The moving average model based on the previous month (i.e., 21 days)
b) The moving average model based on the previous year (i.e., 252 days)
c) The RiskMetrics model
d) The GARCH model and
e) The HAR model.
Hint: For the GARCH and HAR models,you have to estimate the model's parameters. For the sake of easy implementation, we will consider in-sample estimates. That is, you should estimate the parameters of the GARCH and HAR models only once using the complete dataset (in real life, one should re-estimate the models every day as new observations become available).
Section II Models and Forecasting:
Now think about which of the four models does a better job of forecasting future variance. Let's run this regression equation:
σ_(t+1)^2=a+b Pred.Varianc?e(i)?_(t→t+1)+ε_t
where Pred.Varianc?e(i)?_(t→t+1)is the predicted variance according to model i (i=MA, RM, GARCH, HAR) for day t+1 computed using data available up to day t.
The problem here is that we do not observe σ_(t+1)^2. So, what we can do is to replace σ_(t+1)^2 with the best proxy that we know of, which is RV_t. Thus, we can effectively run the following regression:
RV_(t+1)=a+b Pred.Varianc?e(i)?_(t→t+1)+ε_t
Now answer these questions:
Q1: What should the value of the parameters a and b be equal to, for a model to be good? Should we be considering anything else besides the value of a and b?
Q2: Which model do you think is better? Why? Comment on the results. [5 points]
Section III Realized Volatility
In this exercise, you will learn to compute daily realized variance and realized volatility measures. First, Open the file "High-Frequency_Data_Emini.csv." It contains 1-minute observations on the price of the E-mini (the E-minis the futures written on the S&P 500. It is among the most liquid instruments in the U.S. financial markets) from May 2, 2010, to May 10, 2010. There are three columns: Date, Time, Price.
Q1: On each day in the sample, compute the daily realized variance (RV) using 1-minutes returns
Q2: Transform the daily realized variance into annualized realized volatility (realized volatility is simply the "=sqrt()" of the realized variance).
Q3: Plot the time-series of realized volatility.
Q4: What do you notice? What is the average value of the realized volatility? Did something unusual happen? Comment on the results.
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