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!