A Selectivity Corrected Time Varying Beta Estimator
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Transcript of A Selectivity Corrected Time Varying Beta Estimator
A SELECTIVITY CORRECTED TIME-VARYING BETA RISK ESTIMATOR
Robert D. Brooks*, Jonathan Dark*, Robert W. Faff** and Tim R.L. Fry***
* Department of Econometrics and Business Statistics, Monash University ** Department of Accounting and Finance, Monash University *** School of Economics and Finance, RMIT Business
ABSTRACT
This paper explores two issues in beta risk estimation, namely, time variation of risk and the impact of thin trading (data censoring). Within a multivariate GARCH setting, the paper first conducts an analysis of the importance of assumptions made about the correlation structure. Collectively, the results of Monte Carlo analysis and an empirical application to a sample of individual Australia stock returns, demonstrate that it is preferable to allow for time variation in the correlation structure. The paper then develops a selectivity corrected time-varying beta risk estimator. The results of a Monte Carlo experiment demonstrate that the new estimator performs well in handling censored data. Finally, we confirm that when the new model is applied to our Australian dataset it successfully captures the impact of censoring and thin trading on time-varying beta risk. Keywords: Time-varying betas; GARCH; Censoring; Thin trading; Sample selectivity model JEL Classification: G12, C24 Acknowledgements: The authors acknowledge the financial support of Australian Research Council Discovery Grant DP0345680 – A Complex Systems Approach to Modelling Time-varying Risk in the Presence of Market Frictions. The authors wish to thank Diana Maldonado and Michael Gangemi for their work as research assistants on this project. The authors also wish to thank the participants at the 2005 Quantitative Methods in Finance conference and seminars at Deakin and ANU for their helpful comments on earlier versions of this paper. Correspondence to: Robert Faff, Department of Accounting and Finance, Monash University, VIC 3800, Australia, Phone: 61-3-99052387, Fax: 61-3-99052339, Email: [email protected]
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A SELECTIVITY CORRECTED TIME-VARYING BETA RISK ESTIMATOR
ABSTRACT
This paper explores two issues in beta risk estimation, namely, time variation of risk and the impact of thin trading (data censoring). Within a multivariate GARCH setting, the paper first conducts an analysis of the importance of assumptions made about the correlation structure. Collectively, the results of Monte Carlo analysis and an empirical application to a sample of individual Australia stock returns, demonstrate that it is preferable to allow for time variation in the correlation structure. The paper then develops a selectivity corrected time-varying beta risk estimator. The results of a Monte Carlo experiment demonstrate that the new estimator performs well in handling censored data. Finally, we confirm that when the new model is applied to our Australian dataset it successfully captures the impact of censoring and thin trading on time-varying beta risk. Keywords: Time-varying betas; GARCH; Censoring; Thin trading; Sample selectivity model JEL Classification: G12, C24
1. Introduction
The modelling of the risk-return trade off remains a core issue in finance, and for
several decades the workhorse in this area has been the capital asset pricing model
(CAPM), (Sharpe, 1964 and Lintner, 1965). The CAPM is a simple model which
suggests that there is a positive and linear relationship between risk and (expected)
return. More formally, the model says that the expected return of an asset comprises a
risk-free component and a risk premium component. Further, it defines the asset's risk
premium as being equal to the risk of the asset multiplied by the market risk premium.
To complete the picture, according to the CAPM the asset's risk — referred to as β
(beta) risk — is defined as the covariance of returns between the asset and the market,
divided by the variance of market returns.
Despite the controversies associated with it (for example, see, Fama and
French, 1992, 1993 and Daniel and Titman, 1997), the CAPM remains the central
theoretical model of the risk-return relation in asset pricing. Ferson and Harvey
(1991a, 1991b) in their modelling of US stock returns discuss two main problems that
arise in the empirical application of the CAPM, namely, time-varying risk premia and
time-varying betas. The present paper focuses its attention on the problem of
estimating time-varying betas.
There are a number of modelling challenges that arise in the estimation of
time-varying betas. The first, and perhaps most fundamental, of these is around the
choice of model to parameterise the time variation. A range of techniques exist in this
regard including: variable parameter regression models estimated using the Kalman
filter (see Black, Fraser and Power (1992), Wells (1994)); modelling the time
variation in beta as a function of economic variables (see Abell and Kreuger (1989),
Faff and Brooks (1998)); or using a member of the multivariate GARCH family of
models (see, for example, Braun, Nelson and Sunier (1995), Giannopoulos (1995),
2
McClain, Humprehys and Boscan (1996), Brooks, Faff and McKenzie (1998)). The
present paper focuses its attention on the multivariate GARCH approach. In this
regard a number of choices still need to be made around parameterisation, particularly
to keep the number of parameters sufficiently small for estimation (see Pagan (1996)).
These choices typically relate to the degree of flexibility that is allowed in the
correlation structure of the model. The simplest model is that of Bollerslev (1990)
which fixes the correlation to be constant over time. While this simplifies estimation
considerably it is quite restrictive and is often at odds with reality. To provide a more
robust setting, this paper considers the more complex model of Tse and Tsui (2002)
which allows the correlation to evolve over time following an ARMA process.
The second area of empirical challenge that arises in modelling beta risk for
individual stock data is around correcting for thin trading. In the simplest context of
estimating a constant beta, a number of time-series based techniques exist in this
regard (see, for example, Scholes and Williams (1977), Dimson (1979), Fowler and
Rorke (1983)). However, it is very likely that these techniques perform rather poorly
in cases of extreme thin trading where numerous zero return observations are evident
i.e. when a non-trivial data censoring problem occurs.1 We argue that in the current
context it is preferable to utilise a sample selectivity model along the lines of
Heckman (1979) to correct for thin trading/ censoring. Such an approach has been
successfully utilised in estimating constant betas (see Brooks, Faff, Fry and
Maldonado (2004), Brooks, Faff, Fry and Gunn (2005)) and the present paper extends
this analysis to the time-varying beta case.
The plan of this paper is as follows. Section two briefly reviews the relevant
literature on using multivariate GARCH models to estimate time-varying betas, and
1 In related work the percentage of zero returns has been used to construct a liquidity measure – see, for example, Lesmond, Ogden and Trzincka (1999), Lesmond, Schill and Zhou (2004), Lesmond (2005), Bekaert, Harvey and Lundblad (2003).
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on thin trading, censoring and zero returns. Section three sets out the multivariate
GARCH models and then reports the results of a Monte Carlo comparison of the
properties of the time-varying beta estimators, as well as an empirical application of
the models to individual stock data for Australia. Section four then extends the
analysis to incorporate a correction for censoring, and also reports the results of a
Monte Carlo comparison and application of the model to individual stock data for
Australia. Section five contains some concluding remarks.
2. Relevant Literature
2.1 Time-varying Betas
The estimation of time-varying betas for individual stocks and industry portfolios has
proven to be quite popular in the literature. Braun, Nelson and Sunier (1995) use a
bivariate EGARCH specification to model time-varying betas for US industry and
size based portfolios. Despite the presence of leverage effects in the market return,
such effects are absent in the estimated betas. Gonzalez-Rivera (1996) applies a
bivariate GARCH-M model using the parameterisation in Engle and Kroner (1995) to
model the risk-return trade off in US computer industry stocks and find that the
history of the variance of individual stock returns is critical in explaining expected
returns. McClain, Humprehys and Boscan (1996) use the parameterisation in
Bollerslev, Engle and Wooldridge (1988) to model time-varying betas for US mining
industry stocks. They find considerable instability in beta risk, and also document a
role for sample size in detecting significant GARCH effects. Brooks, Faff, Ho and
McKenzie (2000) use the Bollerslev (1990) constant correlation model to examine the
impact of regulatory change on US banking stocks beta risk. They document strong
association between beta risk and regulatory change. Brooks, Faff, Ho and McKenzie
(2000) also observe some extreme beta estimates for individual stocks (as do
4
Gonzalez-Rivera (1996) and McClain, Humprheys and Boscan (1996)). This issue is
then explored in McKenzie, Brooks, Faff and Ho (2000) who find that these extreme
beta events can be linked to economic news about the particular stocks.
The use of multivariate GARCH models to estimate time-varying betas has
also been explored in markets other than the US. Brooks, Faff and McKenzie (1998)
use the Bollerslev (1990) constant correlation model to analyse Australian industry
portfolio time-varying betas and find that the GARCH approach performs well
relative to other models (primarily Kalman filter based). McKenzie, Brooks and Faff
(2000) extend this comparison to consider whether a domestic or world index should
be chosen as the market index, and their results support the use of the domestic index.
Giannopoulos (1995) and Brooks, Faff and McKenzie (2002) have also used the
GARCH family of models to analyse time-varying country beta risk.
2.2 Thin Trading and Data Censoring
A particular challenge in modelling individual stock data is to deal with thin trading
effects – especially in the context of estimating beta risk. Thin trading arises when the
returns on the individual stock moves in a non-synchronous manner relative to the
return on the market index. There are a range of standard techniques designed to
correct for the bias induced in beta estimates by thin trading effects such as those
suggested by Scholes and Williams (1977), Dimson (1979) and Fowler and Rorke
(1983). Each of these techniques deals with the thin trading problem by making a time
series adjustment. They typically involve utilising leads and lags of the market return,
as well as making some allowance for any autocorrelation observed in the market
return. While all these alternative techniques tend to modify the beta estimate in the
appropriate direction (i.e. increase the estimated betas for thinly traded stocks), they
are unlikely to work well (or even adequately) in the case of extreme thin trading in
5
which many zero return observations are occur. In such circumstances we effectively
have a data censoring problem.
The incidence of zero returns has been used to construct a measure of
transaction costs by Lesmond, Ogden and Trizincka (1999) for US data, who find that
such a measure performs well relative to other available measures. Further, Lesmond,
Schill and Zhou (2004) find that such a measure of transaction costs can be important
in testing the profitability of a momentum trading strategy in the US market. Thin
trading, zero returns and transaction costs are found to be even more extreme in the
case of emerging markets (see Lesmond (2005), Bekaert, Harvey and Lundblad
(2003)).
Brooks, Faff, Fry and Gunn (2005) incorporate the impact of thin trading, zero
returns and censoring to estimate selectivity corrected static betas for Australia, while
Brooks, Faff, Fry and Maldonado (2004) do similar for a range of Latin American
markets. These papers use a sample selectivity model following Heckman (1979),
using volume as the key variable in the first stage probit. This produces a upward
correction in the estimated betas that adjusts for the thin trading bias. The key factor
in the sample selectivity approach is choice of the variables used in the first stage, and
the factors chosen need to vary sufficiently over time for an individual stock to
explain its temporal zero return pattern. This is an issue for exploration in the current
paper.
6
3. Time-varying Beta in a Multivariate GARCH Model Setting
3.1 Stage 1: Monte Carlo Estimation of Multivariate GARCH Models
The first part of the analysis of time-varying betas is a Monte Carlo analysis with a
dual-pronged experimental setup. The data generating process (DGP) for the first
experiment is the constant correlation (CC) GARCH(1,1) model of Bollerslev (1990),
while the DGP for the second experiment is the varying correlation (VC)
GARCH(1,1) model of Tse and Tsui (2002). This part of the experimental design
enables an analysis of the potential impact of a restrictive correlation structure in the
generation of time-varying betas.
To define the more complex model let ,i tr represent the return for asset i,
where 1i = for the individual asset and 2i = for the market. The DGP for the VC
GARCH model is given by:
1, 1,
2, 2,
t t
t t
rr
ε
ε
=
= (1)
where
2,2,12
,122,1
,2
,1 ,00
~tt
tt
t
t Nrr
σσ
σσ (2)
and
2 2 2, , 1 , 1i t i i i t i i tσ ω α ε β σ− −= + + 1, 2i = (3)
with
( )1 2 1 1 2 11t t tρ θ θ ρ θ ρ θ ϕ− −= − − + + (4)
where
( )( )2
1, 2,11 2 22 2
1, 2,1 1
t h t hht
t h t hh h
z z
z zϕ − −=
−
− −= =
= ∑∑ ∑
(5)
7
with , , ,i t i t i tzε σ= for 1, 2i = , all the α and β elements are non-negative and both ω
elements are positive. If all the roots of 0I L Lα β− − = are outside the unit circle
(where I is a 2x2 identity matrix, ( )1 2,diagα α α= and ( )1 2,diagβ β β= ), the
volatility process is stationary (Ling and McAleer, 2003). This model nests the CC
GARCH model ( 1 2 0θ θ= = ).
To perform the Monte Carlo experiment we need to set ‘true’ values for all the
parameters of the model. We do this through a combined strategy of taking values
similar to those used by Tse and Tsui (2002), while achieving beta risk values close to
unity (i.e. an average risk stock). Accordingly, our chosen set of parameter values are:
ω1 = 0.4; α1 = 0.2; β1 = 0.7; ω2 = 0.2; α2 = 0.2; β2 = 0.7; ρ = 0.71; θ1 = 0.8; θ2 = 0.1.
For each DGP, 1000 samples are generated, each of length 1000 observations.2 Using
MLE the bias and MSE of the estimates are determined. Our analysis extends the
simulations in Tse and Tsui (2002) by considering the performance of both models,
i.e. for each DGP, both the CC and VC GARCH models are estimated. Table 1
summarises the results via examination of the bias and MSE for all of the key
parameters in the model.
(Insert Table 1 about here)
For the first experiment in which the simple constant correlation structure is
true (Panel A), the most striking observation is that the performance of both models
with respect to the GARCH parameter estimates is comparable. As such, modelling
with a more complex structure than is needed, does not seem to overfit the data in
terms of altering the estimates of the parameters of the GARCH model. This can be
primarily explained via consideration of the 1θ parameter which is the key parameter
2 To avoid start up problems, samples of size 10,000 observations are generated, with the first 9,000 observations being discarded.
8
in driving the behaviour of the time-varying correlation in the VC GARCH model.
Specifically, the 1θ estimate averages 0.5286 and has a very large MSE. In unreported
results, we find that for 293 simulations 1 0.01θ ≤ , while for 216 simulations
1 0.99θ > . While these two cases lie near the boundaries of the parameter space, they
both have the same impact on the time-varying correlations, that is, the estimated
correlations are virtually time invariant. The VC GARCH model is therefore
observationally equivalent to the CC GARCH model. These results suggest that
overfitting of the data via the VC GARCH model is not too problematic.
For the second experiment in which the more complex VC GARCH
correlation structure is true (Panel B), the VC GARCH model is estimated quite
tightly via MLE. The CC GARCH model, however, considerably overestimates (by
around 25%) the iω parameters and underestimates the iα , iβ and ρ parameters. The
underestimation of the ρ parameter is consistent with the results in Tse and Tsui
(2002). These findings generally suggest that the consequences of underfitting are far
more serious than the consequences of overfitting. As such, it would seem that there is
not a great cost in erring on the side of fitting the more complex model.
3.2 Stage 2: Time-varying Betas from Monte Carlo Multivariate GARCH
Models
In the second stage of the Monte Carlo experiment, both models are used to estimate
time-varying betas that can be constructed from the estimated covariances, variances
and correlation parameters:
2,2
,2,1,1 ˆ
ˆˆˆˆt
tttt σ
σσρβ = (6)
9
where tρ̂ = ρ for the CC GARCH model or tρ̂ is estimated according to Equation (4)
in the case of the VC GARCH model.
Table 2 compares the average of the moments of the estimated betas to the
average of the moments of the betas from the DGP. For the first experiment (Panel A,
in which the CC GARCH model is the DGP), the CC GARCH and VC GARCH
models estimate the beta moments equally well, with both means very close to the
‘true’ value (1.0385) and the higher moments very similar for both cases. This is
consistent with the results obtained with respect to the parameter estimates in the first
stage of the Monte Carlo comparison, thus reinforcing the earlier view that there is
not a significant penalty from overfitting the data.
For the second experiment (Panel B, in which the VC GARCH model is the
DGP), the CC GARCH model performs poorly. While the mean value of the time-
varying beta is quite close to its ‘true’ value, the CC GARCH model considerably
understates (by more than 50%) the variability in the betas and substantially
overstates the skewness (> 50%) and kurtosis (> 30%). Once again, this is consistent
with the results obtained with respect to the parameter estimates in the first stage of
the Monte Carlo analysis. These results further emphasise that the consequences of
underfitting are more serious than the consequences of overfitting.
(Insert Table 2 about here)
3.3 Stage 3: Empirical Estimates of Time-varying Betas from Multivariate
GARCH Models
The second part of the analysis of time-varying betas is an empirical analysis of
individual stock data. For these purposes we choose an Australian sample for two
primary reasons: (a) because such data display a wide spread of thin trading effects;
and (b) because the issue of thin trading in the context of static betas has been
extensively analysed in this setting. As such, an Australian sample gives us a good
10
point of comparison. Specifically, data are collected on stocks in the Australian All
Ordinaries Index over the period from January 2000 to December 2002. The All
Ordinaries Index is composed of the 500 largest stocks on the exchange, and market
capitalisation is the only requirement for inclusion. As a major benchmark index that
covers most of the market (in terms of market capitalization), stocks in this index are
of great relevance to investors and fund managers in their investment decision
making. From the DataStream database, daily data were obtained for 441 of the stocks
in this index over the sample period.
In the first stage of the empirical analysis the multivariate CC GARCH and
VC GARCH models were estimated between the returns on the individual stocks in
the AOI and the returns on the AOI itself. In unreported results, the autocorrelation
function for the AOI returns showed no signs of temporal dependence, so a constant
mean equation was fitted. To capture any serial correlation in the stock returns, an
AR(1) mean equation was employed.
Table 3 compares selected parameter estimates for the CC GARCH and VC
GARCH models, over the total number of stocks for which the models converged. 3
While all of the stocks in the analysis are in a main benchmark index, the bottom end
of the index is still characterised by some lowly priced and thinly traded stocks.4
(Insert Table 3 about here)
The results in each panel of Table 3 are reported for a number of different
combinations of the data. First, they are reported for the overall sample of stocks.
3 Several points are worthy of note. First, the models were estimated imposing the covariance stationarity condition above. Second, the CC GARCH model was estimated without the imposition of the restriction. The results and conclusions were unchanged. Third, for 40 of the stocks, the CC GARCH model experienced convergence difficulties, while for 95 of the 441 stocks, the VC GARCH model experienced convergence difficulties. Fourth, only those results where strong convergence was achieved within 100 iterations have been reported. 4 The results in Lange (1999), Brooks, Faff and Fry (2001) and Amilon (2003) suggest that there can even be challenges in the testing and estimation of univariate GARCH models in these cases. As such, one might expect that the estimation of our more complex multivariate models is also going to be problematic. The focus of the analysis is therefore on the results for the remaining stocks where the respective models were able to strongly converge.
11
Second, they are reported in categories according to the degree of censoring
(proportion of zero return observations) in the data. While over a quarter of the
sample has low censoring (less than 10%), there are still around 10% of stocks in this
major Australian benchmark index for which censoring is greater than 50%. For these
stocks, thin trading and the associated transaction costs represent a considerable
problem.5 Third, given the findings of the Monte Carlo experiment that iω is likely to
be over-estimated when correlation dynamics are ignored, results are also reported for
two iω categories. Finally, the Nyblom (1989) test is used to test for time-varying
correlations in the data. The results of this test indicate that around only 10% of the
sample have time-varying correlations. Accordingly, results in Table 3 are also
reported for cases where the test is both significant and insignificant.
Generally, the results reported in Table 3 clearly demonstrate that for both
models as the degree of censoring increases, the estimate of ρ decreases, whilst the
sum 1 1α β+ estimates increases (see Panels A and B). The decrease in ρ is consistent
with expectations, as is the increase in the 1 1α β+ estimates, as one potential impact
of censoring is to introduce a structural break like effect into the data. Consistent with
the Monte Carlo results the 1 1α β+ estimates are much less persistent for cases of
high iω values. Interestingly, for the CC GARCH model (Panel A) the distribution of
the parameter estimates becomes more leptokurtic for the most extreme censoring
category, where the censoring level exceeds 50%.
Finally, with regard to Table 3 consider the dynamic correlation parameters.
The 1θ and 2θ estimates appear unaffected by the degree of censoring or the stability
of the ρ estimate, as measured by the Nyblom (1989) test. The distribution of 1θ is
5 Interestingly, problems in the convergence of the models are spread across the range of censoring categories, suggesting a wider range of factors are important in GARCH parameter estimation.
12
shown in the histogram in Figure 1, as well as the kernel density estimate in Figure 3.
The results are consistent with there being very little difference between the average
of the ρ estimates for the CC and VC GARCH models.
(Insert Figure 1 about here)
Table 4 examines the average of the moments of the time-varying betas
estimated by each model. Results are again presented for all of the same categories
shown in Table 3. It is notable to observe that as the degree of censoring in the data
increases, average betas fall, consistent with the downward bias induced by thin
trading effects (see Scholes and Williams (1977), Dimson (1979), Fowler and Rorke
(1983)). This suggests the need for a sample selectivity type correction along the lines
of Heckman (1979) – an approach has been successfully used in estimating static
betas by Brooks, Faff, Fry and Maldonado (2004), and Brooks, Faff, Fry and Gunn
(2005). This decrease in the average betas is similar to the pattern reported in Table 3
for the ρ estimates. The same patterns are not as apparent in the higher order
moments for the betas.
The overall findings are consistent with the simulation where the DGP is the
VC GARCH model. The CC GARCH model betas have a smaller variance and a
higher degree of skewness and kurtosis than the VC GARCH betas. This result
provides further evidence supporting the VC GARCH model over the CC GARCH
model for the majority of stocks.
(Insert Table 4 about here)
13
4. Time-varying Beta in a Multivariate GARCH Model Setting: The Impact
of Censoring and Thin Trading
4.1 Probit Model Component of Selectivity Model
This section of the analysis extends the modelling outlined in the previous section by
accounting for censoring and thin trading effects in the data. This is achieved by
employing a sample selectivity approach. The first stage of this analysis requires a
decision on which variable(s) need to be included in the probit component of the
sample selectivity model. The candidate variables must plausibility be related to the
factors/forces that we believe underlie why stocks do not trade in any given
observation (i.e. why these stocks have zero returns on some days). Logically, such
variable(s) will have some link to transaction cost or similar metric of trading
impediment.
To this end, for each of the Australian stocks for which returns data were
analysed to calculate the time-varying betas in Section 3, data were also collected on
firm size (measured by market capitalisation), volume (measured by number of shares
traded) and the bid-ask spread from the Datastream database. This produced data on
249 stocks. We consider trading volume as a candidate variable for the probit
regression because such measures have been found to work well in similar sample
selectivity investigations of constant betas by Brooks, Faff, Fry and Maldonado
(2004) and Brooks, Faff, Fry and Gunn (2005). Also, the bid-ask spread as a measure
of transaction costs has been found to be well explained by the transaction costs
measure generated by Lesmond, Ogden and Trizincka (1999) and Lesmond (2005)
using the degree of censoring in the data. Finally, it is plausible that firm size will be
highly correlated with transaction cost and thin trading.
The analysis involves the estimation of probit models for each of the 249
stocks. The dependent variable in the probit model (yi) is defined as yi=1 for cases
14
where a non-zero return is observed on a given day and yi=0 for cases where a zero
return is observed on a given day. In the probit model, as foreshadowed above, three
explanatory variables are considered, namely, the daily market capitalisation, daily
trading volume and daily bid-ask spread. A summary of the results of the probit
estimation is reported in Table 5 in terms of the proportion of cases where particular
variables are found to be significant.
(Insert Table 5 about here)
These results show that the most important variable in explaining the time
series variation in censoring for an individual stock is trading volume. At the 1%
significance level, trading volume is significant in the first stage probit for one third
of the stocks, and this increases to a little over half of the stocks at the 10%
significance level. In contrast, size and bid-ask spread are much less significant in the
first stage probit. This is an interesting result, and is most likely a reflection of the fact
that at the level of a given stock there is insufficient temporal variation in either the
bid-ask spread, or size, to explain the time series of censoring.
The number of cases in the first stage probit where two of the three variables
are significant is much lower with the strongest pair being size and volume which is
jointly significant in 18.9% of cases at the 10% level. In contrast, size and bid-ask
spread is only jointly significant in 6.4% of cases at the 10% level. The number of
cases where all three variables are significant is very small, with only 4% of such
cases at the 10% level.
4.2 Monte Carlo Analysis of a Selectivity-corrected VC GARCH Model
The next step in this stage of the analysis is to specify the sample selectivity model.
The results of the probit analysis indicate that the trading volume variable is clearly
the most significant of the three potential explanatory variables. Thus, the stage one
15
model includes only volume as an explanatory variable. In specifying the model, we
assume that there is independence between the selectivity and GARCH components
of the model. This greatly simplifies the analysis as it separates the likelihood
function into two parts to be maximised.
The first component of the likelihood function is concerned solely with the
selectivity equation, while the second component is concerned with the multivariate
GARCH equations. Effectively, the selectivity correction causes the GARCH
estimation to have two components: the normal cases when individual stock returns
are not censored; and a restricted component when the individual stock returns are
censored which is only concerned with the market return.
The next stage of the analysis is a Monte Carlo comparison of the properties of
the selectivity corrected time-varying beta estimator. Given that the results of the
Monte Carlo comparison in the uncorrected case strongly supported the VC GARCH
model of Tse and Tsui (2002), the Monte Carlo analysis in the selectivity corrected
case is confined to that model. To conduct such a Monte Carlo analysis we need to
define a DGP for the variable that is driving the selectivity component of the model,
namely in the current setting, daily trading volume. An inspection of the histograms
of daily trading volume for the individual stocks suggests that the distribution of this
variable is approximately χ2 with three degrees of freedom. Thus for the purpose of
our Monte Carlo exercise, simulated volume figures are drawn from this distribution.
The selectivity component of our model can be specified as:
iti'it
*it uγz += w (7)
where:
>= ,0 if1
otherwise. 0*it
itz z
Or equivalently,
16
= ,return zero-non if1
return. zero if 0 zit
and wit includes a constant and trading volume as the explanatory variables. Thus in
the Monte Carlo design, the level of censoring is capable of being altered by changing
the threshold value of z*it above which a non-zero return is observed. The analysis
explores two specifications. The first case is a low censoring case where the value of
censoring is set at 20% which corresponds to the median level of censoring in the
stocks analysed in this paper. The second case is a high censoring case where the
value of censoring is 60% which corresponds to the median level of censoring in the
sample analysed by Brooks, Faff, Fry and Gunn (2005). We apply the ‘true’ values to
each parameter as used in the previous section.
The results reported in Table 6 show the impact on the estimated parameters
of the VC GARCH model with and without a censoring correction in both the low
(Panel A) and high (Panel B) censoring cases. As revealed in Panel A, in general the
biases and mean squared errors of the parameter estimates are small in the low
censoring case. There is little material difference between the cases which do and
don’t apply a censoring correction. In contrast, and consistent with expectations,
Panel B shows that there are much larger biases in some of the parameters in the high
censoring setting. This is particularly so for the ω and ρ parameters. Most notably, for
the ρ parameter the bias is large when no selectivity correction is made, however the
bias is significantly reduced after selectivity correction. This is to be expected, as the
zero returns in the individual stocks induce a downward bias in the estimated
correlation, which is also consistent with the empirical results reported in Table 3 of
this paper. Thus, it would appear that selectivity correction has the potential to correct
for this problem in high censoring cases.
(Insert Table 6 about here)
17
4.3 Time-varying Betas from a Monte Carlo Selectivity-corrected VC
GARCH Model
The issue that remains to be considered is whether this impact on the parameter
estimates, in particular the correlation parameter, as shown in Table 6 manifests itself
in the time-varying beta parameters generated from the VC GARCH model. This
issue is explored in the results presented in Table 7.
Table 7 presents results on the first four moments of the distribution of the
estimated time-varying betas in both the low (Panel A) and high (Panel B) censoring
cases. In the low censoring case the absence of a censoring adjustment induces a
slight downward movement in the mean and variance of the estimated betas which is
corrected for by the selectivity model. As clearly seen in Panel B, this effect is much
stronger in the high censoring case where the impact is a significant downward
movement, particularly so for the mean of the estimated betas. Specifically, the mean
beta for the approach that makes no adjustment is 0.3568 – which represents a 60%
downward bias relative to the ‘true’ value. This significant downward movement is
adjusted for completely by the selectivity correction – the mean beta in this latter case
is 0.9158 (compared to the ‘true’ value = 0.8953). Thus, it would appear that the
impact of selectivity correction on ρ noted in Table 6 carries across to have a strong
impact on the estimated betas in Table 7. Thus, it would again appear that where
censoring is high that selectivity correction has a strong role to play. Moreover, the
Monte Carlo results suggest that there is not a great cost in erring on the side of
making the selectivity correction.
(Insert Table 7 about here)
18
4.4 Empirical Estimates of Time-varying Betas from Selectivity-corrected
Multivariate GARCH Models
In the final stage of the empirical analysis, similar to Section 3.3, the selectivity-
corrected CC GARCH and VC GARCH models were estimated between the returns
on individual stocks in the AOI and the returns on the AOI itself. Consistent with our
approach in Sections 4.2 and 4.3, we apply a correction that relies on trading volume
only. This has the added advantage that we can retain the same sample as analysed in
Section 3.3, since trading volume data are available for that same group of stocks –
the advantage being comparability.
The models were again estimated imposing the covariance stationarity
condition. Interestingly, there are less convergence problems once the selectivity
correction is made. There are now only 19 cases where the selectivity-corrected CC
GARCH model fails to converge and only 41 cases where the selectivity-corrected
VC GARCH model fails to converge. This suggests that the selectivity correction is
effective in overcoming the estimation problems that are induced by censoring.6
The results are reported in a similar format to those earlier where the
selectivity correction was not utilised and are contained in Tables 8 and 9 (similar to
Tables 3 and 4). The results in Table 8 compares selected parameter estimates when
the selectivity-corrected CC GARCH and VC GARCH models were estimated over
the total number of stocks for which the models converged for both the uncorrected
and selectivity corrected analysis.
Consistent with the earlier results for the case where no selectivity correction
is made, the findings reported in Table 8 clearly demonstrate that for both models as
the degree of censoring increases, the estimate of ρ generally decreases, even after a
selectivity correction is made. In general, the average values for the moments of ρ are 6 For comparability reasons we restrict our current focus to the same set of stocks for which convergence was achieved in the uncorrected analysis of Section 3.3.
19
similar between the cases when a selectivity correction is made or not in the lower
censoring categories (censoring is below 40%). For the two highest censoring
categories there is a small increase in the estimated correlation parameter after
censoring correction is made. Interestingly, the sum 1 1α β+ estimate remains
relatively steady across low and moderate censoring levels and then dips down
somewhat at higher levels of censoring (contrary to the results reported in Table 3).
This suggests that the models are now less persistent than they were in the
uncorrected case, which is again consistent with censoring inducing a structural break
like effect into the data, thus leading to an overstatement of the persistence in the
estimated GARCH parameters. Further, the excess kurtosis for the CC GARCH
parameter estimates in the most extreme censoring category is now removed. Finally,
with regard to this analysis, the distribution of 1θ in the selectivity-corrected case is
shown in the histogram in Figure 2, as well as the kernel density estimate in Figure 3.
(Insert Table 8 about here)
(Insert Figures 2 and 3 about here)
The results revealed in Table 9 examine the average of the moments of the
time-varying betas estimated by each model. Results are again presented for all of the
same categories as Table 4. In general, the findings are consistent for both the CC
GARCH and VC GARCH estimators. As the degree of censoring in the data
increases, average betas fall when estimated by both selectivity-corrected models. For
the lower censoring categories (censoring below 40%) there is no strong impact on
either the mean or variance of the estimated betas. This is consistent with the lack of
impact on the parameter estimates observed in Table 8. However, for the two higher
censoring categories the mean and variance of the beta estimates are increased
(relative to their uncorrected Table 4 counterparts) consistent with the simulation
results reported in Table 7. This would again appear to be due to the adjustment made
20
by selectivity correction in the correlation parameter reported in Table 8. Finally, in
terms of higher order moments, the selectivity corrected betas generally seem to be
more skewed and leptokurtic for both models as compared to the uncorrected betas
for both models. Collectively, the empirical and simulation results presented in
Section 4 suggest that there is not a great cost in erring on the side of making the
selectivity correction.
(Insert Table 9 about here)
5. Conclusion
This paper has conducted an analysis of two key issues in beta risk estimation,
namely, time variation of risk and the impact of thin trading (data censoring). Our
analysis was performed within the framework of the multivariate GARCH family of
models, but specifically compared the performance of a simple constant correlation
model (CC GARCH) to a more complex varying correlation model (VC GARCH).
The results of both a Monte Carlo comparison and an empirical analysis of individual
stock data for Australia indicated that it is better to overfit the data by making
allowance for time variation in the correlation structure. The results also showed that
the standard multivariate GARCH estimation of time-varying betas is impacted by the
downward bias induced by censoring and thin trading effects.
The analysis was then extended to deal with censoring and thin trading effects.
While three possible variables are considered for the probit component of the model,
trading volume appears to be the most important of these variables. Hence, it was
employed in the analysis. Selectivity-corrected multivariate GARCH models were
then simulated and estimated, and the findings show that selectivity-correction
reduces estimation problems.
21
Accordingly, two basic “takeaways” emanate from our analysis: (1) the
consequences of underfitting (e.g. assuming constant correlation) are more serious
than the consequences of overfitting (e.g. time-varying correlation) in such
approaches that seek to model time-varying beta risk and (2) in such models, there is
not a great cost in erring on the side of making the selectivity correction to counter the
problems of thin trading and data censoring.
22
Table 1: Bias and Mean Squared Error Metric for Simulated Multivariate Constant Correlation (CC) and Varying Correlation (VC) GARCH Models This table reports the outcome of Monte Carlo simulations for the constant correlation (CC) and varying correlation (VC) GARCH models. The results are based on 1,000 simulations of a sample size equal to 1,000. To avoid start up problems, an initial sample of 10,000 observations is generated and the first 9,000 observations are discarded. Using MLE, the bias and mean square error (MSE) are reported in the table. Panel A presents the outcome when the CC GARCH model is ‘true’, while Panel B presents the outcome when the VC GARCH model is ‘true’. Estimated CC GARCH Estimated VC GARCH
True value Bias MSE Bias MSE Panel A: DGP is the CC GARCH Model
ω1 0.4 0.0235 0.0117 0.0225 0.0116 α1 0.2 0.0009 0.0010 0.0027 0.0010 β1 0.7 -0.0072 0.0022 -0.0083 0.0022 ω2 0.2 0.0133 0.0028 0.0127 0.0028 α2 0.2 0.0005 0.0009 0.0023 0.0010 β2 0.7 -0.0088 0.0023 -0.0098 0.0023 ρ 0.71 0.0004 0.0002 0.0088 0.0005
1θ 0.5286 0.4644
2θ 0.0093 0.0003
Panel B: DGP is the VC GARCH Model ω1 0.4 0.0970 0.0301 0.0268 0.0129 α1 0.2 -0.0166 0.0014 0.0008 0.0011 β1 0.7 -0.0149 0.0035 -0.0083 0.0023 ω2 0.2 0.0489 0.0076 0.0136 0.0031 α2 0.2 -0.0176 0.0014 -0.0004 0.0010 β2 0.7 -0.0152 0.0038 -0.0083 0.0025 ρ 0.71 -0.0959 0.0108 -0.0029 0.0012
1θ 0.8 -0.0098 0.0024
2θ 0.1 0.0017 0.0005
23
Table 2: Moments of Time-varying Beta for Simulated Multivariate Constant Correlation (CC) and Varying Correlation (VC) GARCH Models This table reports the first four moments of the distribution of time-varying betas produced from Monte Carlo simulations for the constant correlation (CC) and varying correlation (VC) GARCH models. The results are based on 1,000 simulations of a sample size equal to 1,000. To avoid start up problems, an initial sample of 10,000 observations is generated and the first 9,000 observations are discarded. Panel A presents the outcome when the CC GARCH model is ‘true’, while Panel B presents the outcome when the VC GARCH model is ‘true’.
DGP Estimated CC GARCH Estimated VC GARCH Panel A: DGP is the CC GARCH Model
Mean 1.0385 1.0413 1.0417 Variance 0.0721 0.0730 0.0744 Skewness 1.0459 1.0465 1.0606 Kurtosis 5.7920 5.8688 5.8885
Panel B: DGP is the VC GARCH Model Mean 0.8987 0.8997 0.9004
Variance 0.1242 0.0537 0.1254 Skewness 0.6702 1.0796 0.6647 Kurtosis 4.5029 6.1085 4.5625
24
Table 3: Multivariate Constant Correlation (CC) and Varying Correlation (VC) GARCH Model Parameter Estimates – No Censoring Correction This table reports mean sample moments of selected coefficient estimates for the constant correlation (CC) and varying correlation (VC) GARCH models, in Panels A and B respectively. These estimates come from a sample of constituent stocks in the Australian All Ordinaries Index, subject to data availability and model convergence. The results in each panel are reported for a number of different combinations of the data: (a) overall sample of stocks; (b) categories according to the degree of censoring (proportion of zero return observations); (c) split based on whether the variance equation intercept is greater than or less than unity; and (d) split based on whether the Nyblom statistic testing for time-varying correlation is significant, or not, at the 5% level (Panel B only). No Mean Std dev Skew Kurt Mean Std dev Skew Kurt
Panel A: Selected Parameter Estimates for CC GARCH Model ρ
1 1α β+ Overall 358 0.1760 0.1091 1.4400 6.5321 0.8400 0.1901 -1.7254 5.8867
Censoring 0.1c ≤ 103 0.2482 0.1262 1.1347 5.1220 0.8084 0.2247 -1.7235 5.6149
0.1 0.2c< ≤ 77 0.2046 0.0852 1.2625 6.6541 0.8168 0.2009 -1.2816 3.8195
0.2 0.3c< ≤ 77 0.1541 0.0624 1.0264 7.1284 0.8462 0.1693 -1.3662 4.0159
0.3 0.4c< ≤ 52 0.1138 0.0605 1.2100 5.6284 0.8619 0.1637 -1.6043 4.9491
0.4 0.5c< ≤ 22 0.0684 0.0385 0.1175 2.2292 0.8876 0.1274 -1.3480 4.3626
0.5c > 27 0.0886 0.0944 3.7274 17.456 0.9281 0.0935 -1.9781 6.2904
1 1ω > 104 0.6647 0.2340 -0.6521 2.7631
1 1ω ≤ 254 0.9118 0.1037 -1.7556 6.1664
Panel B: Selected Parameter Estimates for VC GARCH Model
ρ 1 1α β+
Overall 306 0.1874 0.1204 1.2890 5.8831 0.8481 0.1923 -1.9885 6.9781
Censoring 0.1c ≤ 91 0.2476 0.1465 0.8447 4.2589 0.8331 0.2093 -1.9813 6.9452
0.1 0.2c< ≤ 64 0.2099 0.0913 0.1580 3.3090 0.8600 0.1697 -1.7387 5.6452
0.2 0.3c< ≤ 73 0.1787 0.0968 2.1400 10.560 0.8358 0.2120 -1.9091 6.0710
0.3 0.4c< ≤ 48 0.1267 0.0731 1.5804 6.3690 0.8398 0.1904 -1.5765 5.1884
0.4 0.5c< ≤ 18 0.0654 0.0401 0.4684 2.5986 0.9076 0.1207 -1.9763 6.8364
0.5c > 12 0.0904 0.0666 0.8542 2.3128 0.9178 0.0558 -0.5005 2.3586
1 1ω > 86 0.6867 0.2555 -0.8818 2.8390
1 1ω ≤ 220 0.9111 0.1084 -1.8810 6.4243
1θ 2θ Overall 306 0.5902 0.3751 -0.5626 1.7209 0.0395 0.0441 1.4737 5.0275
Censoring
0.1c ≤ 91 0.6665 0.3489 -0.9208 2.3362 0.0367 0.0373 1.7381 6.5679
0.1 0.2c< ≤ 64 0.5591 0.3888 -0.4028 1.5414 0.0406 0.0502 1.2657 3.4930
0.2 0.3c< ≤ 73 0.6173 0.3501 -0.7823 2.1339 0.0355 0.0378 1.3665 5.3709
0.3 0.4c< ≤ 48 0.5274 0.3731 -0.2026 1.4753 0.0546 0.0554 1.1044 3.6580
0.4 0.5c< ≤ 18 0.3649 0.4231 0.4467 1.3493 0.0295 0.0292 0.4527 1.9183
0.5c > 12 0.6019 0.3753 -0.7365 1.9215 0.0337 0.0466 1.3222 3.2200
ρ stability
0.47Nyb > 37 0.6592 0.3411 -0.9654 2.5107 0.0456 0.0406 0.8752 3.1875
0.47Nyb ≤ 269 0.5807 0.3785 -0.5122 1.6527 0.0387 0.0445 1.5494 5.2638
25
Table 4: Moments of Time-varying Beta for Estimated Multivariate Constant Correlation (CC) and Varying Correlation (VC) GARCH Models – No Censoring Correction This table reports the first four moments of the distribution of time-varying betas produced from estimation of the constant correlation (CC) and varying correlation (VC) GARCH models. These estimates come from a sample of constituent stocks in the Australian All Ordinaries Index, subject to data availability and model convergence. The results are reported for a number of different combinations of the data: (a) overall sample of stocks; (b) categories according to the degree of censoring (proportion of zero return observations); (c) split based on whether the Nyblom statistic testing for time-varying correlation is significant, or not, at the 5% level. Estimated CC GARCH Estimated VC GARCH
No Mean Std dev Skew Kurt No Mean Std dev Skew Kurt Overall 358 0.6056 0.1838 1.3767 11.461 306 0.5976 0.3114 0.9084 7.1854
Censoring
0.1c ≤ 103 0.6554 0.1727 1.3470 12.939 91 0.6138 0.2326 0.8479 6.9495
0.1 0.2c< ≤ 77 0.6942 0.2233 1.7431 15.138 64 0.6552 0.2979 1.2330 10.878
0.2 0.3c< ≤ 77 0.6806 0.2076 1.3009 8.5924 73 0.6830 0.3916 0.8987 5.9668
0.3 0.4c< ≤ 52 0.4627 0.1564 1.1971 9.1539 48 0.4972 0.3742 0.5939 4.4012
0.4 0.5c< ≤ 22 0.3785 0.1245 1.2317 8.7785 18 0.3736 0.2587 0.8989 7.5174
0.5c > 27 0.4098 0.1469 1.1257 10.143 12 0.3853 0.3214 0.9666 7.3314
ρ stability
0.47Nyb > 70 0.6584 0.1924 1.0758 8.6345 37 0.6206 0.2864 0.8583 6.8624
0.47Nyb ≤ 288 0.5928 0.1817 1.4498 12.148 269 0.5944 0.3149 0.9153 7.2298
26
Table 5: Summary of Significant Variables in First Stage Probit Model Estimations This table reports a summary of the first stage probit estimation results in terms of the proportion of stocks for which individual variables are found to be significant. Results are reported at the 1%, 5% and 10% significance levels, and for cases where individual variables are significant, pairs of variables are significant, and for where all three variables are significant. Significance Level 1% 5% 10%
Panel A: Percentage of Sample for which Individual Variables Significant Size 12.0 25.7 30.5 Volume 33.7 45.8 53.0 Bid-Ask 5.6 14.1 22.5
Panel B: Percentage of Sample for which Pairs of Variables Significant Size & Volume 5.2 14.5 18.9 Size & Bid-Ask 0.1 3.6 6.4 Volume & Bid-Ask 0.1 8.0 12.1
Panel C: Percentage of Sample for which all Three Variables Significant Size, Volume & Bid-Ask 0.0 1.6 4.0
27
Table 6: Bias and Mean Squared Error Metric for Simulated Multivariate Varying Correlation (VC) GARCH Model, with and without Censoring Correction This table reports the outcome of Monte Carlo simulations for the varying correlation (VC) GARCH model for two alternative cases: without a censoring correction applied and with a censoring correction applied. The results are based on 1,000 simulations of a sample size equal to 1,000. To avoid start up problems, an initial sample of 10,000 observations is generated and the first 9,000 observations are discarded. Using MLE, the bias and mean square error (MSE) are reported in the table. Panel A presents the outcome when a low degree of censoring (20%) is imposed, while Panel B presents the outcome when a high degree of censoring (60%) is imposed. The results only report those simulations where strong convergence was achieved prior to 100 iterations. The number of simulations where strong convergence was achieved is as follows: (a) Low degree of censoring: no correction 923 and with correction 918; (b) High degree of censoring: no correction 867 and with correction 842. Estimated with No Censoring
Correction Estimated with Censoring
Correction True value Bias MSE Bias MSE
Panel A: Low Degree of Censoring (20%) Imposed µ1 0.05 -0.0111 0.0024 -0.0041 0.0029 µ2 0.05 -0.0003 0.0014 -0.0008 0.0014 ω1 0.4 -0.0487 0.0152 0.0303 0.0199 α1 0.2 -0.0477 0.0033 -0.0132 0.0015 β1 0.7 0.0361 0.0046 0.0367 0.0044 ω2 0.2 0.0141 0.0039 0.0146 0.0038 α2 0.2 -0.0091 0.0013 -0.0111 0.0012 β2 0.7 -0.0003 0.0029 0.0008 0.0027 ρ 0.71 -0.0759 0.0076 0.0470 0.0047
1θ 0.8 0.0017 0.0064 0.0023 0.0051
2θ 0.1 -0.0219 0.0009 -0.0146 0.0007
Panel B: High Degree of censoring (60%) Imposed µ1 0.05 -0.0301 0.0023 -0.0015 0.0067 µ2 0.05 0.0003 0.0016 -0.0003 0.0015 ω1 0.4 -0.1519 0.0733 0.2074 0.2833 α1 0.2 -0.1244 0.0166 -0.0142 0.0049 β1 0.7 0.0565 0.0334 0.0679 0.0261 ω2 0.2 0.0235 0.0054 0.0273 0.0053 α2 0.2 -0.0076 0.0014 -0.0132 0.0014 β2 0.7 -0.0065 0.0036 -0.0051 0.0034 ρ 0.71 -0.2755 0.0798 0.0468 0.0133
1θ 0.8 -0.0330 0.0447 -0.1624 0.1210
2θ 0.1 -0.0601 0.0043 -0.0412 0.0028
28
Table 7: Moments of Time-varying Beta for Simulated Multivariate Varying Correlation (VC) GARCH Model, with and without Censoring Correction This table reports the first four moments of the distribution of time-varying betas produced from Monte Carlo simulations for the varying correlation (VC) GARCH model for two alternative cases: without a censoring correction applied and with a censoring correction applied. The results are based on 1,000 simulations of a sample size equal to 1,000. To avoid start up problems, an initial sample of 10,000 observations is generated and the first 9,000 observations are discarded. Panel A presents the outcome when a low degree of censoring (20%) is imposed, while Panel B presents the outcome when a high degree of censoring (60%) is imposed. The results only report those simulations where strong convergence was achieved prior to 100 iterations. The number of simulations where strong convergence was achieved is as follows: (a) Low degree of censoring: no correction 923 and with correction 918; (b) High degree of censoring: no correction 867 and with correction 842.
DGP
Estimated with No Censoring Correction
Estimated with Censoring Correction
Panel A: Low Degree of Censoring (20%) Imposed Mean 0.8953 0.7323 0.8988
Variance 0.3503 0.2695 0.3263 Skew 0.6744 0.7421 0.7399
Kurtosis 4.5068 4.6550 4.6093 Panel B: High Degree of Censoring (60%) Imposed
Mean 0.8953 0.3568 0.9158 Variance 0.3503 0.1136 0.2661
Skew 0.6744 0.6514 0.6685 Kurtosis 4.5068 4.3929 4.6687
29
Table 8: Multivariate Constant Correlation (CC) and Varying Correlation (VC) GARCH Model Parameter Estimates – with Censoring Correction This table reports mean sample moments of selected coefficient estimates for the constant correlation (CC) and varying correlation (VC) GARCH models, in Panels A and B respectively. The censoring correction described in the text has been applied. These estimates come from a sample of constituent stocks in the Australian All Ordinaries Index, subject to data availability and model convergence. The results in each panel are reported for a number of different combinations of the data: (a) overall sample of stocks; (b) categories according to the degree of censoring (proportion of zero return observations); (c) split based on whether the variance equation intercept is greater than or less than unity; and (d) split based on whether the Nyblom statistic testing for time-varying correlation is significant, or not, at the 5% level (Panel B only). No Mean Std dev Skew Kurt Mean Std dev Skew Kurt
Panel A: Selected Parameter Estimates for CC GARCH Model ρ
1 1α β+ Overall 358 0.1794 0.1081 1.3123 6.2242 0.7862 0.2514 -1.4784 4.3633
Censoring 0.1c ≤ 103 0.2510 0.1271 1.1054 5.0391 0.8042 0.2302 -1.6350 5.1414
0.1 0.2c< ≤ 77 0.2048 0.0783 0.7910 4.1585 0.8130 0.1977 -1.2111 3.6305
0.2 0.3c< ≤ 77 0.1584 0.0698 1.0156 6.1902 0.7926 0.2503 -1.4853 4.4066
0.3 0.4c< ≤ 52 0.1153 0.0798 1.4053 5.7432 0.8084 0.2023 -1.3861 4.2286
0.4 0.5c< ≤ 22 0.0724 0.0382 0.4662 3.1123 0.7282 0.2965 -0.8003 2.1383
0.5c > 27 0.1051 0.0537 0.1749 2.6869 0.6282 0.4043 -0.5708 1.5086
1 1ω > 131 0.5853 0.2830 -0.4553 2.1243
1 1ω ≤ 227 0.9022 0.1294 -2.7463 14.077
Panel B: Selected Parameter Estimates for VC GARCH Model ρ
1 1α β+ Overall 306 0.1972 0.1305 1.3053 6.1334 0.8004 0.2402 -1.5429 4.6210
Censoring
0.1c ≤ 91 0.2465 0.1474 0.8234 4.3155 0.8329 0.2076 -1.8913 6.5510
0.1 0.2c< ≤ 64 0.2117 0.0904 0.3804 3.4364 0.7945 0.2267 -1.3768 4.0665
0.2 0.3c< ≤ 73 0.1938 0.1187 2.3294 11.020 0.7984 0.2564 -1.6715 4.9051
0.3 0.4c< ≤ 48 0.1314 0.0992 1.5218 6.0029 0.8026 0.2279 -1.3507 4.0173
0.4 0.5c< ≤ 18 0.1017 0.0931 2.4241 9.2458 0.7519 0.2907 -0.8993 2.2123
0.5c > 12 0.1725 0.1906 1.4664 5.5791 0.6601 0.3232 -0.7126 2.2081
1 1ω > 105 0.6048 0.2791 -0.5287 2.1561
1 1ω ≤ 201 0.9025 0.1294 -2.2926 9.4930
1θ 2θ
Overall 306 0.5363 0.3859 -0.2859 1.4419 0.0399 0.0449 1.5804 6.0620
Censoring 0.1c ≤ 91 0.5950 0.3730 -0.5284 1.6559 0.0345 0.0366 2.1170 9.4973
0.1 0.2c< ≤ 64 0.4921 0.4016 -0.0422 1.2843 0.0366 0.0446 1.5024 4.9770
0.2 0.3c< ≤ 73 0.5578 0.3752 -0.4508 1.6024 0.0396 0.0403 1.1027 3.6471
0.3 0.4c< ≤ 48 0.4761 0.3785 -0.0339 1.4357 0.0544 0.0573 1.3858 5.3905
0.4 0.5c< ≤ 18 0.4013 0.4100 0.3464 1.4032 0.0321 0.0530 1.6243 4.4283
0.5c > 12 0.6396 0.3300 -0.9139 2.6542 0.0557 0.0451 0.2003 1.4164
ρ stability
0.47Nyb > 37 0.5784 0.3549 -0.5152 1.6843 0.0435 0.0395 1.5824 6.5927
0.47Nyb ≤ 269 0.5306 0.3896 -0.2551 1.4177 0.0394 0.0456 1.5858 6.0061
30
Table 9: Moments of Time-varying Beta for Estimated Multivariate Constant Correlation (CC) and Varying Correlation (VC) GARCH Models – with Censoring Correction This table reports the first four moments of the distribution of time-varying betas produced from estimation of the constant correlation (CC) and varying correlation (VC) GARCH models. The censoring correction described in the text has been applied. These estimates come from a sample of constituent stocks in the Australian All Ordinaries Index, subject to data availability and model convergence. The results are reported for a number of different combinations of the data: (a) overall sample of stocks; (b) categories according to the degree of censoring (proportion of zero return observations); (c) split based on whether the Nyblom statistic testing for time-varying correlation is significant, or not, at the 5% level.
Estimated CC GARCH Estimated VC GARCH No Mean Std dev Skew Kurt No Mean Std dev Skew Kurt
Overall 358 0.6140 0.1959 1.6192 15.470 306 0.6056 0.3190 1.0912 9.7734
Censoring 0.1c ≤ 103 0.6579 0.1840 1.5742 17.987 91 0.6257 0.2321 1.0987 12.273
0.1 0.2c< ≤ 77 0.7001 0.2199 1.6931 16.130 64 0.6588 0.2926 1.4056 12.390
0.2 0.3c< ≤ 77 0.6882 0.2171 1.4379 10.501 73 0.6746 0.3900 0.9953 6.8064
0.3 0.4c< ≤ 52 0.4683 0.1789 1.5809 10.485 48 0.5043 0.4003 0.7886 5.4036
0.4 0.5c< ≤ 22 0.3649 0.1338 1.7841 15.306 18 0.4289 0.2730 1.3336 12.918
0.5c > 27 0.4736 0.1953 2.0366 27.895 12 0.4209 0.4309 0.7867 7.6791
ρ stability
0.47Nyb > 64 0.6486 0.1816 1.0089 8.4790 37 0.6442 0.2416 0.7145 5.7687
0.47Nyb ≤ 294 0.6065 0.1990 1.7521 16.992 269 0.6003 0.3297 1.1430 10.324
31
Figure 1: Distribution of 1θ estimates – without Censoring Correction
0
10
20
30
40
50
60
70
0.00 0.25 0.50 0.75 1.00
Series: UNCENSORSample 1 334Observations 334
Mean 0.587529Median 0.740235Maximum 0.999900Minimum 0.000100Std. Dev. 0.376801Skewness -0.537972Kurtosis 1.693986
Jarque-Bera 39.84798Probability 0.000000
32
Figure 2: Distribution of 1θ estimates – with Censoring Correction
0
10
20
30
40
50
60
70
80
90
0.00 0.25 0.50 0.75 1.00
Series: CENSORSample 1 334Observations 334
Mean 0.522093Median 0.604270Maximum 0.999900Minimum 0.000100Std. Dev. 0.389754Skewness -0.225435Kurtosis 1.398221
Jarque-Bera 38.53499Probability 0.000000
33
Figure 3: Kernel Densities of 1θ Estimates
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-0.5 0 0.5 1 1.5
Uncensored Censored
Note: Kernel density estimates obtained via EViews 4.1, where normality is assumed and a Silverman bandwidth with 100 points is applied.
34
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