We incorporate the impact of structural breaks in the unbiased unconditional volatility as proposed by Kumar and Maheswaran with a conditional autoregressive range (CARR) model. The findings of the proposed framework are compared with the findings based on the volatility forecasts of the GARCH model with and without structural breaks in volatility. Our findings based on the analysis on S&P 500, FTSE 100, SZSE Composite and FBMKLCI indices indicate that the proposed framework effectively captures the dynamics of conditional volatility and provides better out-of-sample forecasts relative to GARCH models with and without structural breaks in volatility.
The volatility of assets plays a very important role in investment decisions making, portfolio implementation and management, option pricing and risk measurement. There are various ways to estimate the daily unconditional volatility. Based on the kind of data available, different proxies for the daily volatility are available in the literature. The demeaned squared return and absolute return are the popular proxies of volatility based on daily closing prices of the tradable assets. However, these estimates of daily volatility are noisy in nature [
In this paper, we use AddRS with CARR to conditionally model the AddRS volatility estimator. We also incorporate the adjustment for the presence of structural breaks in the model using exogenous dummy variables representing different regimes. These infrequent regime shifts in volatility may be due to major domestic as well as global financial, macroeconomic and political events [
In this study, we use the framework as proposed by Inclan and Tiao [
The remainder of this paper is organized as follows: Section 2 presents the brief literature review. Section 3 presents the methodology used in this study. Section 4 describes the data and discusses the preliminary analysis. Section 5 reports the empirical results. Section 6 describes the conclusion with a summary of main findings.
The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) family of models including EGARCH, GJR-GARCH and much more are quite popular to conditionally model the squared returns and to capture the dynamics of volatility. However, there exist evidence indicating that the conditional volatility based on the opening, high, low and closing prices perform much better than the GARCH based conditional volatility [
Suppose εt is a zero mean series with unconditional variance σ2. Suppose the variance for each regime is given by τ j 2 , where j = 0 , 1 , ⋯ , N T and NT is the total number of sudden changes in volatility in T observations, and 1 < k 1 < k 2 < ⋯ < k N T < T are the change points.
σ t 2 = τ 0 2 for 1 < t < κ 1 (1a)
σ t 2 = τ 1 2 for κ 1 < t < κ 2 (1b)
σ t 2 = τ N T 2 for κ N T < t < T (1c)
In order to estimate the presence of sudden changes in variance and the time point of each variance shift, we use a cumulative sum of squares procedure. The cumulative sum of the squared observations from the start of the series to the kth point in time is given as:
C k = ∑ t = 1 k ε t 2
where k = 1 , ⋯ , T . The Dk (IT) statistics is given as:
D k = ( C k C T ) − k T , k = 1 , ⋯ , T with D 0 = D T = 0 (2)
where CT is the sum of squared residuals from the whole sample period.
If there are no sudden changes in the variance of the series then the Dk statistic oscillates around zero and when plotted against k. On the other hand, if there are sudden changes in the variance of the series, then the Dk statistics values drift either above or below zero. The 95th percentile critical value for the asymptotic distribution of max k ( T / 2 ) | D k | is ±1.358. If max k ( T / 2 ) | D k | violates the confidence band then a sudden change in variance is identified.
Kumar and Maheswaran [
b t = log ( H t O t )
c t = log ( L t O t )
x t = log ( C t O t )
Let u t = 2 b t − x t and v t = 2 c t − x t . Hence, the bias corrected extreme value estimators are given by:
A d d u x = 1 2 ( u t 2 − x t 2 ) + x t 2 ⋅ 1 { b t = 0 or x t = b t }
and
A d d v x = 1 2 ( v t 2 − x t 2 ) + x t 2 ⋅ 1 { c t = 0 or x t = c t }
Therefore, the unbiased AddRS estimator, as proposed by Kumar and Maheswaran [
AddRS = 1 2 [ A d d u x + A d d v x ] (3)
Chou (2005) proposed the CARR model to study the dynamic nature of the range. Here, we propose the use of AddRS estimator in place of range in CARR model because it is unbiased regardless of the drift parameter. The specification of the standard CARR(p, q) model for the AddRS estimator is given as:
AddRS t = λ t ε t , ε t | I t − 1 ~ exp ( 1 , . )
λ t = ω + ∑ i = 1 q α i AddRS t − i + ∑ j = 1 p β i λ t − j (4)
where AddRSt is the AddRS estimator as given in Equation (3), λt is the conditional mean of the AddRS and εt is the innovation term, that is, the normalized AddRS estimator (εt = AddRSt/λt), which is assumed to follow the exponential distribution with unit mean.
The CARR(p, q) model with volatility regimes based on the AddRS estimator can be expressed as follows:
AddRS t = λ t ε t , ε t | I t − 1 ~ exp ( 1 , . )
λ t = ω + d 1 D 1 + ⋯ + d n D n + ∑ i = 1 q α i AddRS t − i + ∑ j = 1 p β i λ t − j (5)
where D 1 , ⋯ , D n are the dummy variables taking the value of 1 from each point of sudden change in the unconditional variance onwards and 0 elsewhere.
We use weekly opening, high, low and closing prices of Standard & Poor 500 (S&P 500), FTSE 100, SZSE Composite (hereafter, SZSEC) and FBMKLCI which include two developed and two emerging markets. All the data have been obtained from the Bloomberg database. The period of study is from April 1996 to June 2017.
First, we identify the presence of volatility regimes in the AddRS estimator and the squared return using IT-ICSS approach.
We estimate the CARR model based on the AddRS estimator with and without structural breaks in the AddRS estimator. The given models are reported in
AddRS | Return | |||||||
---|---|---|---|---|---|---|---|---|
S&P 500 | FTSE 100 | SZSEC | FBMKLCI | S&P 500 | FTSE 100 | SZSEC | FBMKLCI | |
Mean | 1.270 | 1.444 | 1.755 | 1.132 | 0.119 | 0.063 | 0.156 | 0.039 |
Median | 0.552 | 0.659 | 0.764 | 0.296 | 0.274 | 0.214 | 0.242 | 0.101 |
Min | 0.008 | 0.014 | 0.000 | 0.001 | −20.828 | −15.297 | −25.937 | −13.720 |
Max | 45.713 | 59.976 | 31.891 | 54.265 | 10.182 | 13.588 | 17.675 | 28.109 |
Stdev | 2.634 | 3.016 | 3.039 | 3.308 | 2.363 | 2.390 | 3.704 | 2.938 |
Skewness | 8.703# | 9.901# | 4.709# | 9.100# | −0.956# | −0.441# | −0.601# | 1.269# |
Kurtosis | 116.428# | 154.646# | 33.133# | 112.604# | 10.627# | 7.064# | 8.109# | 21.055# |
JB stat | 607965.2# | 1079767.2# | 46014.6# | 569893.7# | 2851.670# | 797.433# | 1270.789# | 15332.112# |
Q(20) | 1149.728# | 856.365# | 596.870# | 1280.295# | 55.247# | 42.985# | 43.736# | 83.904# |
ARCH(10) | 350.873# | 138.080# | 39.573# | 245.005# | 99.229# | 121.457# | 95.687# | 124.006# |
#means significant at 1% level. Note that Stdev represents the standard deviation, JB stat represents the Jarque Bera statistic, Q(20) indicates the Ljung-Box Q statistic up to 20 lags and ARCH(10) indicates the Lagrange multiplier test for conditional heteroskedasticity up to 10 lags.
Index | Number of Breaks | Break Date Detected | Reason |
---|---|---|---|
S&P 500 | 4 | 26-01-2000 16-04-2003 29-08-2007 15-09-2010 | The internet bubble Post-internet bubble bursting impact Global Financial Crisis Bull rally after global financial crisis |
FTSE 100 | 4 | 09-02-2000 23-04-2003 29-08-2007 08-09-2010 | The internet bubble Post-internet bubble bursting impact Global financial crisis Bull rally after global financial crisis |
SZSEC | 2 | 26-04-2006 23-09-2009 | - Bull rally after global financial crisis |
FBMKLCI | 2 | 04-04-2001 29-07-2009 | Internet bubble Bull rally after global financial crisis |
Equations (4) and (5). The models for incorporating the impact of structural breaks in squared return based on the GARCH model is given as:
ε t = z t σ t , z t ~ N ( 0 , 1 )
σ t 2 = ω + d 1 D 1 + ⋯ + d n D n + α 1 ε t − 1 2 + β 1 σ t − 1 2 , (6)
where D 1 , ⋯ , D n are the dummy variables taking a value of 1 for the given volatility regime and 0 elsewhere.
The GARCH model without any volatility regimes is given as:
ε t = z t σ t , z t ~ N ( 0 , 1 )
σ t 2 = ω + α 1 ε t − 1 2 + β 1 σ t − 1 2 , (7)
Index | Number of Breaks | Break Date Detected | Reason |
---|---|---|---|
S & P 500 | 4 | 15-03-2000 04-06-2003 03-09-2008 14-12-2011 | Internet bubble Post-internet bubble bursting impact Late 2000’s financial crisis European debt crisis |
FTSE 100 | 3 | 19-03-2003 17-10-2007 30-11-2011 | Post-internet bubble bursting impact Sub-prime crisis European debt crisis |
SZSEC | 3 | 07-07-1999 15-11-2006 27-01-2010 | Internet bubble - Bull rally after global financial crisis |
FBMKLCI | 3 | 18-08-1999 15-01-2003 22-07-2009 | Internet bubble Post-internet bubble bursting impact Bull rally after global financial crisis |
The standard errors of the parameters of the CARR models based on the AddRS estimator are smaller in magnitude than the standard errors of the corresponding parameters from the GARCH models which confirms the finding of Brandt and Jones [
w | α1 | b1 | α1 + b1 | LLF | Q(10) | Qs(10) | ARCH(10) | |
---|---|---|---|---|---|---|---|---|
S&P 500 | ||||||||
CARR | 0.074# | 0.382# | 0.580# | 0.962 | −1033.61 | 11.961 | 10.917 | 1.118 |
(0.016) | (0.030) | (0.034) | ||||||
CARR-B | 0.120# | 0.171# | 0.452# | 0.623 | −1014.25 | 11.471 | 11.239 | 1.210 |
(0.024) | (0.025) | (0.054) | ||||||
GARCH | 0.220* | 0.137* | 0.832# | 0.968 | −2413.54 | 12.902 | 6.758 | 0.672 |
(0.111) | (0.061) | (0.059) | ||||||
GARCH-B | 0.747# | 0.110# | 0.602# | 0.711 | −2385.95 | 17.635 | 8.841 | 0.888 |
(0.305) | (0.028) | (0.113) | ||||||
FTSE 100 | ||||||||
CARR | 0.097# | 0.379# | 0.574# | 0.953 | −1198.93 | 12.393 | 11.518 | 0.613 |
(0.017) | (0.037) | (0.039) | ||||||
CARR-B | 0.147# | 0.354# | 0.508# | 0.862 | −1186.43 | 11.923 | 11.819 | 0.711 |
(0.029) | (0.039) | (0.055) | ||||||
GARCH | 0.265* | 0.176# | 0.787# | 0.963 | −2423.61 | 6.658 | 12.371 | 1.305 |
(0.112) | (0.044) | (0.049) | ||||||
GARCH-B | 0.498* | 0.171# | 0.699# | 0.870 | −2412.26 | 9.293 | 10.021 | 1.021 |
(0.232) | (0.043) | (0.085) | ||||||
SZSEC | ||||||||
CARR | 0.037# | 0.101# | 0.877# | 0.978 | −1532.04 | 11.419 | 11.171 | 1.215 |
(0.005) | (0.008) | (0.008) | ||||||
CARR-B | 0.045# | 0.106# | 0.742# | 0.848 | −1516.58 | 13.729 | 10.042 | 1.114 |
(0.009) | (0.010) | (0.012) | ||||||
GARCH | 0.290* | 0.128# | 0.854# | 0.982 | −2886.91 | 28.961# | 3.063 | 0.303 |
(0.130) | (0.024) | (0.025) | ||||||
GARCH-B | 0.314* | 0.117# | 0.843# | 0.959 | −2881.85 | 26.629# | 5.199 | 0.528 |
(0.169) | (0.025) | (0.032) | ||||||
FBMKLCI | ||||||||
CARR | 0.028# | 0.268# | 0.726# | 0.994 | −663.40 | 10.319 | 10.200 | 0.816 |
(0.004) | (0.018) | (0.016) | ||||||
CARR-B | 0.077# | 0.375# | 0.418# | 0.793 | −633.19 | 6.341 | 5.619 | 0.531 |
(0.009) | (0.031) | (0.031) | ||||||
GARCH | 0.038 | 0.103# | 0.896# | 0.999 | −2378.57 | 14.831 | 2.077 | 0.202 |
(0.024) | (0.025) | (0.024) | ||||||
GARCH-B | 0.060* | 0.109# | 0.872# | 0.982 | −2373.83 | 15.416 | 2.383 | 0.233 |
(0.039) | (0.034) | (0.040) |
# and * mean significant at 1% and 5% levels respectively. The terms in the parenthesis (.) represent the standard error of the estimates. LLF represents the log-likelihood function, Q(10) and Qs(10) represent the Ljung Box statistic for standardized residuals and squared standardized residuals (respectively) up to 10 lags. ARCH(10) represent the ARCH-LM statistic for the presence of heteroscedasticity in the standardized residuals up to 10 lags.
In this section, we assess the forecasting performance of the models under study based on 1 step ahead prediction of volatility. The forecasts are generated using rolling windows estimation of the models with fixed window size. We generate 500 forecasts for all the models and for all the indices. We use weekly realized volatility (sum of the square of daily returns) based as a proxy for measured volatility. We use the following four loss functions for evaluating the forecasting performance of models under study.
1) Root mean squared errors (RMSE)
RMSE ( m , h ) = 1 T ∑ t = 1 T ( M V t + h − F V t + h ( m ) ) 2
2) Mean absolute errors (MAE)
MAE ( m , h ) = 1 T ∑ t = 1 T | M V t + h − F V t + h ( m ) |
3) Logarithmic loss function (LL)
LL ( m , h ) = 1 T ∑ t = 1 T ( ln ( M V t + h F V t + h ( m ) ) ) 2
4) Loss implied by Gaussian likelihood (QLIKE)
QLIKE ( m , h ) = 1 T ∑ t = 1 T ( ln ( F V t + h ( m ) ) + M V t + h F V t + h (m) )
where m represents the model (CARR-B, CARR, GARCH-B and GARCH), h is equal to 1 representing 1 step ahead forecasts, MVt represents the measured volatility at time t (realized volatility), FVt(m) represents the predicted volatility based on model m and T represents the number of out-of-sample volatility forecasts. Here, T is 500.
In addition to the error statistics, we also use Mincer and Zarnowitz [
M V t + h = α + β F V t + h ( m ) + ε t
where MVt represents the measured volatility (realized volatility) at time t, FVt(m) is a predicted volatility based on model m and εt represents the error term.
To examine the economic significance of the findings of the study, we implement a trading strategy based on the risk-averse investor who uses predicted volatility to switch investment between a portfolio of risky stocks (given index) and a risk-free asset. The risk-free assets for a country is the 3 months (for the USA and the UK) or 6 months (for China and Malaysia) T-Bills of that economy. For bad news (negative return and if forecasted volatility is greater than average volatility), the investor invests 100% of the capital in the risk-free asset or else he invests in the portfolio of risky stocks.
CARR | CARR-B | GARCH | GARCH-B | |
---|---|---|---|---|
S&P 500 | ||||
MSE | 2.663 | 2.602 | 8.484 | 2.553 |
MAE | 1.103 | 1.063 | 5.686 | 1.411 |
LL | 1.351 | 1.329 | 6.077 | 1.963 |
QLIKE | 2.407 | 2.339 | 15.826 | 3.869 |
FTSE 100 | ||||
MSE | 3.380 | 3.335 | 9.266 | 16.431 |
MAE | 1.323 | 1.279 | 6.405 | 11.388 |
LL | 1.267 | 1.219 | 5.276 | 5.947 |
QLIKE | 2.400 | 2.295 | 12.483 | 17.117 |
SZSEC | ||||
MSE | 2.830 | 2.812 | 22.399 | 22.399 |
MAE | 1.463 | 1.417 | 15.093 | 15.093 |
LL | 1.180 | 1.114 | 7.466 | 7.466 |
QLIKE | 2.465 | 2.329 | 22.060 | 22.060 |
FBMKLCI | ||||
MSE | 0.891 | 0.792 | 4.926 | 4.156 |
MAE | 0.415 | 0.345 | 3.358 | 2.650 |
LL | 1.615 | 1.335 | 7.840 | 6.506 |
QLIKE | 1.998 | 1.279 | 20.024 | 15.076 |
CARR | CARR-B | GARCH | GARCH-B | |
---|---|---|---|---|
S&P 500 | 0.407 | 0.418 | 0.286 | 0.295 |
FTSE 100 | 0.227 | 0.312 | 0.299 | 0.158 |
SZSEC | 0.224 | 0.230 | 0.184 | 0.184 |
FBMKLCI | 0.088 | 0.089 | 0.087 | 0.088 |
CARR | CARR-B | GARCH | GARCH-B | |
---|---|---|---|---|
S & P 500 | 8.396 | 9.732 | 4.961 | 5.193 |
FTSE 100 | 4.106 | 4.817 | −2.167 | 3.851 |
SZSEC | 4.219 | 5.151 | 1.518 | 1.518 |
FBMKLCI | 5.316 | 6.473 | 2.153 | 4.619 |
annualized return based on the GARCH and GARCH-B models are quite low and for the case of FTSE 100, the volatility forecasts based on the GARCH model provides a negative average annualized return for the risk-averse investor.
The findings of the study have implications towards policy maker, regulators, traders, risk managers, portfolio managers and investors. The study highlights the importance of incorporating structural breaks in volatility in modelling and in generating more accurate forecasts of volatility. The findings based on economic return earned by the risk-averse investor provide implication of the study for investors, traders and portfolio managers. Policy makers and regulators can use the unbiased AddRS volatility estimator in presence of structural breaks to understand the periods of stability and turbulence in the market and to implement appropriate policies to deal with the adverse impact of any macroeconomic event. Moreover, more accurate forecasts of volatility in deriving more accurate Value-at-Risk and Expected Shortfall measures to quantify risk and has implications for risk managers.
In this study, we propose the use of the CARR model to model the AddRS estimator and to generate a more accurate forecast of it. We also incorporate the impact of structural breaks in volatility in CARR model while modelling and forecasting the AddRS estimator. The results based on the in-sample estimation and impulse response support the evidence that incorporating the impact of structural breaks in volatility modelling does decrease the volatility persistence. We observe that this decrease in volatility persistence is smooth for CARR-B model. We observe an abrupt decrease in volatility persistence for the GARCH-B model. The results based on out-of-sample volatility forecast evaluation indicate that the CARR-B model provides more accurate forecasts of realized volatility when compared with corresponding volatility forecasts by another model. The economic significance analysis also indicates that the risk-averse investor can earn a higher average annualized return by trading based on the volatility forecasts of the CARR-B model. Overall, our finding indicates that the CARR-B model outperforms other models in generating more accurate forecasts of realized volatility.
Kumar, D. (2018) Volatility Prediction: A Study with Structural Breaks. Theoretical Economics Letters, 8, 1218-1231. https://doi.org/10.4236/tel.2018.86080