Comparing Regime Switching GARCH Models and GARCH Models

Transcripción

60
ANÁLISIS FINANCIERO
Minoo Nazifi Naeini* y Shahram Fatahi**
Comparing Regime Switching GARCH
Models and GARCH Models in Developing
Countries (Case study of IRAN)
La comparación de los modelos GARCH y GARCH de
cambio de régimen en los países en desarrollo
(estudio del caso de Irán)
ABSTRACT
This study is for comparing GARCH models and Markov switching GARCH models in their ability to estimate and
forecasting the volatility of Tehran stock market in some horizon of forecasting. This paper provides an analysis of
regime switching in volatility and out-of-sample forecasting of the IRAN using daily data for the period 1995-2011.
We first model volatility regime switching within a univariate Markov-Switching framework. Then we provide outof-sample forecasts of the TEHRAN daily returns using two competing non-linear models, the GARCH Markov
Switching model and the uniregime GARCH Model. The comparison of the out-of-sample forecasts is done on the
basis of forecast accuracy, using the statistical loss function. . The results, also, shows that SW-GARCH models can
remove the high persistence of GARCH models and separately in each regime of volatility, the persistence are high.
This shows the priority of SW-GARCH models. Another implication is that there is evidence of regime clustering.
JEL classification numbers: C32, C11, C22, C52, C32, C53
Keywords: Volatility, Markov Regime Switching GARCH models¡ forecasting, out of sample, bootstrap
RESUMEN
Este estudio busca comparar la capacidad de estimación y previsión de la volatilidad del mercado de valores de
Teherán para un horizonte de predicción dado, de los modelos GARCH y GARCH con cambio de régimen de Markov. Este trabajo ofrece un análisis de la variación del régimen en la volatilidad y de la previsión fuera de muestra
de Irán a partir de datos diarios para el período 1995-2011. Primero modelamos el cambio de régimen de la volatilidad en un escenario univariante de cambio de Markov. A continuación ofrecemos previsiones fuera de la muestra
de los rendimientos diarios de Teherán por medio de dos modelos no lineales, el modelo GARCH de cambio de régimen de Markov y el modelo GARCH de un solo régimen. La comparación de las previsiones fuera de muestra se
realiza sobre la base de la precisión de los pronósticos, utilizando la función de pérdida estadística. Los resultados,
además, muestran que los modelos SW-GARCH pueden eliminar la alta persistencia de los modelos GARCH y
separadamente en cada régimen de volatilidad, las persistencia son altas. Esto demuestra la prioridad de los modelos SW-GARCH. Otra implicación es que hay evidencias de una agrupación de régimen.
Clasificación JEL: C32, C11, C22, C52, C32, C53
Palabras clave: Volatilidad, Modelos GARCH de cambio de régimen de Markov, previsión, fuera de muestra,
bootstrap
Recibido: 18 de enero de 2012
Aceptado: 8 de junio de 2012
* Minoo Nazifi Naeini. MA in economics.
** Shahram Fatahi .Assistant prof at Razi University.
Minoo Nazifi Naeini y Shahram Fatahi: Comparing Regime Switching GARCH Models and GARCH Models in Developing
La comparación de los modelos GARCH y GARCH de cambio de régimen en los países en desarrollo (estudio del caso de Irán)
Análisis Financiero, n.º 119. 2012. Págs.: 60-68
COMPARING REGIME SWITCHING GARCH MODELS AND GARCH MODELS...
61
1. INTRODUCTION
2. RECENT STUDIES
The volatility of financial markets has been the object of
numerous developments and applications over the past two
decades,. In this respect, the most widely used class of
models is certainly that of GARCH .These models usually
indicate a high persistence of the conditional variance (i.e. a
nearly integrated GARCH process). Hence the estimates of a
GARCH model may suffer from a substantial upward bias in
the persistence parameter. Therefore, models in which the
parameters are allowed to change over time may be more
appropriate for modeling volatility.
Last 20 years: lot of research on modeling volatility in financial
markets using GARCH type models. Schwert (1989): model
where returns switch between high or low variance states
according to a two state Markov process. Hamilton and Susmel
(1994) and Cai (1994): ARCH model with regime switching
parameters. Gray (1994): proposes a class of regime-switching
GARCH (RS-GARCH) models with time-varying probability,
but estimates an approximation to the model. The quality of the
approximation is not known. See also Dueker (1996), Bollen et
al. (2000), Klaassen (2002), and Haas et al. (2004). Evidence of
regime switching in the volatility of stock returns have been
found by Hamilton and Susmel (1994), Hamilton and Lin
(1996), Edwards and Susmel (2001), Coe (2002) and Kanas
(2005). In addition, Engel (1994), Engle and Kim (2001)
Kouretas (2003) are examples of studies that they have estimated Markov-Switching regime models for currency markets..
Gray (1994) presents a tractable Markov-switching GARCH
model and a modification of his model is suggested by
Klaassen (2002); see also Bollen, Gray, and Whaley (2000),
Dueker (1996), Haas, Mittnik, and Paolella (2004), and
Marcucci (2005) for related papers. Schwert (1989) consider a
model in which returns can have a high or low variance, and
switches between these states are determined by a two state
Markov process. Hamilton and Susmel (1994) and Cai (1994)
introduce an ARCH model with Markov-switching parameters
in order to take into account sudden changes in the level of the
conditional variance.
This paper addresses three important issues with respect to
the behavior of stock returns volatility of a recently established emerging of stock market the TEHRAN Stock Exchange
using daily data for the period 1995-2011.
The main findings of the paper are summarized as follows.
First, there is strong evidence in favor of volatility regime
switching modeling of nonlinearities which affects the stock
returns. Second, the estimation of the MRS –GARCH model
accurately describes the two regimes based on the different
pattern of adjustment of the stock returns volatility; and the
estimated model captures all the events that are responsible
for the presence of nonlinear features in the TEHRAN stock
returns during the period 1995-2011. Third, there is a high
probability for regime clustering with the likelihood that a
low volatility regime will be followed by a low volatility
regime greater that the likelihood a high volatility regime
will be followed by a high volatility regime. Finally, the
comparison of the out-of-sample short-run forecasts generated by the two models suggest that on the basis of the forecast accuracy criteria, we conclude that the MRSGARCH
and have priority to univariate GARCH models.
The plan of this paper is as follows. First, there is some introduction and the studies about these issues in section 2 .we
explore the issue of GARCH models in section 3, section 4
is for a description for MRS GARCH models. Then in section 5 the stock market data and methodology are illustrated.
then we compare the forecasting performance of two competing non-linear models the Markov Regime Switching
GARCH (MRS) model and the GARCH models in sectiion5
and finally in section 6 the discussion and conclusion are
sketched.
3. DATA AND METHODOLOGY
3.1. Data
The data set used in this study is the daily closing prices of
value-weighted TEPIX index over the period 29/09/1995
through 03/04/2011. The data set is obtained from the web site
of the Central Bank of Republic of Iran (www.irbours. com).
The procedures are computed numerically by using MATLAB
optimization routines. The data is divided into a ten year insample estimation period (2480 observations) and a subsequent one year out-of-sample forecasting period (400observations) Daily observations are converted into continuously
compounded returns in a standard method as log differences:
62
The plot of return and price series are given in Figure 1 and
Figure 2. TEPIX return index displays usual properties of
financial data series.
3.2. Empirical results
3.2.1. GARCH
Descriptive statistics of return series are represented in Table
1 As table shows, the index has a positive average return
0.156%. Daily standard deviation is 5.05%. The series also
displays a negative skewness of 0.504 and an excess kurtosis
of 22.48. These values indicate that the returns are not normally distributed, namely it has fatter tails. Also, Jarque-Bera
test 2 statistic of 0.25 confirms the –normality of TEPIX
returns. These findings are consistent with other financial
returns’ properties.
Table 2 present estimation results for uniregime GARCH
models. It is clear from the table that almost all parameter
estimates including µ in uniregime GARCH models are
highly significant at 1%. Only the leverage effect ξ of
EGARCH model with normal and GED errors are insignificant. However, the asymmetry effect term ξ in GJR-GARCH
models is significantly different from zero, which indicates
unexpected negative returns imply higher conditional variance as compared to same size positive returns.
The degree of volatility persistence for GARCH models can
be obtained by summing ARCH and GARCH parameters
estimates (α 1 + ξ 1). For EGARCH (1, 1) and GJR-GARCH
(1, 1), persistence is equal to β 1 and (α 1+ξ )/2 + β 1 respectively.. All models display strong persistence in volatility ranging from 0.980 to 0.987, that is, volatility is likely to remain
high over several future periods once it increases.
Conditional mean is rt=ì+ut and conditional variance is
and
and
respectively for GARCH, EGARCH, GJR.
63
If distribution assumptions for standardized errors are compared, it reveals that normality assumption is highly outperformed by other two fat-tailed distributions in terms of loglikelihood values. It is an anticipated result because of fat
tails property of Ian Stock Market. Overall, student-t distribution yields an improvement in fitting the data over the
others and the GJR GARCH model with student-t has the largest log-likelihood among uniregime GARCH models. If a
64
GARCH model is successful at capturing volatility clustering, squared standardized residuals should have no autocorrelation.
3.2.2. MRS-GARCH
Estimation results and summary statistics of SW-GARCH
models are presented in Table 3 Almost all parameter esti-
mates are significantly different from zero at least 95% confidence level. The conditional mean estimates in high volatility regime of SW-GARCH with normal and GED distributions are barely significant at 90% confidence level.
However, ARCH parameters α 1 in both volatility regimes of
SW-GARCH with normal distribution are insignificant. In
order to see existence of different volatility regimes, we
compute the un-conditional variances in each volatility regime. The long term volatility level depends on the estimates
of constant parameter á0. Results in Table 3 are consistent
with this argument and display that there are huge differences between α 0 estimates of each volatility regime. The
parameter estimates á0 in high volatility regimes are nearly
eight times greater than parameter estimates α 0 in low volatility regimes. Moreover, short run dynamics of volatility is
determined by the ARCH parameter α 1 and GARCH parameter α 1. Large estimates of α 1 suggest that effect of shocks
to future volatility die out in a long time, so volatility is persistent. Large values of α 1 display reaction of volatility to
the recent price changes.
65
As expected conditional mean returns in low volatility regime
are higher than that of high volatility regimes for all SWGARCH models. So, the lower uncertainty in TEPIX index
gives chance of higher profit to the practitioners. This shows the
importance of regimes switching models to model volatility.
Comparing the low and high volatility regimes in all SWGARCH models, the former volatility regimes have lower
α 1 estimates but higher β1 estimates than latter volatility
regimes have. So, the GARCH processes in the low volatility
regimes are more reactive but less persistent than that in the
high volatility regime. In addition, it is interesting to notice
that degree of volatility persistence (α 1 + β1) within low
volatility regime is higher compared to the high volatility
regime. Persistence within each regime is calculated as α i1 +
β I where i=1, 2.
3.3. In samples
In addition to the goodness-of-fit statistics, we consider
various statistical loss functions to analyze in-sample estimation performance of the volatility models. We assume daily
squared return as actual volatility. As seen Table 4, one of the
SW-GARCH models obtains the highest ranking according
to all statistical loss functions. Also, first three ranks are
generally shared by SW-GARCH models. Thus, evaluating
in sample estimation results according to loss functions, as
well as goodness-of-fit statistics, we conclude that SWGARCH models perform better than uniregime GARCH
models in describing Iran Stock Market volatility. Lastly, as
seen in third column of Table 4, comparing persistence of
uniregime GARCH models and SW-GARCH models, it is
observed that the high persistence in the former specification
is reduced by latter models. This result is consistent with
Lamoureux and Lastrapes’s (1990) finding that is high persistent in volatility of GARCH is caused by regime shifts in
the volatility process. Among all SW-GARCH models, SWGARCH with student-t2 shows the largest decline in volatility persistence.
4. FORECASTING EVALUATION
In this section, we investigate ability of markov regime switching and uniregime GARCH models to forecast Iran Stock
Market volatility at different forecast horizons. The forecast
horizons 1, 5, 10 and 22 days are considered in this thesis. In
Table 5, we present the forecast error statistics for one day
ahead. The six of seven forecast error statistics suggest that
SW-GARCH models provide the most accurate volatility
forecasts. In terms of MSE, MAPE, R2LOG, MAE1 and
MAE2, the best forecasting performance belongs to the SWGARCH model with t2; As well as short forecast horizons,
we consider forecasting performance of various GARCH
models at longer horizon 22 days (one month). Results are
presented in Table 5 and 6. The rankings for one month horizon are quietly similar to that of the one day horizon.
According to all loss functions except HMSE, SW-GARCH
model with t2 is the best model in forecasting volatility while
SW- GARCH model with student-t ranks second. Following
markov regime switching models, standard uni-regime
66
67
GARH models are ranked as fourth, fifth and sixth. On the
other hand, HMSE suggests that top three volatility forecasters are standard uniregime GARCH models. It is important
to note that there is substantial deference between results of
HMSE and other statistical loss functions if model comparisons are considered. Most of time, result of HMSE are completely opposite to that of others. Marcucci (2005) has confronted with similar results and stated that HMSE loss is not
particularly suitable for evaluating deferent volatility forecasts and it should be expected to give weird results.
sis. First, there is strong evidence in favor of regime switching GARCH modeling of nonlinearities in the stock returns
volatility of the TEHRAN. Second, the estimation of the
MRS GARCH accurately describes the two regimes based on
the different pattern of adjustment of the stock returns volatility; and the estimated model captures all the events that are
responsible for the presence of nonlinear features in the
TEHRAN stock returns. Third, there is a high probability for
regime clustering with the likelihood that a low volatility
regime will be followed by a low volatility regime greater
that the
5. CONCLUSIONS
Likelihood a high volatility regime will be followed by a
high volatility regime. Fourth, we consider two competing
non-linear models to conduct forecasting of the stock returns
volatility. These models are the estimated MRS GARCH
models and univariate GARCH model. By obtaining the 1step 5- step,10- step- 22- step ahead forecast for stock returns
volatility for the out-of-sample period, we compare the outof-sample performance of the two competing models on the
The purpose of this paper is twofold. First, it aims to model
volatility regime switching GARCH for the stock returns
volatility of TEHRAN Stock Exchange using daily data for
the period 1995-2011. To conduct our study, we adopt the
univariate GARCH developed by marcucci(2005). There are
several important findings that stem from the present analy-
68
basis of forecasting accuracy by applying 7 statistical loss
function. The results suggest that on the basis of the forecast
accuracy and the, we could conclude that MRS-GARCH
model have priority in univariate GARCH models in their
forecasting performance.
ACKNOWLEDGEMENTS We thanks Marcucci for providing his MATLAB source codes which estimate SWGARCH models’ parameters and forecast volatility.
REFERENCES
[1]
A. Kanas, & G. Kouretas. «A cointegration approach to the
lead-lag effect among size-sorted equity portfolios,»
International Review of Economics & Finance, Elsevier, vol.
14(2), pages 181-201.( 2005).
[2]
Bollen, Nicolas P.B., Gray, Stephen F., Whaley, Robert E.,
Regime switching in foreign exchange rates: Evidence from
currency option prices, Journal of Econometrics,( 2000), 94,
239-276.
[3]
C. Brunetti , C. Scotti, Mariano, S. Roberto & Tan, H.H.
Augustine, Markov switching GARCH models of currency
turmoil in Southeast Asia, Emerging Markets Review,
Elsevier, vol. 9(2), June. (2008), 104-128.
[4]
C. J. Kim and C. R. Nelson , State-space models with regime
switching Classical and Gibbs-sampling approaches with
applications, MIT Press, Cambridge MA and London,1999.
[5]
Chao, Hui-Ping, 1998. «Regime Switching In Us Livestock
Cycles, 1998 Annual meeting, , American Agricultural
Economics Association,(1998).
[6]
Charles Engel & James D. Hamilton, «Long Swings in the
Exchange Rate: Are they in the Data and Do Markets Know
It?,» NBER Working Papers 3165, National Bureau of
Economic Research, Inc.(1989).
[7]
[8]
[9]
Coe, P. “Power Issues When Testing the Markov Switching
Model with the Sup. Likelihood Ratio Test Using U.S.
Output,” Empirical Economics, 27,(2002).
Engel, Charles & Kim, Chang-Jin. «The Long-Run U.S./U.K.
Real Exchange Rate,» Journal of Money, Credit and Banking,
Blackwell Publishing, vol. 31(3), pages 335-56, August, (1999).
F. Klaassen, Improving GARCH volatility forecasts],
Empirical Economics, 27(2), (2002), 363-394.
[10] G. Schwarz, Estimating the dimension of a model, The Annals
of Statistics,6(2) ,(1989),461-464.
[11] G. Lamoureux & W. Lastrapes. « Heteroscedasticity in Stock
Return Data: Volume versus GARCH Effects,» Journal of
Finance, American Finance Association, vol. 45(1), pages
221-29, March, (1990).
[12] H. Daouk and J. Q. Guo, Switching asymmetric GARCH and
options on a volatility index, Journal of Futures Markets,
24(3), (2004), 251-282.
[13] J. Cai, A markov model of unconditional variance in ARCH,
Journal of Business and Economic Statistics, 12(3), (1994),
309-316.
[14] J. D. Hamilton, , A new approach to the economic analysis of
nonstationary time series and the business cycle,
Econometrica, 57, (1989), 357.384.
[15] J. D. Hamilton, Time Series Analysis, Princeton University
Press, Princeton, New Jersey, 1994.
[16] J. Marcucci, Forecasting Stock Market Volatility with RegimeSwitching GARCH Models, Studies in Nonlinear Dynamics &
Econometrics, Berkeley Electronic Press, vol. 9(4), (2005),
pages 6.
[17] J.D. Hamilton, Regime Switching Models, Palgrave
Dictionary of Economics, 2005.
[18] J.D. Hamilton, and R. Susmel, Autoregressive conditional
heteroskedasticity and changes in regime, Journal of
Econometrics, 64, (1994), 307-333
[19] Kim, Jin Chang , Unobserved-Component Time Series Models
with Markov-Switching Heteroscedasticity: Changes in
Regime and the Link between Inflation Rates and Inflation
Uncertainty , Journal of Business & Economic Statistics, vol.
11, issue 3, (1993), 341-49.
[20] M. Haas, S. Mittnik, and M. S. Paollela, A new approach in
markov- switching GARCH models , Journal Of Financial
Econometrics, 2(4) , (2004), 493-530.
[21] M. J. Dueker, Markov switching in GARCH processes and
mean-reverting stock market volatility, Journal of Business
and Economic Statistics, 15(1), (1996), 26-34.
[22] R. F. Engle and V. K. Ng, Measuring and testing the impact of
news on volatility, Journal of Finance, 48, (1993), 1749-78.
[23] S. Gray, Modelling the conditional distribution of interest rates
as a regimeswitching process, Journal of Financial
Economics, 42(1), (1994), 27-62.
[24] Sebastian Edwards & Raul Susmel, «Volatility Dependence
and Contagion in Emerging Equity Markets,» NBER Journal
of Development Economics, Elsevier, (2001).

Comparing Regime Switching GARCH Models and GARCH Models

Transcripción

Documentos relacionados

Cambio de modelo un