Modeling and Forecasting Stock Return Volatility Using a Random Level Shift Model

MODELING AND FORECASTING STOCK RETURN VOLATILITY USING A RANDOM LEVEL SHIFT MODEL

Lu HongdaYu LuMeng Shuangshuang

Introduction  Main goal : to forecast volatility proxied

by daily squared returns Our approach: extends Starica and

Granger (2005)’s work Apply log-absolute returns and daily

returns rather than intra-daily data

Three main parts

1• The model, the specification

adopted and the estimation procedure

2 • Estimation results and diagnostics

3 • The forecasting comparisons

1. THE MODEL AND THE ESTIMATION METHOD

Lu Hongda

1.1 Model

The model we apply to log-absolute returns is given by

is a constant is the random level shift components is a short-memory process, included to model the remaining noise. The level shift component is specified by:

Where ,is a binomial variable that takes value 1 with probability α and value 0 with probability (1 − α). If it takes value 1, then a random level shift occurs, specified by ∼ i.i.d. N(0,).

1.2 Assumptions

For simplicity, we assume:

a) Short memory: = , ()b) The components and are mutually independentc) Normality assumption for , and (needed to

construct the likelihood function)

1.3 Model Estimation Specify level shift component as a random walk process with innovations distributed according to a mixture of two normally distributed processes: where , and is a Bernoulli random variable that

takes value one with probability α and value 0 with probability 1 − α. By specifying and

Estimation of :

The probability of level shifts is very small in all cases considered. Thus… Given that the shifts occur so infrequently,

the noise component accounts for the bulk of total variation.

Then…

Apply the method of Bai and Perron (2003) to obtain the estimates of the break dates that globally minimize the following sum of squared residuals:

where m is the number of breaks, (i = 1,… , m) are the break dates with and and (i = 1, ..., m + 1) are the means within each regime which can easily be estimated once the break dates are.

1.3 Model Estimation

Continue…

We next specify the model in terms of first-differences of the data:

where We then have the following state space form:

Or more generally

Where, in the case of an AR(p) process

And…

and is a p-dimensional normally distributed random vector with mean zero and covariance matrix

1.4 The estimation method

Similar to Markov regime switching models in estimation methodology

The log likelihood function is:

where

1.4 The estimation method

where

where 1 represents a (4*1 )vector of ones denotes element-by-element multiplication with element, with the element of

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f f t t tti s j Yy y i j YsY s s

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1.4 The estimation method

For when We first focus on the evolution of

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1.4 The estimation method

Therefore, the evolution of is given by:

Or more compactly by:

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: the prediction error : the prediction error

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The best forecast for the state variable and its associated variance conditional on past information and are

We have the measurement equation

ist 1

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, (6)

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1.4 The estimation method

Where Hence, the prediction error and variance

are:

Applying standard updating formulae, we have (given ),

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To reduce the dimension of the estimation problem, we adopt the re-collapsing procedure suggested by Harrison and Stevens (1976), given by

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1.4 The estimation method

By doing so, we make unaffected by the history of states before time t − 1

If we define we then have four possible states corresponding to

with the transition matrix Π as defined in (4)

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1.4 The estimation method

The vector of conditional densities

thus have a more compact representation given by:

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1.5. Apply model and estimation method Four major market indices: the S&P 500,

AMEX, Dow Jones and NASDAQ

Daily returns:

Offset parameter/small constant:

1.5. Apply model and estimation method Specification of short-memory component

a) , ,by Starica and Granger(2005)

b) , process, as a robustness check

1.5. Apply model and estimation method Initialize the state vector and its

covariance matrix by their unconditional expected values(all components of states are stationary), i.e., and

We obtain estimates by directly maximizing the likelihood function:

)2(,);(ln)ln(1 1

T

t ttYyfL

2. EMPIRICAL RESULTS FOR RETURNS ON STOCK MARKET INDICES

Yu Lu

2.1. Estimation results The cases in which the short-memory component

is specified to be white noise and an AR (1) process

the standard deviation of the original seriesa) : the standard deviation of level shift

component, b) α: the probability of a shiftc) :the standard deviation of the stationary

componentd) φ :the autoregressive coefficient when

considering the AR (1) specification for .

Estimation results?

S&P 500 0.7524 0.00198 0.73995(SD:) 0.74290 0.00082 0.73920 0.00030AMEX 0.84579 0.00157 0.72346(SD:) 0.85127 0.00097 0.70671 -0.0146Dow Jones

0.97270 0.00120 0.73291

(SD:) 0.94899 0.00121 0.73196 -0.00087NASDAQ 1.53121 0.00081 0.74280(SD:) 1.53283 0.00079 0.74286 -0.00029

Table 1: Maximum Likelihood Estimates

2.2. Noteworthy features of results:

Firstly, for short-memory componenta) When considering process, estimate of φ is very

small and close to zerob) Adopting either an AR(1) or white noise

specification for yields very similar resultsThus…In subsequent sections we shall only consider results based on the white noise specification for the short-memory component. These results are in agreement with those of Starica and Granger (2005)

2.2. Noteworthy features of results:

Secondly, for level shift componenta) The probability of level shifts is very

small in all cases considered. Thus… Given that the shifts occur so infrequently,

the noise component accounts for the bulk of total variation.

2.3. The effect of level shifts on long-memory and conditional heteroskedasticity Investigating whether the shifts can

explain:

a) the well-documented feature of long-memory

b) the presence of conditional heteroskedasticity

Plot:a) the ACF of the original series b) the ACF of its short-memory

component(obtained by subtracting the fitted trend from )

Whether the level shift can account for the long-memory feature of the series?

2.3.1. Effects of level shift on long-memory

a)the ACF of the original series

The log-absolute returns clearly display an autocorrelation function that resembles that of a long-memory process: it decays very slowly and the values remain important even at lag 300.

b) the ACF of its short-memory component For all practical purposes, we can view the

short-memory component as being nearly white noise.

2.3.2 Effects of level shift on conditional heteroskedasticity GARCH (1, 1) model with Student-t errors

given by, for the demeaned returns process

① i.i.d. Student-t distributed with mean 0 and variance 1

② the parameters of interest and measure the extent of conditional heteroskedasticity present in the data.

CGARCH model

where if t is in regime i, i.e., and 0 otherwise, with (i = 1, ..., m) being the break dates documented in Figure 1 (again and ). The coefficients , which index the magnitude of the shifts, are treated as unknown and are estimated with the remaining parameters, while the number of breaks is obtained from the point estimate of α.

Table2. Parameter EstimatesS&P 500

Coefficient

Estimate Std. Error

t-Statistic

p-value

No level shifts in GARCH

0.057 0.004 13.55 00.938 0.004 214.75 0

No level shiftsin CGARCH

0.058 0.007 7.98 00.939 0.009 95.34 00.9996 0.0003 3250.19 00.027 0.005 5.36 0

Level shiftsin CGARCH

-0.085 0.069 4.73 0.0820.525 0.217 1.54 0.0880.622 0.012 106.98 00.138 0.089 2.47 0.031

2.3.2 Effects of level shift on conditional heteroskedasticity

Both estimates are highly significant for all series. In particular, the value of is quite high.

when we none of the estimates ofare significant.

Use the standard GARCH (1,1) model

allow for level shifts(CGARCH)

Summary:

The level shifts model with white noise errors appears to provide an accurate description of the data.

The level shift component is an important feature that explains both the long-memory and conditional heteroskedasticity features generally perceived as stylized facts.

3 FORECASTINGPerformance of the level shift model &GARCH(1,1)in forecasting volatility

Meng Shuangshuang

3.1 Design of forecasting experiment

Following Starica and Granger(2005) Start forecasting at observations 2,000 Re-estimate every 20 days, up to 200

days Proxy: realized squared returns( )

22)( rE

3.2 Evaluation of forecasting performance

evaluated by the ratio of their MSE:

MSE(p)

where n : no. of forecasting produced

)(/)( PP MSEMSE GARCHLS

= 1n σ (rഥt,p2 −σt,p2 )2 n t=1

rҧt,p2 =σ rt+k2pk=1 : the realized volatility over [t+1,t+p]

σt,p2 = σ σ t+k2pk=1 : σ t2 be a p-step ahead forecast of σt+p2 the variance

of returns rtat time t+p

3.3 construction of the forecasts The level shift model(1) The level shift model an appropriate transformation

where C : a positive constant used to bound returns away from zero

yields

𝐥𝐨𝐠ሺa(𝐫𝐭a(+ 𝐂ሻ= 𝛕𝐭+ 𝐂𝐭

a(𝐫𝐭a(+𝐂= 𝐡𝐭𝟏/𝟐𝛜𝐭 where, ht = e2τt E(e2ct) and 𝛜𝐭 = ect /[E(e2ct)]1/2

3.3 construction of the forecasts The level shift model(2) continue

ignore level shifts when forecasting rare uncertain about timing and

magnitudes

E𝛜𝐭2=1 𝛜𝐭 is independent of ht and i.i.d

2ct~ i.i.d N(0, 4σe2) Eሺe2ctሻ= e0+4σe2/2 = e2σe 2

3.3 construction of the forecasts The level shift model(3)

Continue

the K -period ahead forecast of the squared returns is :

Et(a(rta(+C)2 = Etht+k = exp( 2τt +2σe2) Etrt+k2 = expሺ2τt +2σe2ሻ−2CEta(rt+ka(−C2

Where Et |rt+k | = Etexp (𝛕𝐭 +𝐂𝐭+𝐤) - C=exp (τt +0.5σe2) - C

3.3 construction of the forecasts The GARCH model(1) Student-t GARCH(1,1) model

Transformation:

Rt෪ = rt −u = σtεt σt2 = α1 +α2𝐫𝐭−𝟏 +α3σt−12

εt : i.i.d~N(0,1)

rt : demeaned returns

rt2=α1 + (α2+α3）rt−12 +ωt−α3ωt−1 * where ωt = rt2-σt2 : the forecast error associated with the forecast of rt2 , white noise

3.3 construction of the forecasts The GARCH model(2) assuming < 1, recursive form for the squared

demeaned returns

So for k > 1, the k-period ahead forecast

Where the forecasts for the squared returns

E(rt2)=σ2=α1/(1− α2−α3）

rt2-σt2=(α2+α3)( rt−12 -σ2)+ ωt−α3ωt−1

Etrt+k2 =σ2 + ( α2+α3 )k−1(Etrt+12 -σ2)= σ2 +ሺ α2+α3 ሻk−1(σt+12 −σ2) σt+12 = α1 +α2rt2+α3σt2.

Etrt+k2 = Etrt+k2 + (Etrt+k)2 ≈ Etrt+k2 + μ2

3.4 Forecasting Comparisons

use the fitted means obtained from the full sample to get the mean

re-estimate every 20 observations to get the mean

the random level shifts model the GARCH(1,1)

The comparison indicates that with a precise estimate of the mean of log-absolute returns at a given date, we can obtain much better forecasts from the level shift model than the other models. 

3.4 Forecasting ComparisonsHorizon Horizon

P days S&P 500 P days S&P 500

20 0.5 120 0.51

40 0.44 140 0.54

60 0.44 160 0.56

80 0.46 180 0.59

100 0.48 200 0.62

)(/)( PP MSEMSE GARCHLS)(/)( PP MSEMSE GARCHLS

3.4 Forecasting Comparisons Table: volatility forecasting performance

of the LS model & GARCH(1,1) model ratio < 1 : volatility forecast of the LS

model is more precise than the GARCH(1,1) model

The figure shows an over-all better longer-horizon volatility forecast performance of the LS model

Conclusion

a simple estimation method for a model with a random level shift & a short-memory

component The level shift component : explains the

presence of both the long-memory and conditional heteroskedasticity

a short-memory component: assume to be white noise

impressive results : when model applied to log-absolute returns of many indices

Conclusion

Volatility forecasting with the model: improvement with squared returns as a

proxy Noticeable when mean is well estimated Difficult to obtain precise estimates of the

mean

ENDThank you