Run length and the Predictability of Stock Price Reversals Juan Yao Graham Partington Max Stevenson...
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Transcript of Run length and the Predictability of Stock Price Reversals Juan Yao Graham Partington Max Stevenson...
Run length and the Predictability of Stock
Price Reversals
Juan YaoGraham Partington
Max Stevenson
Finance Discipline, University of Sydney
2
Structure of the paper
Background and motivation Empirical design Data In-sample analysis Out-of-sample evaluation Conclusion
3
Motivation of the study
Evidence of predictability 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.
4
Motivation of the study
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 semi-parametric 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)
5
Some definitions
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
6
Cox regression model
We define the hazard of price state transition to be:
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.
0( ) lim ( , ) /ij ij
sh t p t t s s
7
Cox Proportional Hazards Model
The hazard function for each individual run will be:
The log-likelihood:
0( ) [ ( )]
Xh t h t e
0ln[ ( )] ln[ ( )]h t h t X
1 1 ( )
ln ( ) ln[ ]i
k kXj
i
i i j R t
L X e
8
Cox Proportional Hazards Model
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:
S(t) is the probability that the current run will persist beyond the time horizon t.
0
( ) exp[ ( ) ]t
S t h u du
9
Empirical designTwo models are estimated: a transition from an up-state to a down-state
and a transition from a down-state to an up-state
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
10
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.
11
Forecast evaluation
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:
At individual level: Brier score
1
ˆ( ) ( )q
t i
i
E n S t
12
Brier score
Assessing probabilistic forecasts:
When the price has reversed an = 1, and when it has not an = 0.
A lower Brier score implies better forecasting power
2
1
[ ( ) ] / ,N
n n
n
B p a N
13
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
14
Table 1. Summary Statistics for Run Length.
Run Type Count Min Length
MaxLength
Mean Std. Dev.
Skew. Kurt.
Panel A: Daily Price Changes
Positive (Up state)
1092 1 d 14 d 2.51 1.896 1.928(0.074)
4.944(0.148)
Negative(Down state)
1093 1 d 13 d 2.13 1.537 2.031(0.074)
5.420(0.148)
Panel B: Monthly Price Changes
Positive (Up state)
65 1 m 10 m 2.46 1.846 2.299(0.297)
6.208(0.586)
Negative(Down state)
65 1 m 7 m 1.89 1.301 1.833(0.297)
3.508(0.586)
15
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 32 66
02/1988 08/1989 5* 18
12/1990 10/1991 16 10
10/1992 01/1994 12 15
01/1995 09/1997 12 32
08/1998 06/2001** 11 34
Average Length 15.4 32.9
16
0500
1000150020002500300035004000
26/0
2/71
26/0
2/73
26/0
2/75
26/0
2/77
26/0
2/79
26/0
2/81
26/0
2/83
26/0
2/85
26/0
2/87
26/0
2/89
26/0
2/91
26/0
2/93
26/0
2/95
26/0
2/97
26/0
2/99
26/0
2/01
Date
Pri
ce
Figure 1. All Ordinary Price Index (monthly) Feb. 1971 – Dec. 2001.
17
Figure 2. Return of All Ordinary Price Index (monthly) Mar. 1971 – Dec. 2001.
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
31/0
3/7
1
31/0
3/7
3
31/0
3/7
5
31/0
3/7
7
31/0
3/7
9
31/0
3/8
1
31/0
3/8
3
31/0
3/8
5
31/0
3/8
7
31/0
3/8
9
31/0
3/9
1
31/0
3/9
3
31/0
3/9
5
31/0
3/9
7
31/0
3/9
9
31/0
3/0
1
Date
Retu
rn
18
Table 3. State transition models estimated from monthly data.
Step
number
Variable
entered
Coefficient Standard
error
Wald df Sig. Exponential
coefficient
-2Log
Likelihood
Chi-
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
19
Table 4. State transition models estimated from daily data.
Step
number
Variable
entered
Coefficient Standard
error
Wald df Sig. Exponential
coefficient
-2Log
Likelihood
Chi-
square
Sig.
Panel A: Transition from up to down state
Step5 CHANGES .175 .023 59.361 1 .000 1.191 13335.536 227.895 .000
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
Panel B: Transition from down to up state
Step5 LAG2 .007 .001 176.110 1 .000 1.007 13387.343 238.208 .000
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
20
Out-of-sample
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:
0ˆ ˆ( ) [ ( )]piS t S t
ˆ( )Xip e
21
ComparisonsTwo benchmarks
Naïve forecast: setting the survival probability for time t equal to the proportion of runs survived within sample
Random forecast: the probability of each independent state change is 0.5, the survival probability at t is (0.5)t
22
Figure 5. Comparison of the number of actual and expected runs of varying lengths (positive runs, daily data).
0
20
40
60
80
100
120
140
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Time
Num
ber o
f run
s Actual
Expect
Random
Naïve
23
Figure 6. Comparison of the number of runs of varying lengths (negative runs, daily data).
0
20
40
60
80
100
120
140
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Time
Num
ber o
f run
s Actual
Expect
Random
Naïve
24
Figure 7. Brier Scores for monthly negative runs.
BS for negative runs using monthly data
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5
Time
Bri
er s
core Prediction
Naïve
Random
25
Figure 8. Brier Scores for monthly positive runs.
BS for positive runs using monthly data
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5
Time
Bri
er s
core Prediction
Naïve
Random
26
Figure 9. Brier Scores for daily negative runs.
BS of negative runs using daily data
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5 6 7 8
Time
Bri
er s
core Prediction
Naïve
Random
27
Figure 10. Brier Scores for daily positive runs.
BS of positive runs using daily data
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1 2 3 4 5 6 7 8 9
Time
Bri
er
sco
re
Prediction
Naïve
Random
28
Figure 11. Probability forecasts of positive runs.
Positve runs daily data
0
0.2
0.4
0.6
0.8
1
1 3 5 7 9 11 13 15
Time
Pro
bab
ilit
y
Naïve
Random
29
Figure 12. Probability forecasts of negative runs.
Negative runs daily data
0
0.2
0.4
0.6
0.8
1
1 3 5 7 9 11 13
Time
Pro
bab
ilit
y
Naïve
Random
30
Conclusions 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.
31
Conclusions
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.
32
Conclusions
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?
33
Future Research
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!