TIME SERIES ANALYSIS PROF. MASSIMO GUIDOLIN

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1 Massimo Guidolin Dept. of Finance TIME SERIES ANALYSIS PROF. MASSIMO GUIDOLIN LECTURE 3: UNIVARIATE ARCH MODELS AND EXTENSIONS (ADDITIONAL, OPTIONAL READING) Lecture 3: Univariate ARCH Models Prof. Guidolin

2 OVERVIEW 1) Volatility clustering 2) Exogenous regressors in the conditional variance equation 3) Plain ARCH and GARCH models 4) Estimation and Inference in GARCH models 5) Other GARCH models and News Impact Curves: TARCH, EGARCH, NGARCH, component GARCH, etc. 6) Assessing the performance of GARCH models a.a. 11/12 p. 2 Lecture 3: Univariate ARCH Models Prof. Guidolin

3 THE MAIN ISSUES While most classical finance was built on the assumption that both asset returns and their underlying fundamentals are IID Normal over time, casual inspection of GDP, financial aggregates, interest rates, exchange rates etc. reveals that these series display time varying means, variances, andcovariances Fundamentals = quantities that justify asset prices in a rational framework E.g., dividends for stocks, short term rates for long term rates, macroeconomic and fiscal policies for exchange rates, etc. When the variance and covariances are time varying we speak about conditional HETEROSKEDASTICITY (sometimes spelled heteroscedasticity, CH) a.a. 11/12 p. 3 Lecture 3: Univariate ARCH Models Prof. Guidolin

4 THE MAIN ISSUES Three simple facts to remember and understand: 1 The fact that the conditional variance may change in heteroskedastic fashion, does not necessarily mean the series is non stationary Even though the variance may go through high and low periods, the unconditional (long run, steady state, average) variance may exist and be actually constant 2 Conditional heteroskedasticity implies that the unconditional, long run distribution of asset returns will be non normal 3 Many models of conditional heteroskedasticity, but in the end we care for their forecasting performance a.a. 11/12 p. 4 Lecture 3: Univariate ARCH Models Prof. Guidolin

5 THE PROBLEM, SIMPLY STATED Consider the (dividend corrected) realized returns on a value weighted index (by CRSP) of NYSE, AMEX, and NASDAQ stocks Sample period is 1972: :12, monthly data Value Weighted NYSE/AMEX/NASDAQ Returns 15 Turbulence Turbulence Turbulence Turbulence Volatility clusters : 10 high (low) volatility 5 tends to be follo 0 wed by high (low) 5 volatility Quie et period Quiet period Quiet period The objective is to develop models that can fit the sequence of calm and turbulent periods a.a. 11/12 p. 5 Lecture 3: Univariate ARCH Models Prof. Guidolin

6 A NAÏVE APPROACH: REGRESSION IN VARIANCE and especially forecast them One classical approach to forecasting the variance is to explicitly introduce some independent variables that may help predicting volatility Consider the simple case in which wherex x t drivesnow theconditional variance When x t exhibits positive serial correlation, then the conditional variance will exhibit positive serial correlation In practice, the equation in logarithmic form is easier to estimate and will not imply that when x t 0 returns become deterministic E.g., let s ask whether the short term interest rate (= x t ) a.a. 11/12 p. 6 Lecture 3: Univariate ARCH Models Prof. Guidolin

7 A NAÏVE APPROACH: REGRESSION IN VARIANCE might be driving the variance of US stock returns In Eviews one may simply estimate the equation specifi cation (by LS, to keep things simple): log((vw_crsp_returns)^2) c log(tbill_1m_yield) to find This means that when short term rates increase, the variance of stock returns increases as well a.a. 11/12 p. 7 Lecture 3: Univariate ARCH Models Prof. Guidolin

8 A NAÏVE APPROACH: REGRESSION IN VARIANCE However, both thestatistics and theeconomicsofsucheconomics of such a model are not completely free of concerns Let s reconstruct for instance the square root of the in sample fit of the dependent variables in the previous 12 model 10 In Eviews, it is simply genr fitlog_int=(exp(2.0685) i t ( 8 *exp( )* 6 tbill_1m_yield)) yield))^0.5 Square root of fitted values for the squares of CRSP stock returns 4 We are clearly not picking 2 up the quiet & turbulent 0 sequences seen before However, let one point stick: there may be some good in linking CH to the dynamics of predetermined dvariables a.a. 11/12 p. 8 Lecture 3: Univariate ARCH Models Prof. Guidolin

9 ARCH AND GARCH MODELS We also need a richer, and better fitting class models In finance, such a class is represented by ARCH and GARCH models, which are well known tools to fit the tendency of volatility to cluster ARCH = AutoRegressive Conditional Heteroscedasticity GARCH = Generalized ARCH AutoRegressive means that the current variance depends on its own past, and not (only) on some exogenous regressor The assumption of this class of models is that the variance of returns follows a predictable process They provide a parsimonious specification, with few parameters, that seems to fit the data well ARCH models are less persuasive at predicting when volatility will jump from low to high levels We will worry about forecasting issues later a.a. 11/12 p. 9 Lecture 3: Univariate ARCH Models Prof. Guidolin

10 ARCH AND GARCH MODELS: INTUITION The basic idea that has led Engle (1982, ECMA) to introduce ARCH models is that conditional forecasts aregenerallyvastly superiorto unconditional forecasts Consider the case of a simple AR(1) process, x t+1 = x t + t+1, for which E t [x t+1 ] = x t and E[x t+1 ] = 0 /(1 1 ) It comes from E[x t+1 ] = E[x t+1 ] E[x t+1 ](1 1 ) = 0 TheMSE ofthe conditionalforecast is E [x t+1 ( x t )] 2 = E 2 t+1 = 2 while the MSE of the unconditional forecast is E [x t+1 0 /(1 1 )] 2 = 2 /(1 12 ) Check you understand why in Hamilton s book, p. 53 Unless 1 = 0, 2 /(1 12 ) > 2, which means that the MSE of the unconditional forecast is above the MSE of the conditional one a.a. 11/12 p. 10 Lecture 3: Univariate ARCH Models Prof. Guidolin

11 ARCH AND GARCH MODELS: INTUITION But 1 = 0 means that the conditional and unconditional forecasts are trivially identical So a conditional forecast always produces a smaller MSE This example is of course referred to a conditional mean process However the intuition extends to variances as well: when E 2 t+1 2 but instead (say) E t 2 t+1 = t t p 2 t p+1 then forecasting the variance as E 2 t+1 = 2 will always produce a higher MSE than forecasting with t t p 2 t p+1 p1 A model with structure e 2 t+1 = e 2 t + 2 e 2 t p e 2 t p+1 + v t+1, with v t+1 a zero mean shock is a ARCH(p) p t p+1 t+1, t+1 (p) process a.a. 11/12 p. 11 Lecture 3: Univariate ARCH Models Prof. Guidolin

12 ARCH AND GARCH MODELS: INTUITION The model e 2 t+1 = e 2 t + 2 e 2 t p e 2 t p+1 + v t+1 is written with reference to the estimated residuals e t+1 from some conditional mean model, for instance the AR(1) used before to convey intuition This suggests that one could estimate an ARCH model using the residual from a first pass regression or ARMA model However, this is not the best strategy because the resulting (LS) estimates of the ARCH(p) parameters would be severely inefficient Inefficient means that the LS estimators will display inflated standard errors This happens whenever one opts to adopt a limitedinformation estimator, such as two step LS In fact the best possible estimation approach is a fulla.a. 10/11 p. 12 Lecture 3: Univariate ARCH Models Prof. Guidolin

13 ARCH & GARCH MODELS: GENERAL STRUCTURE information maximum likelihood approach (FIML) What a mouthful! It just means you want to jointly estimate the parameters in the conditional mean and variance equations SIMULTANEOUSLY The basic idea is to regard the series of interest r t as being a sequence of IID r.v. s, given by the product of a standardized r.v. s v t+1 with the factor t : r t+1 I t = μ t+1 t + t+1 t v t+1 = μ t + t+1, v t+1 IID N(0,1) where I t is the information set at time t, I t = r t, r t 1, r t 2,, r 0, and μ t+1 t is the conditional mean function For instance, μ t+1 t = r t As a result, E t [r t+1 ] = E t [μ t+1 t + t+1 t v t+1 ] = μ t+1 t and Var t [r t+1 ] = Var t [μ t+1 t + t+1 t v t+1 ] = Var t E t [r t+1 ] + a.a. 11/12 p t+1 tvar t [v t+1 ] + 2Cov t E t [r t+1 ], t+1 t v t+1 = 2 t+1 t Lecture 3: Univariate ARCH Models Prof. Guidolin

14 ARCH & GARCH MODELS: GENERAL STRUCTURE This formulation is extremely general and allows for a wide variety of models, depending on the conditional variance equation specification Moreover, being the conditional density v t+1 IID N(0,1) assigned, we have an explicit expression for the loglikelihood that makes estimation easy Please bear this in mind when we approach stochastic volatility models, which are not easy at all If one replaces v t+1 IID N(0,1) with iha different parametric assumption (e.g., a marginal t Student, or a GED), obtains other models that are usually equally easy to estimate With ARCH models, we can easily apply MLE because even though the conditional volatility appears to be unobserved, it can be reconstructed using past shocks Lecture 3: Univariate ARCH Models Prof. Guidolin a.a. 11/12 p. 14

15 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES There isalso no problem in generalizing the conditional mean function μ t+1 t to be a linear combination of exogenous variables and lagged endogenous variables, x t, with a vector of unknown parameters, μ t+1 = x t The general structure of a Gaussian ARCH(q) model is: r t+1 I t = μ t+1 + h 1/2 t+1v t+1 = μ t+1 + t+1, v t+1 IID N(0,1) h t+1 = t t p 2 t q Let s examine in detail a simple AR(1) ARCH(1) model: r t+1 = [ + r + [ + 2 t] 1/2 0 1 t ] 0 1 t v t+1,, v t+1 IID N(0,1) As we shall see, constraints for (covariance) stationarity of both the mean and the variance of the process are 1 <1, 0 < 1 < 1, while 0 > 0 keeps variance well defined This has to be compared with a homoskedastic Gaussian AR(1) process, r t+1 = [ r t ] + [ 0 ] 1/2 v t+1, v t+1 IID N(0,1) Lecture 3: Univariate ARCH Models Prof. Guidolin a.a. 11/12 p. 15

16 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES Themodelcaptures CH in returnsby using a moving average of past squared unexpected returns Eg E.g., a large t = r t [ r t 1 ] will lead to a large h t1 t+1 = t and therefore to highly volatile shocks at time t+1, [ t] 1/2 v t+1 This of course is what captures the clustering intuition that large movements in asset prices tend to follow large movements, ofeithersign Conversely, returns near the (conditional) mean level μ t imply lower than average future volatility This effect will be very small if 1 is near zero; if 1 is near 1, the variance will tend to infinity and be very volatile Notice now that the total residual of the process is t [ t 1] 1/2 v t and examine in detail its statistical properties Assume that v t is independent of t 1, t 2,, 0 a.a. 11/12 p. 16 Lecture 3: Univariate ARCH Models Prof. Guidolin

17 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES 1 t have mean zero and are serially ill uncorrelated E[ = E[( + 2 t 1) 1/2 v t = E[( + 2 t 1) 1/2 t ] 0 1 ) t] 0 1 ) ] E[v t ] = E[( t 1) 1/2 ] x 0 = 0 E[ t t j ]= E[( t 1) 1/2 ( t j 1) j 1/2 v t v t j ] = E[( F IID t 1) 1/2 ( t j 1) 1/2 ]E[v t v t j ] = E[( t 1) 1/2 ( t j 1) 1/2 ] x 0 = 0 (j 1) This property is important tbecause it provides guarantees (necessary and sometimes sufficient conditions) to proceed to the estimation of the conditional mean function From IID ness 2 t has a finite unconditional variance of 0 /(1 1 ) E[ 2 t] = E[( t 1)v 2 t] = E[ t 1]E[v 2 t] = E[ 2 t] E[ 2 t] = Var[ t ] = 0 /(1 1 ) This iterates the point: ARCH does not imply nona.a. 11/12 p. 17 Lecture 3: Univariate ARCH Models Prof. Guidolin =1 from v t+1 IID N(0,1)

18 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES stationarity, and in fact a finite long run, average, unconditional variance exists, although it diverges to + as t t 1, t 2, has a zero conditional mean and a conditional variance of t 1 E[ ) t t 1, ] = E t 1 [( t 1) 1/2 v t ] = E t 1 [( t 1) 1/2 ] E t 1 [v t ] = ( 2 1) 1/2 = 2 1) 1/ t 1 E t 1 [v t ] ( t 1 x 0 = 0 E[ 2 t t 1, ] = E t 1 [( t 1)v 2 t] = ( t 1)E t 1 [v 2 t] = ( + 2 t 1) x 1 = ) 0 1 t 1 = Var t 1[ [ t ] This confirms what we have stated early on: in a ARCH model, shocks are serially uncorrelated as E[ t t j ] = 0 but they are not independent because E[ 2 t t 1, ] = t 1 Notice now that under a ARCH(1), the forecast error when predicting i squared residuals is: a.a. 11/12 p. 18 Lecture 3: Univariate ARCH Models Prof. Guidolin

19 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES t 2 t E t 1 [ 2 t] = 2 t h t where h t = t 1. Therefore 2 t = h t + t or 2 t = t 1 + t which is an AR(1) process for squared innovations to financialreturns returns. Thisimpliesimplies 4 The autocorrelogram for the series of squared shocks 2 t implied by a ARCH(1) decays at speed ( 1 ) j Here j is the order of the autocorrelogram, i.e., the lag in Cov[ 2 t, 2 t j]/var[ 2 t] The implication is that unless 1 1, the autocorrelogram of a ARCH(1) will decay very quickly (see example below) To develop the next set of properties, it is useful to recall that, by Wold s representation theorem, any AR(p) process can be represented as an infinite MA process a.a. 11/12 p. 19 Lecture 3: Univariate ARCH Models Prof. Guidolin

20 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES If the return series had started in the sufficiently distant past or, equivalently, when t, this is indeed dan MA( ) process, 2 t = [ 0 /(1 1 )] + t + 1 t t For integer r, the 2r th moment of a ARCH(1) process with 0 > 0, 1 0 exists if and only if a.a. 11/12 p y For r = 1 this gives the standard 1 < 1 for existence of the unconditionalvariance; however for r = 2 (existenceof kurtosis), the condition is < 1 or 1 < [1/3] 1/2 which is quite restrictive Lecture 3: Univariate ARCH Models Prof. Guidolin

21 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES In particular, it is possible to show that if 1 < [1/3] 1/2 then the kurtosis of a ARCH(1) process is given by: Kurt( 2 2 t ) = 3[(1 1 )/(1 3 1 )] The intuition is important: kurtosis is defined as E[ 4 t]/(var[ 2 = 2 t]/(var[ 2 t t ]) Var[ t t ]) and it is clear that, given its connection with Var[ 2 t], kurtosis is a natural measure of the variance of variance What about the skewness of t? Because E[ 3 t] = E[( t 1) 3/2 v 3 t] = E[( t 1) 3/2 ] E[v 3 t] = 0 from the fact that v 3 t IID N(0,1), we have that under a ARCH(1) Skew ( t ) = 0, the presence of ARCH cannot induce any asymmetries in the unconditional density What about tthe statistical ttiti properties of asset returns? a.a. 11/12 p. 21 Lecture 3: Univariate ARCH Models Prof. Guidolin

22 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES 6 r t has a conditional mean of r t 1 and a conditional variance of t 1 This easily follows from our general proof that E t [r t+1 ] = μ t+1 t and Var t [r t+1 ] = 2 t+1 t 7 r t has an unconditional i mean of 0 /(1 1 ) and an unconditional variance of [ 0 /(1 1 )][1/(1 2 1)] Clearly, this shows that while the conditional variance function does not affect the mean (unconditional or condi tional) of the return process, the conditional mean process affects the unconditional variance of returns This is important: sometimes applied finance people seem to imply that if you care for the variance you can forget about specifying a careful conditional mean model However: this occasionally makes sense at very high freqs a.a. 11/12 p. 22 Lecture 3: Univariate ARCH Models Prof. Guidolin

23 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES To prove this property you just need to write down r t exploiting its recursive structure Equivalently, y,you can speak of solving the difference equation represented by the AR(1) process If the return series had started in the sufficiently distant past or, equivalently, when t, then because 1 <1, a.a. 11/12 p. 23 Lecture 3: Univariate ARCH Models Prof. Guidolin

24 ARCH(1): STRUCTURE AND STATISTICAL PROPERTIES where the expressions derive from the fact E[ t ] = 0 and E[ t t j ] = 0 j 1 Important result: given 0, the unconditional variance of returns increases not only as 1 1, but also as 1 1 Finally, let s examine kurtosis and skewness of returns Resulting formulas are really complicated even for this simple case, although it is clear that because a.a. 11/12 p. 24 Lecture 3: Univariate ARCH Models Prof. Guidolin

25 ARCH(Q) skewness will be zero while kurtosis will be once more a function of both 1 and 1 In this case, considerable more information can be obtained by running simulations In many (most) cases, there is evidence of a need for a number of ARCH terms (call it q) that far exceeds 1 E.g., in the case of S&P 500 daily returns, we have that the ACF for squared returns decays only very slowly, an indication of AR(q) a.a. 11/12 p. 25 Lecture 3: Univariate ARCH Models Prof. Guidolin

26 ARCH(Q) ARCH(q) models have beenproposed to deal with such cases: h t = t t q 2 t q which is an AR(q) process for volatility The restrictions are in this case 0,,., q 0 and q < 1 to guarantee positivity and stationarity of the variance process The ARCH(q) will generate dt data with fatter ftt til tails than the normal, but always symmetric distributions (odd moments equal to zero) t has a finite unconditional variance of [ 0/(1 1 2 q)] Originally, in the literature ARCH(q) models with prespecified structures of declining weights were popular, e.g., a.a. 11/12 p. 26 h t = [0.4 2 t t t t 4] Lecture 3: Univariate ARCH Models Prof. Guidolin

27 TESTING FOR ARCH In this case the constraints on the parameters greatly simplify from q + 2 to 3 only, (i) 0, 1 0, (ii) < 1 This is one of main issues with high order ARCH processes: the need to estimate under absurd number of constraints The alternative is to either not impose them, getting gnonsensical estimates, or to resort to pre imposed values, as in Engle (1982, ECMA) How does one test tfor the presence of ARCH effects? Notice that because of the implications of ARCH for the (unconditional) non normality normality of returns, ARCH tests can also be interpreted as tests of the null of normality A first idea has been illustrated already: compute the Portmanteau Q statistic of Box and Pierce (1970), calculated from the first k autocorrelations of either (i) squared returns or (ii) absolute value of returns a.a. 11/12 p. 27 Lecture 3: Univariate ARCH Models Prof. Guidolin

28 TESTING FOR ARCH to test whether 2 k is statistically significant Notice that the asymptotic distribution of the Box Pierce statistic ttiti applies if and only if the returns themselves are serially uncorrelated Important implication: if in the decomposition r t+1 = μ t + t v t+1 you are not absolutely sure that in fact μ t is a constant, watch out because if the residuals are not serially uncorrelated, you cannot simply apply portmanteau tests to uncover the presence of ARCH The classical lalternative consists of Engle s ARCH LM test It consists of two simple steps: Compute residuals from the fitted conditional i mean function, making sure that they are serially uncorrelated Estimate: a.a. 11/12 p. 28 Lecture 3: Univariate ARCH Models Prof. Guidolin

29 TESTING FOR ARCH and use the statistic TR 2 2 q to test that the lagged values of the squared residuals fail to explain the current values of squared residuals A number of papers have in fact shown that in small samples a simple F statistic used to test the null hypothesis H 0 : a 1 = a 2 = a q = 0 that has better power properties Power means that the probability to reject the null of no ARCH effects when there actually are ARCH effect, is higher (i.e., closer to 1, which is ideal) How does one test for ARCH(q 1 ) against ARCH(q 2 ), with q 2 > q 1? Simple set up the usual ARCH LM regression with q 2 lags of squared residuals and then use an F test for the null llof H 0 : a q1+1 = a q1+2 = = a q2 = 0 a.a. 11/12 p. 29 Lecture 3: Univariate ARCH Models Prof. Guidolin

30 ARCH: COMPARING DIFFERENT ASSET CLASSES Notice that the comparisons will be valid in small samples only if all the competing ARCH models have been estimated on the same data sets Implication: use data that starts from the observation indexed to the highest q parameter you intend to consider Just to get additional intuition for these concepts, let s assess the comparative properties and fit of simple AR(1) ARCH(1) ARCH(1) processes for US stocks, bonds, real estate, and cash (1 month T bill) returns Sample period dis , 2009 data dt are monthly Will use E Views as it is one of the most common benchmarks to performarch estimation Later on, we will take a look at other software options Lt Let s start tby testing ti for conditional ht heteroskedasticity kd tiit 30 Lecture 3: Univariate ARCH Models Prof. Guidolin

31 ARCH: COMPARING DIFFERENT ASSET CLASSES We know from lecture 1 that there is strong evidence of serial correlation in squared residuals for all assets Let s perform now some LM tests After cleaning the data from any possible conditional mean effect, by fitting a simplear(1) process AR(1) ARCH(1) works! 31 Lecture 3: Univariate ARCH Models Prof. Guidolin

32 ARCH: COMPARING DIFFERENT ASSET CLASSES The specification search for REIT returns is intriguing There is evidence of AR(2) but this fades under ARCH effects Need for an ARMA(1,1) 1) meanprocess? No, pls. checkyourself There is robust evidence of at least ARCH(2) AR(1) cond. mean AR(2) cond. mean 32 Lecture 3: Univariate ARCH Models Prof. Guidolin

33 ARCH: COMPARING DIFFERENT ASSET CLASSES In the case of REITs, a AR(1) ARCH(3) process seems to be not misspecified In conclusion (v t+1 IID N(0,1)): [Stocks] r t+1 = [ r t t] + (0.22) (0.05) Squared Stdz. Res. from AR(2) ARCH(1) ARCH(1) +[ t] 1/2 v s t+1 (1.14) (0.05) ARMA(1,1) variance? [REITs] r t+1 =[ r t ] (0.20) (0.04) + [ t + (1.18) (0.05) t t 2] 1/2 v r t+1 (0.05) (0.08) 33 Lecture 3: Univariate ARCH Models Prof. Guidolin

34 ARCH: COMPARING DIFFERENT ASSET CLASSES Residuals from AR(1) ARCH(3) ARCH(3) Please look into the ARCH modelling of Treasury bond returns as an exercise 34 Lecture 3: Univariate ARCH Models Prof. Guidolin

35 ARCH: GAME OVER? Note: standardized d dresiduals are v t t /h 1/2 t and should ldbe N(0,1) if the model is correctly specified Although hthey are not directly comparable, these kernel density estimates reveal: Dramaticimprovementsvs improvements vs. the original, raw returns Something else remains to be tried What about that conjecture that a ARMA process may fit the squared residuals? 35 Lecture 3: Univariate ARCH Models Prof. Guidolin

36 ARCH: GAME OVER? Using daily data, it is a bit worse than what the monthly return series allow us to see Squared S&P 500 returns, Let s go back to our daily S&P 500 return series The suspicion of ARMA in va riance is even stronger Let s estimate a AR(1) ARCH(8): e? ARMA(1,1) variance 36 Lecture 3: Univariate ARCH Models Prof. Guidolin

37 ARCH: GAME OVER? S&P 500 dil dailyreturns, Lecture 3: Univariate ARCH Models Prof. Guidolin

38 GARCH(Q,P): FROM AR TO ARMA VARIANCE MODELS The ARCH(q) recognizes the difference between the unconditional and the conditional variance allowing the latter to change over time as a function of past errors But often a very long lag structure is required, or arbitrary simplifications must be imposed Moreover it can generate only symmetric distributions A more flexible structure has been proposed by Bollerslev (1986) with the generalized (G)ARCH model The current level of conditional variance is determined not only by past residuals, but also by past levels of volatility Basically, this is ARCH(q) + which is equivalent to a switchfrom anar(q) process for squared residuals, to a ARMA(q,p) process 38 Lecture 3: Univariate ARCH Models Prof. Guidolin

39 GARCH(1,1) The most widely used GARCH models in applied econometrics is the simple ARMA(1,1) case, a GARCH(1,1): h = + 2 t 0 1 t h t 1 Equivalent to the process for returns given by r = 2 1/2 t+1 t + [ t + 1 h t ] v t+1, v t+1 IID N(0,1) where the conditional mean process t may either consist of a regression or of a ARMA model for the level of returns Why is a simple GARCH(1,1) so popular and successful? This is surprising because one of the problems with ARCH was the need to pick relatively large values of q The reason is simple: a GARCH(1,1) model can be shown to beequivalent ttoanarch( ) model! dl! Notice that by recursive substitution, h t = t h t 1 = t [ t h t 2 ] 39 Lecture 3: Univariate ARCH Models Prof. Guidolin

40 GARCH(1,1) h t = [ ] + 1 [ t t 2 ]+ 1 [ t h t 3 ] = 0 [ ]+ 1 [ 2 t t t 3] + 3 1h t 3 = = 0 0[ [ t 1 1] + 1 [ [ t 1 1 t 2 1 t t ]+ t 1h 0 If the return series had started in the sufficiently distant past or, equivalently, when t, this an ARCH( ) ) with a particular structure of decaying power weights, h t = 0 /(1 1 ) + 1 ARCH( ; 1 ) which reminds us of h t = [0.4 2 t t t ] 2 t 4 with weights replaced by 1, 1, 1, The constraints 0, 1,, q, 1,, p 0 are required to ensure that the conditional variance is never negative Bollerslev (1987, JoE) has shown that the process is covariance stationary if 40 Lecture 3: Univariate ARCH Models Prof. Guidolin

41 GARCH(1,1) which in the (1,1) 1) case just means < 1 This follows from the fact that if t 2 t E t 1 [ 2 t] = 2 t h t where h t = t 1+ 1 h t 1, then 2 t = t 1+ 1 h + t = t 1+ t + 1 ( 2 t 1 t 1 ) = + ( + ) t 1+ t 1 t 1 which is an ARMA(1,1) process From your undergrad stats courses, you know that such a process will be covariance stationary if and only if < 1 By a process of recursive substitution, it is possible to see that 1 t has a finite unconditional variance of 0 /(1 1 1 ) As we shall see, typical estimates are 1 < and 1 > The popularity of this model is due to the fact that it has only 3 parameters to be estimated ( 0, 1 1, 1 ) where positivity and stationarity constraints may be easily imposed 41 Lecture 3: Univariate ARCH Models Prof. Guidolin

42 GARCH(1,1) 2 The autocorrelogram for the series of squared innovations implied by a GARCH(1,1) decays at speed ( ) j 1 As long as 1 > 0 there is a slower decay than in the ARCH case This property derives from the fact that from the properties of ARMA(1,1) 1) processes, Autocorrelation coefficients j It is possible to show that if ( ) < 1, then the kurtosis of a GARCH(1,1) process is given by: Kurt( t) = 3[(1 ( 1+ ) 2 )/(1 ( + ) )] 1 Therefore unless 1 = 1 = 0, Kurt( t ) > 3 and a GARCH(1,1) model will generate the excess kurtosis in the data What about the skewness of t? 42 Lecture 3: Univariate ARCH Models Prof. Guidolin j j

43 TESTING FOR GARCH(1,1) Because E[ 3 t] = E[( t 1+ 1 h t 1 ) 3/2 v 3 t] = E[( t h t 1 ) 3/2 ] E[v 3 t] = 0 from the fact that v 3 t IID N(0,1), we have that under a GARCH(1,1), Skew( t ) = 0 How does one test for the presence of GARCH? What we have said in the ARCH case generally applies However, in this case it is often typical to first test for ARCH Using Ljung Box tests on squared residuals: in this case one should see a rather slow decline of both ACF and PACF Using Engle s LM ARCH test and then test for GARCH by estimating the regression: the statistic TR 2 2 P tests whether the lagged values of ARCH variance explain the current values of squared residuals 43 Lecture 3: Univariate ARCH Models Prof. Guidolin

44 THE STRUCTURE OF GARCH(1,1) MODELS What is the respective role of 1 and 1 in affecting the dynamic properties of volatility under a GARCH(1,1)? Clearly, 1 measures the impact of shocks on predicted variance, 1 the persistence of variance Consider the competing models: Both processes have been simulated Model 1 tends to be spiky but variance reverts back quickly Model 2 implies more zzzzzzzzzzzzzzzzzzzzzzzz pers persistence of weaker shocks 44 Lecture 3: Univariate ARCH Models Prof. Guidolin

45 LA SOLUZIONE RISKMETRICS: EXPONENTIAL SMOOTHING It is a special, highly celebrated and yet (so I am afraid) still used special case of GARCH(1,1): (0,1) which is a special case of a GARCH(1,1) under the restrictions: (i) 0 = 0, (ii) 1 = 1 1 Because under Riskmetrics, 1 = 1, the sum of the restricted GARCH coefficients is 1 The unconditional variance does not exist and E t 1 [h t ] = h t 1 B/c in a GARCH(1,1),, it follows that the forecast at any 1 equals h t+1 When estimating on a large number of assets, RiskMetrics found that the estimates were quite similaracrossacross assets, and they simply set = 0.94 for every asset 45 Lecture 3: Univariate ARCH Models Prof. Guidolin

46 ESTIMATION AND INFERENCE OF GARCH MODELS What follows generallyappliesto applies all types of GARCH models, of which ARCH(q) can be considered a special GARCH(q,0) case The common approach to estimation of GARCH models is the ML method The essential ingredients of ARCH models are conditional density functions, which describe the density of the next return conditional on the current information These densities are often assumed to be normal, although more flexibility is provided by permi ng non normal normal conditional densities Assume that the distribution of returns at time t is conditionally normal with variance h t, and call the vector that collects the unknown parameters (e.g., 0, 1, 1 ) then 46 Lecture 3: Univariate ARCH Models Prof. Guidolin

47 ESTIMATION AND INFERENCE OF GARCH MODELS The log likelihood function is then This function has now to be maximized using numerical methods, often imposing the obvious constaints 0, 1,, q, 1,, p 0 and The inverse of the information matrix, will provide the asymptotic covariances of the estimates. Consistent estimatesof the information matrix are calculated from n observations by: 47 Lecture 3: Univariate ARCH Models Prof. Guidolin

48 ESTIMATION AND INFERENCE OF GARCH MODELS ^ ^ The inverse of this matrix can be used for hypothesis testing by constructing the usual z ratio statistic As usual these z ratios are built as t ratios are, but their distribution is asymptotically normal For example, consider testing the null hypothesis that the parameter 1 tk takes a particular value, 1 *, i.e., H 0 : 1 = 1 * The first step is to find the MLE estimate 1 Second, compute anestimate of the covariance matrix, i.e. Third, define the ratio ^ Element (2,2) 2) of whose asymptotic distribution is N (0, 1), assuming 1 * belongs to the feasible set for 1 48 Lecture 3: Univariate ARCH Models Prof. Guidolin ^

49 ESTIMATION AND INFERENCE OF GARCH MODELS To call this ratio statistic either t or z is irrelevant: compare with the t distribution in small samples, with the normal in large samples These are Wald type tests We can also conduct likelihood ratio (LR) tests to test a constraint ton the true parameter vector or even to test tthe conditional normality against other conditional distributions Let us consider testing the null hypothesis that k parameters I assume a particular value: We have to proceed as follows: Find the value of the constrained log lik Find value of the unconstrained log lik Compute the likelihood ratio (log likelihood difference): where k is the difference in number of parameters 49 Lecture 3: Univariate ARCH Models Prof. Guidolin

50 ESTIMATION AND INFERENCE OF GARCH MODELS A large value of the LR statistic indicates that the alternative hypothesis is much more likely than the null hypothesis to describe the data and then the null is rejected Up to this point, we have assumed normality of the conditional distribution of returns and applied MLE If we want to estimate t a GARCH process with non normal innovations, we can still use MLE corresponding with this particular distribution As one can never be sure that the specified distribution is the correct one, an alternative approach is to ignore the problem and base the likelihood on the normal distribution even when this is not exact The procedure is then called Quasi ML (QML) QML will provide estimates consistent and asymptotically normal, provided that the models for the conditional mean and variance are correctly specified 50 Lecture 3: Univariate ARCH Models Prof. Guidolin

51 ESTIMATION AND INFERENCE OF GARCH MODELS It is easy to fail to sufficiently emphasize this condition: if one applies QML, inference on variance parameters will be correct only if THE CONDITIONAL MEAN FUNCTION IS CORRECTLY SPECIFIED Too much empirical volatility work has been scarcely respectful of the role of the conditional mean function inobtaining valid inferences In the QML case, the asymptotic covariance matrix has to be adjusted accordingly: We have small sample evidence that QML is reasonably accurate and close to ML estimates provided that the distribution of the innovation is symmetric 51 Lecture 3: Univariate ARCH Models Prof. Guidolin ^

52 GARCH MODELS: ONE EXAMPLE In the case of daily S&P 500 returns ( ) 2009), we have seen that not even a rich AR(1) ARCH(8) was correctly specified Both the portmanteau tests on squared residuals and formal ARCH tests made us suspect that some ARMA(1,1) model for volatilitywould be needed But ARMA(1,1) for squared residuals = GARCH(1,1)! Let s estimatesuch such AR(1) GARCH(1 GARCH(1,1) 1) model with z stat of 15.6 in AR(1) ARCH(8) Estimated = vs. Sum of estimated ARCH terms = Lecture 3: Univariate ARCH Models Prof. Guidolin

53 GARCH MODELS: ONE EXAMPLE Residuals in levels v 2 t 2 Heteroskedasticity Test: GARCH Residuals in squares 21,905 x R 2 = p value This output derivesfrom the regression of standardized residuals v 2 t on h 2 t 2 and v t 2 which can be taken as a test for ARCH(2) and GARCH(2) terms 53 Lecture 3: Univariate ARCH Models Prof. Guidolin

54 GARCH MODELS: ONE EXAMPLE Density tests applied to standardized residuals reveal that a GARCH(1,1) does fairly well around the center of the distribution, although the tails remain thicker than what one expects with marginal normal shocks 54 Lecture 3: Univariate ARCH Models Prof. Guidolin

55 GARCH MODELS: ONE EXAMPLE Let s estimate now a GARCH(2,2) 2) and practice our ability to perform tests of hypothesis That s what happens when one does not impose constraints Let s test H 0 : 2 > 0 vs. 2 0; z = ( )/ )/ = which certainly exceeds the Gaussian critical values at any size LR test of GARCH(2,2) 2) vs. (1,1): 1): 2x[ ] = which, with 2 d.f., has a p value of essentially zero 55 Lecture 3: Univariate ARCH Models Prof. Guidolin

56 FILTERING & FORECASTING VARIANCE IN GARCH MODELS The one step ahead forecast of conditional variance is trivial to obtain: =1 E t [ t+1 ] = E t [h t+1 v t+1 ] = E t [h t+1 ]E t [v t+1 ] = E t [h t+1 ] = t + 1 h t More interesting the case of a j step ahead horizon: E 2 j] 2 j] 2 t [ t+j = E t [h t+j v t+j = E t [h t+j ]E t [v t+j j] = E t [h t+j ] = E t [ t+j 1+ 1 h t+j 1 ] = E t [ t+j 1 1] + 1 E t [h t+j 1 ] = 0 + ( )E t [h t+j 1 ] = 0 + ( )E t [ t+j 2+ 1 h t+j 2 ] = 0 [1+( )]+ ( ) 2 E t [h t+j 2 ] = = 0 [1+( )+( ) 2 + +( ) j 1 ]+ ( ) j h t which is easy to compute recursively, E t [h t+1 ] = t+ 1 h t E t [h t+2 ] = 0 + ( )E t [h t+1 ] E t [h t+3 ] = 0 + ( )E t [h t+2 ] 56 Lecture 3: Univariate ARCH Models Prof. Guidolin

57 FILTERING & FORECASTING VARIANCE IN GARCH MODELS Inour daily S&P sample for a GARCH(1,1): 1): 57 Lecture 3: Univariate ARCH Models Prof. Guidolin

58 OTHER GARCH MODELS: t GARCH We shall soon see that our early finding of a GARCH(2,2) 2) has a deeper meaning than what one normally thinks The problem was the violation of the usual constraints; some restrictions have to be imposed As seen before, the data often display higher excess kurtosis than what a GARCH(1,1) 1) can generate One popular solution then combines BOTH strategies to obtain non normalities: normalities: Conditional heteroskedasticity Marginal distributions whichare non Gaussian In the literature the t Student GARCH(1,1) is popular: r 2 1/2 t+1 = t + [ t + 1 h t ] v t+1, v t+1 f v where (v > 2) (e) 58 Lecture 3: Univariate ARCH Models Prof. Guidolin e 2

59 OTHER GARCH MODELS: t GARCH Notice that besides the GARCH(1,1) 1) parameters, in this case we also need to estimate the d.f. parameters, v t Student The estimate of the number of d.f. is highly significant and similar to the estimate of found for US monthly stock returns in lecture 1 Distribution results not better 59 Lecture 3: Univariate ARCH Models Prof. Guidolin duals in squares Resi

60 OTHER GARCH MODELS: INTEGRATED GARCH ^ In the GARCH(1,1) 1) case, we have noticed that = which is close to 1 A Wald test of H 0 : = 1 gives a z statistic of z = (0.996)/( e 06) 1/2 = which has a zero p value However, one is not always so lucky, with other series and/or periods, it may be impossible to reject = 1 The model that results when = 1 is commonly referred to as Integrated GARCH This name is due to the fact tthat t the restriction titi = 1 implies a unit root in the ARMA(1,1) model for v 2 t It follows that the unconditional variance of r t is not finite in this case, i.e. the model is NOT covariance stationary However, the IGARCH(1,1) model may still be strictly stationary provided that: 60 Lecture 3: Univariate ARCH Models Prof. Guidolin ^

61 OTHER GARCH MODELS: INTEGRATED GARCH It is also easy to see that because = 1, h t = 0 +(1 1 ) 2 t 1+ 1 h = (1 1 ) 2 t 1+ 1 (1 1 ) 2 t h t 2 = = [ ] +(1 1 )[ 2 t t t 2 + ] which is a geometric decaying function of past realiza tions of the shock sequence 2 t t Two implications: This is a typical ARCH( ) with restrictions involving 1 = 1 1 and the weights Therefore it can be estimated with the same methods that are applied to all ARCH models Because under a IGARCH(1,1) (pls. check) h t+1 = 0 + h t + (1 1 )( 2 t h t ) 61 Lecture 3: Univariate ARCH Models Prof. Guidolin

62 OTHER GARCH MODELS: INTEGRATED GARCH and E t [ 2 t h t ] = 0, the conditional expectation of variance j periods ahead is: E t [h t+j ]= j 0 +h t Therefore E t [h t2 t+2 ] = 0 + E t [h t1 t+1 ] = h t = h t E t [h t+3 ] = 0 +E t [h t+2 ] = h t = 3 0 +h t, etc. Notice that if 0 = 0, we have aeaa martingale at gaeprocess Even though IGARCH is not covariance stationary Sure thing: the variance does not even exist Nelson (1990, JoE) has shown that if 0 > 0, h t is strictly stationary even though its stationary distribution generally lacks unconditional moments: the tails til are so thick that t no second or higher order moment exists When + 0 = 0, h t = (1 )[ 2 t 1+ 2 t t 2 + ] ]and this particular model is the exponential smoother for variance popularized p by RiskMetrics at JP Morgan Ouch! 0 = 0 long run forecast of variance is E t [h t+j ]= h t 62 Lecture 3: Univariate ARCH Models Prof. Guidolin

63 OTHER GARCH MODELS: INTEGRATED GARCH Can a IGARCH(1,1) 1) represent a better model for S&P 500 daily returns? Let s try to estimate the model imposing 0 = 0 63 Lecture 3: Univariate ARCH Models Prof. Guidolin Squa ared Stdz ed Res.

64 LEVERAGE AND NEWS IMPACT CURVES A well known and important stylized fact is that negative and positive shocks (i.e., t ) impact subsequent volatility (i.e., h t+1 ) depending on their sign Asymmetric NIC In particular, in many asset markets there is now robust evidence that t < 0 affects volatility more than t > 0 This is often called leverage effect: a negative e return rn shock would ldlower the net worth of a company increasing its leverage GARCH A more highlyleveragedleveraged company is riskier and this should be reflected by higher, subsequent variance of stock returns You do see the naivite of such reasoning in a CAPM view We also speak about asymmetries in the conditional variance (CH) function This is described dby the (sample) news impact curve (NIC) 64 Lecture 3: Univariate ARCH Models Prof. Guidolin

65 LEVERAGE AND NEWS IMPACT CURVES The NIC measures how new information is incorporated into volatility, i.e. it shows the relationship between the current shock t and conditional variance one period ahead h t+1, holding constant all other past and current information In a GARCH(1,1) model we have: NIC( t h t = h) = h+ 1 2 t = A+ 1 2 t which is a quadratic function of 2 t and therefore symmetric around 0 (with hintercept A h) Problem: for most return series, the empirical NIC fails to be symmetric After Engle and Ng (1993, JF) on NICs, a considerable literature has beendevoted to develop ARCH models that could produce a NIC matching its empirical asymmetries There are many such models: we will study TARCH (also called GJR), EGARCH, and NGARCH 65 Lecture 3: Univariate ARCH Models Prof. Guidolin

66 ASYMMETRIC ARCH MODELS: EGARCH EGARCH is probably the most prominent asymmetric GARCH As in ARCH models, in GARCH models the negativity of parameters may create difficulties in estimation Nelson (1991, ECMA) proposes a new form of ARCH model, the Exponential ilgarch, in which h positivity iii of the conditional i variance is ensured by the fact that ln(h t ) is modeled Two types of EGARCH(1,1) 1) found in the applied literature; the first type is the one originally proposed by Nelson Letting z [ /(h 1/2 t t t ) ], the log conditional variance is: h t ) h t 1 ) The sequence g(z t) is a zero mean, i.i.d. random sequence: If z t > 0, g(z t ) is linear in z t with slope Lecture 3: Univariate ARCH Models Prof. Guidolin h t 1

67 ASYMMETRIC ARCH MODELS: EGARCH If z t < 0, g(z t ) is linear in z t with slope 1 Thus, g(z t ) is function of both the magnitude and the sign of z t and it allows the conditional variance process to respond asymmetrically to rises and falls in stock price Indeed, it can be rewritten as No restrictions on the parameters are necessary to ensure non negativity it of the conditional variances The term represents a magnitude effect: If 1 > 0 and = 0, the innovations in the conditional variance are positive (negative) when the magnitude of z t is larger (smaller) than its expected value If 1 = 0 and < 0, the innovation in conditional variance is positive (negative) when returns innovations are negative (positive), in accordance with empirical evidence for stock returns 67 Lecture 3: Univariate ARCH Models Prof. Guidolin

68 ASYMMETRIC ARCH MODELS: EGARCH The NIC for this EGARCH model is: t t h t = h) t ) t if t > 0 t if t 0 An alternative version of EGARCH(1,1) is: ln(h t ) = 0 + ln(h t 1 ) + 1 ( t 1 /h t 1 ) + + ( t i /h t 1 ) For instance, this is the version implemented in Eviews The presence of a leverage effect in this case implies < 0; more generally, there will be asymmetries in conditional variance as long as 0 Of course, EGARCH(1,1) may be easily generalized to EGARCH(q,p), ln(h = + p ln(h t j + q t ) 0 j=1 j ) i=1 i i( ( t i /h t i ) + + q i=1 i( t i /h t i ) 68 Lecture 3: Univariate ARCH Models Prof. Guidolin

69 EGARCH MOMENT CONDITIONS Nelson s s (1991) EGARCH has another key advantage In the GARCH case, we know that the parameter restrictions needed to ensure moment existence become increasingly gy stringent as the order of the moment grows For instance, even in the simple case of a ARCH(1), we know that For integer r the 2r th moment exists if and only if For integer r, the 2r th moment exists if and only if Instead, in a EGARCH(p,q) case, if the error process {v t } has all moments and the sum of the squares of the j coefficients does not exceed 1, q < 1 then all moments for the EGARCH process exist Nelson (1991) also recommended dthe use of the so called Generalized Error Distribution (GED) for the errors: a distri bution with fatter tails than the normal one helps to increase the kurtosis 69 Lecture 3: Univariate ARCH Models Prof. Guidolin

70 ASYMMETRIC ARCH MODELS: THRESHOLD ARCH The model introduced by Glosten, Jagannathan and Runkle (1993, JF; GJR) offers alternative method to allow for asymmetric effects of positive and negative shocks The model is obtained from the GARCH(1,1) model by assuming that the parameter of 2 t 1 depends on the sign of the shock, that is: h t = t 1+ I t 1 <0 2 t h t 1 which h means h t = 0 +( 1 + ) 2 t 1+ 1 h t 1, GARCH(1,1) w/coeff. 1 + when t 1 <0 h 2 t = t h t 1, GARCH(1,1) 1) w/coeff. 1 when t 1 0 The conditions for non negativeness of the conditional variance are 0 > 0, 1 + > 0, 1 > 0 The condition for covariance stationarity is < 1 If this condition is satisfied, the unconditional variance is 0 /( ) 70 Lecture 3: Univariate ARCH Models Prof. Guidolin

71 ASYMMETRIC ARCH MODELS: NONLINEAR GARCH The NIC for the GJR TGARCH is equal to: t t h t = h ) t ) 2 t if t > 0 ( 1 + ) 2 t if t 0 NAGARCH is a very simple framework that however has played a key role in applied option pricing: h t = ( t 1 h 1/2 t 1) h t 1 = h 1/2 t 1(v t 1 ) h t 1 The model dlisasymmetric when > 0 b/c while v t 1 0 impacts conditional variance only in the measure (v t 1 ) 2 v 2 t 1, v t 1 <0 impacts conditional variance in the measure (v 2 2 t 1 ) > v t 1 The model is non linear because even with = 0itcan be rewritten as a GARCH(1,1) with time varying ARCH coefficient: h t = 0 + 1,t 1 v 2 t 1+ 1 h t 1 where 1,t 1 1 h 1/2 t 1 71 Lecture 3: Univariate ARCH Models Prof. Guidolin

72 ASYMMETRIC ARCH MODELS: NONLINEAR GARCH Notice that N(A)GARCH can also be re written as: h t = t t 1 h 1/2 t 1 + ( )h t 1 which is a GARCH(1,1) 1) with restricted coefficients and augmented by 1 t 1 h 1/2 t 1 However, notice that E[ 1 t 1 h 1/2 t 1]= 1 E[ t 1 ]h 1/2 t 1 = 0 Therefore the NGARCH persistence is given by 1 (1+ 2 ) + 1 With some algebra you can show that the unconditional variance is 0 /(1 1 (1+ 2 ) 1 ) Notice that differently from the EGARCH case, one still needs to impose restrictions, which are in this case no longer so plain as imposing that 0 > 0 and 1 > 0, because of the presence of the term 1 t 1 h 1/2 t 1 Please try and work out the NIC for NGARCH: it will be timevarying, which is an appealing feature indeed 72 Lecture 3: Univariate ARCH Models Prof. Guidolin

73 ASYMMETRIC ARCH MODELS: TESTS & EXAMPLES Let s look now for ARCH asymmetries for 2 of our 3 risky asset classes, monthly data We first estimate a symmetric Gaussian a AR(1) GARCH(1,1) ( ) Second, we learn how to test for the existence of residual asymmetries in squared standardized residuals There are three types of tests that can be applied, based on one s economic priors on the kind of asymmetries in asset returns: Define D t 1 as a dummy that takes value of 1 when the GARCH residual t 1 < 0, and zero otherwise A test of the null hypothesis a 1 = 0 in the ARCH test type regression is called a Sign Bias Test A test of the null hypothesis a 2 = 0 in the ARCH test type regression 73 Lecture 3: Univariate ARCH Models Prof. Guidolin

74 ASYMMETRIC ARCH MODELS: TESTS & EXAMPLES is called a Negative Size Bias Test; a test of the null hypothesis of a 3 = 0 in the same regression is called Positive Size Bias Test Sign Test Regression: Persistence: 0.96 Unconditional volatility: 5.4% TGARCH? Neg. & Pos. Size Test treg.: 74 Lecture 3: Univariate ARCH Models Prof. Guidolin

75 ASYMMETRIC ARCH MODELS: TESTS & EXAMPLES Unfortunately, in a simple TARCH, 1 < 0 which is inadmissible; therefore we estimate a EGARCH(1,1) Unconditional volatility: 4.5% Because < 0 is statistically significant, the EGARCH model is adequate to capture asymmetries in the conditional volatility 75 Lecture 3: Univariate ARCH Models Prof. Guidolin

76 ASYMMETRIC ARCH MODELS: TESTS & EXAMPLES Let s see if equity REIT returns display any differences SignTest Regression: Neg. & Pos. Size Test Reg.: Persistence: Unconditional volatility: 5.1% TGARCH? 76 Lecture 3: Univariate ARCH Models Prof. Guidolin

77 ASYMMETRIC ARCH MODELS: TESTS & EXAMPLES Unfortunately, in a simple TARCH, 1 < 0 which is inadmissible; therefore we estimate a EGARCH(1,1) Unconditional volatility: 5.1% Because < 0 is statistically significant, the EGARCH model is adequate to capture asymmetries in the conditional volatility However, the economic effect is in this case small 77 Lecture 3: Univariate ARCH Models Prof. Guidolin

78 GARCH IN MEAN In asset pricing it is frequent to write and solve models in which the conditional risk premium is given as: Et[x t+1 ] = (risk aversion) x (quantity of risk at time t) where x t+1 r t+1 r f t is the excess return on the asset r f t is the riskless rate as of time t, think of the yield on a 1 month T bill Risk aversion is often interpreted as constant parameter, while the quantity of risk ikis allowed to vary over time Since the paper by Engle, Lilien and Robins (1987, ECMA), it has become typical to use GARCH variance forecasts as a measure of time varying risk: where h t follows some of the GARCH models examined It is also possible to find papers in which the conditional mean is a function of either h 1/2 t or ln(h t ) 78 Lecture 3: Univariate ARCH Models Prof. Guidolin

79 GARCH IN MEAN: EXAMPLES 79 Lecture 3: Univariate ARCH Models Prof. Guidolin

80 COMPONENT GARCH The basic idea is that both short and long term conditional variance would contain ARCH, with short term CH converging gto long termch Component GARCH model has been introduced by Engle and Lee (1999) with a leverage effect a la GJR/TARCH We also speak of a long memory GARCH model Because GARCH(q,p) is equivalent to an ARMA on squared errors, how can we handle non stationarity? tti it? Notice that extension from ARMA(1,1) to ARIMA(1,1,1), i.e., from 2 2 t = 0 + ( ) t 1 + t 1 t 1 to Δ 2 t = 0 +( ) 2 t 1+ t 1 t 1 2 t= 0 +( ) 2 t 1+ t 1 t 1 won t work because ( ) may generate negative variance Component GARCH assumes instead that the long run error variance q t reverts to q t = 0 + (q t 1 0 ) + ( 2 t 1 h t 1 ) 80 Lecture 3: Univariate ARCH Models Prof. Guidolin

81 COMPONENT GARCH while short run run variance h t follows h t = q t + 1 ( 2 t 1 h t 1 ) + I t 1 <0 ( 2 t 1 h t 1 ) + 1 (h t 1 q t 1 ) Alternatively, the deviations of Squared shocks from short term variance Short term variance from long run variance follow a TGARCH(1,1) process with long run mean 0 Zero long run mean, 0 = 0, implies that as t, h t = q t The component GARCH(1,1) can be shown to be equivalent to a special GARCH(2,2) (set = 0 for simplicity): h t = (1 1 1 )(1 ) 0 + ( 1 + ) 2 t 1 ( 1 + ( ) )) 2 t ( 1 )h t 2 ( 1 ( ) )h t 2 which isugly as hell but it is after all a special GARCH(2,2) 2) with 0 (1 1 1 )(1 ) 0, 1 1 +, 2 ( 1 + ( ) )), 1 1, 2 ( 1 ( ) ) Why is it no longer a problem that 2 ( 1 + ( ) )) < 0? 81 Lecture 3: Univariate ARCH Models Prof. Guidolin

82 COMPONENT GARCH Let s take a new look at our S&P 500 daily data, using a component GARCH(1,1) Short run persistence: 0.96 Long run persistence: 1.02! Unconditional variance: 1.33% per day Short run persistence: 0.93 Long run persistence: 1.02! Unconditional variance: 0.77% per day 82 Lecture 3: Univariate ARCH Models Prof. Guidolin

83 COMPONENT GARCH 83 Lecture 3: Univariate ARCH Models Prof. Guidolin

84 ASSESSING THE PERFORMANCE OF GARCH MODELS We have been doing it routinely, but let s ask again: how do you decide whether a GARCH model is appropriate? How do you compare two alternative GARCH models? ❶ Check the properties of standardized residuals v t t /(h t ) 1/2 where h t comes from an estimated ARCH to make sure that v t IID N(0,1) (or any assumed distribution) ❷ If the model has been estimated without formal restrictions on coefficients, important to think whether h the necessary restrictions (e.g., 0, 1,, q, 1,, p 0) for positivity and N&S for stationarity are satisfied ❸ Models can be compared using the (maximized) loglikelihoodfunction: any ARCH modeloughtto ought to do better than the corresponding constant variance model Alternative ARCH models can be compared via their log liks As we have seen, when the models are nested, this can also be 84 Lecture 3: Univariate ARCH Models Prof. Guidolin

85 ASSESSING THE PERFORMANCE OF GARCH MODELS couched in terms of a formal LR test For instance, one can test GARCH(2,2) against the null of GARCH(1,1) estimating (on identical data!) both models and then computing 2[LR GARCH(2,2) LR GARCH(1,1) ] and comparing this statistic with the critical values at (1 p)% under 2 2 Here p is the levellof significance ifi of the test tand 2 comes from the fact that the difference in the number of parameters btw. GARCH(2,2) )and dgarch(1,1), is indeed 2 Notice that this can be used as a test for the need of a component GARCH model, because imposing parameter restrictions may never increase the log likelihood In the case of our monthly data set: Stock returns: 2[ ( )] 1)] = 12.6 (p value: 0.002) 002) REIT returns: 2[ ( )] = 3.0 (p value: 0.223) Bond returns: 2[ ] = 3.0 (p value: 0.223) 1M T Bill returns: 2[ ] = 12.8 (p value: 0.002) 85 Lecture 3: Univariate ARCH Models Prof. Guidolin

86 ASSESSING THE PERFORMANCE OF GARCH MODELS ❹ Modelscan be comparedusing penalized measuresof fit which trade off in sample fit with parsimony, i.e., decrease as the number of parameters increase Why do we value parsimony? Because in general terms the forecasting performance of a model improves as the number of parameters used to fit the data in sample declines The general feeling is that, given an identical fit, e.g., a GARCH(1,1) model will alsoperform better thana GARCH(2,2) The forecasting analog of Okkam s razor Information criteria are sample statistics derived from the maximized log likelihood thataccomplishexactlythistask Their general structure is: (Maximized Log Lik) + f(dim( )) Three information criteria i are widelyemployed l d The Bayes Schwartz IC (BIC): 2(LogLik/n) + (dim( )ln(n)/n) It is known to select rather parsimonious models; very popular in the applied literature 86 Lecture 3: Univariate ARCH Models Prof. Guidolin

87 ASSESSING THE PERFORMANCE OF GARCH MODELS The Akaike IC (AIC): 2(LogLik/n) + 2(dim( )/n) Also popular because asymptotically unbiased, but known to select too large non linear models in small samples The Hannan Quinn IC (H Q): 2(LogLik/n) + 2[dim( )ln(ln(n))/n] It is has been shown to perform very strongly in small samples and for non linear models; it is a compromise btw. BIC and AIC IMPORTANT, before you ever embarass yourself: information criteria need to be MINIMIZED by the choice of the model, because of the special structure of an IC: (Log Lik) + f(dim( )) For instance, let s ponder our S&P 500 daily data: Model Log Lik dim( ) BIC AIC H Q AR(1) Const. Vol AR(1) ARCH(1) ARCH(1) AR(1) GARCH(1,1) AR(1) EGARCH(1,1) AR(1) CGARCH(1,1) AR(1) TGARCH(1,1) Lecture 3: Univariate ARCH Models Prof. Guidolin

88 ASSESSING THE PERFORMANCE OF GARCH MODELS Yes, EGARCH and TGARCH are close, with the former slightly superior but unfortunately there is no known statistical theory to compare and make inference on differences btw. ICs ❺ Because ARCH models are models for conditional variance, it seems appropriate to try and measure how well they fit the time varying variance in the data Problem: how do you measure variance in the data? (Conditional) dto a) variance a is a (conditional) o moment tand dtherefore e e it is latent, not observable First idea: estimate time t conditional variance using the squared conditional i mean residuals, 2 t At this point one can either compute the in sample R 2 that reveals how well the filtered series h 2 t fits the t or actually build information criteria with structure: (RSS)( ) + f(dim( )) ( where RSS is the residual sum of squares computed as [ 2 t h t ] 2 88 Lecture 3: Univariate ARCH Models Prof. Guidolin

89 ASSESSING THE PERFORMANCE OF GARCH MODELS Alternatively, one can actually test whether a = 0, b = 1, and R 2 is high (ideally 1) in the regression 2 t = a + b h t + t t white noise a = 0 and b = 1 jointly come from the fact that if a ARCH model is correctly specified, then E[ 2 t]= h t The problem is thatt 2 t isa very poor measure of time varyingi conditional variance, with Var t [ t ] = E t [( t h t ) ] = E t [h t (v t 1) ] = h 2 te t [(v 2 t 1) 2 ] = h 2 t[kurt(v t ) 1] When either h t or Kurt(v are high, 2 t ) t will contain a lot of noise To remedy to this problem, Christoffersen (2003) & others have suggested using either daily high and low price implied returns or high h frequency dt data to estimate t time varyingi variance For instance, in the case of S&P 500 daily returns, it seems useful to deepen our comparison of EGARCH(1,1) vs. TGARCH(1,1) given that their ICs were close 89 Lecture 3: Univariate ARCH Models Prof. Guidolin

90 ASSESSING THE PERFORMANCE OF GARCH MODELS EGARCH(1,1) 1) 90 Lecture 3: Univariate ARCH Models Prof. Guidolin

91 ASSESSING THE PERFORMANCE OF GARCH MODELS TGARCH(1,1) 1) 91 Lecture 3: Univariate ARCH Models Prof. Guidolin