Empirical Financial Economics
The Efficient Markets
Hypothesis - Generalized
Method of Moments
Random Walk Hypothesis
Random Walk hypothesis a special case of
EMH
E (rt  )  



1,
2,

2
2
2 2
E (rt  )      
Overidentification of model
Provides a test of model (variance ratio criterion)
Allows for estimation of parameters (GMM
paradigm)
Variance ratio tests
using sample quantities
 1
Var(rt  )
k
VR( ) 
 1  2 (1  )  ( k )
 Var(rt )

k 1
1 n
ˆ   (r 1  ˆ ) 2
n 1
The variance ratio
is asymptotically
n
1
2
Normal
ˆ ( )   (r   ˆ ) 2
n 1
2
ˆ ( ) 
VR
ˆ 2 ( )
Overlapping observations
Non-overlapping
observations
Overlapping observations
t
t+T
ln(pt)
ln(pt)
1 n
ˆ 
(r 1  ˆ ) 2

n  1 1


 unbiassed
 estimators
n
1
 1
ˆ 2 ( )   (r   ˆ ) 2 ; m  (n    1) 1   
m 
 n  
ˆ ( )  1) a N[0, 2 (2  1)(  1) ]
Variance ratio is asymptotically
n (VR
3
Normal
2
Lo, A. and A.C. MacKinley, 1988, Stock market prices do not follow random walks:Evidence from a simple
specification test Review of Financial Studies 1(1), 41-66.
Random walk model and GMM
rt 1  
  1t
rt 2 j 
j 2  j 2  2   2t
rt 2k 
k 2  k 2  2   3t
aggregate into moment conditions:
1
rt 1  

T
1
rt 2 j 

T
1
rt 2k 

T
1
 1t

T
1
j 2  j 2  2   2t
T
1
k 2  k 2  2   3t
T

and express as three observations of a nonlinear regression
model:
y  X   X  2  X 2  w
1
11
21
31
1
y2  X 12   X 22 2  X 32  2  w2
y3  X 13   X 23 2  X 33  2  w3
An Aside on Linear Least
Squares
y  X   u  uˆ  y  X ˆ
Min
uˆAuˆ
ˆ

uˆ
FOC :
Auˆ  0  X Auˆ  0
ˆ
Choose ˆ :
X A  y  X ˆ   0
1
1
ˆ
   X AX  X Ay     X AX  X Au
1
1
ˆ
Var    X AX  X A  Euu AX  X AX 
An Aside on Nonlinear Least
Squares
y  f ( X ,  )  u  uˆ  y  f ( X , ˆ )
 ( y  y0 )  D  ˆ   0 
Min
uˆAuˆ
ˆ

uˆ
FOC :
Auˆ  0  DAuˆ  0
ˆ

Choose ˆ : DA ( y  y0 )  D  ˆ   0    0


1
1
ˆ



   D AD  D A ( y  y0 )   0     D AD  DAu
1
1
ˆ
Var    DAD  DA  Euu AD  DAD 
Generalized method of moment
estimators
 ,
wAw A .
Choose
to minimize
is referred
to as the optimal weighting matrix, equal to
w
the inverse covariance
matrix of
Estimators are asymptotically Normal and efficient
Minimand is distributed as Chi-square with d.f.
number of overidentifying information
A
Methods of obtaining
1. SetA  I
(Ordinary Least Squares). Estimate

1
ˆ
model. A Set
(Generalized Least Squares).
Reestimate .

2. Use analytic methods to infer
GMM and the Efficient Market
Hypothesis
1 asset and 1 instrument:E
 r
j ,t 

 E[rj ,t  | xt , j ]  zt  0
… 1 equation and k unknowns:

0
… m equations and >k
unknowns:
E  rt   E[rt  | xt , ]  Z t
m assets and n instrument:
0

E  rt   E[rt  | xt , ]  zt
m assets and 1 instrument:


… mxn equations and >k
unknowns:
t
f t ( )  E  rt   E[rt  | xt , ]  Z t  ;
gT ( )  T 1  f t ( )
t 1
Hansen, L.P. and K.J. Singleton, 1982, Generalized instrumental variables estimation of
nonlinear rational expectations models Econometrica 50(5), 269-286.
Autocovariances and cross
autocovariances
xt
 xx (k )
 xy (k )   yx (k )
 xy (k )
 yy (k )
t-k
t
yt
t+k
Cross autocovariances are not
symmetrical!
Autocovariances are given by:
 xx (k )  E[( xt   x )( xt k   x )]  E[( xt   x )( xt k   x )]   xx (k )
 yy (k )  E[( yt   y )( yt k   y )]  E[( yt   y )( yt k   y )]   yy (k )
Cross autocovariances are given by:
 xy (k )  E[( xt   x )( yt k   y )]  E[( yt   y )( xt k   x )]   yx (k )
Cross autocovariances and the
weighting function
1 T 
 T  xt 
t 1


For w 
1 T 
 T  yt 
 t 1 
T T
 xt x


1  t 1  1
Cov( w)  2 T T
T 
   yt x
 t 1  1

 xt y 

t 1  1

T T

 yt y 

t 1  1

T
T
Assuming stationarity
 xx  0 

  xx 1
 xx  2 
Cov( x)  
 xx  3
 4
 xx  

and so
 xx 1  xx  2   xx  3  xx  4 
 xx  0   xx 1  xx  2   xx  3
 xx 1  xx  0   xx 1  xx  2 
 xx  2   xx 1  xx  0   xx 1
 xx  3  xx  2   xx 1  xx  0 









T 1

1 T T
T k
 xt x   xx  0   2
 xx  k    xx  0   2  xx  k 

T t 1  1
T
k 1
k 1
Apply this to cross covariances
T 1
T 1
T k
T k
1 T T
 xy  k 
 xy  k   
 xt y   xy  0   

T
T
T t 1  1
k 1
k 1
T 1
T k
T k
 yx  k 
 xy  k   
  xy  0   
T
T
k 1
k 1
T 1
and
T 1
T 1
T k
T k
1 T T
 yx   k 
 yx  k   
 yt x   yx  0   

T
T
T t 1  1
k 1
k 1
T 1
T k
T k
 xy  k 
 yx  k   
  yx  0   
T
T
k 1
k 1
T 1
A simple expression for the
inverse weighting matrix
T T
   x x
1  t 1  1 t 
Cov( w)  2 T T
T 
   yt x
 t 1  1
T

T
 x y 

  A A
T T
2

 yt y 

t 1  1

t 1  1
t
where
T 1
T 1



 

T k
T k

0

2

k

0

2

k

0

2

k

0

2

k




















xx
xy
xy
xx
xy
xy
 xx
  xx

T
T
k 1
k 1
k 1
k 1


A
T 1
T 1



 

T k
T k

0

2

k

0

2

k

0

2

k

0

2

k




















yx
yx
yy
yy
yx
yx
yy
yy

 

T
T
k 1
k 1
k 1
k 1

 

Some applications of GMM
Fixed income securities
CIR model : ln pt   A( , )  B( , )it 
Construct moments of returns based on distribution
of it+ 
 Estimate
by comparing to sample moments

Derivative securities

Construct
 moments of returns by simulating PDE
given
 Estimate
by comparing to sample moments

Asset pricing with time-varying risk premia
CEV Example
dS   AS  (1 )  BS  dt   s S  /2dz
Special cases:
  1: dS is a mean reverting process
dS  K (  St )dt   s St dz ( K   B,    A / B )
  2 : dS / St is a standard Gaussian
dS   Sdiffusion
(  A  B)
t dt   s S t dz
St
Method 1: Solve for
i
i
S


f
( A, B,  )
Define moments of
,
t
S
Compare to sample moments
...
CEV Example
dS   AS  (1 )  BS  dt   s S  /2dz
Special cases:
  1: dS is a mean reverting process
dS  K (  St )dt   s St dz ( K   B,    A / B )
  2 : dS / St is a standard Gaussian
dS   Sdiffusion
(  A  B)
t dt   s St dz
S0
( A,and
B,  )
given starting value
Estimate moments Sof , ˆ i  fˆ i ( A, B,  )
dS
Method 2: Simulate
t
S
Compare to sample moments