ALTERNATIVE MODELS TO EXTRACT ASSET VOLATILITY: A

ALTERNATIVE MODELS TO EXTRACT VOLATILITY: A COMPARATIVE STUDY·

ASSET

Pedro L. Valls Pereira·· Luiz K. Hotta··· Luiz Alvares R. de Souza··· Nuno Miguel C. G. de Almeida···· Abstract This paper presents an empirical comparison of the estimation of the volatil­ ity of three Brazilian financial series: a Brazilian Brady bond (the Cbond) , a stock (Telebras PN) and the Brazilian Real/US Dollar exchange rate, using different modelling methods. The models used are: XARCH family, Stochastic Volatil­ ity (SV) and the switching in the variance model (SWARCH). The comparison is done using three criteria: loss functions, which compare the square of the estimated volatility with the instantaneous volatility, a procedure proposed by Herencia et alii (1998) which used prediction confidence intervals and one-step­ ahead prediction, and a prediction exercise for the last 100 observations. In general the SV model presented the best performance although it is dominated by other models in some criteria. Resumo

Este artigo compara metodos de estima 0, ai,{3i � 0, having normal distribution.

(i

=

l,...,q;j

=

ll""p), E

yt = ¢o +tin yt_l + ... + rpp 'Yt-p +Ut, Ut = ..[Fi:; et, et ht = no + 2:i=l (lj ui_i + E�=l f3j ht-j.

rv

Uj

+ Ef3j ::; 1.

The errors were considered as

N(O,l)

(2) SiIIlilar to (1) but the with Student's t distribution for the errors, with the degree of freedom, with the

restriction 1] > 2, estimated jointly with the parameters.

yt = tPo + lPl Yt-l + ... + tPP Yt_p +Ut, 'Ut = ..(ii;et,et,...., t(1]) ht = 0:0 + E;=l (lj ui_i + E;=l f3j ht-j.

(3) Similar to (1) the estimated model allow for an autoregressive AR(p) structure in the mean always

with a constant, but the AR(p) structure is now in the logarithm of the conditional variance allowing for

asymmetric response to the innovations. It is not necessary to impose any restriction in the parameters and we assume that the errors follow a normal distribution.

Vt =¢o +qn yt_l + ... +¢PYt_p+Ut, Ut = .,[h;et, et '" N(O,l). q In(h,) = 0 + Ei=l ;' 2, estimated jointly with the parameters. Yt

=

Om dt + cJ;o + tP1 Yt._1 +

ht ::::: o�, dt +

,

. .. +ptPP Yt-p + UtI 11t

=

..;Ti:;€t ,€t

'"

t(T/)

ao + L: a:i uL. + L pj ht_j. i:==l

j=1

3.2. Stochastic volatility model The Stochastic Volatility models (SV) were estimated by the quasi-maximum likelihood method. 80

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3.2.1. Estimation of Telebras series by quasi-maximum like­ lihood method The SV models estimated for the Telebnis return series were: - Model I: simple AR(l)-SV, Le. given by (2.14 and 2.12); - Model II: AR(l)-SV with adjustment for days of week in the variance; - Model III: AR(l)-SV with dummies in the 171-Th. and 823-Th. observations, which correspond to the change in the target zone and the Asian crisis. Table 9 Parameter Estimates for Model with AR(2) and Dummy in Mean and Variance (Exchange Rate)

n-GARCfI.AR(2)

Constant

Om

"0

'"

0.0324

·0.0208

0.0003

0.2839

fi,

0.3607

fi, 0

fi3

(0.1115)

0.3097

0"

ry

0.0261

(0.0022)

(0.0379)

(0.0001)

(0.0969)

(0.1063)

(restr.)

n-GARCH-AR(2)

0.0319

·0.0287

0.0002

0.2363

0.4502

0.2754

(0.0023)

(0.0382)

(0.0001)

(0,1134)

(O.1264)

(0.1555)

t-GARCH

0,0267

-0.0405

0.0007

0.4153

0.5671

0.1011

3.0477

(0.0023)

(0.0309)

(0.0003)

(O.1096)

(0.1018)

(0,0490)

(0,1011)

(0.0151) 0.0227 (0.0138)

Table 3.10 presents the parameters estimates and some statistics to measure the quality of the adjustment. The AIC criterion selected Model III as the best one while BIC criterion selected ModelL The estimated persistence in the first two models are equal while in the third model is a slightly larger. The week days dummies of Model II are statistically significant but there is no statistical indication of the presence of the effect of week days. Both outliers of Model III are statistically significant.

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Alternative Models to Extract Asset Volatility

Table 3.10 Goodness of Fit Statistics (Telebnis) Log-Lik' P.E.V.' 0'2

{

0'2 �

')'1

Monday

Model I

Model II

Model III

-1910.79 2.8346 1.5974 [1.0000]3

-1909.26 2.8258 1.5984 [1.0000]

-1905.45 2.8040 1.5954 [1.0000]

0.2016 [0.1262] 0.9765

0.2014 [0.1260] 0.9765 -8.2750 (0.2874)' -8.0774 (0.2874) -8.2747 (0.2876) -8.3285 (0.2883) -8.2686 (0.2874)

0.1889 [0.1184] 0.9783

Tuesday Wednesday Thursday Friday il71 i823 AIC BIC Q(6){residuals} Q( Q(24) residuals Q(36) residuals Q(6){residuals "' ;'""'' Q(12) residUalS' Q(24) residuals' Q(36) residuals' } {

t

!

3.8879 3.8826 (0.1392)' (0.3685) (0.2547) (0.5956) (0.2850) (0.4294) (0.8111) (0.5236)

3.8929 3.9327 (0.1592) (0.3825) (0.2225) (0.5523) (0.3350) (0.4419) (0.7790) (0.5221)

(tg���) (r:8!�a)

3.8812 3.9109 (0.0723) (0.3096) (0.1734) (0.5117) (0.1762) (0.1372) (0.4965) (0.0630)

Notes: (1) Log-Lik represents the log- likelihood. (2) P.E.V. represents prediction error variance. (3) The values in parentheses is the rate between the component variance and the largest variance among the components. The values in brackets are the estimated standard deviation of the esti­ mates. (5) For the Q statistics the values in brackets are the p-values.

(4)

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The Q statistics does not reject the hypothesis of no correlation when applied to the residuals and to the square of the residuals for all lags and models. This indicates that the model was able to explain the nonlinearity, 3.2.2. Quasi-maximum likelihood estimation of the Cbonds return series The stochastic volatility models estimated for the Cbond return series were: - Model I: usual AR(l)-SV , i.e., given by (2.14 and 2.12); - Model II: AR(l)-SV with adjustment for week days in the variance; Table 3.11 presents the parameters estimate and the quality of adjustment statistics. According to the information criteria the best model is model 1. Observe that the persistence in both model are approximately equal. The dummies of the weekdays in model II are statistically significa­ tive but the difference are not, favouring model I, which does not consider the weekdays effect. The Q statistics does not reject the null hypothesis for any lag when applied to the residuals. However, when the test is applied to the square of the residuals the null hypoth­ esis of non correlation is rejected indicating that some non linearity remained. 3.2.3. Quasi-maximum likelihood estimation for the returns of the Real/US dollar exchange rate series The stochastic volatility models estimated for the exchange rate Real/US Dollar were: Revista de Econometria

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Alternative Models to Extract Asset Volatility

- Model I: AR(l )-SV model with local level3 instead of a constant; - Model II: AR(l)-SV dummy for the first 250 observations. Table 3.11 Goodness of Fit Statistics ( Cbond) Model I -2138.02 3.2698 1.7394 [1.0000] 0.1629 [0.0937] 0.9824

Log-Lik1 p.E.v. (1 2 �

2

')'1

Monday

Tuesday Wednesday Thursday Friday

{ {

AIC BIC Q(6){ residuals} Q(12) reSiduaIS Q(24) residuals Q(36) residuals Q(6) { residuals2 Q(12) reSidUalS2 Q(24) residuals2 Q(36) residuals2 3In this model

x,

4.0301 4.0488 (0.5275) (0.4432) (0.5453) (0.5691) (0.0000)** (0.0000)** (0.0000)*' (0.0000)**

j

}

Model II -2136.08 3.2579 1.7398 [1.0000] 0.1628 [0.0936] 0.9825 -9.7313 (0.2989) -9.5232 (0.2973) -9.4979 (0.2975) -9.7334 (0.2974) -9.6803 (0.2977) 4.0340 4.0714 (0.5475) (0.4163) (0.4796) (0.5276) (0.0000) *' (0.0000)** (0.0000)'* (0.0008) *'

was allowed to follow a random walk. The rationality i s to allow the mean

and unconditional variance to change slowly.

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Table 3.12 presents the parameters estimates with some quality of adjustment statistics. Table 3.12 Goodness of Fit Statistics (Exchange Rate)

Log-Lik1 P.E.V.2 a2< a21] a2( /'1 D250 AIC BIC Q(6){residuals} Q(12){residuals} Q (24){residuals} Q(36){residuals} Q(6){residuals2} Q(12){residuals2} Q(24){residuals2} Q(36){residuals2}

Model I -1764.60 2.0467 1.2844 [1.0000] 0.4273 [0.3327] 0.0911 [0.0709] 0.6714 3.5550 3.5749 (0.0234) * (0.1281) (0.0000)** (0.0001)** (0.3672) (0.0141)* (0.0000)** (0.0000)**

Model II -1758.78 2.0228 1.2894 [1.0000] 0.4572 [0.3545] 0.7202 3.3638 (0.1521) 3.5453 3.5699 (0.0102)* (0.1021) (0.0000)** (0.0552) (0.0000)*' (0.0000)'* (0.0000)** (0.0000)**

Model II is the best according to both information criteria. 0 b­ serve that the persistence is small in both models. In model II the dummy which control for the change in the mean and in the condi­ tional variance was statistically significative. Revista de Econometria

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Alternative Models to Extract Asset Volatility

According to the Q statistic for the residuals, only for lag 12 for both models and lag 36 for Model II, the null hypothesis of non correlation is not rejected. The results for the square of the residuals show that the models were not able to filter all the non linearity in the series. 3.3. SWARCH Models We followed Hamilton and Susmel (1994) work and fitted SWARCH models with 1 and 2 lags in the ARCH structure, with Gaussian and Student-t disturbances, with 2 and 3 regimes, with and without leverage effect according to the model presented in sec­ tion 2.2. We also introduced a constant in every model and an autore­ gressive structure of order one or two in the mean process of some models. All the models were estimated using programs obtained from Hamilton & Susmel (1994) and the OPTIMUM program from GAUSS. 3.3.1. Telebras return series The results for Telebras series obtained for the SWARCH models selected by the AIC and BIC criteria are presented in table 3.13. The best model according to both criteria was the n-SWARCH-L(2,2) model. Restricting to the 3 regimes models both criteria selected the n-SWARCH-L(3,2) model The introduction of the autoregressive structure in the mean did not improved any of the statistical criteria. The Q-statistic did not detect significant correlation in the mean and in the variance at 5% level. The estimate of the n-SWARCH-L(2,2) model, which was se­ lected by the BIC criterion is given in the following, with estimated 86

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P.L.Valls Pereira; L.K. Hotta; L.A.R. de Souza & N.M.C.G. de Almeida

standard error of the estimate in parentheses:

Yt = 0.002978 + Ut

(0.000727)

Ut = v'gs, Ut where St

Vt

�

1, 2

yfh; Vt

u; = where

=

N(O, 1) and ht is given by:

ht = 0.0002317 + 7.433

(3.098xlO-") (0.0388)

x

1O - ll uL l + 0.2104ur_2+ 0.3203dt_ lUL l

(0.0495)

(0.0839)

where if Ut- l ::s; 0 if Ut- l > 0,

[

gl

=

1,

g2 = 3.7984 , (0.56325)

and transition matrix given by: p

=

0.979707 0.020293

0.016818 0.983182

]

The values Pij are the transition probability from regime i to regime j. We can observe the probability of the series remain in the same regime is large. The probability of changing from one regime to an­ other is very small with the expected time in regimes 1 and 2 are about 49 and 60 days, respectively.' The scale factor that multiply the volatility in the second regime is ";3.7983912, i.e. the volatility in the second regime is about twice the volatility in the first regime. The effect of the asymmetry is significative and in every model improved 4The expected time in

a

regime is computed by

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Alternative Models to Extract Asset Volatility

the information criteria, i.e. for the Telebnis series the volatility al­ ways increase considerably when the return is negative. This type of behaviour was first observed by Black (1976). Table 3.13 Goodness of Fit Statistics (Telebnis) n-SWARCH-L(2,2)

n-SWARCH-L(3,2)

0

0

2154.9264 -4.3592414 -4.3195043 0.0792 0.1318 0.0583 0.2682 0.6276 0.2211 0.3259 0.5809

2156.7223 -4.3567966 -4.3021579 0.0766 0.1228 0.0569 0.2640 0.6260 0.2916 0.4953 0.7038

AR order Degree of freedom Log-Lik Arc BIC Q(6){residuals} Q(12) {residuals} Q(24){ residuals} Q(36){residuals} Q(6) {residuals2} Q(12) {residuals2} Q(24){residuals2} Q(36) {residuals2}

The fitted n-SWARCH-L(3,2) model is presented in the following:

where Vt ht

88

=

rv

Yt

=

0.002836 + Ut

Ut

=

-IgS, Ut where St = 1, 2, 3

'lit

=

AVt

(0.000723)

N(O, 1) and ht is given by:

0.0001551 + 4.3201 l

(3.341X O-5) (0.0344)

x

10-3 UZ_l + 0.2048uL + 0.3210d t- l uL l (0.0482) 2 (0.0841) Revista de Econometria

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where

if Ut-1 ::; 0 if Ut-1 > 0, gl = 1,

g2 =1.82055, (0.45452)

g3 =5.89178, (1.42258)

and the transition matrix is given by: P=

[

0.985644 0.000000 0.003107 0.014356 0.974657 0.015038 0.000000 0.025343 0.981855

]

3.3.2. Chond return series For the Cbond return series the Schawrz and Akaike criteria se­ lected the t-SWARCH(2,1) and t-SWARCH-L(3,2) models, respec­ tively (see table 3.14). The selected models have only a constant in the mean because the inclusion of an autoregressive structure did not improve the Akaike and Schwarz criteria. None of the proposed models was able to model completely the variance because the Q­ statistics were significant at 5% level when applied to the square of the residuals of both selected models. This problem remained for more complex SWARCH models and was also detected in the 12Th. lag of the square of residuals of t-GARCH model with 1% of significance. The estimate of the of t-SWARCH-(2,1) model, which was se­ lected by the Schwarz criterion is given in the following, with the estimated standard error of the estimates in parentheses:

Yt = 0.001532 + Ut

(0.0003669)

Ut

=

Ut = Revista de Econometria

VgS, Ut where St = 1, 2

..;h; Vt 19 (1) Maio 1999

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Alternative Models to Extract Asset Volatility

where

ht =

Vt

�

t (4.8656) and ht is given by:

1.1656

(1.3496x 10-5)

x

10-4 + 0.1734uL 1 where 91 = 1, 9 = 0.94199 2 (5.89456) (0.0681)

and the transition matrix is given by: p

=

[ 0.007562 0.992438

0.016661 0.983339

]

The average time in regimes 1 and 2 are approximately equal to 132 and 60 days, respectively. The scale factor that multiply the variance in the second regime is 5.894, implying that the volatility in one regime is about 2.5 times higher than in the other. Table 3.14 Goodness of Fit Statistics (Cbond)

AR order Degree of freedom Log-Lik AlC BIC Q(6){residuals} Q(12){residuals} Q(24){residuals} Q(36){residuals} Q (6){residuals2} Q(12){residuals2} Q(24){residuals2} Q(36){residuals2}

90

t-SWARCH(2,1)

t-SWARCH-L(3,2)

0 4.8656303 2993.5720 -5.6191383 -5.5864178 0.4322 0.0538 0.1570 0.2557 0.0122' 0.0004*' 0.0216* 0.1614

0 5.6388254 3005.2523 -5.6317070 -5.5756146 0.2388 0.0287' 0.1065 0.2183 0.0400' 0.0008** 0.0354* 0.1964

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The estimate for the t-SWARCH-L(3,2) model is: Yt = 0.001413 (0.000359)

Ut = ";9S, iLt where St = 1, 2, 3 iLt = where Vt ht =

�

-.jh; Vt

t (5, 6389) and ht is given by:

8.4535

(1. 1488xlO -5 )

x

1O-4 + 0.0200UZ_1 + 0.0813iLL + 0.1962dt-1UL l 2 (0.0933) (0.0461) (0.0445)

where gl =

1,

g2 =

2.13032, (0.3932)

g3 =

7.86738, (1.8270)

and the transition matrix is given by:

.

P=

[

0.993958 0.000000 0.008185 0.006042 0.987229 0.007760 0.000000 0.012771 0.984055

]

3.3.3. Real/US Dollar exchange rate return series We worked with the exchange rate series multiplied by 100 in order to avoid numerical problems during the estimation process as we did before when a XARCH model was fitted. The best models selected by the Schwarz and Akaike criteria were the SWARCH(2,2) and SWARCH(3,2) models, respectively, both with Student's t distri­ bution (see table 3.15). These selected models, like the others, were not able to capture all the features of the series; the Q-statistics were significant both for the mean and the variance, i.e. for the residu­ als and for the square of the residuals. The leverage effect is not statistically significative for this series. Revista de Econometria

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Alternative Models to Extract Asset Volatility

Table 3.15 Goodness of Fit Statistics (Exchange Rate)

AR order Degree of freedom Log-Lik AlC BIC Q(6){residuals} Q(12){residuals} Q(24){residuals} Q(36){residuals} Q(6){residuaIs2} Q(12){residuals2} Q(24){residuals2} Q(36){residuals2}

t-SWARCH(2,2)

t-SWARCH(3,2)

0 2.9571174 884.67895 -1.7657179 -1.7262357 0.0324* 0.0115* 0.0013** 0.0104* 0.9562 0.9993 0.0000** 0.0000**

0 3.3789389 888.86354 -1.7681038 -1.7138158 0.0323* 0.0079** 0.0053** 0.0192* 0.6679 0.3443 0.3874 0.4100

The estimated t-SWARCH(2,2)5 model, selected by the Schwarz criterion is: Yt = 0.023952 + Ut Ut

where Vt

�

=

v'9S, Ut where St = 0, 1

Ut = A Vt t (2, 9571) and h t is given by: h t = 0.00264 + 0.3695uL + 0.3044uL 2

where 91 = 1, 92 = 135.2056 and the transition matrix is given by: p 0.001493 = 0.999124 0.000876 0.998506

[

5

]

Due to nl.lmerical problems the estimated standard deviations of the estimates are not re�

ported. Inference in this case can be misleading.

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The estimated persistence in both regimes are very large and the scale factor is also large, about 11.6 V135.20). Notice also that the values of Pu and P22 are near to 1.0 indicating that once in one regime the probability to changing to another regime is almost equal to zero. These facts are expected because we have a clear distinction of change of regime in the series. Although t-SWARCH(3,2) model does not completely specify the model it will be used in section 4.3 when we compare the volatility estimated by different models. 3.4. Volatility estimated by different methods We first present the graphs of the estimated yearly volatility by the models selected in each class. We also present in the same graph the volatility estimated by the sample standard deviation using a 20 day moving average window. In the Telebnis series we observed that the estimates given by the XARCH and the SV models follow the movements of the moving standard deviation estimates. On the other hand the SWARCH class model gave comparatively very large estimates for the peaks of the volatility and presented a more erratic behaviour. This could be explained by the larger persistence found by the SWARCH class model in both regimes. The Cbond series presents the same features found in the Tele­ bras series, i.e. the volatility estimated by the XARCH and SV mod­ els follow the volatility estimated by the moving standard deviation, which did not occur with the volatility estimated by the SWARCH model. In general the conditional volatility tends to revert quickly to the non conditional volatility level because the estimated coefficient of the ARCH structure is small. Revista de Econometria

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Alternative Models to Extract Asset Volatility

Annual Telebras Volatility

-

n-Swarch (2,2) - Egarch(1,1)+interv. - Std. Deviation (20 day )

SV smoother

Figure 3.4 Annual Cbond Volatilities 120,------, 100 80

-

Std. Deviation (20 days) t-Garch (1,1) + AR(2)

SV - smoother -

I-Swarch (2,1)

Figure 3.5 The estimates of the volatility of the exchange rate series follow the movements of the moving standard deviation estimates only after the 250-Th. observation, and the increase in volatility is larger for the XARCH and SWARCH classes models. However, before the 94

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