Generalized method of moments estimation

1
Generalized method of moments estimation

• FINA790C
• HKUST Spring 2006

2
Outline

• Basic motivation and examples
• First case linear model and instrumental
  variables
• Estimation
• Asymptotic distribution
• Test of model specification
• General case of GMM
• Estimation
• Testing

3
Basic motivation

• Method of moments suppose we want to estimate
  the population mean ? and population variance ?2
  of a random variable vt
• These two parameters satisfy the population
  moment conditions
• Evt - ? 0
• Evt2 (?2?2) 0

4
Method of moments

• So lets think of the analogous sample moment
  conditions
• T-1?vt - ?? 0
• T-1?vt2 ( ??2 ??2 ) 0
• This implies
• ?? T-1?vt
• ??2 T-1?(vt - ??)2
• Basic idea population moment conditions provide
  information which can be used for estimation of
  parameters

5
Example

• Simple supply and demand system
• qtD a pt utD
• qtS ?1nt ?2pt utS
• qtD qtS qt
• qtD, qtS are quantity demanded and supplied pt
  is price and qt equals quantity produced
• Problem how to estimate a, given qt and pt
• OLS estimator runs into problems

6
Example

• One solution find ztD such that cov(ztD, utD)
  0
• Then cov(ztD,qt) acov(ztD,pt) 0
• So if EutD 0 then EztDqt aEztDpt 0
• Method of moments leads to
• a (T-1?ztDqt)/(T-1?ztDpt)
• which is the instrumental variables estimator of
  a with instrument ztD

7
Method of moments

• Population moment condition vector of observed
  variables, vt, and vector of p parameters ?,
  satisfy a px1 element vector of conditions
  Ef(vt,?) 0 for all t
• The method of moments estimator ?T solves the
  analogous sample moment conditions
• gT(?) T-1?f(vt,?T) 0 (1)
• where T is the sample size
• Intuitively ?T ? in probability to the solution
  ?0 of (1)

8
Generalized method of moments

• Now suppose f is a qx1 vector and qgtp
• The GMM estimator chooses the value of ? which
  comes closest to satisfying (1) as the estimator
  of ?0, where closeness is measured by
• QT(?) T-1?f(vt,?)WTT-1?f(vt,?)
  gT(?)WTgT(?)
• and WT is psd and plim(WT) W pd

9
GMM example 1

• Power utility based asset pricing model
• Et ?(Ct1/Ct)-aRit1 1 0 with unknown
  parameters b, a
• The population unconditional moment conditions
  are
• E(?(Ct1/Ct)-aRit1 1)zjt 0 for j1,,q
  where zjt is in information set

10
GMM example 2

• CAPM says
• E Rit1 - ?0(1-bi) biRmt1 0
• Market efficiency
• E( Rit1 - ?0(1-bi) biRmt1 )zjt 0

11
GMM example 3

• Suppose the conditional probability density
  function of the continuous stationary random
  vector vt, given Vt-1vt-1,vt-2, is
  p(vt?0,Vt-1)
• The MLE of ?0 based on the conditional log
  likelihood function is the value of which
  maximizes LT(?) ?lnp(vt?,Vt-1), i.e. which
  solves ?LT(?)/?? 0
• This implies that the MLE is just the GMM
  estimator based on the population moment
  condition
• E ?lnp(vt?,Vt-1)/?? 0

12
GMM estimation

• The GMM estimator ?T argmin? e ?QT(?)
  generates the first order conditions
• T-1??f(vt,?T)/??WTT-1?f(vt,?T) 0 (2)
• where ?f(vt,?T)/?? is the q x p matrix with
  i,j element ?fi(vt,?T)/??j
• There is typically no closed form solution for
  ?T so it must be obtained through numerical
  optimization methods

13
Example Instrumental variable estimation of
linear model

• Suppose we have yt xt?0 ut, t1,,T
• Where xt is a (px1) vector of observed
  explanatory variables, yt is an observed scalar,
  ut is an unobserved error term with Eut 0
• Let zt be a qx1 vector of instruments such that
  Eztut0 (contemporaneously uncorrelated)
• Problem is to estimate ?0

14
IV estimation

• The GMM estimator ?T argmin? e ?QT(?)
• where QT(?) T-1u(?)ZWTT-1Zu(?)
• The FOCs are
• (T-1XZ)WT(T-1Zy) (T-1XZ)WT(T-1ZX)?T
• When of instruments q of parameters p
  (just-identified) and (T-1ZX) is nonsingular
  then
• ?T (T-1ZX)-1(T-1Zy)
• independently of the weighting matrix WT

15
IV estimation

• When instruments q gt parameters p
  (over-identified) then
• ?T (T-1XZ)WT(T-1ZX)-1(T-1XZ)WT(T-1Zy)

16
Identifying and overidentifying restrictions

• Go back to the first-order conditions
• (T-1XZ)WT(T-1Zu(?T)) 0
• These imply that GMM method of moments based on
  population moment conditions ExtztWEztut(?0)
  0
• When q p GMM method of moments based on
  Eztut(?0) 0
• When q gt p GMM sets p linear combinations of
  Eztut(?0) equal to 0

17
Identifying and over-identifying restrictions
details

• From (2), GMM is method of moments estimator
  based on population moments
• E?f(vt,?0)/??WEf(vt,?0) 0 (3)
• Let F(?0)W½E?f(vt,?0)/?? and rank(F(?0))p.
  (The rank condition is necessary for
  identification of ?0). Rewrite (3) as
• F(?0)W½Ef(vt,?0) 0 or
• F(?0)F(?0)F(?0)-1F(?0)W½Ef(vt,?0)0 (4)
• This says that the least squares projection of
  W½Ef(vt,?0) on to the column space of F(?0) is
  zero. In other words the GMM estimator is based
  on rankF(?0)F(?0)F(?0)-1F(?0) p
  restrictions on the (transformed) population
  moment condition W½Ef(vt,?0). These are the
  identifying restrictions GMM chooses ?T to
  satisfy them.

18
Identifying and over-identifying restrictions
details

• The restrictions that are left over are
• Iq - F(?0)F(?0)F(?0)-1F(?0)W½Ef(vt,?0)
  0
• This says that the projection of W½Ef(vt,?0) on
  to the orthogonal complement of F(?0) is zero,
  generating q-p restrictions on the transformed
  population moment condition. These
  over-identifying restrictions are ignored by the
  GMM estimator, so they need not be satisfied in
  the sample
• From (2),
• WT½T-1?f(vt,?T) Iq - FT(?T)FT(?T)FT(?
  T)-1FT(?T)WT½T-1f(vt,?T) where FT(?)
  WT½T-1??f(vt,?)/ ??. Therefore QT(?T) is like
  a sum of squared residuals, and can be
  interpreted as a measure of how far the sample is
  from satisfying the over-identifying
  restrictions.

19
Asymptotic properties of GMM instrumental
variables estimator

• It can be shown that
• ?T is consistent for ?0
• (?T,i ?0,i)/vVT,ii/T N(0,1) asymptotically
  where
• VT (XZWTZX)-1XZWTSTWTZX (XZWTZX)-1
• ST limT?8 VarT-½Zu

20
Covariance matrix estimation

• Assuming ut is serially uncorrelated
• VarT-½Zu ET-½?ztutT-½?ztut
• T-1?Eut2ztzt
• Therefore,
• ST T-1?ut(?T)2ztzt

21
Two step estimator

• Notice that the asymptotic variance depends on
  the weighting matrix WT
• The optimal choice is WT ST-1 to give VT
  (XZ ST-1ZX)-1
• But we need ?T to construct ST This suggests a
  two-step (iterative) procedure
• (a) estimate with sub-optimal WT, get ?T(1),get
  ST(1)
• (b) estimate with WT ST(1)-1
• (c) iterated GMM from (b) get ?T(2), get
  ST(2), set WT ST(2)-1, repeat

22
GMM and instrumental variables

• If ut is homoskedastic varu?2IT then ST
  ?2ZtZt where Zt is a (Txq) matrix with t-th row
  zt and ?2 is a consistent estimate of ?2
• Choosing this as the weighting matrix WT ST-1
  gives
• ?T XZ(ZZ)-1ZX)-1 XZ(ZZ)-1Zy
• XX-1Xy
• where XZ(ZZ)-1ZX is the predicted value of X
  from a regression of X on Z. This is two-stage
  least squares (2SLS) first regress X on Z, then
  regress y on the fitted value of X from the first
  stage.

23
Model specification test

• Identifying restrictions are satisfied in the
  sample regardless of whether the model is correct
• Over-identifying restrictions not imposed in the
  sample
• This gives rise to the overidentifying
  restrictions test
• JT TQT(?T) T-½u(?T)Z ST-1 T-½ Zu(?T)
• Under H0 Eztut(?0) 0, JT asymptotically
  follows a ?2(q-p) distribution

24
(From last time)

• In this case the MRS is
• mt,tj ?j Ctj /Ct -?
• Substituting in above gives
• Etln(Rit,tj)
• -jln? ?Et?ctj - ½ ?2 vart?lnctj -
  ½vartlnRit,tj
• ?covt ?lnctj,lnRit,tj

25
Example power utility lognormal distributions

• First order condition from investor maximizing
  power utility with log-normal distribution gives
• ln(Rit1) ui a?lnCt1 uit1
• the error term uit1 could be correlated with
  ?lnCt1 so we cant use OLS
• However Etuit1 0 means uit1 is uncorrelated
  with any zt known at time t

26
Instrumental variables regressions for returns
and consumption growth

• Return Stage 1 Stage 2 JT test
• r ?lnc ? ? r ?lnc
• CP 0.297 0.102 -0.953 -0.118 0.221
  0.091
• (0.00) (0.15) (0.57) (0.11)
  (0.00) (0.10)
• Stock 0.110 0.102 -0.235 -0.008 0.105
  0.097
• (0.11) (0.15) (1.65) (0.06)
  (0.06) (0.08)
• Annual US data 1889-1994 on growth in log real
  consumption of nondurables services, log real
  return on SP500, real return on 6-month
  commercial paper. Instruments are 2 lags each
  of real commercial paper rate, real consumption
  growth rate and log of dividend-price ratio
  

27
GMM estimation general case

• Go back to GMM estimation but let f be a vector
  of continuous nonlinear functions of the data and
  unknown parameters
• In our case, we have N assets and the moment
  condition is that E(m(xt,?0)Rit-1)zjt-10 using
  instruments zjt-1 for each asset i1,,N and each
  instrument j1,,q
• Collect these as f(vt,?) zt-1?ut(Xt,?) where zt
  is a 1xq vector of instruments and ut is a Nx1
  vector. f is a column vector with qN elements
  it contains the cross-product of each instrument
  with each element of u.
• The population moment condition is
• Ef(vt,?0) 0

28
GMM estimation general case

• As before, let gT(?) T-1?f(vt,?). The GMM
  estimator is ?T argmin? e ?QT(?)
• The FOCs are
• GT(?)WTgT(?) 0
• where GT(?) is a matrix of partial derivatives
  with i, j element dgTi/d?j

29
GMM asymptotics

• It can be shown that
• ?T is consistent for ?0
• asyvar(?T,ii) MSM where
• M (G0WG0)-1G0W
• G0 E?f(vt,?0)/??
• S limT?8 VarT-½gT(?0)

30
Covariance matrix estimation for GMM

• In practice VT MTSTMT is a consistent
  estimator of V, where
• MT (GT(?T)WTGT(?T))-1 GT(?T)WT
• ST is a consistent estimator of S
• Estimator of S depends on time series properties
  of f(vt,?0). In general it is
• S G0 ?( Gi Gi)
• where Gi Eft-E(ft)ft-i-E(ft-i)Eftft-I
  is the i-th autocovariance matrix of ft
  f(vt,?0).

31
GMM asymptotics

• So (?T,i ?0,i)/vVT,ii/T N(0,1)
  asymptotically
• As before, we can choose WT ST-1 and
• VT is then (GT(?T)ST-1GT(?T))-1
• a test of the models over-identifying
  restrictions is given by JT TQT(?T) which is
  asymptotically ?2(qN-p)

32
Covariance matrix estimation

• We can consistently estimate S with
• ST G0(?T) ?( Gi(?T) Gi(?T))
• where Gi(?T) T-1?ft(vt,?T)ft-i(vt-i,?T)
• If theory implies that the autocovariances of
  f(vt,?0) 0 for some lag j then we can exclude
  these from ST
• e.g. ut are serially uncorrelated implies ST
  T-1? ut(vt,?T)ut(vt,?T) ? (ztzt)

33
GMM adjustments

• Iterated GMM is recommended in small samples
• More powerful tests by subtracting sample means
  of ft(vt,?T) in calculating Gi(?T)
• Asymptotic standard errors may be understated in
  small samples multiply asymptotic variances by
  degrees of freedom adjustment T/(T-p) or
  (Nq)T/(Nq)T k where k p((Nq)2Nq)/2