Economic ForecastingAnthony Tay Chapter 10 Page 1

Economic Forecasting Singapore Management UniversityAnthony Tay

Modeling and Forecasting Volatility

Here are plots of the daily values of the Singapore Straits Times Index (STI) and the corresponding firstdifferences of log STI (all data in stii.wf1):

smpl @allline(m) stii dl_stii

Visually, the price (or log price) levels appear very much like a random walk, and there is often little cyclicalitybeyond this. On the other hand, the first differences often display ‘volatility clustering’: lengthy periods of highvolatility alternating with periods of low volatility. Volatility clustering is a feature shared by many financialreturn series, and it suggests that, although price levels are not forecastable, the volatility of returns may beforecastable. This would be useful for risk management purposes, options pricing, among other applications.

The model that we will study in this chapter is the Autoregressive Conditional Heteroskedasticity Model(ARCH), so named because it models the conditional volatility (heteroskedasticity) of a variable – in our casestock returns – as a function of the history of the variable. We will study the simplest ARCH model in somedetail, and then talk about generalizations. Apart from finance, ARCH models are also useful in macroeconomicapplications.

The pure ARCH(1) model ist ty ,

1/2 , ~ (0,1)iid

t t t th u u N

21 1t th .

We can write the pure ARCH(1) model with non-normal errors for tu , but we assume normality for the timebeing (we sometimes refer to this as the Gaussian ARCH(1) model). The equation t ty is the ‘mean equation’.More generally, we can have a non-trivial mean equation. For instance, the following is an AR(1)-ARCH(1)model:

0 1 1t t ty y ,1/2 , ~ (0,1)

iid

t t t th u u N

21 1t th

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For the time being, we focus on volatility, and assume no explanatory variables in the mean equation.

To gain some intuition for the pure ARCH(1) model, we discuss some of its properties.

Conditional Mean1 2 1 2

1/21 2

1/21 2

[ | , ,...] [ | , ,...][ | , ,...]

[ | , ,...]0

t t t t t t

t t t t

t t t t

E y y y E y yE h u y yh E u y y

i.e. the pure ARCH model is not designed to use information in past ty to explain or forecast future values of

ty . This result also implies that the unconditional mean is zero, since the conditional mean is zero. The fact thatthe conditional mean is zero also implies that ty is a serially uncorrelated process:

1 1 1 1 1[ [ | ,...]] [ [ | ,...]] [ .0] 0t t t t t t tE E y y y E y E y y E y .

This pure ARCH(1) model therefore does not describe any sort of cyclicality or predictability in the mean. Themodel, however, does not describe an i.i.d. process. In particular, there is predictability in the variance of theprocess.

Conditional Variance

1 2 1 21/2

1 2

1 2

var[ | , ,...] var[ | , ,...]var[ | , ,...]

var[ | , ,...]

t t t t t t

t t t t

t t t t

t

y y y y y yh u y y

h u y yh

i.e. th is the conditional variance of ty , and this conditional variance depends on the past values of ty , asspecified by the ‘variance equation’ 2

1 1t th . We usually assume and 1 to be positive. This is toensure that we always have positive conditional variances.

We can now see how the ARCH model captures volatility clustering. If we are presently (at time t ) in a volatilestate ( th large), it is likely that the time t realization t ty will be large in absolute value (positive or negative,doesn’t matter). If a large t is indeed realized, then 2

1 1t th will in turn be large, which then meansthat 1ty will again tend to be large in absolute terms, and so on. If by chance, despite a large th , a small valueof 1ty is realized, then the process switches to low volatility period because 2

2 1 1t th will be small.Small volatility means small realizations are more likely, so the low volatility will tend to continue into the nextperiod.

Once again, the pure ARCH model focuses on conditional variance, and doesn’t model conditional mean.ARCH models are usually used in conjunction with conditional mean models, with the error terms described asARCH processes. For example, in the AR(1)-ARCH(1) model given earlier. In that model, the conditional meanof ty given past data is 1 0 1 1[ | ]t t tE y y y , while the conditional variance remains as 2

1 1t th .

Economic ForecastingAnthony Tay Chapter 10 Page 3

Incidentally, the unconditional variance of a pure ARCH model is 1/ (1 ) and is therefore constant andfinite if 1 1 :

2[ ] [ ] [ ]t t tVar y E y E h 2 2 21 1 1 1 1 1[ ] [ ] [ ]t t tE y E y E h

2 2 2 2 21 1 2 1 1 2 1 1 2[ ] [ ] [ ]t t tE y E y E h

...

21 1

1

(1 ...)

1

.

where we have used the fact that 2 2 2 21 1[ ] [ ] [ [ | ]] [ [ | ]] [ ]t t t t t t t t t tE y E h u E E h u y E h E u y E h .

Volatility clustering can also account, at least partially, for a feature of the unconditional distribution of financialreturns called “excess kurtosis” or “fat-tails”. The following is a histogram of the DL_STII series.

dl_stii.hist

The kurtosis coefficient if a variable is the standardized fourth central moment4

2 2[( [ ]) ]

( )E Y E Y

The kurtosis coefficient of a normal distributed random variable is three. In constrast, the empirical distributionof DL_STII displays very large kurtosis. This is partially due to the presence of volatility clustering, whichimplies that we can expect to see large absolute values of ty cluster together. So once we have a largeobservation, we tend to see even more large values of ty , and the number of large observations will end upbeing more than would be expected from a normal distribution. Mathematically, we can show that ARCHprocesses imply excess kurtosis, even if it is conditionally normally distributed. (We omit the proof of thisstatement in this course).

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M ean 0.000212M edian 0.000000M ax im um 0.154809M inim um -0.291826S td. Dev. 0.013707S kewnes s -1.546020K urtos is 44.60801

Jarque-B era 507222.0P robability 0.000000

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Another feature of series that display volatility clustering is that the square of the residuals will often showARMA-type properties. Ignoring the MA(1), which is negligible in the dl_stii series, we regress dl_stii ona constant, and compute the correllogram of the square of the residuals:

equation eq1.ls dl_stii ceq1.correlsq

ARCH(1)-type errors possesses this property. Suppose

0 1 1t t ty y , 1| | 1

1/2t t th u , 2

1 1t th

(In the example above, we didn’t have the AR(1), just a constant.) The variance equation can be rewritten as2 2 2

1 1( )t t t th which takes the form of an AR(1) in squares of 2t . (Add 2

t on both sides, bring

th over, and rearrange; view 2t th as a zero-mean error term.)

To illustrate and further motivate the ARCH(1) model, we carry out a forecasting exercise similar to what wehave been doing up to this point, using data until Dec 2007 and forecasting through to end-Oct 2011.

You can show that log(STII) is clearly a unit root process, sowe work with DL_STII (first difference of logs of STII). Thecorrellogram show weak dynamics, which seem to beadequately captured by an MA(1) although there is almost nofit at all.

smpl @first 12/31/2007equation eq1.ls dl_stii c ma(1)eq1.correl

Dependent Variable: DL_STII residual correllogram

Sample (adjusted): 1/07/1985 12/31/2007

Included observations: 5996 after adjustments

Convergence achieved after 6 iterations

MA Backcast: 1/04/1985

Variable Coefficient Std. Error t-Statistic Prob.

C 0.000286 0.0002 1.43213 0.1522

MA(1) 0.165015 0.01274 12.95216 0

R-squared 0.02449

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The following diagram show the in-sample fit as well as the one-step ahead forecasts

(The commands can be found in forecast_10.prg) The in-sample fit is negligible, as is easily seen in the leftfigure, and in the 2R of the MA(1) regression. Obviously the MA(1) isn’t the most important part of the series.The forecasts show something very interesting, especially with the +/- 2 standard error bounds. There is a littlebit of variation in the movement in the bounds due to the movements in the forecasts (red) but the width of thebounds is more or less equal throughout the forecast sample. This is because our model has assumed a constantvariance – there has been no attempt to forecast volatility. In contrast, the series displays substantial volatilityclustering. This has led to too narrow of a band during volatile periods, and a band that is too wide in calmperiods. We get too many observations outside the band during volatile periods, and ‘too few’ during calmperiods.

The standardized residuals also show evidence of volatility clustering:

In addition to the correllogram of squared residuals, we can test for ARCH errors using Engle’s LM test:Suppose wish to see if the error term displays ARCH (or variant) dynamics. Estimate this mean equation,calculate the residuals t̂ , and regress 2

t̂ on m lags of 2t̂ , i.e., regress 2

t̂ on 21t̂ , 2

2t̂ , …, 2ˆt m andcalculate the TR2 for this equation, where T is the number of observations. This statistic follows a chi-squaredistribution with m degrees of freedom under the null of no conditional heteroskedasticity in the errors.

eq1.archtest

Heteroskedasticity Test: ARCH

F-statistic 574.499 Prob. F(1,5993) 0.000

Obs*R-squared 524.419 Prob. Chi-Square(1) 0.000

Test Equation:

Dependent Variable: RESID^2

Variable Coefficient Std. Error t-Statistic Prob.

C 0.000 1.49E-05 8.305 0.000

RESID^2(-1) 0.296 0.01234 23.969 0.000

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The test clearly rejects the null of no conditional heteroskedasticity. You can try with more lags. Of course,volatility clustering was clearly visible from the plot, and the result of the formal test is of no surprise.

We will proceed to fit an ARCH model to the data, but first a few comments on the estimation and forecastingof ARCH processes. Estimation of the coefficients is by maximum likelihood. Forecasting an ARCH process isnot as straightforward as AR processes. We illustrate with a pure ARCH process:

t ty 1/2 , ~ (0,1)

iid

t t t th u u N 2

1 1t th

The one-step ahead forecast 1|t ty of y is simply 1| 0t ty . To compute the one-step ahead forecast of thevariance of 1ty , we use the conditional 1var[ | ,...]t ty y :

2 21 1 1 1var[ | ,...]t t t t ty y h y

To get the two-step ahead forecast for the variance, note that

2 22

2

2 2

2 2

1

var[ | ,...] var[ | ,...][ | ,...][ | ,...][ | ,...] [ | ,...]

t t t t

t t

t t t

t t t t

y y yE yE h u yE u y E h y

Since 2 22 1 1 1 1 1t t t th h u ,

1 1[ | ,...]t t tE h y h , and2

1[ | ,...] 1t tE u y ,

we have 2 22 1 1 1 1 1 1var[ | ,...] ( ) (1 )t t t t ty y h y y .

In general,1

1 1 1 1 1var[ | ,...] ... (1 ... )k kt k t t k ty y h y ,

and as k , 1var[ | ,...] / (1 )t k ty y as long as 1| | 1 . That is, the k -step ahead forecast of thevariance of ty converges to the unconditional variance as the forecast horizon increases toward infinity.

Note: if there was a conditional mean equation, then the same variance forecast equations apply, exceptthat now we cannot substitute t with ty . Instead, the one-step ahead forecast of the variance is

21 1 1var[ | ,...]t t t ty y h

where conditional mean [ ]t t ty y .

Economic ForecastingAnthony Tay Chapter 10 Page 7

The model that we will fit to the _DL STII data is not a pure ARCH model, but an MA(1)-ARCH(1).

equation.eq2.arch(1,0) dl_stii c ma(1)

Dependent Variable: DL_STIIMethod: ML - ARCH (Marquardt) - Normal distributionSample (adjusted): 1/07/1985 12/31/2007Included observations: 5996 after adjustments

Variable Coefficient Std. Error z-Statistic Prob.C 0.000519 0.000156 3.325219 0.0009MA(1) 0.17774 0.006786 26.19121 0.0000

Variance EquationC 0.000107 7.65E-07 140.2959 0.0000RESID(-1)^2 0.347641 0.011511 30.20077 0.0000

R-squared 0.024123 Mean dependent var 0.000286Adjusted R-squared 0.02396 S.D. dependent var 0.013433S.E. of regression 0.013271 Akaike info criterion -6.031292Sum squared resid 1.055603 Schwarz criterion -6.026823Log likelihood 18085.81 Hannan-Quinn criter. -6.02974Durbin-Watson stat 2.036721

Inverted MA Roots -0.18

There are no important changes to the estimate of the mean equation. The coefficient of 21t̂ in the variance

equation is clearly significant. The implied, or ‘fitted’ standard deviation of ty is shown on the left, and thefitted and actuals are shown on the right with the +/- 2 standard deviations.

eq2.makegarch h_hat genr dl_stii_hat = dl_stii-residline h_hat^0.5 genr upp = dl_stii_hat + 2*h_hat^0.5

genr low = dl_stii_hat - 2*h_hat^0.5line dl_stii dl_stii_hat upp low

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The one-step ahead forecasts are shown below.

There is some improvement in that we are capturing some of the changes in volatility, but clearly the model stilldoes not properly describe and forecast volatility – there does not seem to be enough flexibility, and for longperiods the intervals are still far too wide. We will turn to a better model in a moment. For now, we ask: howdo we evaluate the fit of ARCH model?

The 2R says nothing about the volatility fit – it only tells us how well the mean equation fits the data. Note thatthe 2R here is actually smaller than in the pure MA(1) estimation output. In fact, with pure ARCH models wewill usually get negative 2R s (why?) To evaluate how well the ARCH model fits the volatility, we make useof the fact that in ARCH models,

1/2 , ~ (0,1)iid

t t t th u u N

Our ARCH model produces estimates of 1/2th (this is our h_hat^0.5 series plotted earlier). We also have the

residuals t̂ from the MA(1) model that was fit to the data. We can therefore compute

1/2

ˆˆ ˆt

tt

uh

which are called ‘standardized residuals’. If our volatility was well-estimated (if t̂h are good estimates of thetrue th ), and assuming the mean equation is correctly specified, then the standardized residuals should be iid.The correllogram of the standardized residuals, and especially their squares, should not show any dynamics. Wehave in our example:

eq2.correl eq2.correlsqcorrellogram of standardized residuals correllogram of squared residuals

It appears the mean equation has done a reasonable job in describing the conditional mean of the process (lookat the Q-stats on the left) whereas the volatility equation could use some improvements (again Q-stats, on theright-hand figure) although the correlations appear small.

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The histogram of the standardized residuals also tells us something useful about the model.

eq2.hist

We see that kurtosis is much smaller than for the raw data. Thus some excess kurtosis was removed bystandardization by 1/2

th . At least part of the kurtosis that was due to volatility clustering, and standardizing thevariance by dividing throughout by the standard deviation therefore reduces the kurtosis.

The standardized residuals are still not normally distributed – there is still excess kurtosis which suggests thatthe assumption of normality might not be appropriate.

GARCH Models As in our example, the ARCH(1) specification for the variance term is not usually notrich enough to capture all the dynamics in volatility. While we can increase the number of lags in the varianceequation, there is a modification that we can make that enables much richer patterns of volatility, and that is byadding lagged volatility to the specification. This gives the Generalized ARCH(1,1) model, or the GARCH(1,1).

t ty

1/2 , ~ (0,1)iid

t t t th u u N 2

1 1 1 1t t th h

Consider the variance equation for the GARCH(1,1) process2

1 1 1 1t t th h

Recursively substituting backwards gives2 2

1 1 1 1 2 1 22 2 2

1 1 1 1 1 2 1 2

2 2 2 21 1 1 1 1 1 2 1 2 3

( )(1 )

...(1 ...) ...

t t t t

t t t

t t t

h hh

so the conditional variance is allowed to depend on an infinite number of lags, instead of one lag, with just oneadditional parameter. As with the pure-ARCH model, the pure-GARCH(1,1) model is a zero-mean processes(conditionally and unconditionally), and the conditional variance of ty given past data is th . The unconditionalvariance of the GARCH(1,1) process is 1 1/ (1 ) . We can, of course, add more lags of squared errors, ormore lags of th , or both, if necessary.The output for a GARCH(1,1) model is given below, together with the correllogram of the squared standardizedresiduals, and the histogram of the standardized residuals. There is some improvement in the correllogram of

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M ean -0.017997M edian -0.022214M ax im um 14.30354M inim um -10.44520S td. Dev. 0.999893S kewnes s -0.104807K urtos is 18.23639

Jarque-B era 58009.18P robability 0.000000

Economic ForecastingAnthony Tay Chapter 10 Page 10

the squared standardized residuals, and from the histogram of the standardized residuals we see that includingthe GARCH dynamics in the volatility has further reduced the kurtosis slightly, although it is still substantiallyabove 3.

equation eq3.arch(1,1) dl_stii c ma(1)

Dependent Variable: DL_STIIMethod: ML - ARCH (Marquardt) - Normal distribution eq3.correlsqSample (adjusted): 1/07/1985 12/31/2007Included observations: 5996 after adjustmentsConvergence achieved after 36 iterations

Variable Coefficient Std. Error z-Statistic Prob.C 0.001 0.000 4.712 0.000MA(1) 0.145 0.014 10.079 0.000

Variance EquationC 5.35E-06 2.51E-07f 21.342 0.000RESID(-1)^2 0.139 0.004 34.735 0.000 eq3.histGARCH(-1) 0.838 0.004 206.811 0.000

R-squared 0.024 Mean dependent var 0.000Adjusted R-squared 0.023 S.D. dependent var 0.013S.E. of regression 0.013 Akaike info criterion -6.181Sum squared resid 1.056 Schwarz criterion -6.175Log likelihood 18534.360 Hannan-Quinn criter. -6.179Durbin-Watson stat 1.973 Inverted MA Roots -0.150

The one-step ahead forecasts over the forecast sample shows a substantial improvement:

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Jarque-B era 45237.79P robability 0.000000

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GARCH-in-Mean

Another important extension is the GARCH-M or the GARCH-in-Mean model

1 2 2, 1 1,...t t k k t k t ty x x h

1/2 , ~ (0,1)iid

t t t th u u N

2 21 1 1 1... ...t t p t p t q t qh h h

This allows the conditional mean of ty to be correlated with the conditional variance. For instance, the returnto the stock might be higher during periods of higher volatility, as the risk premium in holding the stockincreases.

In the following, we include the standard deviation 1/2th as a regressor in the mean equation:

equation eq3.arch(1,1, archm=sd) dl_stii c ma(1)

Dependent Variable: DL_STIIMethod: ML - ARCH (Marquardt) - Normal distributionSample: 1/01/2004 10/26/2011Included observations: 2040Convergence achieved after 12 iterations

Variable Coefficient Std. Error z-Statistic Prob.@SQRT(GARCH) 0.000 0.060 0.008 0.994C 0.001 0.001 1.327 0.185MA(1) 0.008 0.025 0.309 0.758

Variance EquationC 0.000 0.000 3.851 0.000RESID(-1)^2 0.106 0.010 10.176 0.000GARCH(-1) 0.889 0.010 86.502 0.000R-squared -0.001 Mean dependent var 0.000Adjusted R-squared -0.002 S.D. dependent var 0.013S.E. of regression 0.013 Akaike info criterion -6.380Sum squared resid 0.321 Schwarz criterion -6.363Log likelihood 6513.238 Hannan-Quinn criter. -6.374Durbin-Watson stat 1.968 Inverted MA Roots -0.010

It does not appear that the variance and mean of DL_STII are correlated.

Many other extensions exist: e.g., asymmetric GARCH which allows negative shocks to affect volatilitydifferently from positive shocks. This has been found useful in capturing a ‘leverage effect’, where +ve/-veshocks changes the debt-equity ratio of a firm and so changes the riskiness of the firm. These are left for moreadvanced courses.

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Exercises

1. Take the AR(1)-ARCH(1) model

0 1 1t t ty y

1/2 , ~ (0,1)iid

t t t th u u N 2

1 1t th

Derive the conditional mean and conditional variance of ty given lags of ty .

2. Show that the square of GARCH(1,1) residuals follow an ARMA(1,1) process.

3. Use the data in stii.wf1 to estimate the following models to _DL STII over the sample period 1/04/1985to 12/31/2007:

mean equation: MA(1) with constant termvariance equation:

ARCH(1), ARCH(2), GARCH(1,1), GARCH(1,2), GARCH(2,1), GARCH(2,2)distribution: Normal, t -distribution.

In other words, fit twelve different models: MA(1)-ARCH(1) normal, MA(1)-ARCH(2) t -dist, etc...

(a) Show the SIC for each of the twelve models. Which model does the SIC select? Show details of theestimation output for this model only, and comment on the squared residual correllogram for this model.

(b) Which change among the twelve models produces the largest fall in the SIC?

(c) Produce and plot the one-step ahead forecasts for the period 1/1/2008 to 10/26/2011, with +/- 2 standarderrors, from the model chosen in (a).

(d) Produce and plot the dynamic forecasts for the period 1/1/2008 to 10/26/2011, with +/- 2 standard errors,from the model chosen in (a). Comment on the forecasts.