Part B: (30%)
The purpose of this part is to evaluate the forecast performances of GARCH, CARR and CARR with range replaced by two alternative volatility estimators. They are called GK1 and GK2 as they are originally proposed by Garman and Klass (1980). (You don’t need to read this paper, just to let you know what GK means.)
I will follow the notations used in Ruey Tsay’s lecture note 6 (page 5-7) to define the two volatility estimators. All (weekly) price data used are in natural logarithm and as in my original paper I multiply each of the volatility estimators by 100 for percentage expression. The first estimator is called GK1t and it is defined as the following
GK1t = 100* where sigma1t = (Ot – Ct-1)2 + (Ct – Ot)2.
The second estimator is called GK2t and it is defined as
GK2t = 100* where sigma2t = 0.5(Ht – Lt)2 – 0.386 (Ct – Ot)2.
Note that in the CARR model, the range is computed by RANGEt = 100* (Ht – Lt). We then define the Conditional Autoregressive RANGE (CARR) model as
RANGEt = tt
t = RANGEt-1t.
We will also use the same formula to estimate and forecast volatility using GK1 and GK2. Specifically, we define the Conditional Autoregressive GK1 (CAGK1) model as
GK1t = tt
t = GKt-1t.
Similarly we define the Conditional Autoregressive GK2 (CAGK2) model as
GK2t = tt
t = GKt-1t.
Hence CAGK1 and CAGK2 can be estimated using the same method as in CARR. We will only consider exponential distribution as the conditional density function of the errors.
For empirical analyses in this part, I will ask you to use the same S&P500 price index data but with different sample periods, each covering 19 years. Use your school number () to determine the sample periods for your analyses:
January 1st, 1990 to December 31, 2008 if the last digit of your school number is 0;
January 1st, 1991 to December 31, 2009 if the last digit of your school number is 1;
January 1st, 1992 to December 31, 2010 if the last digit of your school number is 2;
January 1st, 1993 to December 31, 2011 if the last digit of your school number is 3;
January 1st, 1994 to December 31, 2012 if the last digit of your school number is 4;
January 1st, 1995 to December 31, 2013 if the last digit of your school number is 5;
January 1st, 1996 to December 31, 2014 if the last digit of your school number is 6;
January 1st, 1997 to December 31, 2015 if the last digit of your school number is 7;
January 1st, 1998 to December 31, 2016 if the last digit of your school number is 8;
January 1st, 1999 to December 31, 2017 if the last digit of your school number is 9;
Follow Part A to properly specify the in-sample and out-of-sample periods such that 100 out-of-sample forecasts are made.
The results should include (very similar to Part A unless specified):
Tables
Table B1: Add to the table summary statistics for GK1 and GK2.
Table B2a: Same as in original paper the estimation for CARR model.
Table B2b: Same as B2a but for CAGK1 model.
Table B2c: Same as B2a but for CAGK2 model.
Table B3: Out-of-sample forecast Comparison for CARR, GARCH, CAGK1 and CAGK2 using SSDR as the “measured” volatility (ignore WRSQ WRNG AWRET)
Table B4: Out-of-sample predictive power for CARR, GARCH, CAGK1 and CAGK2 using SSDR as the “measured” volatility (ignore WRSQ WRNG AWRET)
Table B5: Encompassing tests for the four model forecasts
(Again, ignoring WRSQ WRNG AWRET regress SSDR on the forecast volatilities of each of the four models and report the parameter estimates.)
SSDR = a + b1 FV(GARCH) + b2 FV(CARR) + b3 FV(CAGK1) + b4 FV(CAGK2)
Figures
Figure 1: Add plots for weekly GK1 and weekly GK2 in additional to the weekly return and weekly range.
Figure 2a: volatility forecasts SSDR and GARCH (take 104 observations for 2007-2008)
Figure 2b: volatility forecasts SSDR and CARR (same as 2a)
Figure 2c: volatility forecasts SSDR and CAGK1 (same as 2a)
Figure 2d: volatility forecasts SSDR and CAGK2 (same as 2a)