Library Class 09/10
Meets in
Doe Library
Room 105
Honor’s Thesis Elements
1. Why is this interesting?
2. Economic Theory (model)
3. Can one refute the model (theory?)—
testable implications
4. Hypothesis
5. Test
6. Results
7. Conclusion
Efficient Markets
September 2010
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Axis Title
Can You Predict Stock Prices?
spindx
1600
1400
1200
1000
800
600
spindx
Linear (spindx)
400
200
0
Why is this interesting?
• An empirical puzzle led to Efficient
Markets theory
• Empirical “fact”
• Stock prices look like random walks
• Why?
Malkiel’s Definition
• A capital market is said to be efficient if it fully
and correctly reflects all relevant information in
determining security prices. Formally, the market
is said to be efficient with respect to some
information set…if security prices would be
unaffected by revealing that information to all
participants. Moreover, efficiency with respect to
an information set….implies that it is impossible
to make economic profits by trading on the basis
of [that information set].
Great (Bert) Malkiel, How do
you test it?
• Here’s Eugene Fama’s famous method
• And a good model for you to follow for a
thesis
Can one test the Efficient
Market Hypothesis?
• Here’s Fama’s formalization (which is very
good)
• All economics comes down to demand,
supply, and equilibrium conditions
• For a stock the supply is especially simple
• SUPPLY---fixed
• DEMAND---is more complicated
– demand slopes down in the price plane
– but, what determines the intercept?
Intercept & Equilibrium
• Intercept: Here’s a simple robust trading
rule for an individual
– if the discounted Expected Asset Payoff next
period is greater than the current price, then
buy the asset, i.e., in symbols
λ E ( St +1 + dt +1 ) f St ,
−1
– then buy. If the discounted expected payoff is
less, then sell.
• Equilibrium: λ
−1
E ( S t +1 + d
t +1
) = St
Let’s Make it Operational: theorists
get your hands dirty
• Expectations aren’t observable
– Assumption 1: The Expectation is the
mathematical expectation (Rat-X), then the
realization
St +1 + dt +1 ≡ E ( St +1 + dt +1 ) + vt +1
– equals the expectation plus a mean zero error
independent of information at time t
• Slick—yes very slick, not sick
Now econometricians, what are
you going to do about the
unobservable discount factor?
• What’s the discount factor? The discount
factor is the risk free factor plus a risk
premium. What’s the risk free factor?
What’s the risk premium? Since we don’t
have a good answer, choose the simplest
one—the discount factor is constant!
– not slick, down and dirty
– but, maybe a good approximation
Now we have a testable model
• In observable terms—the payoff next
period equals the compounded price plus
a mean zero independent error.
E ( St +1 + dt +1 ) + vt +1 ≡ St +1 + dt +1 = λ St + vt +1
• This is a random walk with drift
• For econometric reasons let’s normalize
this equation,
Normalization to an
econometrically Friendly Model
• Transform payoffs into returns which are
stationary
St +1 − St + dt +1 St
vt +1
Rt +1 ≡
+ =λ+
St
St
St
vt +1
rt +1 ≡ Rt +1 − 1 = λ − 1 +
≡ r + et +1
St
Properties of Constant Expected
Return and Random Walk Model
• Stock Payoffs (price plus dividend) follow
a random walk with drift
– stock prices are unpredictable except for drift
• Expected returns are constant
– returns are unpredictable except for the
constant
• Model satisfies definition of an efficient
market and is testable
An Alternative: Mean-Reversion
rt +1 = r − α (rt − r ) + ut +1 ; rt
0<α < 1, u ~ N (0, σ )
2
Properties of Mean Reverting Walk
• Expected returns are predictable and not
constant=>expected excess returns > r
• Mean-reverting model does not satisfy
simple definition of an efficient market
Test Constant Expected Returns
• If one can predict
future returns, then
the model and (this
form of the) efficient
markets hypothesis
are rejected
• Here X is anything
known at time t
rt +1 = a + b ' X t + ut +1
Project: Test the Constant
Returns Model
• See the assignment on the class webpage
• If it rejects (an a few will), then (not this
problem set) an honors thesis should ask:
– Is this statistical (you reject 5% of the time
when the null is true)
– Is the model wrong or is the theory wrong
– Is it economically significant? How much
money can one make betting against the
random walk model?
We’ll test Weak-form Efficiency
• Information Sets
• Weak-form Efficiency: The information
set includes only the history of prices or
returns.
• Semistrong-form Efficiency: The
information set includes all publicly
available information.
• Strong-form Efficiency: The information
set includes all (public and private)
information.
Surprisingly some resist testing
Warning: Statistical Tests require
Formalization
• Usually that means a Mathematical model
that
– Captures essential characteristics
– Necessarily simplifies complex reality
• If you reject are you rejecting the
economic hypothesis, or the simplified
model?
Shiller: Theory is Wrong
• Rejects Constant Expected Returns Model
• Interprets the rejection as a rejection of
• The Efficient Markets Hypothesis
Shiller’s picture
Shiller Believes the Theory is
wrong
• He believes that Assumption 1, Rat-X, is
wrong and expectations are based on
psychology not statistics
• Alternative
– Assumption 2, constant discount factor, is
wrong
• In either case the model is wrong, but
under the alternative the theory might be
correct
Theory Correct—test wrong:
Malkiel
• Mutual Fund Performance
• Model? Professional managers should
beat amateurs
– this is really clever—if there is predictability,
then the pros should make excess returns
over a buy and hold strategy
Malkiel’s Results
What to take away from today
• Model for a thesis
1. Why is this interesting?
2. Economic Theory (model)
3. Can one refute the model (theory?)—
4. Test hypothesis
5. Results
• Resisting testing can restrict your
degrees of freedom
Next Week
• Library Class: Doe 105 (be on time!)
• Email me a citation and 100 word
summary of the paper, or news letter, that
you want to present. Presentations begin
the following week.
– here’s an example of how to find papers
– http://www.frbsf.org/economics/