Transcript A Test on Predictability of Stock Price Reversal

Run length and the
Predictability of Stock
Price Reversals
Juan Yao
Graham Partington
Max Stevenson
Finance Discipline, University of Sydney
Structure of the paper
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Background and motivation
Empirical design
Data
In-sample analysis
Out-of-sample evaluation
Conclusion
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Motivation of the study
Evidence of predictability
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McQueen and Thorley (1991,1994)
Low (high) returns follow runs of high (low) returns –
probability that a run ends declines with the length;
Maheu and McCurdy (2000)
Markov-switching model – probability that a run ends
depends on the length of the run in the markets;
Ohn, Taylor and Pagan (2002)
The turning point in a stock market cycle is not a purely
random event.
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Motivation of the study
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The function form of the occurrence of such events
is not known, and the baseline hazard can be given
many parametric shapes
Cox’s proportional hazard approach is a semiparametric techniques – doesn’t need to specify the
exact form of the distribution of event times
Successful forecasting of price reversal in property
market index by Partington and Stevenson (2001)
The technique seems to also work on consumer
sentiment index (a work is currently on going)
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Some definitions
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Events: reversal of price
Event time versus calendar time
Up-state and down-state:
Up-state: positive runs, when Pt – Pt-1>0
Down-state: negative runs, when Pt - Pt-1<0
State transition
Probability of transition
Not predicting a price reversal, but the probability of
a reversal
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Cox regression model
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We define the hazard of price state transition
to be:
hij (t )  lim pij (t , t  s) / s
s 0
where pij(t,t+s) is the probability that the
price in state i at time t will be in state j at
time t+s.
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Cox Proportional Hazards
Model
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The hazard function for each individual run
will be:
h(t )  [h 0(t )]e
X
ln[h(t )]  ln[h0(t )]  X 

The log-likelihood:
k
k
ln L(  )   Xi    ln[
i 1
i 1
e
Xj 
]
jRi ( t )
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Cox Proportional Hazards
Model
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The cumulative survival probability is defined
as: S(t) = P(T>t)
where T is time of the event
S(t) can be calculated from:
t
S (t )  exp[   h(u )du ]
0

S(t) is the probability that the current run will
persist beyond the time horizon t.
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Empirical design
Two models are estimated:
 a transition from an up-state to a down-state
and a transition from a down-state to an upstate
Covariates:
 lagged price changes up to 12 lags for
monthly data, 30 lags for daily data
 the number of state transitions in the previous
period
 a dummy variable to distinguish a bull and
bear market
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Identification of Bull and Bear
Market
Pagan and Sossounov (2003) criterion
 Identify the peaks and troughs over a window of
eight months
 The minimum lengths of bull and bear states are
four months.
 The complete cycle has minimum length of sixteen
months
 The minimum four months for a bull or bear state
can be disregarded if the stock price falls by 20% in
a single month. This enables the accommodation of
dramatic events such as October 1987.
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Forecast evaluation
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At aggregate level:
Sum the estimated probability of survival to
time t for each run in holdout sample to
obtain the expected number of runs survive
beyond t: q
E (nt )   Sˆi (t )
i 1
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At individual level:
Brier score
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Brier score
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Assessing probabilistic forecasts:
N
B  [  ( pn  an ) ] / N ,
2
n 1
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When the price has reversed an = 1, and
when it has not an = 0.
A lower Brier score implies better forecasting
power
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Data
Monthly All Ordinary Price Index
 Feb. 1971 – Dec. 2001
 holdout sample: last five years
Daily All Ordinary Price Index
 31st Dec. 1979 – 30th Jan. 2002
 holdout sample: last two years
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Table 1. Summary Statistics for Run Length.
Run Type
Count
Min
Max
Length Length
Mean
Std.
Dev.
Skew.
Kurt.
Panel A: Daily Price Changes
Positive
(Up state)
1092
1d
14 d
2.51
1.896
1.928
(0.074)
4.944
(0.148)
Negative
(Down state)
1093
1d
13 d
2.13
1.537
2.031
(0.074)
5.420
(0.148)
Panel B: Monthly Price Changes
Positive
(Up state)
65
1m
10 m
2.46
1.846
2.299
(0.297)
6.208
(0.586)
Negative
(Down state)
65
1m
7m
1.89
1.301
1.833
(0.297)
3.508
(0.586)
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Table 2. Bull and Bear Market Identification.
Trough
Peak
Bear
(months)
Bull
(months)
11/1971
01/1973
N.A.
14
09/1974
11/1980
20
74
03/1982
09/1987
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66
02/1988
08/1989
5*
18
12/1990
10/1991
16
10
10/1992
01/1994
12
15
01/1995
09/1997
12
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08/1998
06/2001**
11
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15.4
32.9
Average Length
15
26/02/01
26/02/99
26/02/97
26/02/95
26/02/93
26/02/91
26/02/89
26/02/87
26/02/85
26/02/83
26/02/81
26/02/79
26/02/77
26/02/75
26/02/73
26/02/71
Price
Figure 1. All Ordinary Price Index (monthly)
Feb. 1971 – Dec. 2001.
4000
3500
3000
2500
2000
1500
1000
500
0
Date
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Return
-0.05
-0.1
31/03/01
31/03/99
31/03/97
31/03/95
31/03/93
31/03/91
31/03/89
31/03/87
31/03/85
31/03/83
31/03/81
31/03/79
31/03/77
31/03/75
31/03/73
31/03/71
Figure 2. Return of All Ordinary Price Index (monthly)
Mar. 1971 – Dec. 2001.
0.1
0.05
0
-0.15
-0.2
-0.25
-0.3
Date
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Table 3. State transition models estimated from monthly data.
Step
Variable
number
entered
Coefficient
Standard
Wald
df
Sig.
error
Exponential
-2Log
Chi-
coefficient
Likelihood
square
Sig.
Panel A: Transition from up to down state
Step 1
CHANGES
.542
.171
10.044
1
.002
1.720
420.023
10.287
.001
Step 2
CHANGES
.549
.174
10.002
1
.002
1.732
417.258
14.179
.001
LAG2
-.002
.001
4.217
1
.040
.998
Panel B: Transition from down to up state
Step 1
LAG2
.009
.003
10.778
1
.001
1.009
422.706
10.716
.001
Step 2
LAG2
.009
.003
10.083
1
.001
1.009
417.971
15.859
.000
LAG3
.007
.003
4.797
1
.029
1.007
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Table 4. State transition models estimated from daily data.
Step
Variable
Coefficient Standard Wald
number entered
df Sig. Exponential -2Log
error
coefficient
Chi-
Sig.
Likelihood square
Panel A: Transition from up to down state
Step5
CHANGES .175
.023
59.361
1
.000 1.191
LAG1
-.002
.001
8.345
1
.004 .998
LAG2
-.005
.000
99.859
1
.000 .995
LAG3
-.002
.000
24.152
1
.000 .998
LAG14
.001
.000
6.504
1
.011 1.001
13335.536 227.895 .000
Panel B: Transition from down to up state
Step5
LAG2
.007
.001
176.110 1
.000 1.007
LAG3
.003
.001
37.092
1
.000 1.003
LAG4
.002
.001
9.354
1
.002 1.002
LAG5
.001
.001
5.596
1
.018 1.001
CHANGES .146
.022
44.089
1
.000 1.158
13387.343 238.208 .000
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Out-of-sample
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Monthly: 21 completed negative price runs and 21
complete positive runs.
Daily:121 negative price runs and 121 positive price
runs.
The out-of-sample survival functions are estimated
according to:
ˆi (t )  [ S
ˆ 0(t )] p
S
pe
( Xi ˆ )
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Comparisons
Two benchmarks
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Naïve forecast: setting the survival probability
for time t equal to the proportion of runs
survived within sample
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Random forecast: the probability of each
independent state change is 0.5, the survival
probability at t is (0.5)t
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Figure 5. Comparison of the number of actual and expected
runs of varying lengths (positive runs, daily data).
140
Number of runs
120
100
Actual
80
Expect
60
Random
40
Naïve
20
0
1 2
3 4 5
6 7
8 9 10 11 12 13 14
Time
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Figure 6. Comparison of the number of runs of
varying lengths (negative runs, daily data).
140
Number of runs
120
100
Actual
80
Expect
60
Random
40
Naïve
20
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Time
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Figure 7.
Brier Scores for monthly negative runs.
BS for negative runs using monthly data
Brier score
0.35
0.3
0.25
Prediction
0.2
Naïve
0.15
Random
0.1
0.05
0
1
2
3
4
5
Time
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Figure 8.
Brier Scores for monthly positive runs.
BS for positive runs using monthly data
Brier score
0.35
0.3
0.25
Prediction
0.2
Naïve
0.15
Random
0.1
0.05
0
1
2
3
4
5
Time
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Figure 9. Brier Scores for daily negative runs.
BS of negative runs using daily data
Brier score
0.35
0.3
0.25
Prediction
0.2
Naïve
0.15
Random
0.1
0.05
0
1
2
3
4
5
6
7
8
Time
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Figure 10. Brier Scores for daily positive runs.
BS of positive runs using daily data
Brier score
0.35
0.3
0.25
Prediction
0.2
Naïve
0.15
Random
0.1
0.05
0
1
2
3
4
5
6
7
8
9
Time
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Figure 11. Probability forecasts of positive runs.
Positve runs daily data
Probability
1
0.8
0.6
Naïve
0.4
Random
0.2
0
1
3
5
7
9
11
13
15
Time
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Figure 12. Probability forecasts of negative runs.
Negative runs daily data
Probability
1
0.8
0.6
Naïve
0.4
Random
0.2
0
1
3
5
7
9
11
13
Time
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Conclusions
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Lagged price changes and the previous
number of transitions are significant predictor
variables.
In an up-state, the lagged positive (negative)
changes decreases (increases) the possibility
of reversal; in a down-state, the lagged
positive (negative) changes increases
(decreases) the possibility of reversal.
State of the market, bull or bear is not
significant.
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Conclusions
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Predictive power exists at the aggregate
level.
For individual runs, the model forecasts less
accurate than naïve and random-walk
forecasts.
The random-walk and naïve forecasts are
almost identical.
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Conclusions
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Are the price changes random?
-No, price reversals are related to
information in previous prices,
specifically, the signs and magnitude
of lagged price changes as well as the
previous volatility.
Is the market efficient?
- Probably, model forecasts poorly in
out-of-sample, no profitable trading..
Or, it is a bad specification?
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Future Research
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Test for duration dependence
(whether the hazard function is a constant, or
the density is exponential)
Examine the runs of the individual stocks
(choice of stocks? Frequency of the data?)
Any suggestions are welcome!
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