Description
We investigate the effect standard time series
β-adjustments have on the OLS-β. We report
that most changes are not statistically
significant and the β-adjustments appear to
have no relationship to the extent of thin
trading. Researchers using β face the difficult
choice of using an estimate known to be biased
by thin trading, or making an adjustment that
may not be statistically significant.
Accounting Research Journal
The Impact of Thin Trading Adjustments on Australian Beta Estimates
Sinclair Davidson Thomas J osev
Article information:
To cite this document:
Sinclair Davidson Thomas J osev, (2005),"The Impact of Thin Trading Adjustments on Australian Beta Estimates",
Accounting Research J ournal, Vol. 18 Iss 2 pp. 111 - 117
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
111
The Impact of Thin Trading Adjustments on
Australian Beta Estimates
Sinclair Davidson and Thomas Josev
School of Economics and Finance
RMIT
Abstract
We investigate the effect standard time series
?-adjustments have on the OLS-?. We report
that most changes are not statistically
significant and the ?-adjustments appear to
have no relationship to the extent of thin
trading. Researchers using ? face the difficult
choice of using an estimate known to be biased
by thin trading, or making an adjustment that
may not be statistically significant.
1. Introduction.
Despite well-known problems, the CAPM
remains the “work horse” of finance theory
and, arguably, practice. Not least amongst these
problems is the issue of thin trading. While it
may be debated whether thin trading is an
economic problem, it does pose econometric
problems in the estimation of beta (see
Campbell, Lo and MacKinlay 1997). In short,
estimated betas will generally be understated
for thinly traded stocks and overstated for
(very) thickly traded stocks.
1
Numerous studies, for example Scholes and
Williams (SW) (1977), Dimson (1979), Fowler
and Rorke (FR) (1983), “adjust” estimated
betas to approximate their so-called “true” or
unbiased value. In addition, there is a small
literature that compares various adjustment
techniques to each other and OLS betas on
Keywords: Beta, Thin trading
JEL Classification: G10, G12
Acknowledgments: We thank John Fowler for his research
assistance and Robert Faff, Richard Heaney, an anonymous
referee and Tim Brailsford (the editor) for helpful
comments.
1 We acknowledge that the ?-estimate could be biased in
either direction. The problem is normally discussed in
terms of a downward bias. Further our data shows a
downward bias.
econometric grounds. For example, McInish
and Wood (1986), using daily US data, find
that Dimson-? estimates are the least biased,
but that their “improvement” is less than
spectacular (only 29%). Bartholdy and Riding
(1994), using daily New Zealand data, find that
OLS-? estimates are less biased than Dimson-
?s and SW-?s. Researchers making use of ?-
adjustments, however, tend to be ad hoc in
their approach. Fama and French (1992), for
example, use Dimson ?s to correct for thin
trading and not FR-?s. They justify this choice
by calculating the serial autocorrelation of the
index, which is not statistically significantly
different from zero (making the Dimson-?
estimate equivalent to the FR-? estimate).
They, however, do not justify Dimson-?s vis-a-
vis, say, SW-?s. On the other hand, Amihud
and Murgia (1997) employ SW-?s in their
study of the German capital market and have
no justification or comment on their choice of
the ? adjustment.
Using Australian data, Brailsford, Faff and
Oliver (1997: 25) compare adjusted ?-estimates
to OLS ?-estimates and provide evidence of
“substantial variation” in ?-estimates. They do
not, however, indicate whether these variations
are either statistically or economically
significant. This paper investigates whether
thin trading adjustments are statistically
significant and attempts to model those
adjustments. The results are counter-intuitive
and not at all consistent with what we would
have expected. The various ?-estimates are not
different from each other and do not appear to
be related to measures of thin trading.
2. Data and Method.
We select our data from Datastream over the
five-year period January 1995 to December
1999 (1266 daily observations). Our final
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 2 (2005)
112
sample consists of 481 firms. We arrive at this
sample size as follows: There are 1419 firms in
Datastream for our sample period. For 938 of
these firms, data are incomplete (either the firm
was suspended for a period of time, or its return
index was incomplete or size data was
unavailable or volume data was unavailable).
Continuously compounded returns are
calculated from daily prices and are also
adjusted for capitalization changes and
dividends. Public holidays and weekends have
also been removed from the data. We define a
‘zero-return’ as either the stock did not trade or
the stock traded at an unchanged price. We are
able to determine whether a zero-return
constitutes thin trading (i.e. the stock did not
trade) or a stock trading at an unchanged price.
We define a ‘Zero Trade’ as the stock not
trading. The difference between a zero-return
and a Zero Trade is the stock trading at an
unchanged price. Using the All Ordinaries
Accumulation Index as our proxy for the
market portfolio the market model is used to
estimate OLS-?s. Given that thin trading may
be related to firm size as measured by market
capitalization (Dimson and Marsh 1983), we
also rank the data by size
2
and segment it into
quartiles (from smallest, Q1, to largest, Q4).
Table 1 shows some descriptive statistics.
Table 1 clearly shows a size effect in the
data. The OLS-? increases as the average size
of stocks increase. Similarly the proportion of
zero returns and proportion of zero trades
decrease with size. We now estimate the
following linear regressions for the entire
sample (size is the average value of the firm
over the sample period)
3
,
OLS-? = 0.264 + 0.063lnSize + ?
(0.00) (0.00) adj-R
2
= 0.113 (1)
OLS-? = 0.851 - 0.008
Proportion of Zero Returns + ?
(0.00) (0.00) adj-R
2
= 0.272 (2)
OLS-? = 0.780 - 0.009
Proportion of Zero Trades + ?
(0.00) (0.00) adj-R
2
= 0.411 (3)
2 We have ranked the data by starting market value, i.e.
the market capitalisation in January 1995.
3 Results are quantitatively similar if start or end size is
used in the regressions.
OLS-? = 1.012 - 0.025lnSize - 0.009
Proportion of Zero Returns + ?
(0.00) (0.05) (0.00) adj-R
2
= 0.278 (4)
OLS-? = 0.837 - 0.011lnSize - 0.001
Proportion of Zero Trades + ?
(0.00) (0.23) (0.00) adj-R
2
= 0.421 (5)
(With White heteroskedasticity consistent
p-values shown in parentheses).
This exercise shows a clear relationship
between OLS-?s and firm size and the
proportion of zero returns and proportion of no
trades. As can be expected, there is a positive
relationship with size and a negative
relationship with the proportion of zeros and
proportion of no trades. Given that ? is a
measure of systematic risk it should not be
related to characteristics such as size and the
proportion of zeros in the trading data. The
results clearly indicate that the OLS-?s are
biased. What is of interest is the sign reversal
on size in equations (4) and (5) relative to
equation (1).
4
While size and the proportion of
Zero Returns are highly correlated (? = -
0.7517) multicollinearity is not the cause of the
sign reversal (variance inflation factor = 2.30).
Outliers can also be eliminated as a cause as
there are no outliers more than two standard
deviations from the mean.
The various beta-adjustments usually
involve making use of leads and lags in the
market index. The question then is the number
of leads and lags that should be employed. In
order to answer this question, we calculated the
serial correlation of returns in the All
Ordinaries Accumulation Index. The Ljung-
Box Q statistics indicate, however, no
statistical significance in the lags. Despite this
result, we employ up to two leads and lags in
the data.
We calculate a trade-to-trade beta and three
beta-adjustments, the Scholes-Williams (1977)
for one and two leads and lags, the Dimson
(1979) for one and two leads and lags and the
Fowler-Rorke (1983) for one and two leads and
lags. The actual formulae are shown below.
4 Note, however, that size in equation (5) is not
statistically significant.
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
113
Scholes-Williams (1977) (one lead and one
lag):
SW-?
i
1
1
i
0
i
1
i
2 1
b b b
? +
+ +
=
+ ?
and (two leads and lags)
SW-?
i
2 1
2
i
1
i
0
i
1
i
-2
i
2 2 1
b b b b b
? + ? +
+ + + +
=
+ + ?
where,
it 2 mt
2
i i it
2
i
e R b a R : b + + =
?
? ?
b R a b R e
i it i i mt it
? ?
?
= + +
1 1
1
:
it mt
0
i i it
0
i
e R b a R : b + + =
b R a b R e
i it i i mt it
+ +
+
= + +
1 1
1
:
it 2 mt
2
i i it
2
i
e R b a R : b + + =
+
+ +
?
1
and ?
2
= first and second order serial
correlation coefficient for the market return.
Dimson (1979) (for 2 leads and lags):
D-?
i
!
? =
=
2
2 n
n
i
b
where the beta estimates used are derived from
the following regression equation,
mt
o
i 1 mt
1
i 2 mt
2
i i it
R b R b R b a R + + + + =
?
?
?
?
it 2 mt
2
i 1 mt
1
i
e R b R b + + +
+
+
+
+
The Fowler-Rorke (1983) ?-adjustment is a
variation of the Dimson ?-adjustment. Instead
of the sum of the ?s being equally weighted
they are weighted as follows:
1
1
1
2 1
1
w
? +
? +
=
for one lead and lag and
2 1
2 1
2
2 2 1
1
w
? + ? +
? + ? +
=
for two leads and lags.
Table 1
Descriptive characteristics of data.
Characteristic All Q1 Q2 Q3 Q4
OLS Beta
Average 0.518 0.479 0.437 0.463 0.695
Standard Deviation 0.387 0.479 0.346 0.336 0.313
Minimum -0.619 -0.619 -0.047 -0.048 -0.070
Maximum 1.961 1.961 1.387 1.827 1.367
Proportion Zero Returns (%)
Average 43.850 64.215 54.581 40.502 15.933
Standard Deviation 26.702 18.848 19.647 21.261 18.562
Minimum 0.000 18.167 0.000 0.000 0.000
Maximum 99.368 99.368 98.736 95.340 87.046
Proportion Zero Trades (%)
Average 28.457 46.000 36.140 24.003 7.541
Standard Deviation 26.993 26.799 24.287 21.985 17.549
Minimum 0.079 1.106 0.553 0.158 0.079
Maximum 97.867 97.867 97.551 90.837 84.281
Market Value ($millions)
Average 454.985 3.905 15.528 56.811 1747.455
Standard Deviation 1962.421 2.031 5.185 27.449 3645.137
Minimum 0.140 0.140 8.190 27.190 122.980
Maximum 31764.940 8.100 26.000 122.760 31764.940
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 2 (2005)
114
Table 2
Summary Statistics
Table shows summary statistics for ? estimates. OLS-? is an ordinary least squares estimation of ?.
TT-? is a trade-to-trade ?. SW-? is a Scholes-Williams ?. D-? is a Dimson ?. FR-? is a Fowler-
Rorke ?. Also shown are corresponding standard errors.
All OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.5184 0.5747 0.6266 0.5735 0.6278 0.5676 0.6284 0.661
Std. Dev. 0.3874 0.4424 0.5515 0.4395 0.5535 0.4333 0.523 1.001
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.1434 0.2465 0.3200 0.2425 0.3207 0.3026 0.4703 0.3194
Std. Dev. 0.203 0.3498 0.4545 0.344 0.4556 1.0922 2.0341 0.5921
Q1 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.4789 0.5616 0.6409 0.5599 0.6425 0.5542 0.6442 0.6529
Std. Dev. 0.4794 0.5873 0.8371 0.5823 0.8407 0.5719 0.7905 1.3358
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.2494 0.4293 0.5574 0.4222 0.5587 0.5987 0.9711 0.7298
Std. Dev. 0.3688 0.6355 0.8259 0.6251 0.8278 2.1433 4.0127 0.9564
Q2 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.4367 0.5258 0.5916 0.5240 0.5933 0.5170 0.5941 0.7118
Std. Dev. 0.3456 0.4339 0.4940 0.4306 0.4962 0.4251 0.4734 1.4176
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.1471 0.2522 0.3273 0.2481 0.3281 0.2727 0.4090 0.3087
Std. Dev. 0.0694 0.1204 0.1561 0.1184 0.1565 0.1597 0.2144 0.4671
Q3 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.4628 0.5014 0.5461 0.5006 0.5468 0.4957 0.5509 0.5579
Std. Dev. 0.3360 0.3367 0.3736 0.3352 0.3742 0.3308 0.3526 0.3529
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.1082 0.1861 0.2415 0.1831 0.2421 0.1960 0.2949 0.1526
Std. Dev. 0.0605 0.1040 0.1350 0.1023 0.1354 0.1235 0.1652 0.1061
Q4 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.6954 0.7100 0.7279 0.7097 0.7282 0.7037 0.7244 0.7214
Std. Dev. 0.3131 0.3368 0.3438 0.3357 0.3444 0.3326 0.3287 0.3021
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.0679 0.1169 0.1517 0.1150 0.1520 0.1403 0.2021 0.0830
Std. Dev. 0.0462 0.0794 0.1031 0.0781 0.1033 0.1022 0.1289 0.0809
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
115
Table 3
Analysis of changes in Beta value
Panel A shows the number of ?-estimates significantly different from zero. Panel B shows the
number of Adjusted ?-estimates significantly different from the OLS-?.
Panel A OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
ALL 338 291 261 294 262 280 209 344
Q1 55 43 27 43 27 30 13 60
Q2 75 61 55 64 55 60 34 76
Q3 97 80 74 80 75 82 60 98
Q4 111 107 105 107 105 108 102 110
Panel B OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
ALL - 41 49 37 52 38 42 14
Q1 - 11 13 9 13 9 11 10
Q2 - 10 7 9 7 8 5 2
Q3 - 7 12 7 12 7 8 1
Q4 - 13 17 12 20 14 18 1
We also calculate empirical standard errors
for the adjusted-?s. Estimating the appropriate
parameters through the use of a system of
equations does this.
5
The method uses the
ordinary least squares approach of minimising
the sum of squared residuals while adjusting
for any cross equation restrictions on the
parameters of the system. These system
equations result from re-arranging the separate
estimating equations allowing the system to
estimate the adjusted-? along with its standard
error.
Once the appropriate ?-adjustments are
made summary statistics are calculated. Table 2
shows the mean and standard deviations of the
?-estimates and the estimated standard errors of
those estimates. The data are broken up by ?-
estimates and size. It is worth noting that the
adjusted ?-estimates tend to be larger than the
OLS estimates. This is consistent with the
notion that thin trading leads to an under-
estimate of the “true” ?. The other thing worth
noting is that the OLS standard errors tend to
be smaller than the adjusted ?-estimate
standard errors. We also observe a “u-shaped”
pattern in the ?-estimates with Q2 and Q3
5 We thank Robert Faff for suggesting this approach. The
exact derivation is available on request.
estimates being lower than Q1 and Q4. Finally
the table shows that the standard error declines
as firm size increases. To the extent that firm
size and trading are positively correlated this
implies that thin trading affects not only the
OLS ?-estimate but also the precision of the
estimated correction.
We compare each estimate to zero and then to
the OLS - ? in Table 3. Panel A of Table 3
shows the number of ?-estimates that are
statistically significantly different from zero.
To the extent that any ?-estimate is empirically
useful it must pass this first hurdle. A ?-
estimate equal to zero indicates that a known
risky asset has no risk. This, we believe, is a
very easy test. Unfortunately, many of the ?-
estimates fail this simple test. There is a clear
size effect. For those firms with smaller size, it
is more likely that the ?-estimate will not be
statistically significantly different from zero. It
is only for firms in Q4 that a high proportion of
firms have non-zero ?-estimates. The large
standard errors for the adjusted ?-estimates
becomes problematic. Except for the trade-to-
trade ?-estimates the number of non-zero
adjusted ?-estimates is less than the OLS ?-
estimates. In essence the standard error grows
faster than the “adjusted” value of the ?-
estimate. In panel B we compare the adjusted
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 2 (2005)
116
?-estimates to the OLS - ?. Here we establish
how many adjusted ?-estimates are
significantly different from the OLS -
?-estimates. Overall the numbers are small. For
the smallest stocks (most likely to be plagued
by thin trading) there are, at most, 13 stocks
with a significantly different ?-estimate. For all
stocks there are, at most, 52 significantly
different ?-estimates. What is surprising is that
there are more significant adjustments for
stocks in Q4 – we would have expected less for
that group.
Finally we investigate the ?-adjustments in a
regression framework. Table 4 contains the
results. The percentage change between each
adjusted-? and the OLS-? is used as the
dependent variable. Independent variables are
the average size of the firm and the extent of
thin trading (measured by the proportion of
Zero Trades). We also include the interaction
of these two variables. We would expect, if the
?-adjustments are adding any value, that the
smaller the firm and/or the greater the
proportion of no trades the greater the
percentage change in ? there should be. We are
assuming that the effect of thin trading is to
downwardly bias the estimated beta. Looking
at Table 4 we see that the adjusted-R
2
statistics
are very low. The ?-adjustments are smaller the
larger the firm. This is not unexpected. What is
surprising is that, in some instances, the higher
the proportion of Zero Trades the smaller the ?-
adjustment. This effect seems to impact on
those adjustments with two leads and lags. The
interaction effects are negative but never
statistically significant.
Table 4
Regression results for percentage change in beta
?SW-? is the percentage change between a Scholes-Williams beta and an OLS beta, ?D-? is the
percentage change between a Dimson beta and an OLS beta and ?FR-? is the percentage change
between a Fowler-Rorke beta and an OLS beta. White heteroskedasticity consistent p-values are
shown in parenthesis.
?D-?1 ?D-?2 ?FR-?1 ?FR-?2 ?SW-?1 ?SW-?2 ?TT-?
c 80.2429 174.9608 79.0781 176.3507 74.6277 155.0003 -33.4636
(0.0131) (0.0011) (0.0128) (0.0011) (0.0151) (0.0040) (0.8039)
ln(Size) -10.9999 -22.2966 -10.8480 -22.4849 -10.3570 -19.3111 -3.4232
(0.0183) (0.0035) (0.0179) (0.0035) (0.0195) (0.0124) (0.8048)
Proportion Zero Trade -0.3512 -2.2867 -0.3524 -2.2868 -0.3029 -1.6701 4.9826
(0.6023) (0.0517) (0.5947) (0.0533) (0.6358) (0.1455) (0.3310)
Adj-R
2
0.0039 0.0240 0.0040 0.0236 0.0040 0.0127 0.0102
?D-?1 ?D-?2 ?FR-?1 ?FR-?2 ?SW-?1 ?SW-?2 ?TT-?
C 70.6714 179.5079 69.4913 180.7538 65.8547 166.1418 -171.1879
(0.0202) (0.0003) (0.0199) (0.0003) (0.0226) (0.0005) (0.5524)
ln(Size) -8.7889 -23.3470 -8.6334 -23.5020 -8.3304 -21.8849 28.3913
(0.0312) (0.0005) (0.0309) (0.0005) (0.0314) (0.0005) (0.5366)
Proportion Zero Trade 0.0973 -2.4998 0.0969 -2.4931 0.1082 -2.1923 11.4374
(0.9125) (0.0793) (0.9112) (0.0812) (0.8976) (0.0753) (0.3908)
Size*Proportion Zero Trade -0.1473 0.0700 -0.1475 0.0678 -0.1350 0.1715 -2.1196
(0.4946) (0.8294) (0.4859) (0.8358) (0.5104) (0.5810) (0.4576)
Adj-R
2
0.0030 0.0221 0.0031 0.0217 0.0030 0.0114 0.0158
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
117
3. Conclusion
In this paper we have investigated the most
common ?-adjustments made to correct for
problems of thin trading. Our initial expectation
was that thin trading would have a downward
bias on the OLS-?. The thin trading
adjustments would lead to an increase in the
size of the ?-estimate. In addition, the change
in ?-estimate would be related to the extent of
thin trading. Our results, however, indicate that
these adjustments do not operate, as our initial
expectations would indicate. First, many of the
adjustments do not add any statistical value.
Second, these adjustments appear to be
unrelated to thin trading. As best we can
determine, the adjusted ?-estimates have a
relationship with firm size. Researchers are left
in quandary. Complicated adjustments do not
appear to add much to the OLS ?-estimate, yet
that estimate is known to be problematic in the
presence of thin trading. Consideration such as
this may explain the ad hoc approach to ?-
adjustments.
References
Amihud, Y., and Murgia, M. (1997), ‘Dividends, Taxes,
and Signalling: Evidence from Germany’, The Journal
of Finance, vol. 52, pp. 397-408.
Bartholdy, J. and Riding, A. (1994), ‘Thin trading and the
estimation of betas: The efficacy of alternative
techniques’, The Journal of Financial Research,
vol. 17, pp. 241-254.
Brailsford, T., Faff, R. and Oliver, B. (1997), Research
issues in the estimation of beta, Sydney, McGraw-Hill.
Campbell, J., Lo, A. and MacKinlay, C. (1997), The
econometrics of financial markets, Princeton
University Press, New Jersey.
Dimson, E. (1979), ‘Risk measurement when shares are
subject to infrequent trading’, Journal of Financial
Economics, vol. 7 pp. 197-226.
Dimson, E. and Marsh, P. (1983), ‘The stability of UK risk
measures and the problem of thin trading’, Journal of
Finance, vol. 38, pp. 753-783.
Fama, E. and French, K. (1992), ‘The cross-section of
expected stock returns’, Journal of Finance, vol. 47,
pp. 427-467.
Fowler, D. and Rorke, C. (1983), ‘Risk measurement when
shares are subject to infrequent trading: Comment’,
Journal of Financial Economics, vol. 12, pp. 279-283.
McInish, T. and Wood, R. (1986), ‘Adjusting for beta bias:
An assessment of alternative techniques: A note’,
Journal of Finance, vol. 41 pp. 277-286.
Scholes, M. and Williams, J. (1977), ‘Estimating betas from
non-synchronous data’, Journal of Financial
Economics, vol. 5 pp. 309-327.
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doc_337033677.pdf
We investigate the effect standard time series
β-adjustments have on the OLS-β. We report
that most changes are not statistically
significant and the β-adjustments appear to
have no relationship to the extent of thin
trading. Researchers using β face the difficult
choice of using an estimate known to be biased
by thin trading, or making an adjustment that
may not be statistically significant.
Accounting Research Journal
The Impact of Thin Trading Adjustments on Australian Beta Estimates
Sinclair Davidson Thomas J osev
Article information:
To cite this document:
Sinclair Davidson Thomas J osev, (2005),"The Impact of Thin Trading Adjustments on Australian Beta Estimates",
Accounting Research J ournal, Vol. 18 Iss 2 pp. 111 - 117
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
111
The Impact of Thin Trading Adjustments on
Australian Beta Estimates
Sinclair Davidson and Thomas Josev
School of Economics and Finance
RMIT
Abstract
We investigate the effect standard time series
?-adjustments have on the OLS-?. We report
that most changes are not statistically
significant and the ?-adjustments appear to
have no relationship to the extent of thin
trading. Researchers using ? face the difficult
choice of using an estimate known to be biased
by thin trading, or making an adjustment that
may not be statistically significant.
1. Introduction.
Despite well-known problems, the CAPM
remains the “work horse” of finance theory
and, arguably, practice. Not least amongst these
problems is the issue of thin trading. While it
may be debated whether thin trading is an
economic problem, it does pose econometric
problems in the estimation of beta (see
Campbell, Lo and MacKinlay 1997). In short,
estimated betas will generally be understated
for thinly traded stocks and overstated for
(very) thickly traded stocks.
1
Numerous studies, for example Scholes and
Williams (SW) (1977), Dimson (1979), Fowler
and Rorke (FR) (1983), “adjust” estimated
betas to approximate their so-called “true” or
unbiased value. In addition, there is a small
literature that compares various adjustment
techniques to each other and OLS betas on
Keywords: Beta, Thin trading
JEL Classification: G10, G12
Acknowledgments: We thank John Fowler for his research
assistance and Robert Faff, Richard Heaney, an anonymous
referee and Tim Brailsford (the editor) for helpful
comments.
1 We acknowledge that the ?-estimate could be biased in
either direction. The problem is normally discussed in
terms of a downward bias. Further our data shows a
downward bias.
econometric grounds. For example, McInish
and Wood (1986), using daily US data, find
that Dimson-? estimates are the least biased,
but that their “improvement” is less than
spectacular (only 29%). Bartholdy and Riding
(1994), using daily New Zealand data, find that
OLS-? estimates are less biased than Dimson-
?s and SW-?s. Researchers making use of ?-
adjustments, however, tend to be ad hoc in
their approach. Fama and French (1992), for
example, use Dimson ?s to correct for thin
trading and not FR-?s. They justify this choice
by calculating the serial autocorrelation of the
index, which is not statistically significantly
different from zero (making the Dimson-?
estimate equivalent to the FR-? estimate).
They, however, do not justify Dimson-?s vis-a-
vis, say, SW-?s. On the other hand, Amihud
and Murgia (1997) employ SW-?s in their
study of the German capital market and have
no justification or comment on their choice of
the ? adjustment.
Using Australian data, Brailsford, Faff and
Oliver (1997: 25) compare adjusted ?-estimates
to OLS ?-estimates and provide evidence of
“substantial variation” in ?-estimates. They do
not, however, indicate whether these variations
are either statistically or economically
significant. This paper investigates whether
thin trading adjustments are statistically
significant and attempts to model those
adjustments. The results are counter-intuitive
and not at all consistent with what we would
have expected. The various ?-estimates are not
different from each other and do not appear to
be related to measures of thin trading.
2. Data and Method.
We select our data from Datastream over the
five-year period January 1995 to December
1999 (1266 daily observations). Our final
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 2 (2005)
112
sample consists of 481 firms. We arrive at this
sample size as follows: There are 1419 firms in
Datastream for our sample period. For 938 of
these firms, data are incomplete (either the firm
was suspended for a period of time, or its return
index was incomplete or size data was
unavailable or volume data was unavailable).
Continuously compounded returns are
calculated from daily prices and are also
adjusted for capitalization changes and
dividends. Public holidays and weekends have
also been removed from the data. We define a
‘zero-return’ as either the stock did not trade or
the stock traded at an unchanged price. We are
able to determine whether a zero-return
constitutes thin trading (i.e. the stock did not
trade) or a stock trading at an unchanged price.
We define a ‘Zero Trade’ as the stock not
trading. The difference between a zero-return
and a Zero Trade is the stock trading at an
unchanged price. Using the All Ordinaries
Accumulation Index as our proxy for the
market portfolio the market model is used to
estimate OLS-?s. Given that thin trading may
be related to firm size as measured by market
capitalization (Dimson and Marsh 1983), we
also rank the data by size
2
and segment it into
quartiles (from smallest, Q1, to largest, Q4).
Table 1 shows some descriptive statistics.
Table 1 clearly shows a size effect in the
data. The OLS-? increases as the average size
of stocks increase. Similarly the proportion of
zero returns and proportion of zero trades
decrease with size. We now estimate the
following linear regressions for the entire
sample (size is the average value of the firm
over the sample period)
3
,
OLS-? = 0.264 + 0.063lnSize + ?
(0.00) (0.00) adj-R
2
= 0.113 (1)
OLS-? = 0.851 - 0.008
Proportion of Zero Returns + ?
(0.00) (0.00) adj-R
2
= 0.272 (2)
OLS-? = 0.780 - 0.009
Proportion of Zero Trades + ?
(0.00) (0.00) adj-R
2
= 0.411 (3)
2 We have ranked the data by starting market value, i.e.
the market capitalisation in January 1995.
3 Results are quantitatively similar if start or end size is
used in the regressions.
OLS-? = 1.012 - 0.025lnSize - 0.009
Proportion of Zero Returns + ?
(0.00) (0.05) (0.00) adj-R
2
= 0.278 (4)
OLS-? = 0.837 - 0.011lnSize - 0.001
Proportion of Zero Trades + ?
(0.00) (0.23) (0.00) adj-R
2
= 0.421 (5)
(With White heteroskedasticity consistent
p-values shown in parentheses).
This exercise shows a clear relationship
between OLS-?s and firm size and the
proportion of zero returns and proportion of no
trades. As can be expected, there is a positive
relationship with size and a negative
relationship with the proportion of zeros and
proportion of no trades. Given that ? is a
measure of systematic risk it should not be
related to characteristics such as size and the
proportion of zeros in the trading data. The
results clearly indicate that the OLS-?s are
biased. What is of interest is the sign reversal
on size in equations (4) and (5) relative to
equation (1).
4
While size and the proportion of
Zero Returns are highly correlated (? = -
0.7517) multicollinearity is not the cause of the
sign reversal (variance inflation factor = 2.30).
Outliers can also be eliminated as a cause as
there are no outliers more than two standard
deviations from the mean.
The various beta-adjustments usually
involve making use of leads and lags in the
market index. The question then is the number
of leads and lags that should be employed. In
order to answer this question, we calculated the
serial correlation of returns in the All
Ordinaries Accumulation Index. The Ljung-
Box Q statistics indicate, however, no
statistical significance in the lags. Despite this
result, we employ up to two leads and lags in
the data.
We calculate a trade-to-trade beta and three
beta-adjustments, the Scholes-Williams (1977)
for one and two leads and lags, the Dimson
(1979) for one and two leads and lags and the
Fowler-Rorke (1983) for one and two leads and
lags. The actual formulae are shown below.
4 Note, however, that size in equation (5) is not
statistically significant.
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
113
Scholes-Williams (1977) (one lead and one
lag):
SW-?
i
1
1
i
0
i
1
i
2 1
b b b
? +
+ +
=
+ ?
and (two leads and lags)
SW-?
i
2 1
2
i
1
i
0
i
1
i
-2
i
2 2 1
b b b b b
? + ? +
+ + + +
=
+ + ?
where,
it 2 mt
2
i i it
2
i
e R b a R : b + + =
?
? ?
b R a b R e
i it i i mt it
? ?
?
= + +
1 1
1
:
it mt
0
i i it
0
i
e R b a R : b + + =
b R a b R e
i it i i mt it
+ +
+
= + +
1 1
1
:
it 2 mt
2
i i it
2
i
e R b a R : b + + =
+
+ +
?
1
and ?
2
= first and second order serial
correlation coefficient for the market return.
Dimson (1979) (for 2 leads and lags):
D-?
i
!
? =
=
2
2 n
n
i
b
where the beta estimates used are derived from
the following regression equation,
mt
o
i 1 mt
1
i 2 mt
2
i i it
R b R b R b a R + + + + =
?
?
?
?
it 2 mt
2
i 1 mt
1
i
e R b R b + + +
+
+
+
+
The Fowler-Rorke (1983) ?-adjustment is a
variation of the Dimson ?-adjustment. Instead
of the sum of the ?s being equally weighted
they are weighted as follows:
1
1
1
2 1
1
w
? +
? +
=
for one lead and lag and
2 1
2 1
2
2 2 1
1
w
? + ? +
? + ? +
=
for two leads and lags.
Table 1
Descriptive characteristics of data.
Characteristic All Q1 Q2 Q3 Q4
OLS Beta
Average 0.518 0.479 0.437 0.463 0.695
Standard Deviation 0.387 0.479 0.346 0.336 0.313
Minimum -0.619 -0.619 -0.047 -0.048 -0.070
Maximum 1.961 1.961 1.387 1.827 1.367
Proportion Zero Returns (%)
Average 43.850 64.215 54.581 40.502 15.933
Standard Deviation 26.702 18.848 19.647 21.261 18.562
Minimum 0.000 18.167 0.000 0.000 0.000
Maximum 99.368 99.368 98.736 95.340 87.046
Proportion Zero Trades (%)
Average 28.457 46.000 36.140 24.003 7.541
Standard Deviation 26.993 26.799 24.287 21.985 17.549
Minimum 0.079 1.106 0.553 0.158 0.079
Maximum 97.867 97.867 97.551 90.837 84.281
Market Value ($millions)
Average 454.985 3.905 15.528 56.811 1747.455
Standard Deviation 1962.421 2.031 5.185 27.449 3645.137
Minimum 0.140 0.140 8.190 27.190 122.980
Maximum 31764.940 8.100 26.000 122.760 31764.940
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 2 (2005)
114
Table 2
Summary Statistics
Table shows summary statistics for ? estimates. OLS-? is an ordinary least squares estimation of ?.
TT-? is a trade-to-trade ?. SW-? is a Scholes-Williams ?. D-? is a Dimson ?. FR-? is a Fowler-
Rorke ?. Also shown are corresponding standard errors.
All OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.5184 0.5747 0.6266 0.5735 0.6278 0.5676 0.6284 0.661
Std. Dev. 0.3874 0.4424 0.5515 0.4395 0.5535 0.4333 0.523 1.001
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.1434 0.2465 0.3200 0.2425 0.3207 0.3026 0.4703 0.3194
Std. Dev. 0.203 0.3498 0.4545 0.344 0.4556 1.0922 2.0341 0.5921
Q1 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.4789 0.5616 0.6409 0.5599 0.6425 0.5542 0.6442 0.6529
Std. Dev. 0.4794 0.5873 0.8371 0.5823 0.8407 0.5719 0.7905 1.3358
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.2494 0.4293 0.5574 0.4222 0.5587 0.5987 0.9711 0.7298
Std. Dev. 0.3688 0.6355 0.8259 0.6251 0.8278 2.1433 4.0127 0.9564
Q2 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.4367 0.5258 0.5916 0.5240 0.5933 0.5170 0.5941 0.7118
Std. Dev. 0.3456 0.4339 0.4940 0.4306 0.4962 0.4251 0.4734 1.4176
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.1471 0.2522 0.3273 0.2481 0.3281 0.2727 0.4090 0.3087
Std. Dev. 0.0694 0.1204 0.1561 0.1184 0.1565 0.1597 0.2144 0.4671
Q3 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.4628 0.5014 0.5461 0.5006 0.5468 0.4957 0.5509 0.5579
Std. Dev. 0.3360 0.3367 0.3736 0.3352 0.3742 0.3308 0.3526 0.3529
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.1082 0.1861 0.2415 0.1831 0.2421 0.1960 0.2949 0.1526
Std. Dev. 0.0605 0.1040 0.1350 0.1023 0.1354 0.1235 0.1652 0.1061
Q4 OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
Mean 0.6954 0.7100 0.7279 0.7097 0.7282 0.7037 0.7244 0.7214
Std. Dev. 0.3131 0.3368 0.3438 0.3357 0.3444 0.3326 0.3287 0.3021
OLS-?SE D-?1SE D-?2SE FR-?1SE FR-?2SE SW-?1SE SW-?2SE TT-?SE
Mean 0.0679 0.1169 0.1517 0.1150 0.1520 0.1403 0.2021 0.0830
Std. Dev. 0.0462 0.0794 0.1031 0.0781 0.1033 0.1022 0.1289 0.0809
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
115
Table 3
Analysis of changes in Beta value
Panel A shows the number of ?-estimates significantly different from zero. Panel B shows the
number of Adjusted ?-estimates significantly different from the OLS-?.
Panel A OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
ALL 338 291 261 294 262 280 209 344
Q1 55 43 27 43 27 30 13 60
Q2 75 61 55 64 55 60 34 76
Q3 97 80 74 80 75 82 60 98
Q4 111 107 105 107 105 108 102 110
Panel B OLS-? D-?1 D-?2 FR-?1 FR-?2 SW-?1 SW-?2 TT-?
ALL - 41 49 37 52 38 42 14
Q1 - 11 13 9 13 9 11 10
Q2 - 10 7 9 7 8 5 2
Q3 - 7 12 7 12 7 8 1
Q4 - 13 17 12 20 14 18 1
We also calculate empirical standard errors
for the adjusted-?s. Estimating the appropriate
parameters through the use of a system of
equations does this.
5
The method uses the
ordinary least squares approach of minimising
the sum of squared residuals while adjusting
for any cross equation restrictions on the
parameters of the system. These system
equations result from re-arranging the separate
estimating equations allowing the system to
estimate the adjusted-? along with its standard
error.
Once the appropriate ?-adjustments are
made summary statistics are calculated. Table 2
shows the mean and standard deviations of the
?-estimates and the estimated standard errors of
those estimates. The data are broken up by ?-
estimates and size. It is worth noting that the
adjusted ?-estimates tend to be larger than the
OLS estimates. This is consistent with the
notion that thin trading leads to an under-
estimate of the “true” ?. The other thing worth
noting is that the OLS standard errors tend to
be smaller than the adjusted ?-estimate
standard errors. We also observe a “u-shaped”
pattern in the ?-estimates with Q2 and Q3
5 We thank Robert Faff for suggesting this approach. The
exact derivation is available on request.
estimates being lower than Q1 and Q4. Finally
the table shows that the standard error declines
as firm size increases. To the extent that firm
size and trading are positively correlated this
implies that thin trading affects not only the
OLS ?-estimate but also the precision of the
estimated correction.
We compare each estimate to zero and then to
the OLS - ? in Table 3. Panel A of Table 3
shows the number of ?-estimates that are
statistically significantly different from zero.
To the extent that any ?-estimate is empirically
useful it must pass this first hurdle. A ?-
estimate equal to zero indicates that a known
risky asset has no risk. This, we believe, is a
very easy test. Unfortunately, many of the ?-
estimates fail this simple test. There is a clear
size effect. For those firms with smaller size, it
is more likely that the ?-estimate will not be
statistically significantly different from zero. It
is only for firms in Q4 that a high proportion of
firms have non-zero ?-estimates. The large
standard errors for the adjusted ?-estimates
becomes problematic. Except for the trade-to-
trade ?-estimates the number of non-zero
adjusted ?-estimates is less than the OLS ?-
estimates. In essence the standard error grows
faster than the “adjusted” value of the ?-
estimate. In panel B we compare the adjusted
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 2 (2005)
116
?-estimates to the OLS - ?. Here we establish
how many adjusted ?-estimates are
significantly different from the OLS -
?-estimates. Overall the numbers are small. For
the smallest stocks (most likely to be plagued
by thin trading) there are, at most, 13 stocks
with a significantly different ?-estimate. For all
stocks there are, at most, 52 significantly
different ?-estimates. What is surprising is that
there are more significant adjustments for
stocks in Q4 – we would have expected less for
that group.
Finally we investigate the ?-adjustments in a
regression framework. Table 4 contains the
results. The percentage change between each
adjusted-? and the OLS-? is used as the
dependent variable. Independent variables are
the average size of the firm and the extent of
thin trading (measured by the proportion of
Zero Trades). We also include the interaction
of these two variables. We would expect, if the
?-adjustments are adding any value, that the
smaller the firm and/or the greater the
proportion of no trades the greater the
percentage change in ? there should be. We are
assuming that the effect of thin trading is to
downwardly bias the estimated beta. Looking
at Table 4 we see that the adjusted-R
2
statistics
are very low. The ?-adjustments are smaller the
larger the firm. This is not unexpected. What is
surprising is that, in some instances, the higher
the proportion of Zero Trades the smaller the ?-
adjustment. This effect seems to impact on
those adjustments with two leads and lags. The
interaction effects are negative but never
statistically significant.
Table 4
Regression results for percentage change in beta
?SW-? is the percentage change between a Scholes-Williams beta and an OLS beta, ?D-? is the
percentage change between a Dimson beta and an OLS beta and ?FR-? is the percentage change
between a Fowler-Rorke beta and an OLS beta. White heteroskedasticity consistent p-values are
shown in parenthesis.
?D-?1 ?D-?2 ?FR-?1 ?FR-?2 ?SW-?1 ?SW-?2 ?TT-?
c 80.2429 174.9608 79.0781 176.3507 74.6277 155.0003 -33.4636
(0.0131) (0.0011) (0.0128) (0.0011) (0.0151) (0.0040) (0.8039)
ln(Size) -10.9999 -22.2966 -10.8480 -22.4849 -10.3570 -19.3111 -3.4232
(0.0183) (0.0035) (0.0179) (0.0035) (0.0195) (0.0124) (0.8048)
Proportion Zero Trade -0.3512 -2.2867 -0.3524 -2.2868 -0.3029 -1.6701 4.9826
(0.6023) (0.0517) (0.5947) (0.0533) (0.6358) (0.1455) (0.3310)
Adj-R
2
0.0039 0.0240 0.0040 0.0236 0.0040 0.0127 0.0102
?D-?1 ?D-?2 ?FR-?1 ?FR-?2 ?SW-?1 ?SW-?2 ?TT-?
C 70.6714 179.5079 69.4913 180.7538 65.8547 166.1418 -171.1879
(0.0202) (0.0003) (0.0199) (0.0003) (0.0226) (0.0005) (0.5524)
ln(Size) -8.7889 -23.3470 -8.6334 -23.5020 -8.3304 -21.8849 28.3913
(0.0312) (0.0005) (0.0309) (0.0005) (0.0314) (0.0005) (0.5366)
Proportion Zero Trade 0.0973 -2.4998 0.0969 -2.4931 0.1082 -2.1923 11.4374
(0.9125) (0.0793) (0.9112) (0.0812) (0.8976) (0.0753) (0.3908)
Size*Proportion Zero Trade -0.1473 0.0700 -0.1475 0.0678 -0.1350 0.1715 -2.1196
(0.4946) (0.8294) (0.4859) (0.8358) (0.5104) (0.5810) (0.4576)
Adj-R
2
0.0030 0.0221 0.0031 0.0217 0.0030 0.0114 0.0158
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The Impact of Thin Trading Adjustments on Australian Beta Estimates
117
3. Conclusion
In this paper we have investigated the most
common ?-adjustments made to correct for
problems of thin trading. Our initial expectation
was that thin trading would have a downward
bias on the OLS-?. The thin trading
adjustments would lead to an increase in the
size of the ?-estimate. In addition, the change
in ?-estimate would be related to the extent of
thin trading. Our results, however, indicate that
these adjustments do not operate, as our initial
expectations would indicate. First, many of the
adjustments do not add any statistical value.
Second, these adjustments appear to be
unrelated to thin trading. As best we can
determine, the adjusted ?-estimates have a
relationship with firm size. Researchers are left
in quandary. Complicated adjustments do not
appear to add much to the OLS ?-estimate, yet
that estimate is known to be problematic in the
presence of thin trading. Consideration such as
this may explain the ad hoc approach to ?-
adjustments.
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