Financial Times Series. Lecture 8

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1 Financial Times Series Lecture 8

2 Nobel Prize Robert Engle got the Nobel Prize in Economics in 2003 for the ARCH model which he introduced in 1982 It turns out that in many applications there will be many parameters to estimate when trying to fit an ARCH Maybe there is a more convenient way capture past observations?

3 GARCH, Bollerslev 1996 In the famous GARCH(1,1) model the evolution of the volatility, σ t is governed by σ t 2 = ω + αr t βσ t 1 2 May be considered as an ARCH( ) We note that old volatilities and old squared returns are captured in the beta term

4 GARCH It is possible to fit a GARCH(p,q) but it turns out that in many applications a GARCH(1,1) is sufficient As for the ARCH, we assume that r t = σ t z t where is WN, typically N(0,1) or t with degrees of freedom between 3 and 6

5 GARCH properties The GARCH(1,1) is (weakly) stationary with Cov r s, r t = 0 for s t iff α + β < 1 (proof in Bollerslev 1986) The 2m-th unconditional moments of r t exist iff m m j=0 j a j α j β m j < 1 where a 0 = 1, a j = j i=1 2j 1, j = 1,

6 Given existence The unconditional mean of r t is zero (same proof as for ARCH) The unconditional variance of r t is (same proof as for ARCH) ω 1 α β The unconditional kurtosis is 3 1 α + β 2 1 β 2 2αβ 3α 2 > 3

7 Example, using garchfit in matlab Returns from N225 (note Tsunami/Fukushima extreme event )

8 Example, using garchfit in matlab Is it correct to fit a model to this data and to use it to predict values now? To what extent does the extreme event affect the parameter estimates? We will assume N 0,1 noise and use garchfit in matlab which is an ML based method with loglilkelihood functions as in the notes for lecture 7

9 Example, using garchfit in matlab If we use the whole data set as is, we get σ t 2 = r t σ t 1 2 If we instead use data from obs 400 (which is after the extreme event) and forward, we get σ t 2 = r t σ t 1 2

10 Example, using garchfit in matlab So, we see that the extreme event from more than two years ago greatly affects the parameter estimates We may use the output from garchfit for further analysis

11 Volatility fits with/without the extreme event With, jumpy, nervous Without, more calm

12 Devolatized returns If the model fit is ok, we want devolatized returns zt = r t /σ t to act like white noise For the series N225 with the extreme event, we get

13 Devolatized returns We see that the extreme event is still extreme For the series without the extreme event, we get

14 Devolatized returns We may use Ljung-Box, lbqtest in matlab Gives p-value , which is satisfactory

15 Extensions of the GARCH There are numerous extensions of the GARCH Among the more well known are EGARCH (Nelson 1991) and GJR (Glosten et al 1993) Both models allow for volatility skewness or leverage effects and are available in matlab econometrics

16 EGARCH E means exponential and the model for the conditional variance may be written lnσ 2 t = ω + α r t 1 + γr t βlnσ σ t 1 t 1 The parameter γ accounts for skewness We fit the model to the N225 data without the extreme event

17 EGARCH vs. GARCH

18 Devolatization with EGARCH p-value of Ljung-Box is

19 GJR The Glosten-Jagannathan-Runkle GARCH may be written as σ t 2 = ω + α + φi t 1 r t βσ t 1 2 where I t 1 = 0 if r t 1 0 and I t 1 = 1 if r t 1 < 0, so that the parameter φ accounts for skewness

20 GJR philosophy Bad news gives higher volatility than good news

21 GJR fit (N225 without extreme event)

22 Devolatization with GJR p-value for Ljung-Box is

23 Comparison For the N225 without the tsunami there does not seem to be an improvement, at least not in devolatizing properties, using the more advanced models On the other hand, we have not yet used a statistical test procedure to compare the models Below we try the three models for NASDAQ data and look at a statistical test for comparing the models

24 NASDAQ returns

25 Volatility fits

26 Devolatization of NASDAQ Ljung-Box p-value for GARCH is Ljung-Box p-value for EGARCH is Ljung-Box p-value for GJR is

27 Evaluating predictions We may us squared returns as a proxy and compute MSE:s as 1 T T t=1 r t 2 σ t 2 2 For the NASDAQ data, we get , and for the GARCH, EGARCH and GJR respectively

28 Evaluating predictions Another way of evaluating predictions, again with squared returns as proxy, is to regress squared returns on squared volatility predictions and hope for a slope close to one and R 2 1 For the GARCH, EGARCH and GJR we have slopes , and and R-squares , and which is not so satistisfactory, however it can be shown theoretically that for a GARCH(1,1) that R-squares close to one are highly unlikely

29 Squared returns is a noisy proxy What if we instead use realized variance over 30 days and compare to 30 day squared volatility forecasts? The 30 day realized variance for is given by t+29 i=t r i 2

30 Squared returns is a noisy proxy Our 30 day volatility predictions will just be the sums of the daily volatility estimates of the past 30 days Using the 30 day framework, we get, for the GARCH, EGARCH and GJR slopes 2.37(!), and 1.06 and R-squares , and which is more satisfactory, but the slope for the GARCH is not reasonable

31 Diebold-Mariano If we choose a loss function and a proxy, there is a test proposed by Diebold and Mariano (1995) for evaluating if one prediction method is significantly better than another The null hypothesis is that both methods have the same accuracy

32 Diebold-Mariano Define d t = L ε At L ε Bt where L denotes the loss function ε At and ε Bt denote the prediction errors from method A and B, respectively The test statistic is d LRV/T ~N(0,1) where LRV = Var d t + 2 Cov d t, d t j j=1

33 Diebold-Mariano Note that you have to keep track of which error is to the left and to the right of the minus sign in order to tell which method is better A DM test using L x = x 2, i.e. squared loss, is available at matlab central It also accounts for the length of the forecast horizon

34 Diebold-Mariano For our three models of 30 day NASDAQ volatility, the observed values of test statistic are for GJR vs. EGARCH, for GJR vs. GARCH and for EGARCH vs. GARCH. So, p-values are , and At 0.05 signicance level, no model is significantly better than the other, but of course this decision depends on the choice of loss function

35 If high-frequency data is available We might be able to evaluate daily volatility predictions/estimates against realized variances based on intra-day data It is not easy to get hold of high frequency data but when you start working in a bank you should try using it If you really can t wait, Mattias might let you in at the finance lab at Handels

Financial Times Series. Lecture 6

Financial Times Series. Lecture 6 Financial Times Series Lecture 6 Extensions of the GARCH There are numerous extensions of the GARCH Among the more well known are EGARCH (Nelson 1991) and GJR (Glosten et al 1993) Both models allow for