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 appropriate. 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 sequence{ } 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 heteroscedasticity 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. It is easy to see that the unconditional variance in also increasing in α1 (and in the absolute value of a1). The ARCH process presented in Equation 5.11 can be extended in a number of ways. The most straight forward is considering the higher-order ARCH( q) process

q 2 εt = vt α0 + ∑ αiεt i (5.22) s i=1 −

5.3 GARCH Processes

The ARCH idea was extended in Bollerslev (1986) to allow an ARMA process embedded in the conditional variance. Let the error process be

εt = vt ht (5.23) where σ 2 = 1, and p q p 2 ht = α0 + ∑ αiεt 1 + ∑ βiht 1 (5.24) i=1 − i=1 −

The conditional and unconditional means of εt are both zero because vt is a { } white-noise process. The key point is that the conditional variance of εt is given 2 by Et 1(εt ) = ht , which is the ARMA process given in Equation 5.24. This− heteroscedastic variance that allows autoregressive and moving average components is called GARCH( p,q), where the G in GARCH denotes generalized. Notice that a GARCH(0,1) is just the ARCH model in Equation 5.11. The impor- tant restriction in a GARCH process is that all coefficients in Equation 5.24 must be positive and must ensure that the variance is finite (i.e., its characteristic roots must lie inside the unit circle). A simple procedure to know if a series yt follows a GARCH process is to { } estimate the best fitting ARMA process and then obtain the fitted errors εˆt and the 2 { } squares of the fitted errors εˆt . White the ACF and the PACF of the fitted errors should be consistent with a white{ } noise, the squared fitted errors should indicate that they follow an ARMA process. It is also useful that besides the ACF and PACF, we use the Ljung-Box Q-statistic. Let’s follow the previous suggested steps in Stata for the previously generated process Y1, under the assumption that we know that it follows an ARMA(1,0) process with no constant (otherwise we need to search for the optimal ARMA( p,q)). arima Y1, arima(1,0,0) nocons predict eserro, res gen eserro2 = eserroˆ2 corrgram eserro2, lags(20) -1 0 1 -1 0 1 LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] 50 5 Autoregressive Conditional Heteroskedasticity Models

------1 0.6590 0.6594 66.452 0.0000 |----- |----- 2 0.6541 0.3899 132.36 0.0000 |----- |--- 3 0.5579 0.0811 180.64 0.0000 |---- | 4 0.3386 -0.3243 198.54 0.0000 |-- --| 5 0.3055 -0.0239 213.22 0.0000 |-- | 6 0.1378 -0.0922 216.22 0.0000 |- | 7 0.0662 0.0044 216.92 0.0000 | | 8 -0.0413 -0.1291 217.2 0.0000 | -| 2 The Ljung-Box Q-statistics of εˆt show strong evidence that the yt follows a GARCH process. { } { } A more formal test is the LM (Lagrange Multiplier) test for ARCH errors de- veloped in McLeod and Li (1983). The idea in this test is to estimate the most 2 appropriate ARMA model and obtain the squared ﬁtted errors εˆt . Then, estimate the following equation { }

2 2 2 2 εˆt = α0 + α1εˆt 1 + α2εˆt 2 + + αqεˆt q + ut (5.25) − − ··· −

If the estimated coefﬁcients are all jointly equal to zero, α0 = α1 = α2 = = αq = 0. This can be easily checked with an F test with q degrees of freedom.··· An 2 2 alternative is to use TR , which for large samples (large T) converges to a χq . For our Y1 series we have: reg eserro2 l.eserro2 l2.eserro2

Source | SS df MS Number of obs = 148 ------+------F( 2, 145) = 78.70 Model | 173533.28 2 86766.64 Prob > F = 0.0000 Residual | 159865.35 145 1102.51965 R-squared = 0.5205 ------+------Adj R-squared = 0.5139 Total | 333398.63 147 2268.01789 Root MSE = 33.204

------eserro2 | Coef. Std. Err. t P>|t| [95% Conf. Interval] ------+------eserro2 | L1. | .4021842 .0764732 5.26 0.000 .251038 .5533304 L2. | .3898682 .0764729 5.10 0.000 .2387226 .5410139 | _cons | 3.29114 2.906213 1.13 0.259 -2.452873 9.035152 ------where we clearly reject the null of H0 : α0 = α1 = α2 = = αq = 0. Hence, there ··· is evidence that yt follows a GARCH process. The the q can be selected using any optimal lag selection{ } criteria.

5.4 Maximum-Likelihood Estimation of GARCH Models

If we assume that εt follow a normal distribution with mean zero and constant 2 { } variance σ , the likelihood of εt is

2 1 εt Lt = exp − (5.26) √2πσ 2 2σ 2 5.4 Maximum-Likelihood Estimation of GARCH Models 51

Because εt are independent, the likelihood function of the of the joint realizations { } of ε1,ε2,..., εT is T 1 ε2 L = exp − t . (5.27) ∏ 2 2 t=1 √2πσ 2σ Taking the logs we obtain the log-likelihood function

T T T 2 1 2 log L = log (2π) log σ 2 ∑(εt ) (5.28) − 2 − 2 − 2σ t=1

If we want to estimate the simple model yt = βxt + εt , the maximum log-likelihood function becomes

T T T 2 1 2 log L = log (2π) log σ 2 ∑(yt βxt ) . (5.29) − 2 − 2 − 2σ t=1 −

Then, the maximum likelihood estimates of β and σ 2 can be obtained from the following ﬁrst-oder conditions

T ∂ log L T 1 2 2 = 2 + 4 ∑(yt + βxt ) = 0 (5.30) ∂σ −2σ 2σ t=1 T ∂ log L 1 2 = 2 ∑(yt xt + βxt ) = 0 (5.31) ∂β σ t=1

2 ˆ 2 which yield the familiar OLS results σˆ2 = ∑εt /T and β = ∑xt yt /∑xt . A close form solution is possible because the FOC are linear. How do we introduce heteroscedastic errors in this familiar maximum likelihood procedure? Let’s assume that yt follows an ARCH(1) process. Hence, εt is

εt = vt ht (5.32) p where the conditional variance of εt is ht . Then the likelihood function becomes

T 1 ε2 L = ∏ exp − t (5.33) t=1 √2πht 2ht and the log-likelihood function is

T 2 T 1 1 2 log L = log (2π) ∑ log ht ∑(εt /ht ). (5.34) − 2 − 2 t=1 − 2 t=1

2 With εt = yt βxt and ht = α0 + α1εt 1 we have − − T T 1 1 2 log L = − log (2π) ∑ log (α0 + α1εt 1) (5.35) − 2 − 2 t=2 − 52 5 Autoregressive Conditional Heteroskedasticity Models

2 1 2 2 ∑[( yt βxt ) /(α0 + α1εt 1)] −2 t=2 − − T T 1 1 2 = − log (2π) ∑ log (α0 + α1(yt 1 βxt 1) ) − 2 − 2 t=2 − − − 2 1 2 2 ∑[( yt βxt ) /(α0 + α1(yt 1 βxt 1) )] , −2 t=2 − − − − where the ﬁrst observation is lost because of the lag. While the FOC yield complicated nonlinear equations that do not have a close-form solution, this log-likelihood function can be maximized using numerical methods.

5.4.1 Estimation of GARCH Models in Stata

A simple GARCH process in Stata has the following form

yt = ARMA( p,q) + εt (5.36)

2 2 2 var( εt ) = σt = α0 + α1,1εt 1 + α1,2εt 2 + (5.37) − − ··· 2 2 + α2,1σt 1 + α2,2σt 2 + − − ··· while both equations (5.36 and 5.37) can be estimated iteratively using the ARMA and OLS methods we already know, Stata has a speciﬁc command to obtain the estimates in a single step using maximum likelihood. For example, if we want to estimate the ARCH(1) process in the simulated series Y1, the command is arch Y1, arima(1,0,0) arch(1) nocons

(setting optimization to BHHH) Iteration 0: log likelihood = -360.36487 Iteration 8: log likelihood = -300.84557

ARCH family regression -- AR disturbances

Sample: 1 - 150 Number of obs = 150 Distribution: Gaussian Wald chi2(1) = 3237.07 Log likelihood = -300.8456 Prob > chi2 = 0.0000

------| OPG Y1 | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------+------ARMA | ar | L1. | .9048536 .0159038 56.90 0.000 .8736826 .9360245 ------+------ARCH | arch | L1. | 1.127475 .2373718 4.75 0.000 .662235 1.592715 | 5.4 Maximum-Likelihood Estimation of GARCH Models 53

_cons | .6363727 .1757892 3.62 0.000 .2918322 .9809133 ------That can be written as

Y1t = 0.90 Y1t 1 + εt (5.38) − 2 2 σt = 0.64 + 1.13 εt 1 (5.39) − For an AR(1) with GARCH(1,1) we need arch Y1, arima(1,0,0) arch(1) garch(1) nocons (output omitted) which yields

Y1t = 0.90 Y1t 1 + εt (5.40) − 2 2 2 σt = 0.65 + 1.13 εt 1 0.03 σt 1. (5.41) − − −

5.4.2 Volatility Breaks

A large literature indicates that volatility in macroeconomic variables in industri- alized economies decrease in early 1984. For example, Stock and Watson (2002) report that the volatility of the U.S. real GDP growth was smaller after 1984. Fig- ure 5.4 shows the real GDP and the real GDP growth rates from the first quarter of 1947 to the first quarter of 2008. While this figure appears to provide some evidence in favor of higher volatility prior 1984, GARCH models can provide a formal test to verify this claim. Consider the following specification for the variance

2 2 σt = exp (λ0 + λ1xt ) + αε t 1 (5.42) − where xt is a variable that affects the conditional variance of yt . For our case, let this variable xt a dummy variable I[t>1984 q1] that is equal to one if after the ﬁrst quarter of 1984, zero otherwise. Then, to estimate this model in Stata we need to type use rgdp.dta, clear tsset date

gen y = log(rgdp/l.rgdp) gen dum = 0 replace dum = 1 if date >= 149 // 149 is 1984q1

arch y, arima(1,0,0) arch(1) het(dum)

(setting optimization to BHHH) Iteration 0: log likelihood = 834.7398 Iteration 6: log likelihood = 835.59237

ARCH family regression -- ARMA disturbances and mult. heteroskedasticity

Sample: 2 - 246 Number of obs = 245 Distribution: Gaussian Wald chi2(1) = 22.20 Log likelihood = 835.5924 Prob > chi2 = 0.0000

------54 5 Autoregressive Conditional Heteroskedasticity Models

Real GDP and Real GDP Growth Rate 12 10 8 6 Real GDP Real 4 2

01jan1950 01jan1960 01jan1970 01jan1980 01jan1990 01jan2000 01jan2010 Date .04 .02 0 Real GDP Growth Rate GDP Real Growth -.02

01jan1950 01jan1960 01jan1970 01jan1980 01jan1990 01jan2000 01jan2010 Date

Fig. 5.4 Real GDP and Real GDP Growth Rate (1947q1 - 2008q1)

yt = 0.008 + 0.321 yt 1 + εt (5.43) − 2 2 σt = exp ( 9.06 1.70 I[t>1984 q1]) + 0.085 εt 1. − − −

The coefficient on the dummy variable I[t>1984 q1] is negative and statistically signif- icant. This means that the volatility is significantly lower after the first quarter of 1984. 5.5 ARCH-in-mean Models 55 5.5 ARCH-in-mean Models

The ARCH-in-mean (ARCH-M) models were introduced by Engel, Lilien, and Robins (1987). The idea is to allow for the mean of a sequence to depend on its conditional variance. If the riskiness of an asset can be measured by the variance of its returns, the risk premium will be an increasing function of the conditional variance of returns. This can be formally modeled with ARCH-M models. Engel, Lilien, and Robins (1987) write the excess return of holding a risky asset as yt = µt + εt (5.44) where yt is the excess return from holding a long-term asset relative to a one-period treasury bill, µt is the risk premium necessary to induce the risk-averse agent to hold the long-term asset rather than the one-period bond, and εt is the unforecastable shock to the excess return on the long-term asset. The expected excess return from holding the long-term asset is equal to the risk premium Et 1(yt ) = µt . (5.45) − We can model the risk premium as a function of the conditional variance of ε. Let ht be the conditional variance of εt . Then the risk premium can be modeled to depend on the conditional variance

µt = β + δht , δ > 0 (5.46) and where the conditional variance ht follows an ARCH( q) process

q 2 ht = α0 + ∑ αiεt i (5.47) i=1 − Hence, the ARCH-M model is the combination of Equations 5.44, 5.46, and 5.47. If α1 = α2 = = αq = 0, the ARCH-M models becomes the traditional constant risk premium··· model. To get an intuition of how volatility affects the mean of a series, we simulate the 150 realizations of a white-noise process εt and then construct the conditional { } 2 variance that follows the ﬁrst-order ARCH process ht = 1 + 0.75 εt 1. The upper panel in Figure 5.5 illustrates the white-noise process, while the lower− panel shows the constructed conditional variance. From this ﬁgure is it easy to see how higher volatility in εt (large positive or negative realizations) translates into larger con- { } ditional variance (a large value of ht ). clear set obs 150 set seed 1002 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) 56 5 Autoregressive Conditional Heteroskedasticity Models

White-noise and Conditional Variance 4 2 0 white-noise -2 -4 0 50 100 150 time 15 10 5 conditional variance conditional 0 0 50 100 150 time

2 Fig. 5.5 A white-noise process and conditional variance ht = 1 + 0.65 εt 1. −

gen vari = 0 replace vari = 1 + 0.75 *(l.white)ˆ2 if time > 1 twoway line vari time, scheme(sj) /// ytitle( "conditional variance" ) saving(gvari, replace)

gr combine gwhite.gph gvari.gph, col(1) /// iscale(0.7) fysize(100) /// title( "White-noise and Conditional Variance" )

Figure 5.5 shows two simulated ARCH-M processes. The upper panel simulates the series yt = 4 + 4ht + εt (5.48) − that is, β = 4 and δ = 4, while the lower panel simulates the series with β = 1 and δ = 1, − − yt = 1 + 1ht + εt . (5.49) − In the upper panel we can see that during periods of higher volatility are associated with higher values of yt . For example, around t = 75 higher volatility is moving { } the mean of yt to the positive side. The lower panel has smaller effects for the ARCH-M effects{ } because of the smaller β and δ coefﬁcients. Notice that if δ = 0, there will be no ARCH-M effects.

gen Y1 = -4 + 4 *vari + white gen Y2 = -1 + 1 *vari + white

twoway line Y1 time, scheme(sj) /// ytitle( "Y1" ) saving(gy1, replace) 5.5 ARCH-in-mean Models 57

Simulated ARCH-M Processes 40 30 20 Y1 10 0 -10 0 50 100 150 time 10 5 Y2 0 -5 0 50 100 150 time

Fig. 5.6 Upper panel: Y1t = 4 + 4ht + εt . Lower panel: Y2t = 1 + 1ht + εt . − −

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-M Processes" )

The LM test we described before to test for the presence of an ARCH process can also be used here. However, this test will not help in knowing whether the ARCH process is an ARCH-M. The Stata estimation of an ARCH-M process is straight forward. The process for yt can be extended from Equation 5.36 to include the ARCH-M component

p 2 yt = ARMA( p,q) + ∑ φig(σt i) + ε (5.50) i=1 − and the conditional variance can follow the same form as before

2 2 2 var( εt ) = σt = α0 + α1,1εt 1 + α1,2εt 2 + (5.51) − − ··· 2 2 + α2,1σt 1 + α2,2σt 2 + − − ··· Consider the following example that tests whether demand uncertainty affects mean demand realizations use airlines.dta, clear 58 5 Autoregressive Conditional Heteroskedasticity Models

tsset date arch demand, arima(1,0,0) archm arch(1)

(setting optimization to BHHH) Iteration 0: log likelihood = 163.58772 Iteration 9: log likelihood = 182.39822

ARCH family regression -- AR disturbances

Sample: 1 - 126 Number of obs = 126 Distribution: Gaussian Wald chi2(2) = 142.56 Log likelihood = 182.3982 Prob > chi2 = 0.0000

------| OPG demand | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------+------demand | _cons | .7233085 .0227503 31.79 0.000 .6787187 .7678982 ------+------ARCHM | sigma2 | 4.559358 4.980237 0.92 0.360 -5.201728 14.32044 ------+------ARMA | ar | L1. | .6990354 .0604844 11.56 0.000 .5804882 .8175825 ------+------ARCH | arch | L1. | .2915877 .1572736 1.85 0.064 -.016663 .5998383 | _cons | .0024397 .0003445 7.08 0.000 .0017646 .0031149 ------The resulting estimated mean equation is

2 demand t = 0.723 + 0.699 demand t 1 + 4.55 σt + εt (5.52) − and the conditional variance equation is

2 2 σt = 0.002 + 0.292 εt 1 (5.53) − Notice that the φ coefﬁcient is not statistically signiﬁcant. Hence, demand uncertainty as measured by the conditional variance of demand does not affect average demand.

5.6 GARCH Models with Asymmetry

5.6.1 TARCH Models

Glosten, Jaganathan, and Runkle (1994) propose a methodology that allows for an asymmetric effect of shocks on volatility. When modeling stock prices, whether εt is positive (good news) or negative (bad news), the effect on volatility may be different. Consider the following threshold-GARCH (TARCH) process 5.6 GARCH Models with Asymmetry 59

2 2 ht = α0 + α1εt 1 + λ1dt 1εt 1 + β1ht 1 (5.54) − − − − where dt 1 is a dummy variable that is equal to one if εt 1 < 0 and it is equal to zero otherwise.− − The intuition in this model is simple. If εt 1 < 0, then dt 1 = 1 and the effect of − 2 − the εt 1 shock on volatility ht is given by (α1 + λ1)εt 1. On the other hand, if the − − shock is positive, then εt 1 0 and dt 1 = 0, and the effect of the εt 1 shock on 2 − ≥ − − volatility ht is α1εt 1. Hence, if λ1 > 0, negative shocks will have a larger effect on − volatility than positive shocks. The statistical signiﬁcance of the λ1 coefﬁcient will determine whether there is a threshold effect on the conditional volatility. The implementation of TARCH models in Stata has the following form use http://www.stata-press.com/data/r11/wpi1, clear arch D.ln_wpi, ar(1) ma(1 4) arch(1) garch(1) tarch(1)

(setting optimization to BHHH) Iteration 45: log likelihood = 403.871

ARCH family regression -- ARMA disturbances

Sample: 1960q2 - 1990q4 Number of obs = 123 Distribution: Gaussian Wald chi2(3) = 279.50 Log likelihood = 403.871 Prob > chi2 = 0.0000

------| OPG D.ln_wpi | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------+------ln_wpi | _cons | .0092552 .0042914 2.16 0.031 .0008442 .0176662 ------+------ARMA | ar | L1. | .8462294 .0696347 12.15 0.000 .7097478 .982711 | ma | L1. | -.3919203 .1084576 -3.61 0.000 -.6044932 -.1793474 L4. | .2199699 .0886465 2.48 0.013 .046226 .3937138 ------+------ARCH | arch | L1. | -.0658397 .0631152 -1.04 0.297 -.1895432 .0578637 | tarch | L1. | .509789 .2881884 1.77 0.077 -.0550498 1.074628 | garch | L1. | .7253014 .1278951 5.67 0.000 .4746316 .9759713 | _cons | .0000108 7.14e-06 1.52 0.129 -3.16e-06 .0000248 ------

Let yt = δ log (wpi t ), then the estimated equations are written as

yt = 0.009 + 0.846 yt 1 0.392 εt 1 + 0.219 εt 4 + εt (5.55) − − − −

2 2 ht = 0.000 0.066 εt 1 + 0.509 dt 1εt 1 + 0.725 ht 1 (5.56) − − − − − where the threshold effect is statistically signiﬁcant at a 10% level. 60 5 Autoregressive Conditional Heteroskedasticity Models 5.6.2 EGARCH Models

A second model that allows for asymmetries in the effect of εt is the exponential- GARCH (EGARCH) model, as proposed in Nelson (1991). One constraint with the regular GARCH model is that it requires that all estimated coefﬁcients are positive. The EGARCH specify the conditional variance in the following way

log (ht ) = α0 + α1(εt 1/ ht 1) + λ1 εt 1/ ht 1 + β1 log (ht 1). (5.57) − − | − − | − There are three features aboutp this speciﬁcation: p

1. Because of the log (ht ) form, the implied value of ht can never be negative (re- gardless of the values of the coefﬁcients). 2. Instead of using εt 1, EGARCH uses a standardized value εt 1/√ht 1 (which is a unit free measure).− − − 3. EGARCH allows for leverage effects. If εt 1/√ht 1 > 0, the effect of the shock − − on the log of the conditional variance is α1 + λ1. If εt 1/√ht 1 < 0, the effect is − − α1 + λ1. − The Stata manual has the following example for the estimation of an EGARCH model. use http://www.stata-press.com/data/r11/wpi1, clear

arch D.ln_wpi, ar(1) ma(1 4) earch(1) egarch(1)

(setting optimization to BHHH) Iteration 0: log likelihood = 227.5251 Iteration 23: log likelihood = 405.31453

ARCH family regression -- ARMA disturbances

Sample: 1960q2 - 1990q4 Number of obs = 123 Distribution: Gaussian Wald chi2(3) = 156.03 Log likelihood = 405.3145 Prob > chi2 = 0.0000

------| OPG D.ln_wpi | Coef. Std. Err. z P>|z| [95% Conf. Interval] ------+------ln_wpi | _cons | .0087348 .0034005 2.57 0.010 .0020699 .0153997 ------+------ARMA | ar | L1. | .7692179 .0968355 7.94 0.000 .5794238 .9590121 | ma | L1. | -.3554589 .1265699 -2.81 0.005 -.6035313 -.1073865 L4. | .2414647 .0863819 2.80 0.005 .0721593 .4107701 ------+------ARCH | earch | L1. | .406411 .1163508 3.49 0.000 .1783677 .6344543 | earch_a | L1. | .246747 .1233373 2.00 0.045 .0050103 .4884837 | egarch | L1. | .8417259 .0704077 11.96 0.000 .7037294 .9797225 5.7 Multivariate GARCH Models 61

| _cons | -1.488427 .6604365 -2.25 0.024 -2.782859 -.1939957 ------

With yt = δ log (wpi t ), we have

yt = 0.009 + 0.769 yt 1 0.355 εt 1 + 0.241 εt 4 + εt (5.58) − − − −

log (ht ) = 1.488 + 0.406 (εt 1/ ht 1) (5.59) − − − +0.247 (εt 1/ ht 1)p 2/π + 0.842log (ht 1) | − − − | − p p Under normally distributed errors ( εt ), εt /√ht follows a standard normal distribution. The results indicate a strong leverage effect. The positive α1 coefﬁcient im- plies that positive innovations (unanticipated price increases) are more destabilizing than negative innovations. This effect is larger than the symmetric effect ( λ1). No- tice that the coefﬁcients can be directly compared because of the standardization of εt 1/√ht 1. − −

5.7 Multivariate GARCH Models

If more than one variable appears on the analysis, the conditional volatilities can be modeled jointly. This helps because there may exist contemporaneous shocks that affect the volatility of more than one of the variables. Multivariate GARCH modeling allows for this, in addition to allowing for volatility spillovers where volatility shocks to one variable might affect the volatility of a related variable. If we want to jointly model k variables we can deﬁne the symmetric variance- covariance matrix of the shocks as

h11 t h21 t ... hk1t h12 t h22 t ... hk2t H =  . . . . ......    h h ... h   1kt 2kt kkt    where hi jt are just the time t conditional variance of the shock i if i = j, and it is the conditional covariance if i = j (i.e., hi jt = Et 1(εit ε jt )). Let H be the determinant of H. Then the log-likelihood6 function to estimate− the coefﬁcients| | is

T T k 1 1 log L = log (2π) ∑(log Ht + εt0Ht− εt ), (5.60) − 2 − 2 t=1 | | where εt in this case is the k vector εt = ( εt1,εt2,..., εtk ). To keep things simple suppose that there are only two variables, y1t and y2t . Let the error processes be 62 5 Autoregressive Conditional Heteroskedasticity Models

ε1t = v1t h11 t (5.61)

ε2t = v2t ph22 t . (5.62) p We assume that var( v1t ) = var( v2t ) = 1. The vech model is the construction of multivariate GARCH(1,1) process where all volatility terms are allowed to interact with each other. That is,

2 2 h11 t = c10 + α11 ε1t 1 + α12 ε1t 1ε2t 1 + α13 ε2t 1 (5.63) − − − − +β11 h11 t 1 + β12 h12 t 1 + β13 h22 h 1 − − −

2 2 h12 t = c20 + α21 ε1t 1 + α22 ε1t 1ε2t 1 + α23 ε2t 1 (5.64) − − − − +β21 h11 t 1 + β22 h12 t 1 + β23 h22 h 1 − − −

2 2 h22 t = c30 + α31 ε1t 1 + α32 ε1t 1ε2t 1 + α33 ε2t 1 (5.65) − − − − +β31 h11 t 1 + β32 h12 t 1 + β33 h22 h 1 − − − These equations show that each conditional variance depends on its own past, the conditional covariance between the two variables, the lagged square errors, and the product of lagged errors. As simple as the model in Equations 5.63, 5.64, and 5.65 appears to be, it is actually difﬁcult to estimate because of the following reasons 1. There is a large number of parameters that need to be estimated. In the 2 variable case there are 21 parameters plus the parameters in the mean equations. 2. There is no analytical solution for the maximization problem detailed in log- likelihood function of Equation 5.60. Numerical methods do not always ﬁnd the solution. 3. Because the conditional variance need to be positive, we need to impose restric- tions that are more complicated than in the univariate case. A number of solutions have been proposed to circumvent these problems. A popular solution is to use a diagonal system such that hi jt contains only lags of itself and the cross products of εit ε jt . For example,

2 h11 t = c10 + α11 ε1t 1 + β11 h11 t 1 (5.66) − − h12 t = c20 + α22 ε1t 1ε2t 1 + β22 h12 t 1 (5.67) − − − 2 h22 t = c30 + α33 ε2t 1 + β33 h22 t 1 (5.68) − − While this speciﬁcation is easier to estimate, there are no interactions among the variances. Another popular solution is the constant-conditional-correlation-GARCH (CCC- GARCH). This model restricts the correlation coefﬁcients to be constant. Hence, for each i = j, the model assumes hi jt = ρi j hiit h j jt . While the variance terms are 6 not diagonalized, the covariance terms are always proportional to h h . For p ii j j jt example, building on the model in Equations 5.63, 5.64, and 5.65, p 5.7 Multivariate GARCH Models 63

h12 t = ρ12 h11 t h22 t (5.69)

This makes the the covariance equation consistp of only one parameter instead of seven. 1 Consider the estimation of the following diagonal vech multivariate GARCH model is Stata. Following the Stata manual, we have data on a secondary market rate of a six-month U.S. Treasury bill, tbill , and on Moody’s seasoned AAA cor- porate bond yield, bond . We model the ﬁrst-differences of both, tbill and bond in a VAR(1) with an ARCH(1) term, use http://www.stata-press.com/data/r11/irates4

dvech (D.bond D.tbill = LD.bond LD.tbill), arch(1)

Getting starting values (setting technique to bhhh) Iteration 0: log likelihood = 3569.2723 Iteration 8: log likelihood = 4221.6577

Diagonal vech multivariate GARCH model

Sample: 3 - 2456 Number of obs = 2454 Wald chi2(4) = 1183.52 Log likelihood = 4221.6577 Prob > chi2 = 0.0000

------| Coef. Std. Err. z P>|z| [95% Conf. Interval] ------+------D.bond | bond | LD. | .2967674 .0247149 12.01 0.000 .2483271 .3452077 | tbill | LD. | .0947949 .0098683 9.61 0.000 .0754533 .1141364 | _cons | .0003991 .00143 0.28 0.780 -.0024036 .0032019 ------+------D.tbill | bond | LD. | .0108373 .0301501 0.36 0.719 -.0482558 .0699304 | tbill | LD. | .4344747 .0176497 24.62 0.000 .3998819 .4690675 | _cons | .0011611 .0021033 0.55 0.581 -.0029612 .0052835 ------+------Sigma0 | 1_1 | .004894 .0002006 24.40 0.000 .0045008 .0052871 2_1 | .0040986 .0002396 17.10 0.000 .0036289 .0045683 2_2 | .0115149 .0005227 22.03 0.000 .0104904 .0125395 ------+------L.ARCH | 1_1 | .4514942 .0456835 9.88 0.000 .3619562 .5410323 2_1 | .2518879 .036736 6.86 0.000 .1798866 .3238893 2_2 | .843368 .0608055 13.87 0.000 .7241914 .9625446 ------The estimated equations are

1 Engel (2002) implemented a generalized CCC-GARCH that allowed the correlations to change over time. This is called the dynamic-conditional-correlation-GARCH (DCC-GARCH). Stata 12 can estimate CCC-GARCH, DCC-GARCH and VCC-GARCH models, the VCC stands for vary- ing conditional correlation. 64 5 Autoregressive Conditional Heteroskedasticity Models

∆bond = 0.0004 + 0.296 ∆bond t 1 + 0.095 ∆tbill t 1 + ε1t (5.70) − − ∆tbill = 0.0011 + 0.011 ∆bond t 1 + 0.434 ∆tbill t 1 + ε2t (5.71) − − 2 0.0048 ε1t 1 + 0.4515 h11 t 1 0.0040 ε1t 1ε2t 1 + 0.2519 h21 t 1 H = − − −2 − − . 0.0040 ε1t 1ε2t 1 + 0.2519 h12 t 1 0.0115 ε2t 1 + 0.8434 h22 t 1 − − − − − or

2 h11 t = 0.0048 ε1t 1 + 0.4515 h11 t 1 (5.72) − − h12 t = 0.0040 ε1t 1ε2t 1 + 0.2519 h12 t 1 (5.73) − − − 2 h22 t = 0.0115 ε2t 1 + 0.8434 h22 t 1 (5.74) − −

5.8 Supporting .do ﬁles

For Figure 5.4: use rgdp.dta, clear tsset date gen y = log(rgdp/l.rgdp) label variable rgdp "Real GDP" label variable y "Real GDP Growth Rate" label variable statadate "Date" twoway line rgdp statadate, scheme(sj) /// ytitle( "Real GDP" ) saving(grgdp, replace) twoway line y statadate, scheme(sj) /// ytitle( "Real GDP Growth Rate" ) saving(gy, replace) gr combine grgdp.gph gy.gph, col(1) /// iscale(0.7) fysize(100) /// title( "Real GDP and Real GDP Growth Rate" )