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Efficiently ARMA-GARCH Estimated Trading Volume
Characteristics in Thinly Traded Markets
by
Per Bjarte Solibakke Department of Business
Administration and Economics, Molde University
College P.O.Box 308, N-6401 Molde,
Norway. e-mail per.b.solibakke_at_himolde.no
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Motivation

• A Changing Volatility Specifications of Asset
  Series in Illiquid Markets
• report misspecifications (Solibakke, 2001)
• Volatility Studies suggest a stochastic and
  constant volatility
• irrespective of trading/non-trading in an open
  market (Solibakke, 2000c)
• A Stochastic Volatility Specifications of Asset
  Series in Illiquid Markets
• report success (Solibakke, 2000d)
• Campbell et al. (1997) interpret non-trading as
  temporal aggregation of
• asset returns implying a need for a smoother
  non-trading modelling

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Investigation Approach

• Classical (discrete time) ARMA-GARCH
  specification with Diagnostics
• We hypothesise Temporal Aggregation of Asset
  Returns Count the number
• of non-trading days between the observed return
  series
• Apply a Continuous Time ARMA-GARCH lag
  specification where we
• explicitly account for the number of
  non-trading days for individual assets
• Apply it for both continuously and thinly traded
  asset series

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Literature Review

• ARMA-GARCH estimation
• Engle, 1982 Bollerslev, 1986, 1987, 1988
• Glosten et al., 1993, Drost and Werker, 1993,
  1996
• Bayes Information Criterion Schwartz, 1978
• Data Dependence Test Statistics
• Q-test Ljung Box, 1976
• ARCH test Engle, 1982
• RESET-test Ramsey, 1969
• BDS-test Brock et al., 1988, 1991
• Scheinkman, 1990

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Classical ARMA-GARCH lag specification
yt (1) ut (2) et ? N(0,
ht ) og D(0, ht,w) (3) ht (4)
Applying the BFGS / BHHH algorithm for iterative
optimisation.
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Bayes Information Criterion (Schwarz, 1978) for
lag specification in both ARMA and GARCH
The preferred model in the majority of
series An ARMA (1,0) GARCH (1,1) lag
specification
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Data and Adjustment Procedures

• Oslo Børs Informasjon/DNB Markets
• Daily Closing prices for the time period
  01.01.1984 to 01.01.1995
• Calculated as ln(St/St-1)
• Adjustment procedure for systematic day of the
  week, weekend,
• (sub)-month and trends
• Regress return on exogenous variables (keep
  residuals)
• Regress ln(res2) on same exogenous variables
  (keep residuals)
• Retain the mean and volatility from original
  series applying
• the equation R a b Radj, where a and b is
  determined
• by the Goal seek tool in Excel 2000.

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Table 1. Portfolio characteristics for the
Norwegian Equity Market
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Table 2. Summary statistic for adjusted daily
returns
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Table 4. ARMA (p,q) coefficients for six return
series. We estimate the model Ri,t ai
fi..Ri,t-1 qi,1 . ei,t-1 qi,2 . ei,t-2
ei,t, where i is four asset series and two index
series. Rit is the return series. ai is a
constant parameter, fi is the auto-regressive
parameter and qi,1 and qi,2 is the moving average
parameters. ei is model residuals.
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Table 5. Summary characteristics from an ARMA
(p,q) specification
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Table 7. An ARMA-GARCH-in-Mean specification for
Portfolio returns
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Table 8. BDS and ARCH test statistics for i.i.d.
residuals
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Table 9. Simple and Joint bias test for model
misspecification
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Table 10. Number of days for half of a shock to
have dissipated
Table 11. (Un-)Conditional Volatility
Characteristics
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Empirical Findings (1)

• Conditional Mean
• A positive and significant drift seems to
  exclusive for continuously traded assets
• Autocorrelation is found for all series
• Positive serial correlation for continuously
  traded assets
• Negative serial correlation for thinly traded
  assets
• Predictability
• Cross-autocorrelation exists from continuously
  traded assets to more thinly traded
• assets.
• The continuously traded assets seem to lead the
  market
• The in-Mean coefficient is insignificant for all
  series. The volatility does not indicate
• mean returns.

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Empirical Findings (2)

• Conditional Volatility
• Conditional Heteroscedasticity and Volatility
  Clustering is present.
• Volatility serial correlation is strongest for
  the thinnest traded assets.
• The persistence is strongest for the thinnest
  traded assets
• The weight to the long-run average volatility
  seem highest for the most frequently
• traded assets.
• Asymmetric Volatility is present for all series
  except the thinnest traded assets

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Empirical Findings (3)

• Specification tests
• The thinnest traded series report model
  misspecification
• gt non-linear dependence in asset series because
  of non-synchronous trading
• Bias-tests suggest that especially bad news are
  not very well predicted by the
• asymmetric ARMA-GARCH-GJR model.
• Continuously traded assets may apply the term
  structure form GARCH-models in
• the Norwegian market

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Summaries and Conclusions

• BIC-preferred ARMA-GARCH specifications seem to
  model the Norwegian market well. However, the
  most thinly traded assets show model
  misspecification.

• Illiquidity induces more complex changing
  volatility models
• Option pricing implications for all assets

1. Replace s with a GARCH process over the
option's life
2. The Tracking Portfolio may therefore not show
a perfect track

• We have to embed the security in a model of
  economic equilibrium,
• with specific assumptions about investors'
  preferences and their
• investment opportunity set (non-spanning)

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Summaries and Conclusions (2)

• Non-syncronous trading seem to imply an extra
  challenge for model
• specifications
• Temporal aggregation or Stochastic Volatility
  may be applied to model
• non-synchronous trading in the Norwegian
  market.