Robust Volatility Forecasts and Model Selection

7/30/2019 Robust Volatility Forecasts and Model Selection

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Robust Volatility Forecasts and Model Selection in

Financial Time Series

Luigi Grossi and Gianluca Morelli Dipartimento di Economia, Universita di Parma, Italy

Abstract

In order to cope with the stylized facts of financial time series, many modelshave been proposed inside the GARCH family (e.g. EGARCH, GJR-GARCH,QGARCH, FIGARCH, LSTGARCH) and the stochastic volatility models (e.g.SV). Generally, all these models tend to produce very similar results as concerns

forecasting performance. Most of the time it is difficult to choose which is themost appropriate specification. In addition, all these models are very sensitiveto the presence of atypical observations. The purpose of this paper is to providethe user with new robust model selection procedures in financial models whichdownweight or eliminate the effect of atypical observations. The extreme case iswhen outliers are treated as missing data. In this paper we extend the theoryof missing data to the family of GARCH models and show how to robustify theloglikelihood to make it insensitive to the presence of outliers. The suggestedprocedure enables us both to detect atypical observations and to select the bestmodels in terms of forecasting performance.

Keywords: GARCH models, extreme value, robust estimation.JEL classification: C16, C22, C53, G15.

1 Introduction

Financial returns are generally characterized by small first-order autocorrelation, kur-tosis much higher than that of the normal distribution, slow decay of the autocor-relations of squared observations towards zero and clusters of high volatility (see forexample Franses and van Dijk (2000) or Rossi and Gallo (2006)). During the last 20years many models have been proposed to cope with these stylized facts. The most

often used are the generalized autoregressive conditional heteroscedasticity (GARCH)models, introduced independently by Bollerslev (1986) and Taylor (1986) generalizinga specification proposed by Engle (1982) and the autoregressive stochastic volatilitymodel also proposed by Taylor (1986).

Despite being the result of a joint work, the computational part should be attributed to GianlucaMorelli, while Luigi Grossi developed the methodological plan of the paper.

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Another stylized fact which is often observed in high frequency financial returns isthe asymmetric response of volatility to positive and negative changes in prices. Thefirst model which was introduced to cope with this effect is the so called exponentialGARCH model introduced by Nelson (1991). This approach has been further devel-oped by Glosten, Jagannathan, and Runkle (1993) and Sentana (1995) who proposedrespectively the GJR-GARCH and Quadratic GARCH (QGARCH) specifications. Fi-nally, other extensions of GARCH models are IGARCH (Engle and Bollerslev 1986),FIGARCH (Baillie, Bollerslev, and Mikkelsen 1996), GARCH in mean (Engle, Lilien,and Robins 1987) and LSTGARCH (Gonzalez-Rivera 1998).

In practice all these models tend to produce very similar results as concerns forecast-ing performance. Sometimes, it is difficult to choose which is the most appropriate. Inaddition, all these specifications are very sensitive to the presence of particular observa-tions. In the last years there have been some papers dealing with outliers in stochasticvolatility models (e.g. Muler and Yohai 2002; Franses and Lucas 2004; Zhang 2004;Battaglia and Orfei 2005; Charles and Darne 2005).

The purpose of this paper is to develop new methods that can help the user toselect among different similar alternative specifications in the galaxy of GARCHmodels. This procedure, which is based on a forward search algorithm (Atkinson

and Riani 2000 or Atkinson, Riani, and Cerioli 2004), is robust to the presence ofatypical observations. A distinction between this contribution and the previous workson robust GARCH models is that the emphasis was on aggregate statistics and onrobustification of standard quantities. For example, Park (2002) suggested replacingiterative LS estimation with least absolute deviation estimation. Muler and Yohai(2002) proposed to replace mean square errors of the standardized observations withthe square of a robust -scale estimate. In this paper we are concerned with methodswhich show the effect individual observations (outliers or not) exert on the fitted model.The procedure is based on a series of fits of subsets of increasing size which treat theobservations which are outside the subsets as missing. Given that in financial timeseries all the data are always available and the problem of missing values is absent, this

argument has never received particular attention and the software which is regularlyused to estimate GARCH models does not allow the possibility of dealing with missingobservations (e.g. the Finmetrics module of S-plus). The issue of missing values,however, arises when we have detected some observations as atypical and we do notwant them to affect the out-of-sample volatility forecasts, the parameter estimates andso on.

The structure of the paper is as follows. In section 2 we briefly review linearand non linear GARCH models with particular attention to the QGARCH and GJR-GARCH specifications. In section 3 we show how to robustify the parameter estimatesof the previous models and provide a unified treatment of missing values in stochastic

volatility models. In section 4 we show the additional insight the suggested procedureprovides in terms of robust model selection. In section 5 we construct robust confidenceenvelopes which act as calibratory backgrounds for judging the eventual significanceof the jumps we observe during the forward search and show the robustness of thesuggested approach when the data are contaminated with outliers. Section 6 containsconclusions and extensions for further research.

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2 Linear and nonlinear GARCH models

Let rt be an observed time series of returns, such that rt = log(pt/pt1) where pt isa stock price or a stock market index. As it is well known, GARCH models wereintroduced to capture the volatility clustering of financial returns which is observedon the conditional variance of returns or of residuals in a time series model applied toreturns. Formally, we can write the observed time series of returns as the sum of apredictable and an unpredictable part

rt = E[rt|t1] + t (1)

where t1 is the set of all relevant information arrived on the market up to andincluding time t 1; t is conditionally heteroscedastic, that is

t = ztt, (2)

where zt iid(0, 1), and E[2t|t1] =

2t

. The linear GARCH(1,1) model can bewritten as

2t = 0 + 12t

1

+ 12t

1

, (3)

with 0 > 0, 1 > 0 and 1 0 for nonnegativity of conditional variance and 1+1 < 1for covariance stationarity.

For stock returns, it has been observed that volatile periods are often initiated bya large negative shock which suggests that negative and positive shocks have a differ-ent impact on conditional volatility of subsequent times. This phenomenon called theleverage effect is not captured by the linear GARCH models introduced above, be-cause conditional volatility depends only on the squares of the shocks so that positiveand negative shocks of the same magnitude have the same effect on the conditionalvolatility. In this paper we consider two nonlinear models which are able to capturethe leverage effect: the model introduced by Glosten, Jagannathan, and Runkle (1993)

called GJR-GARCH and the quadratic GARCH (called QGARCH) introduced by Sen-tana (1995). The GJR-GARCH(1,1) model is obtained from the GARCH(1,1) model(3) with a correction which links the parameter of 2

t1 to the sign of the shock, that is

2t

= 0 + 12t1(1 I[t1 > 0]) + 1

2t1I[t1 > 0] + 1

2t1, (4)

where I[] is an indicator function which equals 1 when the event inside the bracketsis true. For nonnegativeness of variance the following conditions must be satisfied:0 > 0, (1 + 1)/2 and 1 > 0. For covariance stationarity (1 + 1)/2 + 1 < 1.

The QGARCH(1,1) model is an alternative way to cope with asymmetric effects ofshocks on volatility and is specified as follows:

2t = 0 + 1t1 + 12t1 + 1

2t1, (5)

where the conditions for covariance stationarity are the same as the correspondingconditions in the GARCH(1,1) model. The additional term 1t1 makes possible anasymmetric effect of positive and negative shocks on the conditional variance. When1 < 0 the effect of negative shocks on

2t

will be larger than the effect of positiveshocks of the same size. Furthermore, the effect depends on the size of the shock.

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When t is assumed to be normally distributed, the conditional log-likelihood forthe t-th observation is given by:

t() = 1

2log(2)

1

2log(2

t)

2t

22t, t = 3, . . . , T , (6)

where the vector contains the parameters of the specified model. For example, in thecontext of the QGARCH model (5), = (0, 1, 1, 1)T.

The optimal s-step-ahead out-of-sample forecast of the conditional variance can becomputed recursively from:

2(T+s)|T = 0 + (1 + 1) 2(T+s1)|T GARCH (7)

2(T+s)|T = 0 + [(1 + 1)/2 + 1] 2(T+s1)|T GJR GARCH (8)

2(T+s)|T = 0 + (1 + 1) 2(T+s1)|T QGARCH. (9)

In practice all these specifications tend to produce very similar results as concernsforecasting performance. Moreover, it is clear from the previous equations that theforecasts can be strongly affected by the presence of atypical observations whose effectpropagates recursively. In the next section we show how to robustify the previous mod-els and at the same time not to lose the efficiency of maximum likelihood estimators.

3 Robustification of linear and non linear

GARCH models

In order to robustify the estimates of the parameters of models (3), (4) and (5), werepeatedly fit the forward search algorithm in the way suggested by Atkinson and Riani(2000) and extended to time series by Riani (2004) and Grossi (2004). The algorithm isboth efficient and robust. It is efficient because it makes use of the Gaussian likelihood

machinery underlying model (6). It is robust because the outliers enter in the last stepsof the procedure and their effect on the statistics of interest is clearly depicted. Moregenerally, this approach allows evaluation of the inferential effect each time period,either outlying or not, exerts on the fitted model. The key features of the forwardsearch applied to linear and non linear GARCH models can be summarized as follows.

Choice of the initial subset. We take periods of contiguous observations as thebasic sets of our algorithm. These blocks are intended to retain the autocorrelationstructure of the whole time series. Confining attention to subsets of continuous observa-tions ensures that the parameters can be consistently estimated within each block. Theinitial subset can be obtained through least median or least trimmed squares appliedto these blocks.

Progressing in the search and diagnostic monitoring. The transformedmodel is repeatedly fitted to subsets of increasing sizes ignoring contiguity and selectedin such a way that outliers are included only at the end of the search. For this reason,in each step of size m, we take as the new subset that formed by the smallest squaredone step ahead prediction errors. One major advantage of the forward search over otherhigh-breakdown techniques is that a number of diagnostic measures can be computedand monitored as the algorithm progresses. Given that one of the main purposes of

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financial models is to forecast the volatility, it seems natural to monitor the out-of-sample h-step ahead prediction errors as the subset size grows. In each step of thesearch the observations not forming the subset are treated as if they were missing.The most natural way to replace a missing observation consists in using its optimalpredictor (see Harvey and Pierce 1984). In the context of state space models thisis equivalent to omit the Kalman filter updating equations for the conditional meanand the conditional variance. The generalization of this argument to the family oflinear and non linear GARCH models implies that we have to replace 2t with

2t

, theconditional expectation of 2

t. Thus, given a subset of size m (say S

m), if observation

t does not belong to the subset, the conditional variance at time t + 1 is iterativelycomputed as follows, depending on the underlying model:

2t+1|S

m

= 0 + (1 + 1)2t|S

m

GARCH (10)

2t+1|S

m

= 0 + [(1 + 1)/2 + 1]2t|S

m

GJR GARCH (11)

2t+1|S

m

= 0 + sgn(t1)12t|S

m

+ (1 + 1)2t|S

m

QGARCH (12)

Finally, when the t-th observation is missing, skipping the equation of the condi-

tional variance implies modifying the conditional log-likelihood (6) so that log(2t ) = 0and

2t

22t

= 12 . In other words, the resulting log-likelihood for the t-th observation when

it does not belong to the subset is:

t() = 1

2log(2)

1

2. (13)

4 Model selection through robust comparison of

the forecasting performance

In this section we show how the suggested procedure can help the user to select, in arobust way, the best model belonging to the GARCH family. In order to illustrate thedifficulties we encounter in model selection even when the data do not contain outlierswe start with an example with simulated data.

4.1 Simulated data

We have generated a series of 205 observations from a GARCH(1,1) model with pa-rameters 0 = 0.1, 1 = 0.1, 1 = 0.6. To these data we have fitted a GARCH(1,1),GJR-GARCH(1,1) and QGARCH(1,1) specification1. The additional parameter 1turned out to be significant in both cases. However, given that the true model isGARCH, we expect that this specification should outperform that of the other models.Table 1 shows the ratios of the out-of sample forecasting performance of GJR-GARCHand QGARCH evaluated using MAPE (Mean Absolute Prediction Error) with respectto that of the GARCH model. This index relays on models estimated on the basis ofrolling windows of 100 observations (for weekly data it is equivalent to rolling windows

1From now on, when we refer to these models, we will drop the suffix (1,1).

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Table 1: Simulated GARCH data: comparison of out-of sample forecasting performanceof GARCH, GJR-GARCH and QGARCH using rolling windows MAPE index for 1-to-5steps forecast horizons and average. Base: GARCH=100

Forecast step GARCH GJR-GARCH QGARCH

1 100 102.0 101.12 100 102.0 103.0

3 100 99.8 100.14 100 100.3 100.05 100 99.5 100.2

Average 100 100.7 100.8

of two years). For example, with a sample of 205 observations, we start with the sub-sample ranging from the first 100 observations. The fitted models are then used toobtain 1-to-5-steps-ahead forecasts of the conditional volatility, that is the conditionalvolatility of observations 101-105. Next the window is moved 1 step into the future,by deleting the observation at time 1 and adding observation at time 101. The various

models are re-estimated on this sample, and are used to obtain forecasts for 2t for

time 102 until time 106. This procedure is repeated until the final estimation sampleconsists of observations from time 101 until time 200. In this way we obtain 100 1-to-5-steps-ahead forecasts of the conditional variance (see Franses and Van Dijk, 2000for more details). To evaluate and compare the forecasts from the different models,the MAPE is computed, with true volatility measured by the squared realized returns.Table 1 shows the ratio of the MAPE of the nonlinear GARCH models to those of theGARCH model for different forecast horizons. The last row reports the ratio of theMAPE averaged through the different horizons. For example, the value 100.7 in thelast row of the column of GJR-GARCH means that the average MAPE from this modelis 0.7% greater than the corresponding criteria for forecasts from the linear GARCH

model. This table clearly shows that even if the data have been generated by a GARCHmodel these three specifications give a similar forecasting performance leaving the userwith unclear ideas about the best specification. In other words, we have no reasons forconsidering one model better than the other.

Figure 1 shows the monitoring of one step forecast error (top panel) and two stepsahead forecast error (bottom panel) for observations 201 and 202 for the three alter-native specifications. This figure clearly shows that throughout the search the bestperformance is given by the GARCH model. The QGARCH model, even if at theend of the search has the best forecasting performance both in terms of one step andtwo steps forecast horizon, has a curve which always lies above that of the other two

models.The message which comes from the analysis of these simulated data is that evenif the series under study does not contain outliers, if we compute the forecasting per-formance on rolling windows the different specifications are likely to have similar fore-casting performance or, even worse, can suggest a wrong model. In the next sectionwe will analyze 3 real financial time series where the above stylized facts are present.

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Subset size m

Onestep

160 170 180 190 200

2

4

6

8

10

14

GARCH

QGARCH

GJRGARCH

Subset size m

Twosteps

160 170 180 190 200

5

10

15

GARCH

QGARCH

GJRGARCH

Figure 1: Monitoring of one step ahead forecast error (top panel) and two steps aheadforecast error (bottom panel) for GARCH, QGARCH and GJR-GARCH model.

4.2 Real financial data

In this section we apply our method of model selection to a series of data sets makinguse of the expertise and insights gained in analyzing the simulated data. The data weconsider are monthly stock prices indices for Italy (MIB storico generale), Japan (TSETOPIX) and USA (NYSE composite), gained from the OECD database MEI (MainEconomic Indicator) downloaded from the website http://lysander.sourceoecd.org .The monthly indices are averages of daily closing quotations. Time series cover the

period January 1988 - June 2005 which gives 209 observations. In order to compareprice trends in different countries, data are transformed to obtain index numbers withbase 2000=100. As can be seen from Figure 2, Japan stock index followed an oppositepath with respect to US and Italian stock indexes from 1988:1 to 1998:1, while in thesubsequent period began a path very similar to the other stock indexes essentially fol-lowing the US indexes as the indexes in the majority of the developed countries. Asit is well known, from 1998 a sharp bull trend started with a high peak in the firstmonths of 2000. In 2001 indexes followed a bearish trend with a relative minimum inSeptember (twin towers attack), while a relative minimum took place in the first fewmonths of 2003. It is interesting to note that the US index shows in the middle of 2005a level higher than the maximum reached in 2000 while the Japanese and the Italian

indexes are only at 70-80% of the 2000 maximum. The corresponding plot of returns(Figure 3) shows that in the Italian case the volatility is larger than that in the othercountries. This is confirmed by the presence of a large number of extreme returns inthe Italian series (nine months above 10% and two months under -15%) and by themean of the squared returns which for the Italian index is 29.8 against 10.1 and 21.7for USA and Japan, respectively. Thus, we expect a worse forecasting performancein the Italian case. We applied to the series of returns the three models presented in

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1988 1990 1992 1994 1996 1998 2000 2002 2004 200620

0

10

ITALY

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

20

0

10

US

1988 1990 1992 1994 1996 1998 2000 2002 2004 200620

0

10

JAPAN

Figure 3: Returns on stock prices index of Italy (top panel), USA (middle panel) andJapan (bottom panel) during the period February 1988 - June 2005

Table 2: Monthly index number of stock prices of Italy, US and Japan: comparison ofprediction errors of GARCH, QGARCH and GJR-GARCH models through the MAPEindex using all observations (end of the search). Averages for 1-to-4 and 1-to-5 stepsforecast horizons. Base: GARCH=100

GARCH QGARCH GARCH-GJRCountry MAPE 1-to-5 steps ahead

Italy 100 102.2 100.0USA 100 94.6 130.0Japan 100 100.2 133.0

MAPE 1-to-4 steps aheadItaly 100 101.6 100.0USA 100 89.2 117.0

Japan 100 102.3 128.1

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Table 3: Italian stock index: comparison of out-of sample forecasting performance ofGARCH, GJR-GARCH and QGARCH using rolling windows MAPE index for 1-to-5steps forecast horizons and average. Base: GARCH=100

forecast step GARCH QGARCH GARCH-GJR1 100 91.3 91.7

2 100 89.1 91.03 100 89.3 91.04 100 87.5 90.45 100 86.6 89.7

Average 100 88.7 90.8

Table 4: US stock index: comparison of out-of sample forecasting performance ofGARCH, GJR-GARCH and QGARCH using rolling windows MAPE index for 1-to-5steps forecast horizons and average. Base: GARCH=100

forecast step GARCH QGARCH GARCH-GJR1 100 102.6 115.72 100 100.6 113.83 100 101.4 115.04 100 100.7 120.45 100 105.6 123.6

Average 100 102.2 117.7

Table 5: Japanese stock index: comparison of out-of sample forecasting performance ofGARCH, GJR-GARCH and QGARCH using rolling windows MAPE index for 1-to-5steps forecast horizons and average. Base: GARCH=100

forecast step GARCH QGARCH GARCH-GJR1 100 104.5 105.0

2 100 101.6 104.53 100 101.4 103.84 100 100.9 103.55 100 97.7 102.5

Average MAPE 100 101.2 103.9

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Subset size m

Averageforecast

error

160 170 180 190 200

100

200

300

400

500

600

GARCH

QGARCH

GJRGARCH

Figure 4: Stock prices index of Italy: monitoring of average 1-to-4 step ahead squaredvolatility forecast errors for GARCH, QGARCH and GJR-GARCH model.

Subset size m

On

estep

160 170 180 190 200

0

100

200

300 GARCH

QGARCH

GJRGARCH

Subset size m

Tw

osteps

160 170 180 190 200

200

400

600

GARCH

QGARCH

GJRGARCH

Subset size m

Threesteps

160 170 180 190 200

200

400

600

800

GARCH

QGARCH

GJRGARCH

Subset size m

Foursteps

160 170 180 190 200

200

400

600

GARCH

QGARCH

GJRGARCH

Figure 5: Stock prices index of Italy: monitoring of one, two, three and four stepssquared volatility forecast errors for GARCH, QGARCH and GJR-GARCH model.

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section when we have analyzed simulated data. Figure 5 shows the monitoring of theforecasting performance for the three models for 1, 2, 3 and 4 steps ahead forecasthorizon. In the final step all the curves are very similar making us wrongly think thatthese three models are equivalent. The forward search on the other hand shows that:

1. The prediction performance of the GJR-GARCH model is always worse than thatof the other two specifications;

2. The difference of forecasting performance between the GJR-GARCH and theother two models seems to increase when the forecast horizon increases;

3. The GARCH and QGARCH specifications seem to provide similar prediction er-rors even if the QGARCH seems slightly better.

4. The small difference in performance between GARCH and QGARCH becomesnegligible when the forecast horizon increases. As a matter of fact, it is interest-ing to notice that in the central part of the search the solid line associated with

the GARCH models becomes closer and closer to the dotted line of the QGARCHspecification when the forecast horizon increases.

Let us now consider the series of stock prices indexes of US and Japan. For theUnited States Table 2 and Table 4 show that the QGARCH model outperforms theGARCH specification at the end of the search, while the rolling windows MAPE in-dicate that the GARCH and the QGARCH specifications are substantially equivalent.The worst fit seems to be given by the GJR-GARCH model. Figures 6 and 7, whichshow respectively the monitoring of average 1-to-4 step ahead absolute forecast errorsand the detail of 1, 2, 3 and 4 forecast errors clearly confirm these conclusions. The

curve associated with the Q-GARCH specification is always virtually below that of theother two curves throughout the search. This example has been given to show thatsometimes the results which come from the application of traditional methods coincidewith what the robust analysis reveals. On the other hand, as in the final example weconsider (stock prices index of Japan) the forward search shows that the conclusionswhich come from the analysis of the final step of the search are not supported by themajority of the data. As concerns Japan, Table 2 and Table 5 makes us conclude thatthe best specification should be the GARCH model. Figure 8 clearly shows that theaverage forecast errors of the QGARCH model are always below those of the GARCHexcept in the final step. The monitoring of the detail of the prediction errors in the first4 steps clearly shows that even if in the central part of the search the curve associated

with the QGARCH is lower, the forecasting performance seems equivalent when it isbased on all the observations. Notice that in the final step of the search the two curvesassociated with GARCH and QGARCH cross in the case of 1 step and 2 step predictionerrors, while they become very close to each other for 3 and 4 steps prediction errors.

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Subset size m

Averageforecast

error

160 170 180 190 200

20

40

60

80

GARCH

QGARCH

GJRGARCH

Figure 6: Stock prices index of USA: monitoring of average 1-to-4 step ahead squaredvolatility forecast errors for GARCH, QGARCH and GJR-GARCH model.

Subset size m

On

estep

160 170 180 190 200

0

2

4

6

8

10 GARCH

QGARCH

GJRGARCH

Subset size m

Tw

osteps

160 170 180 190 200

20

40

60

80

100

GARCH

QGARCH

GJRGARCH

Subset size m

Threesteps

160 170 180 190 200

0

5

10

15

20

GARCH

QGARCH

GJRGARCH

Subset size m

Foursteps

160 170 180 190 200

50

100

150 GARCH

QGARCH

GJRGARCH

Figure 7: Stock prices index of USA: monitoring of one, two, three and four stepssquared forecast errors for GARCH, QGARCH and GJR-GARCH model.

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Subset size m

Averageforecast

error

160 170 180 190 200

50

100

1

50

200

GARCH

QGARCH

GJRGARCH

Figure 8: Stock prices index of Japan: monitoring of average 1-to-4 step ahead squaredvolatility forecast errors for GARCH, QGARCH and GJR-GARCH model.

Subset size m

On

estep

160 170 180 190 200

0

20

4

0

60

80

GARCH

QGARCH

GJRGARCH

Subset size m

Tw

osteps

160 170 180 190 200

50

150

250

350

GARCH

QGARCH

GJRGARCH

Subset size m

Threesteps

160 170 180 190 200

0

50

100

200 GARCH

QGARCH

GJRGARCH

Subset size m

Foursteps

160 170 180 190 200

0

50

150

250 GARCH

QGARCH

GJRGARCH

Figure 9: Stock prices index of Japan: monitoring of one, two, three and four stepssquared forecast errors for GARCH, QGARCH and GJR-GARCH model.

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Subset size m

Onestep

160 170 180 190 200

0

5

10

15

20

Subset size m

Twosteps

160 170 180 190 200

0

5

10

15

20

Subset size m

Threesteps

160 170 180 190 200

0

5

10

15

20

Subset size m

Foursteps

160 170 180 190 200

0

5

10

15

20

Figure 10: One, two, three and four steps ahead squared forecast errors for contami-

nated data with 1%, 5%, 50%, 95%, 99% simulation envelopes

5 Envelopes of h-step ahead prediction errors for

outlier detection

In order to evaluate the trajectories of the forecast errors which come from the forwardsearch we need to superimpose a calibratory background in order to judge the even-tual significance of jumps. To this purpose we have constructed forward simulationenvelopes for different combinations of parameters values, different sample sizes andthe three different models described in section 3. In more detail, for a particular sam-ple size and a set of parameter values we have performed 1000 independent forwardsearches. The data in each simulation have been generated assuming the same specifi-cation. With this calibratory background we can check if the forward search curve forour real data stays inside the bands and in which step it eventually goes out.

In order to better understand how the procedure works we have contaminated theseries used in the previous example, adding a level shift of size 5 to 4 consecutiveobservations in the middle of the sample (observations 150-153). Figure 10 shows one,two, three and four steps ahead forecast errors with 1%, 5%, 50%, 95%, 99% simulationenvelopes. In the central part of the search, it is possible to notice that the observedforecast error lies very close to the line representing the median of the envelope. As

soon as the first outlier enters the subset (subset size m = 197) there is an upwardjump which, for example, in the monitoring of one step forecast error is significant atthe 1% level. At the end of the search it is possible to observe the well-known maskingeffect with a sudden decrease of the forecast error. The top left panel of Figure 10shows that the final value is even below the lower threshold suggesting that there issomething wrong.

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6 Conclusions

There is an appreciable literature on financial models and the set of models whichhave been proposed is very wide (e.g. see the books of Gourieroux (1997) and Fransesand van Dijk (2000) and the references contained). Generally, all these specificationsgive the same forecasting performance and it is difficult to choose among them usingtraditional indexes (e.g. h-step ahead out-of-sample prediction errors or MAPE indexbased on rolling windows). In this paper we have suggested robust and efficient tools

for model selection in stochastic volatility models which show which is the model withthe best forecasting performance. The procedure is efficient, because it always usesmaximum likelihood estimators and is robust, because is not affected by the presenceof atypical observations. Finally we have provided robust envelopes so that the usercan have formal tests about the presence of atypical observations. Simulated andreal data showed that the procedure can help in selecting the model with the bestforecasting performance avoiding the effect of extreme observations. Further researchwill be devoted to extend the robust model selection procedure to a wider class ofnonlinear and to the application to high frequency data.

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Robust Volatility Forecasts and Model Selection