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Autoregressive Conditional Heteroskedasticity Models Chapter 5 Autoregressive Conditional Heteroskedasticity Models 5.1 Modeling Volatility In most econometric models the variance of the disturbance term is assumed to be constant (homoscedasticity). However, there is a number of economics and finance series that exhibit periods of unusual large volatility, followed by periods of relative tranquility. Then, in such cases the assumption of homoscedasticity is not appropri- ate. For example, consider the daily changes in the NYSE International 100 Index, April 5, 2004 - September 20, 2011, shown in Figure 5.1. One easy way to understand volatility is modeling it explicitly, for example in yt+1 = εt+1xt (5.1) where yt+1 is the variable of interest, εt+1 is a white-noise disturbance term with 2 variance σ , and xt is an independent variable that is observed at time t. If xt = xt 1 = xt 2 = = constant, then the yt is a familiar white-noise process − − ··· { } with a constant variance. However, if they are not constant, the variance of yt+1 conditional on the observable value of xt is 2 2 var (yt 1 xt ) = x σ (5.2) + | t If xt exhibit positive serial correlation, the conditional variance of the yt se- quence{ } will also exhibit positive serial correlation. We can write the model{ in loga-} rithm form and introduce the coefficients a0 and a1 to have log (yt ) = a0 + a1 log (xt 1) + et (5.3) − where et = log (εt ). 43 44 5 Autoregressive Conditional Heteroskedasticity Models 15 10 5 0 -5 Percent change Percent -10 -15 01jul2004 01jul2005 01jul2006 01jul2007 01jul2008 01jul2009 01jul2010 01jul2011 01jan2004 01jan2005 01jan2006 01jan2007 01jan2008 01jan2009 01jan2010 01jan2011 Date Fig. 5.1 Daily Changes in the NYSE International 100 Index, April 5, 2004 - September 20, 2011 5.2 ARCH Processes Engel (1982) shows that is is possible to simultaneously model the mean and the variance of a series. To understand the key idea in Engel’s methodology we need to see how the conditional forecast is superior to the unconditional forecast. Consider the ARMA model yt = a0 + a1yt 1 + εt . (5.4) − The conditional mean of yt+1 is given by Et (yt+1) = a0 + a1yt (5.5) If we use the conditional mean to forecast yt+1, the forecast variance is 2 2 2 Et [( yt 1 a0 a1yt ) ] = Et (ε ) = σ (5.6) + − − t+1 However, if we use the unconditional forecast, which is always the long-run mean of the yt sequence, that is, equal to a0/(1 a1), the unconditional forecast error variance{ is} − a0 2 2 3 2 Et yt+1 = Et [( εt+1 + a1εt + a1εt 1 + a1εt 2 + ) ] (5.7) − 1 a1 − − ··· − h i 5.2 ARCH Processes 45 σ 2 = 1 a2 − 1 2 Because 1 /(1 a1) > 1, the unconditional forecast has a greater variance than the conditional forecast.− Hence, conditional forecasts are preferable. In the same way, if the variance of εt is not constant, you can estimate any { } tendency using an ARMA model. For example, if εˆt are the estimated residuals { } of the model yt = a0 + a1yt 1 + εt , the conditional variance of yt+1 is − 2 var (yt+1 yt ) = Et [( yt+1 a0 a1yt ) ] (5.8) | 2− − = Et (εt+1) We do not assume that it is constant as in Equation 5.6. Let’s say we can estimate it using the conditional variance as an AR( q) process using squares of the estimated residuals 2 2 2 2 εˆt = α0 + α1εˆt 1 + α2εˆt 2 + + αqεˆt q + v (5.9) − − ··· − where v is a white-noise process. If α1 = α2 = = αn= 0, the estimated variance 2 · is simply a constant α0 = σ . Otherwise we can use Equation 5.9 to forecast the conditional variance at t + 1, 2 2 2 2 Et (εˆt+1) = α0 + α1εˆt + α2εˆt 1 + + αqεˆt+1 q (5.10) − ··· − This is why Equation 5.9 is called an autoregressive conditional heteroscedasticity ARCH model. The linear specification in 5.9 is actually not the most convenient. Models for yt and the conditional variance can be estimated simultaneously using maximum likelihood.{ } In most cases v enters in a multiplicative fashion. Engel (1982) proposed the following multiplicative conditional heteroscedastic- ity model 2 εt = vt α0 + α1εt 1 (5.11) − q 2 where vt is a white-noise process with σv = 1, vt and εt 1 are independent of each − other, α0 > 0, and 0 α1 1. ≤ ≤ Consider the properties of εt . 1. The unconditional expectation of εt is zero 2 1/2 E(εt ) = E[vt (α0 + α1εt 1) ] (5.12) − 2 1/2 = E(vt )E[( α0 + α1εt 1) ] = 0 − 2. Because E(vt vt 1), − E(εt εt 1) = 0 i = 0 (5.13) − 6 46 5 Autoregressive Conditional Heteroskedasticity Models 3. The unconditional variance of εt is 2 2 2 E(εt ) = E[vt (α0 + α1εt 1)] (5.14) − 2 2 = E(vt )E(α0 + α1εt 1) − 2 because σv = 1 and the unconditional variance of εt is equal to the unconditional variance of εt 1, hence − 2 α0 E(εt ) = (5.15) 1 α1 − 4. The conditional mean of εt is zero 2 1/2 E(εt εt 1,εt 2,... ) = Et 1(vt )Et 1(α0 + α1εt 1) = 0 (5.16) | − − − − − 5. The conditional variance of εt is 2 2 E(εt εt 1,εt 2,... ) = α0 + α1εt 1. (5.17) | − − − which just means that the conditional variance of εt depends on the realized value 2 of εt 1. − In Equation 5.17, the conditional variance follows a first-order autoregressive process denoted by ARCH(1). As opposed to the usual autoregression, the coeffi- cients α0 and α1 have to be restricted to make sure that the conditional variance is never negative. Both have to be positive, and in addition, to ensure stability of the process we need to have the restriction 0 α1 1. ≤ ≤ The key point in an ARCH process is that even though the process εt is serially { } uncorrelated (i.e., E(εt εt s) = 0, s = 0), the errors are not independent. They are − ∀ 6 related through their second moment. The heteroscedasticity in ε results in yt being heteroscedastic. Then the ARCH process is able to capture{ periods} of relative{ } tranquility and periods of relative high volatility in the yt series. To understand the intuition behind an ARCH process,{ } consider the simulated white-noise process presented in the upper panel of Figure 5.2. While this is cer- tainly a white noise εt , the lower panel shows the generated heteroscedastic errors { } 2 εt = vt 1 + 0.8εt 1. Notice that when the realized value εt 1 is far from zero, the − − varianceq of εt tends to be large. The Stata code to obtain these graph is (you can try obtaining the graph with different seeds): clear set obs 150 set seed 1001 gen time=_n tsset time gen white=invnorm(uniform()) twoway line white time, m(o) c(l) scheme(sj) /// ytitle( "white-noise" ) saving(gwhite, replace) gen erro = 0 replace erro = white *(sqrt(1+0.8 *(l.erro)ˆ2)) if time > 1 twoway line erro time, scheme(sj) /// 5.2 ARCH Processes 47 White-noise and Heteroscedastic Errors 3 2 1 0 white-noise -1 -2 0 50 100 150 time 20 10 0 error -10 -20 0 50 100 150 time 2 Fig. 5.2 A white-noise process and the heteroscedastic error εt = vt 1 + 0.8εt 1 − q ytitle( "error" ) saving(gerror, replace) gr combine gwhite.gph gerror.gph, col(1) /// iscale(0.7) fysize(100) /// title( "White-noise and Heteroscedastic Errors" ) The panels in Figure 5.3 show two simulated ARMA processes. The idea is to illustrate how the error structure affect the yt sequence. The upper panel shows { } the simulated path of yt when a0 = 0.9, while the lower panel shows the simulated { } path of yt when a0 = 0.2. Notice that when a0 = 0, the yt sequence is the same { } { } as the vt sequence depicted in Figure 5.2. However, the persistence of the series { } increases with a0. Moreover, notice how the volatility in yt is increasing in the { } value of a0 (it also increase with the value of α1). gen Y1 = 0 gen Y2 = 0 replace Y1 = +0.90 *l.Y1 + erro if time > 1 replace Y2 = +0.20 *l.Y2 + erro if time > 1 twoway line Y1 time, scheme(sj) /// ytitle( "Y1" ) saving(gy1, replace) twoway line Y2 time, scheme(sj) /// ytitle( "Y2" ) saving(gy2, replace) gr combine gy1.gph gy2.gph, col(1) /// iscale(0.7) fysize(100) /// title( "Simulated ARCH Processes" ) 48 5 Autoregressive Conditional Heteroskedasticity Models Simulated ARCH Processes 40 30 20 Y1 10 0 -10 0 50 100 150 time 20 10 Y2 0 -10 0 50 100 150 time Fig. 5.3 Upper panel: Y1t = 0.9Y1t 1 + εt . Lower panel: Y2t = 0.2Y2t 1 + εt − − Formally, the conditional mean and variance of yt can be written as { } Et 1(yt ) = a0 + a1yt 1 (5.18) − − and 2 var (yt yt 1,yt 2,... ) = Et 1(a0 + a1yt 1) (5.19) − − − − | 2 = Et 1(ε) − 2 = α0 + α1(εt 1) − For a nonzero realization of εt 1, the conditional variance is positively related to − α1. For the unconditional variance, recall that the solution (omitting the constant A) for the difference equation in 5.4 is ∞ a0 i yt = + ∑ a1εt i (5.20) 1 a1 − − i=0 Because E(εt ) = 0, the unconditional expectation is E(yt ) = a0/(1 a1). Moreover, − because E(εt εt i) = 0, i = 0, the unconditional variance is − ∀ 6 ∞ 2i var (yt ) = ∑ a1 var (εt i) (5.21) i=0 − 5.3 GARCH Processes 49 α0 1 = 2 1 α1 1 a1 − − where the last equality follows from the result in Equation 5.15.
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