Bootstrap Prediction Intervals for Conditional ...

JSM 2013 - Business and Economic Statistics Section

Bootstrap Prediction Intervals for Conditional Heteroskedastic Models with Cyclically Varying Unconditional Variance Malaka Thilakaratne1, V.A. Samaranayake1 1

Missouri University of Science and Technology, Rolla, MO 65401

Abstract A computationally fast method for obtaining bootstrap-based prediction intervals for returns and volatilities in ARCH and GARCH processes was introduced in a recent publication. This method utilizes a constrained least squares approach to estimating the GARCH parameters. We extend this technique to ARCH type processes having unconditional variance that changes cyclically. Results of a Monte Carlo study that looks at the finite sample properties of the proposed intervals are presented. Simulation results indicate that the intervals for returns provide reasonable coverage probabilities in most circumstances, but that the intervals for volatilities are liberal.

Key Words: Volatility Forecasting; Financial Time Series; Bootstrap; Cyclical Volatility; Daily Volatility

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1. INTRODUCTION It is well known that certain empirical time series exhibit conditional heteroskedasticity. Financial time series involving attributes such as stock returns are especially prone to such behavior. The standard models used to describe such volatility assume that the unconditional variance is constant over time. In this paper we propose a model whose unconditional variance changes cyclically, such as over the days of the week. In addition, a bootstrap-based method is proposed to obtain prediction intervals. Researches in mathematical finance usually employ stochastic volatility models to explain time series with bursts of high volatility followed by calmer periods. Alternatively, the Autoregressive Conditional Heteroskedastic (ARCH) models introduced by Engle (1982) and Generalized Conditional Heteroskedastic (GARCH) models proposed by Bollerslev (1986) have been commonly used to characterize such behavior (Engle, Robert F. (2001). The use of ARCH and GARCH models naturally led to the task of obtaining point forecasts for returns as well as volatility. For example, see Baillie and Bollerslev (1992), Andersen and Bollerslev (1998), Andersen (2001), Engle and Patton (2001), and Poon (2005). Bootstrap-based Prediction intervals for returns and volatilities were first introduced by Miguel and Olave (1999). Reeves (2005) observed that the previous bootstrap method introduced by Miguel and Olave does not incorporate the uncertainty due to parameter estimation and suggested that an additional step of re-estimating the ARCH parameters for each bootstrapped realization of returns be introduced to remedy this problem. Pascual (2006) extended these procedures to GARCH models and obtained prediction intervals with good coverage probabilities. Chen, Gel, Balakrishna and Abraham (2011) introduced a least squares based bootstrap method for obtaining prediction intervals for both volatilities and returns. Their method is computationally very fast due to the fact that the estimation of ARCH/GARCH parameters is done using a constrained least squares procedure rather than the maximum likelihood. While their intervals for returns provide near nominal coverage probabilities, the coverage for volatility is somewhat liberal. All of the previous bootstrap-based prediction intervals apply to models with unconditional variance that remains constant over seasons or the day of the week. Such an assumption is not valid in some situations. For example, Osborn and Smith (1989) States that “The fact that many economic times series have one season that exhibits a higher volatility than other seasons is often overlooked. This behavior is found in monthly production series; the variability of the index of production is higher for the month with the lowest level of production.” In addition to volatility that changes across what we normally consider to be the “seasons,” volatility that varies according to the day of the week have caught the attention of several researchers. For example, Harvey and Huang (1991) noted that interest rate and foreign exchange futures market showed higher volatilities on Fridays. Ederington and Lee (1993), Jones (1998) and Berument and Kiymaz (2001) found similar evidence in the bond and stock markets. Choudhry (2000)

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investigated the day-of-the-week effect on data from seven Asian stock markets and found evidence of such effects on both stock returns and volatility, but the patterns were not identical across the countries under study. A similar study on the stock markets in Latin America by Abril (2012) found Monday (lower than expected) and Friday (higher than expected) effects. Day-of-the-week effects were also found in the natural gas market by Le (2008) where volatility is lower on Fridays than on Wednesdays and all other days except Thursday. Edirisinghe (2011) modeled hourly electricity prices using a modified ARCH process that allowed for both day-of-the-week as well as seasonal changes in volatility. We proposed a similar model and adopt a modified version of the bootstrap-based approach of Chen et al. (2011) to obtain prediction intervals for both returns and volatility. This paper proceeds as follows. In Section 2, the properties of the ARCH model, autoregressive (AR) representation of the ARCH process, and cyclically varying unconditional variance are discussed. Section 3 is devoted for the discussion of the bootstrap algorithm. Results of the simulation study are presented in Section 4. Section 5 provides a discussion of our findings and we conclude in Section 6.

2. THE ARCH MODEL Our model is developed for ARCH processes but it can be extended to GARCH processes as well. The proposed method is based on the AR representation of the ARCH process and a brief introduction to this formulation is given before introducing the modifications that lead to cyclical changes in the unconditional variance.

2.1 Model Characteristics Let  u t t 1 be an ARCH ( p) process with p  1 . T

Assume that,

ut   t  t ,

  a0  2 t

p

a u i 1

i

2 t i

,

where  t t 1 is a sequence of independent, identically distributed (i.i.d) random variables T

 

with mean equal to zero, variance equal to one and E  t4  . Note that a0 and ai are unknown parameters that satisfy

a0  0 , ai  0, for i  1,..., p with  2t and  t being uncorrelated.

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We also assume that

ut Tt1

satisfies the strict stationary conditions given in Bougerol

and Picard (1992a, b). The reader is referred to Chen et al. (2011) for details.

2.2 Autoregressive Representation of the ARCH Model

 

Let  t  ut2   t2 . Then u t2

T t 1

has an autoregressive form of order p given by

p

u t2  a0   ai u t2i   t . i 1

Specifically, if  u t t 1 follows an ARCH (1) model, the above equation is reduced to an T

AR (1) process given by: ut2  a0  a1ut21   t .

2.3 Derivation of the Unconditional Variance of an ARCH  p  Model We now derive the unconditional variance of the process  u t  . We first note that

 

var ut   E ut2 and that ut   t  t . Then ,

 



E u t2  E E (u t2 | Ft 1 )



p    E  a 0   ai u t2i  i 1  

 

p

 a 0   ai E u t2i . i 1

Now, since u t  is covariance stationary,

   

E u t2  E u t2i

which implies Thus,

 

E u t2

for i  1,..., p,

p   1   ai  E u t2  a 0 . i 1   a0  , p   1   ai  i 1  

 

which is the unconditional variance of  u t  . Note that it is independent of t.

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2.4 Introducing a Model with Unconditional Variance that Changes Cyclically While the volatility can be made to change with the season or the day of the week using formulations such as that proposed by Koopman et al. (2007), such an approach makes the unconditional variance intractable. Instead, we use a simpler approach proposed by Edirisinghe (2011). In this approach we start with a regular ARCH  p  process  u t  , and define the returns  wt , to be given by

wt   s t ut , t  1,2,..., T . where st  takes the values 1,2,..., s depending on the “season” and 1 , 2 ,..., s are constants that define the “inflation” or the “deflation” of the unconditional variance due to the “seasonal” effect. We will assume without loss of generality that  s  1. In this paper we concentrate on the special case where s  5 and the “seasons” are the 5 working days of the week. Let

 w  be the set containing all the observations corresponding to the k 2 t (k )

th

day, k = 1,

2, 3, 4, 5. So, we have,

wt2( k )   k ut2( k ) and





for t  1,..., T / 5.

 

E wt2( k )   k E u t2k   k

a0 p   1   ai  i 1  

.

Clearly, the unconditional variance of wt  depends on st  and is given by

 s t 

a0 p   1   ai   i 1 

.

3. Bootstrap Procedure for the modified ARCH ( p ) Process The following steps are similar to those Chen et al. (2011) proposed. We modify them by introducing the additional steps needed to account for the “adjustment” due to the terms 1 ,  2 ,  3 ,  4 and  5 .

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Also note that

 k a0

ˆk 

   1  a     a  E w  2 t (k )

Ew

1

2 t (5)

5

0

1  a1 

 Thus, ˆk  k ,

k

where 0  k  5 .

,

5

where 0  k  5 .

5

Observe that w.l.o.g. we assumed that  5  1. Step 1: Divide the data set

w 

2 T t t 1

into five sub data sets, such that the Subdata Set 1

contains the all of Monday data, Subdata Set 2 contains all of Tuesday data and so on. Obtain the estimates ˆ1 ,ˆ2 ,ˆ3 , and ˆ4 by dividing the sample mean of each sub dataset by the sample mean of Subdata Set 5. Then divide each wt2 of

w 

2 T t t 1

by the

corresponding ˆi , with ˆ5  1 for i  1, 2, 3, 4, 5.

 

Denote the new dataset obtained at the end of step 1 by y t2

T t 1

.

 

Then y t2 can be approximately represented by, p

yt2   0    i yt21   t , for t  1,...T , i 1

where  0  0,  1  0,  t  y   2 t

p

2 t

and    0    i yt21 . 2 t

i 1

Step 2: Using a constrained least squares (LS) method estimate the AR coefficients ˆ 0 ,

ˆ 1 ,…, ˆ p . The constraints are that  0  0 and

p

 i 1

i

 1.

Step 3: Estimate the residuals vˆt t  p 1 by T

p

vˆt  y  ˆ 0  ˆ i yt2i , for t  p  1, ...T , 2 t

i 1

where vˆt  0, for t  1,... p .

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Step 4: Center the residuals using T   1 v~t   vˆt  vˆt , for t  p  1, ...T .  T  p t  p 1  

T The empirical distribution of the centered residuals v~t t  p 1 is

Fˆv ,T  y  

T

1

t  p 1

v~t  y 

.

 

Step 5: Obtain the bootstrap error process vt*

T  200 t 1

by sampling with replacement from

Fˆv,T  y  .

 

Step 6: Construct a bootstrap sample yt2*

T  200 t 1

recursively by

p

y

2* t

 ˆ 0   ˆ i yt2*i  vt* , for t  1, 2,..., T  200, i 1

where y k2* equals to the mean of yt2 obtained at the end of Step 1 . If yt2*  0 , then replace it with zero. Note that we generate T+200 yt2* values. Discard the first 200 to minimize the effect of the initial conditions. We re-label the index of the remaining yt2* to go from 2 to T.

 

Step 7: Using y t2*

T t 1

* * * , estimate the coefficients ˆ 0 , ˆ1 , ..., ˆ p using constrained LS.

 

Step 8: Obtain the bootstrap prediction error process vT*  h

s

h 1

, where s  1 , by

sampling with replacement from Fˆv,T  y  . Step 9: The h -step –ahead forecast of volatility and the squared return are obtained respectively by, p

 T2* p  ˆ 0*   ˆ i* yT2* h i , and i 1

p

yT2* h  ˆ 0*   ˆ i* yT2* h i  vT*  h , for h  1, 2, ..... . i 1

Note that we use y k2* = y k2 for k  T .

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Step 10: Repeat Steps 5 – 9 B times, where B = number of bootstrap runs. Denote the

yT2* h and  T2* p obtained in bootstrap run j by y 2j ,*T h and  2j ,*T  p respectively, j  1, 2, ..., B

 

Step 11: Divide the original data set wt2

T t 1

into five sub data sets as described in Step

1. a) Obtain a sample with replacement from each data set and calculate the respective means of each sample. b) Divide each mean by the mean corresponding to the Friday data and obtain

ˆi* for i  1, ... ,5 . Note that ˆ5* =1. Repeat Step11, B times to obtain ˆi*, j , where i  1, ... ,5 and j  1,..., B . 2* 2* Step 12: Multiply y j ,T  h and  j ,T  p obtained in Step 9 by the corresponding

ˆi*, j , where i  s(T  h)  1, 2, 3, 4, 5 and j  1,..., B. 2* As described by Chen et al. (2011) we use the bootstrap distributions of wT2* h and  T  p 2 to approximate the unknown distributions of wT2  h and  T  p for h  1, ... , s .

Thus, a 1001   % prediction interval for wT2  h is given by

0, H

* T h

1    , h  1, ..., s ,

where H T*  h 1    is the 1    quantile of the distribution of wT2* h . * Now, a 1001   % prediction interval for wT  h ( PI w ) is

Q

* T h

 / 2, QT*h 1   / 2 ,

* * * where QT h  / 2   H T  h 1    and QT  h 1   / 2 

H T* h 1    .





2 * Also, the 1001   % prediction interval of  T  h ( PI  2 ) is 0, K T*  h 1    ,

where KT*  h 1    is the 1    quantile of the distribution of  T  p . 2*

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3.2 Computational Details for Estimating Coverage Probabilities and Interval Lengths We carry out the following tasks to compute coverage probabilities and other statistics. 1.

Simulate and generate R=1,000 future values wT  h and  T2 h for h  1,...s .

2.

* * Construct the 1001   % PI w and PI  2 for h  1, ... , h respectively by

L*,T mh, y  QT* h 1   / 2  QT* h  / 2 and L*,T m h, 2  K T  h 1    . 3.

* * Estimate the coverage of PI w and PI  2 for h  1, ... , h respectively by

CT*,mh, w 

1 R 1 R *,m and 1 C  . * *, r * 2  1 2*,r * T  h , R r 1 QT  h  / 2   wT  h  QT  h 1 / 2  R r 1 0   T  h  KT  h 1 

4. Repeat steps 1 – 3 M = 1,000 times.

* Compute the average and the standard deviation of the coverage of PI w by



M

CT* h   CT*,mh, w / M and std CT* h, w



m 1

M   CT*,mh, w  CT* h, w  m1





1

2

2 /M , 

* and those of the length of PI y by

M

* T h

L

 L m 1

*,m T h,w



* T  h,w

/ M and std L



M   L*,T m h, w  LT*  h, w  m1





2

1

2 /M . 

Similarly, the averages and the standard deviations of the coverage and the length of

PI * 2 are calculated.

4. Monte-Carlo Simulation A Monte-Carlo simulation study was carried out with standard normal error distribution using the models described below. In order to incorporate the day of the week effect in the ARCH models, we used 1  2,  i  0.5, for i  2,3,4 and  5  1.

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The Models we consider are: Table I. Models used in the Simulation Study.

Model Label

Model Formulation

1

yt   t  t

2

 t2  0.1  0.1yt21 yt   t  t

3

 t2  0.1  0.4 yt21 yt   t  t

4

 t2  0.5  0.4 yt21 yt   t  t

5

 t2  0.1  0.2 yt21  0.15 yt22 yt   t  t  t2  0.5  0.2 yt21  0.15 yt22

The results of the simulation study carried out using the method described in Section 3 are presented below for T=300, T=1,000, T=3,000. Please note that due to space restriction we present the results for only Model 2 and 4.

Table II: Simulation Results for T = 300 and a0  0.1; a1  0.4

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Table III: Simulation Results for T = 1,000 and a0  0.1; a1  0.4

Table IV: Simulation Results for T = 3,000 and a0  0.1; a1  0.4

Table V: Simulation Results for T = 300 and a0  0.1; a1  0.2 ; a2  0.15

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Table VI: Simulation Results for T = 1,000 and a0  0.1; a1  0.2 ; a2  0.15

Table VII: Simulation Results for T = 3,000 and a0  0.1; a1  0.2 ; a2  0.15

5. Discussion of Results In this study we constructed bootstrap based prediction intervals under the scenario where the unconditional variance of the returns changes depending on the day of the week. We looked at nine different combinations of model type and sample size. In light of space limitations, we report only a subset of results, in Tables II – VII. The complete results of the simulation study are available upon request from the first author. Results of the Monte Carlo study show that the coverage probabilities for returns are very close to the nominal level of 0.95 for all parameter and sample size combinations, under the standard normal error distribution. The coverage probabilities for the volatility are, however, liberal, but these coverage probabilities are similar to what Chen et. al (2007) obtained for the non-seasonal case. Also, the coverage probabilities for the first day ahead volatility are higher than nominal in all cases except when the sample size is 300.

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For instance, the coverage for the first day ahead volatility when the sample size is 1,000,  0  0.1, and 1  0.4 is 0.9610 as shown in Table II. This is in contrast to the coverage probabilities close to 0.94 for the other four days. The main reason for this is that while the first day ahead bootstrap predictions, which are used in computing the prediction limits, vary from each other due to the varying bootstrap estimates of model parameter as well as from the varying bootstrapped ˆi*, j values, the true future values for the first day ahead volatility do not vary (within a simulation run) due to the fact that it is calculated using the true values of the parameters and the previous day’s squared return. Thus, the prediction interval for the first day ahead volatility is wider than necessary to capture 95% of the true future first day ahead volatility values. This overestimation does not happen for the other days because starting from the second day ahead the true future value for the volatility depends on the previous day’s value for the return, which is not constant from one bootstrap run to the next. Another important observation is the fact that the coverage probabilities for both returns and volatilities do not change much as the value of the intercept parameter  0 changes for both ARCH(1) and ARCH(2) models. The effect of the sample size on coverage probability is discernible as a slight improvement in coverage, for both returns and volatility, as it increases from 300 to 1,000. This increase in coverage is more pronounced for volatility intervals, but an unforeseen result of this increase is the conservative intervals produced for day ahead predictions when sample size is 3,000.

6. Conclusion Time series with varying unconditional variance is observed in certain empirical series. An ARCH type model that accounts for such variation is introduced and used to obtain bootstrap-based prediction intervals for returns and volatility for situation where the day of the week is associated with different unconditional volatilities. The proposed method is a computationally efficient technique and is an extension of an existing procedure for the case with content unconditional variance. Monte-Carlo simulation results show that the coverage probabilities for the returns are very reasonable for all the models considered. Even though the coverage probabilities for the volatility are not close to 0.95, mostly for small sample sizes, the results obtained are comparable to those obtained by existing procedures developed for the constant unconditional variance case.

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