1 2. ARCH and GARCH PROCESSESThe GARCH Model 2. ARCH and GARCH PROCESSES
2 Other Methods Let et = demeanded daily return. One method is to use 30-day moving average 𝜀 𝑡 /30 Implicit volatility Logs can stabilize volatility
3 One simple strategy is to model the conditional variance as an AR(q) process using squares of the estimated residuals In contrast to the moving average, here the weights need not equal1/30 (or 1/N). The forecasts are:
4 Properties of the Simple ARCH ModelSince vt and et-1 are independent: Eet = E[ vt(a0 + a1et-12 )]1/2 ] = 0 Et-1et = Et-1vtEt-1 [a0 + a1et-12 ]1/2 ] = 0 Eet et-i = 0 ( i ≠ 0) Eet 2= E[ vt 2(a0 + a1et-12 )] = a0 + a1E(et-1) 2 = a0/( 1 - a1 ) Et-1et 2= Et-1 [ vt 2(a0 + a1(et-1 ) 2 )] = a0 + a1(et-1) 2
5 ARCH Interactions with the MeanConsider: yt =a0 + a1yt–1 + et Var(ytyt–1, yt–2, …) = Et–1(yt – a0 – a1yt–1)2 = Et–1(et)2 = a0 + a1(et–1)2 Unconditional Variance: Since:
6 Figure 3.7: Simulated ARCH ProcessesWhite Noise Process vt yt = 0.2yt-1 + εt yt = 0.9yt-1 + εt
7 Other Processes ARCH(q) GARCH(p, q)The benefits of the GARCH model should be clear; a high-order ARCH model may have a more parsimonious GARCH representation that is much easier to identify and estimate. This is particularly true since all coefficients must be positive.
8 Testing For ARCH Step 1: Estimate the {yt} sequence using the "best fitting" ARMA model (or regression model) and obtain the squares of the fitted errors . Consider the regression equation: If there are no ARCH effects a1 = a2 = … = 0 All the coefficients should be statistically significant No simple way to distinguish between various ARCH and GARCH models
9 Testing for ARCH II Examine the ACF of the squared residuals:Calculate and plot the sample autocorrelations of the squared residuals Ljung–Box Q-statistics can be used to test for groups of significant coefficients. Q has an asymptotic c2 distribution with n degrees of freedom
10 Engle's Model of U.K. InflationLet p = inflation and r = real wage pt = pt– pt–4 – 0.404pt– rt–1 + et the variance of t is ht = 8.9 x 10-5 pt = pt– pt–4 – 0.325pt– rt–1 + et ht = 1.4 x 10– ( L L L3 ) (et-1 )2 (8.5 x 10-6) (0.298)
11 4. THREE EXAMPLES OF GARCH MODELSA GARCH Model of Oil Prices Volatility Moderation A GARCH Model of the Spread 4. THREE EXAMPLES OF GARCH MODELS
12 A GARCH Model of Oil PricesUse OIL.XLS to create pt = 100.0*[log(spott) − log(spott−1)]. The following MA model works well: pt = t t− t−3 The McLeod–Li (1983) test for ARCH errors using four lags: The F-statistic for the null hypothesis that the coefficients 1 through 4 all equal zero is With 4 numerator and 1372 denominator degrees of freedom, we reject the null hypothesis of no ARCH errors at any conventional significance level.
13 The GARCH(1,1) Model pt = 0.130 + t + 0.225t−1ht = (t−1) ht−1
14 Volatility ModerationUse the file RGDP.XLS to construct the growth rate of real U.S. GDP yt = log(RGDPt/RGDPt1). A reasonable model is: yt = yt1 + t (6.80) (6.44) After checking for GARCH errors yt = yt1 + t (7.50) (6.76) ht = 1.10x – 8.76x105Dt (7.87) (2.89) (–6.14) The intercept of the variance equation was 1.10 104 prior to 1984Q1 and experienced a significant decline to 2.22 105 (= 1.10 104 – 8.76 105) beginning in 1984Q1.
15 Figure 3.8: Forecasts of the Spread_____ 1-Step Ahead Forecast ± 2 Conditional Standard Deviations
16 Section 5 A Garch Model of RISK
17 Holt and Aradhyula (1990) The study examines the extent to which producers in the U.S. broiler (i.e., chicken) industry exhibit risk averse behavior. The supply function for the U.S. broiler industry takes the form: qt = a0 + a1pet - a2ht - a3pfeedt-1 + a4hatcht-1 + a5qt-4 + ε1t qt = quantity of broiler production (in millions of pounds) in t; pet = Et-1pt = expected real price of broilers at t ht = expected variance of the price of broilers in t pfeedt-1 = real price of broiler feed (in cents per pound) at t-1; hatcht-1 = hatch of broiler-type chicks in t-1; ε1t = supply shock in t; and the length of the time period is one quarter.
18 Note the negative effect of the conditional variance of price on broiler supply.The timing of the production process is such that feed and other production costs must be incurred before output is sold in the market. Producers must forecast the price that will prevail two months hence. The greater pte, the greater the number of chicks that will be fed and brought to market. If price variability is very low, these forecasts can be held with confidence. Increased price variability decreases the accuracy of the forecasts and decreases broiler supply. Risk-averse producers will opt to raise and market fewer broilers when the conditional volatility of price is high.
19 The Price equation qt = 2.767pet - 0.521Et-1ht - 4.325pfeedt-1 ( L L L L4)pt = ε2t ht = ε2t ht-1 The paper assumes producers use these equations to form their price expectations. The supply equation: qt = 2.767pet Et-1ht pfeedt-1 hatcht pt-4 + ε1t
20 Section 6: THE ARCH-M MODELEngle, Lilien, and Robins let yt = mt + et where yt = excess return from holding a long-term asset relative to a one-period treasury bill and mt is a time-varying risk premium: Et–1yt = mt The risk premium is: mt = b + dht , d > 0
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22 7. Additional ProPerties of GARCH ProcessDiagnostic Checks for Model Adequacy Forecasting the Conditional Variance 7. Additional ProPerties of GARCH Process
23 Forecasting with the GARCH(1, 1)The 1-step ahead forecast can be calculated directly. Etht+j = 0 + (1 + 1)Etht+j–1
24 Volatility PersistenceLarge values of both 1 and 1 act to increase the conditional volatility but they do so in different ways.
25 Assessing the Fit Standardized residuals: AIC' = –2 ln L + 2nSBC' = –2ln L + n ln(T) where L likelihood function and n is the number of estimated parameters.
26 Diagnostic Checks for Model AdequacyIf there is any serial correlation in the standardized residuals--the {st} sequence--the model of the mean is not properly specified. To test for remaining GARCH effects, form the Ljung–Box Q-statistics of the squared standardized residuals.
27 Maximum likelihood estimationSection 8 Maximum likelihood estimation
28 Maximum Likelihood and a RegressionUnder the usual normality assumption, the log likelihood of observation t is: With T independent observations: We want to select b and s2 so as to maximize L Nonlinear Estimation
29 The Likelihood Function with ARCH errorsFor observation t Now substitute for et and ht There are no analytic solutions to the first-order conditions for a maximum. Nonlinear Estimation
30 OTHER MODELS OF CONDITIONAL VARIANCESection 9 OTHER MODELS OF CONDITIONAL VARIANCE
31 IGARCH Etht+1 = 0 + ht and Etht+j = j0 + htThe IGARCH Model: Nelson (1990) argued that constraining 1 + 1 to equal unity can yield a very parsimonious representation of the distribution of an asset’s return. Etht+1 = 0 + ht and Etht+j = j0 + ht
32 RiskMetrics ht = a(et-1)2 + ( 1 - a )(ht-1) ; a > 0.9RiskMetrics assumes that the continually compounded daily return of a portfolio follows a conditional normal distribution. The assumption is that: rt|It-1 ~ N(0, ht) ht = a(et-1)2 + ( 1 - a )(ht-1) ; a > 0.9 Note: (Sometimes rt-1 is used). This is an IGARCH without an intercept. Suppose that a loss occurs when the price falls. If the probability is 5%, RiskMetrics uses 1.65ht+1 to measure the risk of the portfolio. The Value at Risk (VaR) is: VaR = Amount of Position x 1.65(ht+1)1/2 and for k days is VaR(k) = Amount of Position x 1.65(k ht+1)1/2
33 Why Do We Care About ARCH Effects?We care about the higher moments of the distribution. The estimates of the coefficients of the mean are not correctly estimated if there are ARCH errors. Consider 3. We want to place conditional confidence intervals around our forecasts (see next page) Nonlinear Estimation
34 Example from Tsay Suppose rt = 0.00066 – 0.0247rt-2 + etht = (et-1) (ht-1) Given current and past returns, suppose: ET(rT+1) = and ET(hT+1) = The 5% quantile is – *( )1/2 = The VaR for a portfolio size of $10,000,000 with probability 0.05 is ($10,000,000 )( ) = $287,700 i.e., with 95% chance, the potential loss of the portfolio is $287,700 or less.
35 Models with Explanatory VariablesTo model the effects of 9/11 on stock returns, create a dummy variable Dt equal to 0 before 9/11 and equal to 1 thereafter. Let ht = 0 + 1 + 1ht–1 + Dt
36 TARCH and EGARCH Glosten, Jaganathan and Runkle (1994) showed how to allow the effects of good and bad news to have different effects on volatility. Consider the threshold-GARCH (TARCH) process
37 Expected Volatility (Etht+1)Figure 3.11: The leverage effect Expected Volatility (Etht+1) c b a t New Information
38 The EGARCH Model 1. The equation for the conditional variance is in log-linear form. Regardless of the magnitude of ln(ht), the implied value of ht can never be negative. Hence, it is permissible for the coefficients to be negative. 2. Instead of using the value of , the EGARCH model uses the level of standardized value of t–1 [i.e., t–1 divided by (ht–1)0.5]. Nelson argues that this standardization allows for a more natural interpretation of the size and persistence of shocks. After all, the standardized value of t–1 is a unit-free measure. 3. The EGARCH model allows for leverage effects. If t–1/(ht–1)0.5 is positive, the effect of the shock on the log of the conditional variance is 1 + 1. If t–1/(ht–1)0.5 negative, the effect of the shock on the log of the conditional variance is –1 + 1. 4. Although the EGARCH model has some advantages over the TARCH model, it is difficult to forecast the conditional variance of an EGARCH model.
39 Testing for Leverage Effects1. If there are no leverage effects, the squared errors should be uncorrelated with the level of the error terms 2. The Sign Bias test uses the regression equation of the form where dt–1 is to 1 if et-1 < 0 and is equal to zero if et-1 0. 3. The more general test is dt–1st–1 and (1 – dt–1)st–1 indicate whether the effects of positive and negative shocks also depend on their size. You can use an F-statistic to test the null hypothesis a1 = a2 = a3 = 0.
40 ESTIMATING THE NYSE U.S. 100 INDEXSection 10 ESTIMATING THE NYSE U.S. 100 INDEX
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42 The estimated model rt = et – 0.058rt-1 0.038rt AIC = , SBC = (2.82) (–3.00) (1.91) ht = ( 𝜀 𝑡 ) ht-1 (4.91) (9.59) (98.31) Instead, if we use a t-distribution, we obtain rt = et – 0.062rt-1 0.045rt AIC = , SBC = (5.24) (–3.77) (2.64) ht = ( 𝜀 𝑡 ) ht-1 (3.21) (8.58) (95.24)
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45 Section 11 Multivariate GARCH
46 11. MULTIVARIATE GARCH If you have a data set with several variables, it often makes sense to estimate the conditional volatilities of the variables simultaneously. Multivariate GARCH models take advantage of the fact that the contemporaneous shocks to variables can be correlated with each other. Equation-by-equation estimation is not efficient Multivariate GARCH models allow for volatility spillovers in that volatility shocks to one variable might affect the volatility of other related variables
47 Suppose there are just two variables, x1t and x2tSuppose there are just two variables, x1t and x2t. For now, we are not interested in the means of the series Consider the two error processes 1t = v1t(h11t)0.5 2t = v2t(h22t)0.5 Assume var(v1t) = var(v2t) = 1, so that h11t and h22t are the conditional variances of 1t and 2t, respectively. We want to allow for the possibility that the shocks are correlated, denote h12t as the conditional covariance between the two shocks. Specifically, let h12t = Et-11t2t.
48 The VECH Model A natural way to construct a multivariate GARCH(1, 1) is the vech model h11t = c10 + 11 (1t-1)2 + 121t-12t-1 + 13 (2t-1)2 + 11h11t–1 + 12h12t–1 + 13h22t-1 h12t = c20 + 21 (1t-1)2 + 221t-12t-1 + 23 (2t-1)2 + 21h11t–1 + 22h12t–1 + 23h22t-1 h22t = c30 + 31 (1t-1)2 + 321t-12t-1 + 33 (2t-1)2 + 31h11t–1 + 32h12t–1 + 33h22t-1 The conditional variances (h11t and h22t) and covariance depend on their own past, the conditional covariance between the two variables (h12t), the lagged squared errors, and the product of lagged errors (1t-12t-1). Clearly, there is a rich interaction between the variables. After one period, a v1t shock affects h11t, h12t, and h22t.
49 ESTIMATION Multivariate GARCH models can be very difficult to estimate. The number of parameters necessary can get quite large. In the 2-variable case above, there are 21 parameters. Once lagged values of {x1t} and {x2t} and/or explanatory variables are added to the mean equation, the estimation problem is complicated. As in the univariate case, there is not an analytic solution to the maximization problem. As such, it is necessary to use numerical methods to find that parameter values that maximize the function L. Since conditional variances are necessarily positive, the restrictions for the multivariate case are far more complicated than for the univariate case. The results of the maximization problem must be such that every one of the conditional variances is always positive and that the implied correlation coefficients, ij = hij/(hiihjj)0.5, are between –1 and +1.
50 The diagonal vech One set of restrictions that became popular in the early literature is the so-called diagonal vech model. The idea is to diagonalize the system such that hijt contains only lags of itself and the cross products of itjt. For example, the diagonalized version of (3.42) (3.44) is h11t = c10 + 11(1t-1)2 + 11h11t–1 h12t = c20 + 221t-12t-1 + 22h12t-1 h22t = c30 + 33(2t-1)2 + 33h22t–1 Given the large number of restrictions, model is relatively easy to estimate. Each conditional variance is equivalent to that of a univariate GARCH process and the conditional covariance is quite parsimonous as well. The problem is that setting all ij = ij = 0 (for i j) means that there are no interactions among the variances. A 1t-1 shock, for example, affects h11t and h12t, but does not affect the conditional variance h2t.
51 THE BEKK Engle and Kroner (1995) popularized what is now called the BEK (or BEKK) model that ensures that the conditional variances are positive. The idea is to force all of the parameters to enter the model via quadratic forms ensuring that all the variances are positive. Although there are several different variants of the model, consider the specification Ht = C'C + A't-1t-1'A + B'Ht-1B where for the 2-variable case If you perform the indicated matrix multiplications you will find
52 THE BEK II In general, hijt will depend on the squared residuals, cross-products of the residuals, and the conditional variances and covariances of all variables in the system. The model allows for shocks to the variance of one of the variables to “spill-over” to the others. The problem is that the BEK formulation can be quite difficult to estimate. The model has a large number of parameters that are not globally identified. Changing the signs of all elements of A, B or C will have effects on the value of the likelihood function. As such, convergence can be quite difficult to achieve.
53 The BEKK (and Vech) as a VAR
54 Constant Conditional Correlations (CCC)As the name suggests, the (CCC)model restricts the correlation coefficients to be constant. As such, for each i j, the CCC model assumes hijt = ij(hiithjjt)0.5. In a sense, the CCC model is a compromise in that the variance terms need not be diagonalized, but the covariance terms are always proportional to (hiithjjt)0.5. For example, a CCC model could consist of (3.42), (3.44) and h12t = 12(h11th22t)0.5 Hence, the covariance equation entails only one parameter instead of the 7 parameters appearing in (3.43).
55 EXAMPLE OF THE CCC MODELBollerslev (1990) examines the weekly values of the nominal exchange rates for five different countries--the German mark (DM), the French franc (FF), the Italian lira(IL), the Swiss franc (SF), and the British pound (BP)--relative to the U.S. dollar. A five-equation system would be too unwieldy to estimate in an unrestricted form. For the model of the mean, the log of each exchange rate series was modeled as a random walk plus a drift yit = i + it (3.45) where yit is the percentage change in the nominal exchange rate for country i, Ljung-Box tests indicated each series of residuals did not contain any serial correlation
56 hijt = ij(hiithjjt)0.5 (i j)Next, he tested the squared residuals for serial dependence. For example, for the British pound, the Q(20)-statistic has a value of ; this is significant at any conventional level. Each series was estimated as a GARCH(1, 1) process.The specification has the form of (3.45) plus hiit = ci0 + ii (it-1)2 + iihiit–1 (i = 1, …, 5) hijt = ij(hiithjjt) (i j) The model requires that only 30 parameters be estimated (five values of i, the five equations for hiit each have three parameters, and ten values of the ij). As in a seemingly unrelated regression framework, the system-wide estimation provided by the CCC model captures the contemporaneous correlation between the various error terms.
57 RESULTS The estimated correlations for the period during which the European Monetary System (EMS) prevailed are DM FF IL SF 0.932 0.886 0.876 SW 0.917 0.866 0.816 BP 0.674 0.678 0.622 0.635 It is interesting that correlations among continental European currencies were all far greater than those for the pound. Moreover, the correlations were much greater than those of the preEMS period. Clearly, EMS acted to keep the exchange rates of Germany, France, Italy and Switzerland tightly in line prior to the introduction of the Euro.
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59 For the second step, you should check the squared residuals for the presence of GARCH errors. Since we are using daily data (with a five-day week), it seems reasonable to begin using a model of the form The sample values of the F-statistics for the null hypothesis that 1 = … = 5 = 0 are 43.36, 89.74, and for the Euro, BP and SW, respectively. Since all of these values are highly significant, it is possible to conclude that all three series exhibit GARCH errors. The sample values of the F-statistics for the null hypothesis that 1 = … = 5 = 0 are 43.36, 89.74, and for the Euro, BP and SW, respectively. Since all of these values are highly significant, it is possible to conclude that all three series exhibit GARCH errors.
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65 Appendix: The Log Likelihood Functionwhere 12 is the correlation coefficient between 1t and 2t; 12 = h12/(h11h22)0.5. Now define where t = ( 1t, 2t )', and | H | is the determinant of H.
66 Now, suppose that the realizations of {t} are independent, so that the likelihood of the joint realizations of 1, 2, … T is the product in the individual likelihoods. Hence, if all have the same variance, the likelihood of the joint realizations is
67 MULTIVARIATE GARCH MODELSFor the 2-variable The form of the likelihood function is identical for models with k variables. In such circumstances, H is a symmetric k x k matrix, t is a k x 1 column vector, and the constant term (2) is raised to the power k.
68 The vech Operator The vech operator transforms the upper (lower) triangle of a symmetric matrix into a column vector. Consider the symmetric covariance matrix vech(Ht) = [ h11t, h12t, h22t ] Now consider t = [1t, 2t]. The product tt = [1t, 2t][1t, 2t] is
69 If we now let C = [ c1, c2, c3 ], A = the 3 x 3 matrix with elements ij, and B = the 3 x 3 matrix with elements ij, we can write vech(Ht) = C + A vech(t-1t-1) + Bvech(Ht-1) it should be clear that this is precisely the system represented by (3.42) (3.44). The diagonal vech uses only the diagonal elements of A and B and sets all values of ij = ij = 0 for i j.
70 Constant Conditional CorrelationsNow, if h11t and h22t are both GARCH(1, 1) processes, there are seven parameters to estimate (the six values of ci, ii and ii and 12).
71 Dynamic Conditional CorrelationsSTEP 1: Use Bollerslev’s CCC model to obtain the GARCH estimates of the variances and the standardized residuals STEP 2: Use the standardized residuals to estimate the conditional covariances. Create the correlations by smoothing the series of standardized residuals obtained from the first step. Engle examines several smoothing methods. The simplest is the exponential smoother qijt = (1 )sitsjt + qijt-1 for < 1. Hence, each {qiit} series is an exponentially weighted moving average of the cross-products of the standardized residuals. The dynamic conditional correlations are created from the qijt as rijt = qijj/(qijtqjjt)0.5