Volatility. Gerald P. Dwyer. February Clemson University

Transcription

1 Volatility Gerald P. Dwyer Clemson University February 2016

2 Outline 1 Volatility Characteristics of Time Series Heteroskedasticity Simpler Estimation Strategies Exponentially Weighted Moving Average Use of Shorter-term Variance Use of intraday path to estimate daily volatility ARCH Properties of ARCH(1) Generalizations of ARCH(1) GARCH Testing and Estimation IGARCH GARCH-M Asymmetric GARCH Variables Affecting Volatility EGARCH Stochastic volatility Summary

3 Economic time series: Characteristic features (Stylized facts) Trend in level (rgdp) A high level of persistence (dlrgdp) Volatility not constant over time (rgdp) Series may have a random walk with drift component (rgdp) Series share co-movements (m2sl and ambsl; exchange rates)

4 Financial time series: Characteristic features (Stylized facts) Trend in level (CRSP and exchange rates) A high level of persistence (dlcrsp and exchange rates) Volatility not constant over time (dlcrsp and exchange rates) Series may have a random walk with drift component (dlcrsp and exchange rates) Series share co-movements (exchange rates) Skewed (crsp) High excess kurtosis (crsp)

5 CRSP daily index 1985 to 2014 vwcrspd_

6 CRSP daily returns 1985 to vwretd

7 Absolute value of CRSP daily returns 1985 to VWRETD_ABS

8 Square of CRSP daily returns 1985 to VWRETD_SQ

9 CRSP daily returns summary statistics 1985 to ,500 3,000 2,500 2,000 1,500 1, Series: VWRETD Sample 1/02/ /31/2014 Observations 7564 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability

10 Exchange rate Eurozone and U.S. 1/4/1999 to 1/29/2016 fxuseu

11 Change in log of exchange rate Eurozone and U.S. 1/4/1999 to 1/29/2016 DLFXUSEU

12 Absolute value of change in log of exchange rate Eurozone and U.S. 1/4/1999 to 1/29/ DLFXUSEU_ABS

13 Square of change in log of exchange rate Eurozone and U.S. 1/4/1999 to 1/29/ DLFXUSEU_SQ

14 Change in log of exchange rate summary statistics Eurozone and U.S. 1/4/1999 to 1/29/ Series: DLFXUSEU Sample 1/04/1999 1/29/2016 Observations 4129 Mean 1.08e-05 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability

15 Heteroskedasticity over time These graphs suggest heteroskedasticity over time Time-varying volatility of returns Of interest in itself to characterize returns Matters for prices of options and some other financial instruments Risk of holding assets VaR Volatility clustering These graphs are suggestive but don t tell us too much Using individual observations on squared changes and absolute value to estimate variance and standard deviation as it changes Similar to using each individual observation to estimate mean as it changes Can t forecast anything going forward

16 Exponentially weighted moving average of variance Exponentially weighted average assumes today s variance forecast for tomorrow is a weighted average of variance today and variance forecasted for today σ 2 t+1 = (1 λ) (r t r) 2 + λ σ 2 t And forecasted variance for today is a weighted average of variance the day a period ago and variance forecasted a period ago, and so on σ 2 t = (1 λ) (r t 1 r) 2 + λ σ 2 t 1 λ = 0.94 daily frequency has been suggested

17 Use shorter-term returns to estimate variance over longer periods I Use daily variance in the month to calculate variance for the month, called realized volatility Let rt m be the monthly log return in month t Let r t,i be the daily log return on day i in month t Suppose that daily returns are serially uncorrelated and the daily variance is constant Then and r m t = n r t,i i=1 Var [r m t ] = n Var [r t,i ] Var [r t,i ] = n i=1 (r t,i r t ) 2 n 1 where r t is the mean of the daily returns

18 Use shorter-term returns to estimate variance over longer periods II The estimated monthly variance thus is ( σ m t ) 2 = n n i=1 (r t,i r t ) 2 n 1

19 Daily variance to estimate monthly variance I The estimated monthly variance is simple to calculate ( σ m t ) 2 = n n i=1 (r t,i r t ) 2 n 1 This becomes more complicated if the daily returns are serially correlated, but it s still manageable If daily log returns have high excess kurtosis and serial correlations, then this estimator may not be consistent

20 Garman-Klass estimator of daily variance Use high, low, opening, and closing prices to estimate variance Can estimate daily variance just knowing opening, high, low and closing prices Assume that price follows a random walk Let c t be the logarithm of the closing price so r t = c t c t 1 Conventional estimator is based on closing price σ t 2 = E [(c t c t 1 ) 2] Using only closing price High H t, low L t, and open O t also often are available Lower case indicates logarithms Can estimate daily variance of price (not log price) from σ 2 GK = 0.12 (o t c t 1 ) 2 f (h t l t ) (c t o t ) 2 1 f where f is the fraction of the day that the market is closed Minimum variance unbiased estimator for a random walk with no drift

21 Yang and Zhang estimator Use high, low, opening, and closing prices to estimate variance of log prices over a longer period Define o t = ln O t ln O t 1, h t = ln H t ln O t 1 l t = ln L t ln O t 1, c t = ln C t ln O t 1 Monthly variance based on n days of trading is σ 2 YZ = σ o + k σ c + (1 k) σ rs where σ o and σ c are is the estimated variances of o t and c t and σ 2 rs = 1 n [h t (h t c t ) + l t (l t c t )] k = (n + 1) / (n 1) and k was chosen to minimize the variance of the estimator σ 2 YZ

22 Annualization Volatilities sometimes annualized Multiply variance by T (T is number of trading days per year) Daily returns often not annualized Will want to annualize returns when comparing them to annual volatility Why not annualize in general? Magnitudes would be ridiculous A 1 percent return in one day is a 252 percentage point log return per year A 2 percent return in one day is a 504 percentage point log return per year

23 Serial correlation Change in logarithm of value-weighted CRSP index measures continuously compounded return Serial correlation of squared changes in logarithm of value-weighted CRSP index Serial correlation of absolute values of change in logarithm of value-weighted CRSP index

24 Simple way to deal with serial correlation Suppose the mean equation is y t = α 0 + α 1 y t 1 + ε t Might think just to estimate a qth-order autoregression for the squared residuals ε 2 t = b 0 + b 1 ε 2 t b 1 ε 2 t 1 q + v t Not as tractable as another setup y t = α 0 + α 1 y t 1 + ε t E ε t = 0, E ε 2 t = σ 2 t, E ε t ε s = 0 t = s σ 2 t = γ 0 + γ 1 ε 2 t 1

25 Autoregressive conditional heteroskedasticity (ARCH) Simple ARCH model y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 where y t is the variable being examined, σt 2 is the variance of ε t conditional on past values of the squared innovations, ε 2 t 1 y t = α 0 + α 1 y t 1 + ε t is the mean equation for y t σt 2 = γ 0 + γ 1 ε 2 t 1 is the variance equation for y t ε t is the innovation in y t

26 Autoregressive conditional heteroskedasticity (ARCH) with AR(1) Further explore AR(1) with ARCH(1) and its properties Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1

27 Autoregressive conditional heteroskedasticity (ARCH) with AR(1) Further explore AR(1) with ARCH(1) and its properties Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 Differences from textbook s use of letters Lag coefficients in mean equation are α s (not a) γ s are the coefficients in the variance equation (not α)

28 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 Changing variance has no effect on the predictability of the series y t Both unconditional expected value and conditional expected value (i.e. forecast at t 1 of y t ) Changing variance with deterministic equation σ 2 t = γ 0 + γ 1 ε 2 t 1 implies there is predictability in the variance given recent values In sum, predictable variance but the predictable variance does not help to predict the level of the series Will show this by looking at conditional expectations

29 Conditional Expectation Define conditional expectation at t 1 as E t 1 x t E [x t F t 1 ] where F t 1 is the set of all information available up through period t 1, which includes for example y t 1, ε t 1, σ t 1, and ν t 1

30 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Want variance σt 2 positive Sufficient to guarantee this by imposing γ 0 > 0 and γ 1 > 0

31 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 Expected value of innovations ε t E ε t = E σ t ν t = E = E = E = 0 [ ] (γ0 + γ 1 ε 2 ) 1 2 t 1 ν t [ ] (γ0 + γ 1 ε 2 ) 1 2 t 1 E ν t [ ] (γ0 + γ 1 ε 2 ) 1 2 t 1 0

32 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Conditional expectation of the innovations ε t ( E t 1 ε t = E t 1 σ t ν t = E t 1 γ0 + γ 1 ε t 1) ν t [ ] (γ0 = E t 1 + γ 1 ε 2 ) 1 2 t 1 E t 1 ν t [ ] (γ0 = E t 1 + γ 1 ε 2 ) 1 2 t 1 0 = 0

33 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Unconditional and conditional expected values E ε t = 0 E t 1 ε t = 0 Note that E t 1 ε t = 0 implies E t 1 y t = E t 1 [α 0 + α 1 y t 1 + h t ν t ] = α 0 + α 1 E t 1 y t 1 + E t 1 ε t = α 0 + α 1 y t 1

34 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Variance of the innovations ε t E ε 2 t = E σt 2 νt 2 = E ( γ 0 + γ 1 ε 2 t 1) ν 2 t = E ( γ 0 + γ 1 ε 2 t 1) E ν 2 t = ( γ 0 + γ 1 E ε 2 t 1) 1 = ( γ 0 + γ 1 E ε 2 ) t and therefore E ε 2 t = γ 0 1 γ 1 E ε 2 t > 0 if γ 0 > 0 and 0 γ 1 < 1

35 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Conditional variance of the innovations ε t E t 1 ε 2 t = E t 1 σt 2 νt 2 ( = E t 1 γ0 + γ 1 ε 2 t 1) ν 2 t ( = E t 1 γ0 + γ 1 ε 2 t 1) Et 1 νt 2 = ( γ 0 + γ 1 E t 1 ε 2 t 1) 1 = γ 0 + γ 1 ε 2 t 1 and therefore E t 1 ε 2 t = γ 0 + γ 1 ε 2 t 1 E t 1 ε 2 t > 0 and stable difference equation if γ 0 > 0 and γ 1 < 1 But want γ 1 > 0 for positive variance and therefore 0 γ 1 < 1

36 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 Unconditional and conditional expected values E ε t = 0 E t 1 ε t = 0 Variance in ARCH model with γ 0 > 0 and 0 γ 1 < 1 E ε 2 t = γ 0 1 γ 1 E t 1 ε 2 t = γ 0 + γ 1 ε 2 t 1

37 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Third moment If ν t is symmetric, then σ t ν t is symmetric Fourth moment Assume ν t is normally distributed Do we get fatter tails in ε t than from the normal distribution?

38 Kurtosis Kurtosis is the fourth moment of a distribution divided by the variance squared K = µ 4 (σ 2 ) 2 where µ 4 is the fourth moment about the mean Could also define conditional kurtosis Kurtosis for a normal distribution is three Excess kurtosis is kurtosis minus three K e = µ 4 (σ 2 ) 2 3

39 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 If ν t is normally distributed, the unconditional kurtosis ε t with normally distributed ν t is K = E ε4 t Var [ε t ] 2 = 3 1 γ γ 2 1 E ε 4 t > 0 and Var [ε t ] 2 > 0 and therefore γ 1 must satisfy ( 1 3γ 2 1 ) > 0 and therefore 0 γ 2 1 < 1 3 Therefore, E ε 4 t Var [ε t ] 2 > 3 Fatter tails than for a normal distribution with 0 γ 2 1 < 1 3

40 Properties of ARCH(1) Representation is y t = α 0 + α 1 y t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Third moment If ν t is symmetric, then σ t ν t is symmetric Fourth moment If ν t is normally distributed, then ε t has excess kurtosis

41 Generalizations of ARCH(1) Generalizations ARMA model for series would be y t = α 0 + α (L) y t 1 + β (L) ε t 1 + ε t ε t = σ t ν t ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 Fairly easy to see this point so we ll just stick to a simple AR(1)

42 Generalizations of ARCH(1) Generalizations more lags in ARCH y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 + γ 2 ε 2 t γ p ε 2 t q Sufficient condition to be sure σ 2 t is positive is that γ i 0, i = 0,..., q

43 GARCH ARCH models can require many lags Reduce lags in mean equations by using ARMA models In level equation, MA terms can substitute for several AR terms Including something like AR terms in ARCH equation can reduce number of lags

44 GARCH GARCH (Generalized ARCH) model ARCH Model σ 2 t = γ 0 + γ 1 ε 2 t γ qε 2 t q Instead, try GARCH, here a GARCH(p,q) (order of lags often not consistent across authors) σ 2 t = γ 0 + γ 1 ε 2 t γ qε 2 t q + δ 1 σ 2 t δ k σ 2 t p Lag lengths are q for the part analogous to the moving average and p for the part analogous to an autoregression

45 GARCH GARCH for AR(1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t γ q ε 2 t q + δ 1 σ 2 t δ k σ 2 t p Sufficient conditions to be sure σ 2 t is positive are γ i 0 and δ i 0 with max(p,q) i=1 (γ i + δ i ) < 1 Just as with an ARMA model, this can reflect more complicated dynamics with fewer parameters than only adding more lagged squared innovations

46 Properties of GARCH models σ 2 t = γ 0 + γ 1 ε 2 t γ q ε 2 t q + δ 1 σ 2 t δ p σ 2 t p Restrictions on parameters max(q,p) γ i > 0, δ i > 0, i=1 (γ i + δ i ) < 1

47 Properties of GARCH models σ 2 t = γ 0 + γ 1 ε 2 t γ q ε 2 t q + δ 1 σ 2 t δ p σ 2 t p Restrictions on parameters max(q,p) γ i > 0, δ i > 0, i=1 (γ i + δ i ) < 1 Variance E [ ε 2 t ] = γ 0 1 max(q,p) i=1 (γ i + δ i )

48 Properties of GARCH models σ 2 t = γ 0 + γ 1 ε 2 t γ q ε 2 t q + δ 1 σ 2 t δ p σ 2 t p Restrictions on parameters max(q,p) γ i > 0, δ i > 0, i=1 (γ i + δ i ) < 1

49 Properties of GARCH models σ 2 t = γ 0 + γ 1 ε 2 t γ q ε 2 t q + δ 1 σ 2 t δ p σ 2 t p Restrictions on parameters Kurtosis max(q,p) γ i > 0, δ i > 0, i=1 (γ i + δ i ) < 1

50 Properties of GARCH models σ 2 t = γ 0 + γ 1 ε 2 t γ q ε 2 t q + δ 1 σ 2 t δ p σ 2 t p Restrictions on parameters Kurtosis max(q,p) γ i > 0, δ i > 0, i=1 (γ i + δ i ) < 1 For GARCH(1,1) with 1 (γ 1 + δ 1 ) 2 2γ1 2 > 0, then E [ ε 4 ] [1 t (E [ε 2 t]) 2 = 3 (γ 1 + δ 1 ) 2] 1 (γ 1 + δ 1 ) 2 > 3 2γ1 2

51 Limitations of GARCH models 1 Symmetric effects of shocks. This is too restrictive for stock returns, where negative shocks have a larger effect on future variance than positive shocks 2 Returns, for example, tend to have some clusters of high and low variance, whereas GARCH models tend to predict slow decay to mean variance from any current variance 3 Restrictive parametrizations, e.g. 0 γ1 2 < 1 3 for kurtosis to be well defined for ARCH(1) 4 Deterministic equation for variance; no error term in σ 2 t = γ 0 + γ 1 ε 2 t 1 5 Provides no evidence on source of changes in variance

52 Estimating a GARCH model A simple GARCH model y t = α 0 + α 1 y t 1 + ε t, ε t = h t ν t, ν t iid (0, 1) σt 2 = γ 0 + γ 1 ε 2 t 1 + δ 1 σt 1 2 Steps in estimating a GARCH model 1 Estimate a model for the mean equation 2 Use the residuals of the mean equation to test for ARCH effects 3 Specify a variance model with ARCH effects if it seems warranted 4 Check the fitted model and refine as suggested by diagnostic statistics

53 Estimating mean equation In general, there is no reason the mean equation can t be as complicated as we like y t = µ t + ε t y t can be a complicated ARMA(p,q) or can have variables included y t is stationary in mean y t may be first difference of original series for example y t = Y t Y t 1 Original estimates may be mis-specified if ignore conditional heteroskedasticity of ε t

54 Three tests for ARCH first McLeod-Li test Box-Ljung test applied to squared residuals, ε 2 t, for some pre-specified number of lags k T (T + 2) k l=1 ρ ( ε 2 t) 2 l T l a χ 2 k where ρ ( ε 2 t ) l is the lth serial correlation of the squared residuals and reject the null hypothesis that all of the first k autocorrelations are zero if p-value less than 0.05 for test statistic

55 Three tests for ARCH second and third Engle test based on a regression for the squared residuals ε 2 t = γ 0 + γ 1 ε 2 t 1 + γ 2 ε 2 t γ k ε 2 t k + e t (1) where e t is the error term in the regression for squared residuals Test whether γ 1 = γ 2 =... = γ k using T R 2 a χ 2 k where T is the number of observations and R 2 is the R 2 of equation (1) Can be very informative with just one lag F-test for regression (1) (SSR 0 SSR 1 ) /k SSR 1 / (T k 1) F k,t k 1 where T is the number of observations in the estimated equation (1), SSR 0 = ε 2 t and SSR 1 = e 2 t

56 Estimation of GARCH model Steps in estimating a GARCH model 1 Estimate a model for the mean equation DISCUSSED 2 Use the residuals of the mean equation to test for ARCH effects DISCUSSED 3 Specify a variance model with GARCH effects if it seems warranted 4 Check the fitted model and refine as suggested by diagnostic statistics

57 Estimation of GARCH model Steps in estimating a GARCH model 1 Estimate a model for the mean equation DISCUSSED 2 Use the residuals of the mean equation to test for ARCH effects DISCUSSED 3 Specify a variance model with GARCH effects if it seems warranted 4 Check the fitted model and refine as suggested by diagnostic statistics

58 Estimation by maximum likelihood Maximize likelihood function based on y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 + δ 1 σ 2 t 1 Nonlinear and involves a latent variable σt 2 which is not observed but is estimated for each observation Not as complicated as it could be because σt 2 is a deterministic function of ε 2 t 1 and past σ2 t 1 Still cannot be done by a linear algorithm Nonlinear maximization Use computed information matrix as basis of variance-covariance matrix and standard errors

59 Distributions for maximum likelihood Distributions Normal distribution t-distribution with a small number of degrees of freedom Generalized error distribution

60 Quasi-maximum likelihood Quasi-maximum likelihood estimation Consistent estimates of parameters under fairly general conditions Issue of correct standard errors of coefficients

61 Estimation by maximum likelihood with normal distribution Likelihood function if normally distributed y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t IIDN (0, 1) σ 2 t = γ 0 + γ 1 ε 2 t 1 + δ 1 σ 2 t 1 Maximize likelihood function with respect to parameters L (θ ν) = T t=1 ( ) 1 exp ε2 t 2πσt 2σt 2 This can be written ln L (θ ν) = T 2 T ln (2π) 1 t=1 2 T ( ) 2 ln σ t 1 t=1 2 Not remotely the same as minimizing T ε 2 T t or vt 2 t=1 t=1 T ε 2 t t=1 σt 2

62 Estimation by maximum likelihood with t distribution Likelihood function if distributed Student s t, which provides fatter tails, is L (θ ν) = T t=1 Γ ((df + 1) /2) Γ (df /2) (df 2) π ( ) (df +1)/2 1 + ν2 t df > 2 df 2 where df represents the degrees of freedom in the distribution and Γ (x) is the gamma function Γ (x) = 0 z x 1 e z dy Maximize likelihood function with respect to parameters L (θ ν) = T t=1 Γ ((df + 1) /2) Γ (df /2) (df 2) π ( ) (df +1)/2 1 + ν2 t df > 2 df 2 df can be estimated or can be specified in advance commonly 3 to 6

63 Practical issues with GARCH estimation Nonlinear maximization (or minimization of -ln L (θ y)) Can be something of an art form to be sure have reached maximum but generally not a problem with low-order GARCH and a well-specified mean equation Redundant parameters can create serious problems for convergence Generally speaking, it is hard to estimate more than a few lags Fairly common to use GARCH(1,1) for financial data

64 IGARCH What if the variance is very persistent? σ 2 t = α 0 + α 1 ε 2 t α q ε 2 t q + δ 1 σ 2 t δ p σ 2 t p with max(q,p) i=1 (γ i + δ i ) 1, suggesting something like a unit root in the variance process Actually pretty common with returns Change in logarithm of value-weighted CRSP index 1/2/1984 to 12/31/2013 dlnvwcrsp = â t σ t 2 = ât σ2 t 1 standard errors of coefficients are , and sum of coefficients is

65 Properties of IGARCH model IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1

66 Properties of IGARCH model IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Constant term is similar to constant term for a random walk with drift, which has a trend

67 Properties of IGARCH model IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Constant term is similar to constant term for a random walk with drift, which has a trend A nonzero constant term suggests a trend in variance for a random walk

68 Properties of IGARCH model IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Constant term is similar to constant term for a random walk with drift, which has a trend A nonzero constant term suggests a trend in variance for a random walk y t = α 0 + y t 1 + ε t

69 Properties of IGARCH model IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Constant term is similar to constant term for a random walk with drift, which has a trend A nonzero constant term suggests a trend in variance for a random walk y t = α 0 + y t 1 + ε t y t+1 = 2α 0 + y t 1 + ε t+1 + ε t

70 Properties of IGARCH model IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Constant term is similar to constant term for a random walk with drift, which has a trend A nonzero constant term suggests a trend in variance for a random walk y t = α 0 + y t 1 + ε t y t+1 = 2α 0 + y t 1 + ε t+1 + ε t y t+2 = 3α 0 + y t 1 + ε t+2 + ε t+1 + ε t

71 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t }

72 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t } Suppose that know σ 2 t and ε 2 t

73 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t } Suppose that know σ 2 t and ε 2 t E t σ 2 t+1 = γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t

74 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t } Suppose that know σ 2 t and ε 2 t E t σ 2 t+1 = γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t E t σ 2 t+2 = γ 0 + (1 δ 1 ) E t ε 2 t+1 + δ 1 E t σ 2 t+1

75 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t } Suppose that know σ 2 t and ε 2 t E t σ 2 t+1 = γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t E t σ 2 t+2 = γ 0 + (1 δ 1 ) E t ε 2 t+1 + δ 1 E t σ 2 t+1 Already know E t σ 2 t+1 and best forecast of ε2 t+1 is E t ε 2 t+1 = E t[σ 2 t+1 ν2 t+1 ] = E t σ 2 t+1, so

76 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t } Suppose that know σ 2 t and ε 2 t E t σ 2 t+1 = γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t E t σ 2 t+2 = γ 0 + (1 δ 1 ) E t ε 2 t+1 + δ 1 E t σ 2 t+1 Already know E t σ 2 t+1 and best forecast of ε2 t+1 is E t ε 2 t+1 = E t[σ 2 t+1 ν2 t+1 ] = E t σ 2 t+1, so E t σ 2 t+2 = γ 0 + E t σ 2 t+1

77 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t } Suppose that know σ 2 t and ε 2 t E t σ 2 t+1 = γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t E t σ 2 t+2 = γ 0 + (1 δ 1 ) E t ε 2 t+1 + δ 1 E t σ 2 t+1 Already know E t σ 2 t+1 and best forecast of ε2 t+1 is E t ε 2 t+1 = E t[σ 2 t+1 ν2 t+1 ] = E t σ 2 t+1, so E t σ 2 t+2 = γ 0 + E t σ 2 t+1 E t σ 2 t+2 = 2γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t

78 Trend in variance with IGARCH Does a nonzero constant term indicate a trend in variance? σ 2 t = γ 0 + (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1 Let E t σ 2 t+h be the h-step-ahead forecast at t of {σ2 t } Suppose that know σ 2 t and ε 2 t E t σ 2 t+1 = γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t E t σ 2 t+2 = γ 0 + (1 δ 1 ) E t ε 2 t+1 + δ 1 E t σ 2 t+1 Already know E t σ 2 t+1 and best forecast of ε2 t+1 is E t ε 2 t+1 = E t[σ 2 t+1 ν2 t+1 ] = E t σ 2 t+1, so E t σ 2 t+2 = γ 0 + E t σ 2 t+1 E t σ 2 t+2 = 2γ 0 + (1 δ 1 ) ε 2 t + δ 1 σ 2 t γ 0 = 0 will introduce a predictable trend in the variance

79 IGARCH with no constant term in variance equation IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = (1 δ 1 ) ε 2 t 1 + δ 1 σ 2 t 1, 0 < δ 1 < 1

80 IGARCH with no constant term in variance equation IGARCH(1,1) y t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σt 2 = (1 δ 1 ) ε 2 t 1 + δ 1 σt 1, 2 0 < δ 1 < 1 Unconditional variance E σt 2 is undefined but {y t } is strictly stationary

81 GARCH-M GARCH-M is GARCH in mean Simple GARCH-M is r t = α 0 + cσ 2 t + ε t, ε t = σ t ν t σ 2 t = γ 0 + γ 1 ε 2 t 1 + δ 1σ 2 t 1 r t is a return on an asset and c is called a risk premium parameter Could use σ t or ln σ t instead of σt 2 in mean equation Question: When will the variance of an asset s return reflect its risk?

82 Asymmetric GARCH Glosten, Jagannathan and Runkle (1993) Represent the asymmetry in returns that negative shocks associated with higher future variance Includes GARCH-M r t = α 0 + cσ 2 t + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + γ 1 ε 2 t 1 + κ 1 I t 1 ε 2 t 1 + δ 1 σ 2 t 1 I t 1 = 1 if ε t 1 < 0 and I t 1 = 0 if ε t 1 0 I t 1 is an indicator function (dummy variable) Greater effect of negative shocks if estimate κ > 0

83 TGARCH Threshold GARCH TGARCH(p,q) r t = α 0 + α 1 y t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + q i=1 (γ i + κ i I t i ) ε 2 t i + p j=1 β j σ 2 t j I t 1 = 1 if ε t 1 < 0 and I t 1 = 0 if ε t 1 0 This also allows for bigger effects of negative shocks

84 ARCH with variables affecting volatility With GARCH(1,1) r t = α 0 + α 1 r t 1 + ε t, ε t = σ t ν t, ν t iid(0,1) σ 2 t = γ 0 + γ 1 ε 2 t 1 + δ 1 σ 2 t 1 + κ 1 x t 1 Using r t here just for consistency variable need not be a return x t 1 is an exogenous variable Issue of positive variance at every observation or guaranteed For example, guaranteed if x t 1 0 and κ 1 0

85 EGARCH Exponential GARCH allows for asymmetry and does not require positive coefficients An EGARCH(1,1) general representation r t = α 0 + ε t, ε t = σ t ν t, ν t iid(0,1) ( ) ln σt 2 εt 1 = γ 0 + g + δ 1 ln σ t 1 σ t 1 Nelson argued that ε t 1 /σ t 1 is more interpretable and therefore preferable

86 EGARCH by Nelson Exponential GARCH An EGARCH(1,1) with asymmetry r t = α 0 + ε t, ε t = σ t ν t, ν t iid(0,1) ( ) ln σt 2 εt 1 = γ 0 + κ 1 + κ 2 ε t 1 + δ 1 ln σ t 1 σ t 1 σ t 1

87 EGARCH explained Exponential GARCH An EGARCH(1,1) r t = α 0 + ε t, ε t = σ t ν t ( ) ln σt 2 εt 1 = γ 0 + κ 1 + κ 2 ε t 1 + δ 1 ln σ t 1 σ t 1 σ t 1 ln σt 2 on left of variance equation means that coefficients need not be positive Allows for leverage effects ε t 1 /σ t 1 > 0, then effect of one-unit lagged standardized innovation on this-period s variance is κ 1 + κ 2 ε t 1 /σ t 1 < 0, then effect of one-unit lagged standardized innovation on this-period s variance is -κ 1 + κ 2 κ 1 < 0, then consistent with a leverage effect

88 Test for asymmetric volatility Estimate a mean equation and get residuals û t Define I t 1 = 1 if ε t 1 < 0 and I t 1 = 0 if ε t 1 0 Run regression ε 2 t = b 0 + b 1 I t 1 + e t If b 1 > 0, then there is an asymmetric effect of negative shocks

89 Stochastic Volatility ARCH and GARCH are restrictive Evolution of variance is deterministic except for influence of innovations in mean equation Stochastic volatility The evolution of volatility is not a deterministic function of only past volatility and innovations to the mean equation Innovations to the variance affect variance independent of mean equation

90 Relatively simple example of stochastic volatility r t = α 0 + α 1 r t 1 + ε t = α 0 + α 1 r t 1 + σ t ν t, ln ( σt 2 ) = γ0 + δ 1 ln ( σt 1 2 ) + ση η t α > 0, ε t N(0, 1), η t N(0,1) Parameters are not the same as in ARCH of course Two innovations, ε t and η t, for every observation r t How can that be? The innovations reflect different aspects of the series Consider a representation where we cannot identify two innovations r t = α 0 + ε 1,t + ε 2,t ε 1,t N(0, σ1 2), ε 2,t N(0, σ2 2) Never able to tell how much of variance of r t is due to ε 1,t and how much is due to ε 2,t if σ1 2 and σ2 2 and unknown and estimated from the data

91 Summary I Economic and financial time series have common characteristics Serial correlation of series Serial correlation of volatility Heteroskedasticity Simple ways of dealing with heteroskedasticity Use of higher frequency data to estimate changes in variance for lower frequency data ARCH and GARCH Variance a deterministic function of squared innovations Properties of ARCH and GARCH Series can be have an unpredictable mean but predictable variance ARCH and GARCH can help to account for excess heteroskedasticity ARCH and GARCH do not generate asymmetric distributions of series from symmetric innovations Estimate ARCH and GARCH by maximum likelihood All the desirable properties that go with maximum likelihood if distribution of innovations correctly specified

92 Summary II Even if distribution of innovations not correctly specified, estimator still consistent under general conditions If only quasi-maximum likelihood, estimators of standard errors may well be wrong Testing for ARCH or GARCH in time series This is important because nonlinear estimation with redundant parameters can fail to converge without it being obvious why it fails to converge Most common tests McLeod-Li Q statistics on squared innovations Engle s T R 2 test IGARCH is a common specification Volatility clustering IGARCH(1,1) particularly common GARCH-M allows for effect of volatility on level of variable Asymmetries in volatility common Various specifications common TGARCH Stochastic volatility appealing but more complex