1 An Empirical Analysis of High Frequency Intraday Stock Returns: Evidence from Dow-Jones and NASDAQ Indices Thomas C. Chiang, 1 Hai-Chin Yu, 2 and Ming-Chya Wu 3 1 Department of Finance, Drexel University, Philadelphia, PA 19104, U.S.A. 2 Department of International Business, Chung-Yuan University, Chungli, 32023, Taiwan 3 Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan Abstract This paper presents empirical evidence on the properties of high frequency data of stock returns. We employ both time series and panel data from the Dow Jones Industrial Average (DJIA) and the NASDAQ 10-minute intraday intervals from Aug. 1, 1997, to Dec. 31, 2003. Our findings indicate that the statistical properties are highly influenced by the opening returns that contain overnight and non-regular information. The stylized fact of high opening returns tends to generate significant autocorrelation. While examining the AR(1)-GARCH(1,1) pattern across time and frequency, we find consistent negative AR(1) of 10-minute and 30-minute frequencies for the DJIA, positive AR(1) for the NASDAQ stock returns, and no obvious pattern beyond the 30-minute return series. By examining the dynamic conditional correlation coefficients between the DJIA and the NASDAQ over time, we find that the correlations are positive and fluctuate mainly in the range of 0.6 to 0.8. The variance of the correlation coefficients has been declining and appears to be stable for the post-2001 period. By checking the conditions for a stable Lévy distribution, both the DJIA and the NASDAQ can converge to their respective equilibriums after shocks without relying on external regulation or intervention. (Version: September 30, 2005) JEL: C22 G10 G11 Keywords: High Frequency; Probability Distribution; Financial Markets; Dynamic Conditional Correlation; Panel Data. E-mail address: Thomas C. Chiang ([email protected]) Hai-Chin Yu ([email protected]) Ming-Chya Wu ([email protected]) Corresponding author: Thomas C. Chiang Department of Finance, Drexel University, Philadelphia, PA 19104, U.S.A E-mail: [email protected]TEL: +1-215-895-1745 FAX: +1-609-265-0141
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An Empirical Analysis of High Frequency Intraday Stock Returns: Evidence from Dow-Jones and NASDAQ Indices
Thomas C. Chiang,1 Hai-Chin Yu,2 and Ming-Chya Wu3
1Department of Finance, Drexel University, Philadelphia, PA 19104, U.S.A. 2Department of International Business, Chung-Yuan University, Chungli, 32023, Taiwan
3Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
Abstract This paper presents empirical evidence on the properties of high frequency data of stock returns. We
employ both time series and panel data from the Dow Jones Industrial Average (DJIA) and the NASDAQ
10-minute intraday intervals from Aug. 1, 1997, to Dec. 31, 2003. Our findings indicate that the statistical
properties are highly influenced by the opening returns that contain overnight and non-regular information.
The stylized fact of high opening returns tends to generate significant autocorrelation. While examining the
AR(1)-GARCH(1,1) pattern across time and frequency, we find consistent negative AR(1) of 10-minute
and 30-minute frequencies for the DJIA, positive AR(1) for the NASDAQ stock returns, and no obvious
pattern beyond the 30-minute return series. By examining the dynamic conditional correlation coefficients
between the DJIA and the NASDAQ over time, we find that the correlations are positive and fluctuate
mainly in the range of 0.6 to 0.8. The variance of the correlation coefficients has been declining and
appears to be stable for the post-2001 period. By checking the conditions for a stable Lévy distribution,
both the DJIA and the NASDAQ can converge to their respective equilibriums after shocks without relying
on external regulation or intervention.
(Version: September 30, 2005)
JEL: C22 G10 G11
Keywords: High Frequency; Probability Distribution; Financial Markets; Dynamic Conditional Correlation; Panel Data. E-mail address: Thomas C. Chiang ([email protected])
In this paper, we investigate the statistical properties of high frequency data on stock
returns. The analysis applies to both the Dow Jones Industrial Average (DJIA) and the
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NASDAQ indices. The evidence indicates that the empirical regularities are highly
influenced by opening returns that contain overnight and other information of irregular
length. The statistics show that both the NASDAQ and the DJIA have excessively high
returns during opening time, although the NASDAQ, on average, has a higher return than
the DJIA. The higher returns in the NASDAQ are matched by higher volatilities over the
entire trading intervals over the business day. The evidence also shows that the
high-frequency-return variances for a given scale produce a smile curve and the curvature
is seen to be increasing with the time scale.
By examining the AR(1) pattern across time and frequencies, we often find that the
coefficients are somehow affected by data that include the opening interval. However,
when we fit the intraday stock returns into an AR(1)-GARCH(1,1) model, we find
consistent results for both 10-minute and 30-minure return horizons. The evidence shows
that the DJIA returns are negatively autocorrelated, while the NASDAQ returns are
positively autocorrelated, meaning that investors in the DJIA behave as positive feedback
traders, whereas investors in the NASDAQ are a negative feedback group.
By examining the dynamic correlation coefficients between the DJIA and the
NASDAQ returns over time, the return correlations are positive and fluctuate mainly in
the range of 0.6 to 0.8. The statistics show that the correlation coefficients are time
varying, reflecting some sort of dynamic portfolio allocations among different financial
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assets. By inspecting the time series path of conditional correlation coefficients, we find
that the variations of the coefficients are declining and appear to be more stable for the
post-2001 period. This suggests that both markets are sensitive to or driven by some
common factors, such as systematic risk, macroeconomic announcements, Fed policy, or
investor psychology. This also implies that the benefit of diversifying by holding a
combination of DJIA and NASDAQ stocks has declined in recent years.
By checking for conditions of a stable Lévy distribution, we find that both the DJIA
and the NASDAQ can converge to a stable equilibrium after shocks without relying on an
external regulation or intervention. This implies that both markets appear to be stable and
mature and are governed by a self-stabilizing mechanism, especially the DJIA market.
Table 1 Summary statistics across different frequencies of return and volatility series This table summarizes the return ( ,tRτ ) statistics across different frequencies of intraday, interday, and interweek for both the DJIA and the NASDAQ. The sample period covers Aug. 1, 1997, to Dec. 31, 2003, for a total of 60,884 10-minute observations.
DJIA NASDAQ
τ Mean Med. Max. Min. Std. Dev. Skew. Kurt. J.-B. Prob. Mean Med. Max. Min. Std. Dev. Skew. Kurt. J.-B. Prob. Observ.
Table 2 Time series estimates of AR(1) for the DJIA and the NASDAQ across different time frequencies Total sample adjusted is 6 out of 60,884, which included 60,879 observations for the 10-minute series. DJIA NASDAQ
Notes: There is only one observation in 390-min interval per day; hence, there is no observation after
excluding the opening interval in the 390-min and daily frequency series. The AR(1)-GARCH(1,1) modelis
, , 1 ,
2 2 2, , 1 , 1
t t t
t t t
R Rτ τ τ τ τ
τ τ τ τ τ τ
δ φ ε
σ ω α ε β σ−
− −
= + +
= + +
where τ represents different frequencies.
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Table 5 Time series analysis of dynamic conditional correlation coefficients at various time scales
________________________________________________________________________ Coefficient t,τρ∆ (1-day) t,τρ∆ (30-min) t,τρ∆ (10-min) ___________________ ___________________ ___________________ before crisis after crisis before crisis after crisis before crisis after crisis ________________________________________________________________________ Panel A: Mean and standard deviation of t,τρ
b. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively. The
numbers in parentheses are standard errors.
c. t,τρ∆ (10-min) denotes change in conditional correlation coefficient for the (10-minute) series,
etc.
d. LB(10) is the Ljung-Box statistics test for autocorrelation up to the 10th lag.
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Figure 1. Time series plots of 10-minute frequencies of DJIA and NASDAQ indices
(Aug. 1, 1997 - Dec. 31, 2003).
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Figure 2. Time series plots of stock returns for intraday and interday DJIA data.
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Figure 3 (a) Panel data for intraday returns of the DJIA and the NASDAQ, and (b)
panel data for intraday volatility of the DJIA and the NASDAQ without
opening time.
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Figure 4 (a) DJIA 10-minute panel data for intraday volatility, (b) NASDAQ 10-minute
panel data for intraday volatility, (c) DJIA 30-minute panel data for intraday volatility, and (d) NASDAQ 30-minute panel data for intraday volatility.
Figure 5 Dynamic conditional correlation between the DJIA and the NASDAQ: (a) Daily
dynamic conditional correlation between the DJIA and NASDAQ, (b) 10-minute dynamic conditional correlation between the DJIA and the NASDAQ, (c) 30-minute dynamic conditional correlation between the DJIA and the NASDAQ.
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Figure 6 Probability distributions of return changes of (a) the DJIA, and (b) the
NASDAQ for intraday data with time sampling intervals of multiples of 10
minutes.
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Figure 7 (a) Probability of return variation ( ), 0tP Zτ = as a function of the time
sampling intervals τ. The slope of the best-fit straight line is 0.58 0.01− ±for DJIA, and 0.59 0.01− ± for NASDAQ. Scaled plot of the probability distributions with (b) 1.73α = for theDJIA, and (c) 1.70α = for the NASDAQ.
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Endnotes 1 Alternatively, Engle and Russell (1998) have developed the autoregressive conditional duration (ACD)
model to investigate high-frequency stock market data. In the ACD model the expected duration between
trades depends on past durations. Here we follow Andersen and Bollerslev’s approach to investigate the
time series properties of two high frequency stock returns. However, unlike Andersen and Bollerslev
(1997) and Mian and Adam (2001), we do not omit the close-to-open returns. Rather, we keep them in the
data to conduct sensitivity analyses. After eliminating the omitted days for which all of the 10-minute
values of the index were not available, a total of 1,543 trading days with 60,177 observations of 10-minute
index values was obtained. 2 Examination of their dynamic correlations between two series can be found in section 5. 3 We do not plot the NASDAQ to save space. 4 It may be seen that most of the interday intervals show left skewness, especially the DJIA. As noted by
Andersen and Bollerslev (1997), the negative skewness may be interpreted as evidence of the “leverage”
and /or volatility feedback effects discussed by Black (1976), Campbell and Hentschel (1992) and
Bekaert and Wu (2000). 5 The statistics show that daily returns have a skewness of -0.09 and kurtosis of 5.87 for the DJIA and
skewness of 0.197 and kurtosis of 5.56 for the NASDAQ. 6 Evidence of the U-shape pattern of intraday volatility can be found in Wood et al. (1985), Andersen and
Bollerslev (1997), and Ito et al. (1998), among others. The theoretical models on the U-shaped pattern
appear in Foster and Viswanathan (1990) and Slezak (1994). 7 The smile curve of volatility reflects the fact that the highest point of return volatility occurs around the
opening time 9:30 interval (0.006216804), followed by the 10:00, 9:40, and 9:50 intervals, respectively,
and hits the lowest point around the 12:20 interval (0.001240272), followed by the second lowest at the
13:00 interval and third lowest at the 12:10 interval, respectively. 8 By including the opening intervals, there are 39 cross-sectional data to be covered in the total panel
(unbalanced) with 62,906 observations. However, without including the opening series at 9:30, there are 38
cross-sections with 61,294 total panel (balanced) observations. 9 The parameters will be estimated by the log-maximum likelihood method. The density function in
Nelson (1991) is given by
/ 21/ 2
,, , 3 / 2
,,
[ (3 / )] (3 / )( , , ) exp(1/ )2[ (1/ )]
tt t
tt
fν ν
ττ τ
ττ
εν ν νµ σ νσ νν σ−
Γ Γ = − ΓΓ
where ( )Γ ⋅ is the gamma function and ν is a scale parameter or degree of freedom to be estimated. For
ν =2, the GED yields the normal distribution, while for ν =1 it yields the Laplace or double-exponential
distribution. Given initial values of ,tτε and 2,tτσ , the parameter vector ( , , , , , )τ τ τ τ τδ φ ω α β νΘ ≡ can be
estimated by log-maximum likelihood method (log-MLE) over the sample period. The log-maximum
likelihood function can be expressed as
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, ,1
( ) log ( , , )T
t tt
L f τ τµ σ ν=
Θ = ∑
where ,tτµ is the conditional mean and ,tτσ is the conditional standard deviation. Since the
log-likelihood function is non-linear, the numerical procedure is used to derive estimates of the parameter
vector. 10 As argued by Sentana and Wadhwani (1992) and expounded by Antoniou et al. (2005), positive
feedback traders buy stocks after prices rise and sell stocks after prices fall. Shiller (1989) found a main
reason that prompted investors to sell their stocks in October 1987 was that stock prices had fallen, thus
inducing a fear of contagion in other investors. In contrast, the negative feedback traders sell stocks after
prices increase and buy stocks after prices decline. Shiller argued that feedback models suggest that price is
determined in part by its own lagged values, increases in price tending at times to foster further increases.
However, as argued by Shiller, there is little, even a negative, serial correlation of price changes (Shiller,
1989, p. 375). 11 A variety of papers have documented the fact that correlations across major stock markets change over
time. King, Sentana, and Wadhwani (1994) find the covariances of stock returns change over time.
Kaplanis (1988) compares the matrices of returns across 10 markets and finally rejects the constant
correlations hypothesis. Koch and Koch (1991) use Chow tests to examine stock returns in 1972, 1980, and
1987 and find higher correlations in more recent years. Some evidence shows that correlations tend to
increase during unstable periods (Forbes and Rigobon, 2002). Longin and Solnik (1995) find that
correlations between the major stock markets rise in periods of high volatility. Karolyi and Stulz (1996)
report that covariances are high while returns on the national indices are high and when “markets move a
lot” All these papers are based mainly on daily data. Very few attempts have been devoted to analyzing
dynamic conditional correlations in high frequency data. Moreover, we are interested in exploring the
results from varying different scales of data in the context. 12 An alternative definition of the correlation coefficient (more precisely, cross-correlation coefficient, see Laloux et al., 1999; Plerou et al., 1999) denoted by , ,ij tCτ is defined as the statistical overlap of the
fluctuations ( ), , , , , ,i t i t i tR R E Rτ τ τδ = − between the two stocks i and j , that is, ( ), , , ,
, ,, , , ,
i t j tij t
i t j t
E R RC τ ττ
τ τ
δ δ
σ σ= ,
where ,tRτ is the logarithmic return, and ( )22, , , ,i t i tE Rτ τσ δ = . The average ( )E ⋅ is over a time period
T . We are interested in exploring whether different scales of data types would cause different results in
dynamic cross correlations. Based on this equation, we can perform two analyses: one with T fixed to
one day, and the other with T fixed to a certain number of events. Using two ways (with and without
deleting opening intervals) to investigate the DCC, we found that there is no difference between the two,
and we do not report it here. 13 In our case, we fixed 38T = after removing the 09:30 data point. The discontinuation of the
correlation coefficients in the figures is due to missing data for the sample period from Feburary 9, 2002, to
May 9, 2002.
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14 In their analysis of intraday foreign exchange rate and S&P 500 futures, Andersen and Bollerslev also
find that the intraday returns display an MA(1)-GARCH pattern.