Chapter 1 Nonlinear econometrics for finance
GENERALIZED METHOD of MOMENTS
A structural approach to asset pricing
Consider a representative investor.
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The investor has utility over consumption today and tomorrow. Write
U(ct, ct+1) = u(ct) + βu(ct+1).
β is a subjective discount factor (β < 1).
We will assume an increasing u(.) reflecting non-satiation.
We will also assume a concave u(.) reflecting decreasing marginal utility from con- sumption.
For example, but this is only an example, u(c ) = 1 c1−γ , a CRRA utility. t 1−γt
We are interested in the optimization problem:
max [u(ct) + βEt(u(ct+1))] ,
subject to
ct =et−ptθ, ct+1 = et+1 + xt+1θ,
where et is the first period’s endowment, et+1 is the second period’s endowment, θ is the number of shares purchased in the first period, and xt+1 is the asset’s payoff in period 2, i.e., pt+1 + dt+1 (price plus dividends, when dividends play a role).
Chapter 1 Nonlinear econometrics for finance
• The consumer either consumes or invests. Less consumption today implies more investment, and a probability of more consumption, tomorrow. The consumer “dies” after period 2. Hence, everything needs to be consumed in period 2.
• We could just plug in the expressions for consumption in the two periods: max [u(et − ptθ) + βEt(u(et+1 + xt+1θ))] .
• Let us solve. From the first-order conditions of the optimization problem: −ptu′ (ct) + βEt(u′ (ct+1)(pt+1 + dt+1)) = 0
Interpretation:
ptu (ct) = βEt(u (ct+1)(pt+1 + dt+1)), (1)
u′ (ct+1)
(pt+1+dt+1). (2) u(ct)
• Eq. (1) says that the investor invests to the point where the loss of utility (of consumption) from investing one more unit today is equal to the expected (dis- counted) utility gain from that unit tomorrow. The investor continues to buy (or sell) securities until the marginal loss equals the marginal gain.
• Eq. (2) is an asset pricing relation: the price today is equal to expected discounted future cash flows.
u′(ct+1)
• The discounting i.e., mt+1 = β u′ (ct) is stochastic.
• Ignoring dividends, prices are martingales under a new measure determined by the ′
u (ct+1) stochastic adjustment provided by β u′ (ct ) .
Chapter 1 Nonlinear econometrics for finance
pt = Et(mt+1(pt+1 + dt+1)), (3)
where mt+1 is generally called a “stochastic discount factor”. Note: any asset pricing model implies Eq. (3) and a risk adjustment mt+1. In fact, as we discussed in class, prices are always discounted (by mt+1) expectations of future cash flows. What we are doing in this section is solely char- acterizing the stochastic discount factor in terms of marginal utility of consumption. Said differently, we are considering a specific model for mt+1 .
• In terms of returns, by simply dividing both the left-hand side and the right-hand side of Eq. (3) by pt, we also have the general expression:
pt+1 +dt+1
= Et(mt+1(1 + Rt+1)). (4)
risk correction oughts to take place. Write
• Once more, any asset should satisfy Eq. (3), for prices, and Eq. (4), for returns. Different models simply imply different stochastic discount factors or, equivalently, different risk adjustments.
• Interpret, again: if there were no uncertainty about future cash flows, then pt =
+ d ). Since there is uncertainty, and investors are risk-averse, some t+1
Et(mt+1(pt+1 + dt+1))
Ct(mt+1, pt+1 + dt+1) + Et(mt+1)Et(pt+1 + dt+1)
1 + Rf Et(pt+1 + dt+1) + Ct(mt+1, pt+1 + dt+1), (5)
where the last equality derives from Eq. (4) by noting that the stochastic discount
factor should price the risk-free asset as well. In this case, 1 = Et(mt+1(1 + Rf )) ⇒
(1 + Rf ) = 1 . Et (mt+1 )
• Eq. (5) clarifies the nature of risk adjustments. Assets which pay-off when the 3
Chapter 1 Nonlinear econometrics for finance
marginal utility is high (i.e., when consumption is low, in our model) are assets which sell at a premium (their price is relatively higher). Why? In our model, because investors want to hedge fluctuations in their consumption levels.
1.1 Expected return-beta representation: the consumption CAPM
• We are used to asset pricing models written in terms of excess returns. Let us derive one. Assume existence of N assets. Denote a generic asset by the superscript i with i = 1,...,N. Write,
1=E(m (1+Ri ))=C(m ,1+Ri )+E(m )E(1+Ri ).
• Similarly,
Et (mt+1 )
C(m ,1+Ri ) −t t+1 t+1.
t t+1 t+1 t t+1
t+1 t t+1 t t+1
E(1+Ri)= t t+1
C(m ,1+Ri ) ) = 1 + Rf − t t+1 t+1
Et (mt+1 )
Et (mt+1 ) 1+Rf =1/Et(mt+1).
C(m ,1+Ri ) E(Ri )−Rf=−t t+1 t+1.
t t+1 Et (mt+1 )
C(m ,1+Ri )V(m )
)−Rf =− t t+1 t+1 Vt (mt+1 )
t t+1 Et (mt+1 )
where βi,m = Ct(mt+1,1+Rti+1) and λm = Vt(mt+1).
E(Ri )−Rf=−β λ, t t+1 i,m m
Vt (mt+1 )
Et (mt+1 )
Chapter 1 Nonlinear econometrics for finance
Interpret: assets whose returns are high when the marginal utility (of consumption, in our model) is high are good for you. People require relatively lower returns to hold these stocks.
Given our representative agent on page 1, this formulation leads to a consumption- CAPM (CCAPM) model. Because consumption enters the utility function of the representative agent, people care about fluctuations in consumption rather than about fluctuations in market returns.
Compare this logic to the classical CAPM from your investment classes ...
Testing asset pricing models: GMM
E(m (1+Ri ))=1 t t+1 t+1
E(m (1+Ri )−1)=0 t+1 t+1
by “conditioning down”.
This is a “moment condition” that oughts to hold for each asset i with i = 1, ..., N . Consider, now, N assets. Stack the corresponding moment conditions to obtain:
E(mt+1(1 + Rt+1) − 1) = 0, whereR =[R1 ,...,RN ]⊤.
t+1 t+1 t+1
Now, notice that mt+1 depends on parameters. In the case of the CCAPM with
1 1−γ ct+1 −γ
CRRA utility: u(ct) = 1−γ ct and mt+1 = β ct . The two parameters are
the subjective discount factor β and the coefficient of relative risk aversion γ.
Chapter 1 Nonlinear econometrics for finance
• Hence, write
E(mt+1(θ)(1 + Rt+1) − 1) = 0.
to make the dependence on the parameter vector θ apparent.
• Estimation: Intuitively, if the pricing model is right, then the vector mt+1(θ)(1 + Rt+1) and the vector 1 should be close to each other (on average). This observation gives us a way to “choose” a value for θ. Once a value is chosen, testing can be conducted.
• Strategy: (a) It seems natural to select the θ value that would give the model the best chance. GMM estimates θ by minimizing the empirical difference between
E(mt+1(θ)(1 + Rt+1)) and 1. (b) Given an estimate for θ, θT say, GMM evaluates the size of the pricing errors, i.e., the size of the difference between the empirical
mean of mt+1(θT )(1 + Rt+1) and 1.
• Write mt+1(θ)(1+Rt+1)−1 = g(Xt+1,θ) to signify that the time t+1 pricing errors
depend on data at time t + 1 and a parameter vector θ.
• Then, the model implies:
• Empirically:
is an N-vector.
• Estimation of θ :
E(g(Xt+1, θ)) = 0.
should be as close as possible to zero for the model to be right. Notice that gT (θ)
arg min gT (θ)⊤WT gT (θ) θ
arg min [QT (θ)] . θ
gT(θ)= g(Xt+1,θ)
Chapter 1 Nonlinear econometrics for finance
• Assume the dimension of the vector θ is d. Typically, we cannot choose θ to make the errors exactly zero (N ≥ d). However, we want to make the pricing errors as small as possible. The weight matrix WT tells you how much emphasis you are putting on specific moments (i.e., on specific assets). If WT = IN , i.e., the identity matrix, then you are effectively treating all assets symmetrically. In this case, the criterion minimizes the sum of the squared pricing errors.
• GMM is, of course, a general estimation procedure. We motivated it here with the CCAPM but its logic applies generally to broad classes of nonlinear estimation problems.
2.1 Consistency and asymptotic normality in GMM
• Let WT →p W . Assume N moment conditions and d parameters with N ≥ d.
• Assume there is no dependence in the data (this is, of course, restrictive and will
be relaxed).
• By Taylor’s expansion, stopped at the first order, around the true value θ0:
∂ Q ( θ ) ∂ Q ( θ ) ∂ 2 Q ( θ )
TT−T0=T0θ−θ.
⊤T0
d×d matrix
• Now, notice that ∂QT (θT ) ≈ 0 since we are minimizing the criterion with respect to ∂θ
d×1 vector
θT to find θ0. • Hence,
∂2QT(θ0)−1 ∂QT(θ0)
• We begin with consistency. We will (later) show that the first term, i.e., ∂2QT (θ0)
∂θ∂θ⊤ converges in probability to a finite number (c.f., Eq. (7)). What about the second
term,i.e., ∂QT(θ0)? ∂θ
• Notice that
Nonlinear econometrics for finance
WT converges in probability by assumption.
• By the WLLN, however,
gT (θ0) = T
• Because, E(g(Xt+1, θ0) = 0, we have
or, equivalently,
θ T − θ 0 →p 0 ,
∂QT (θ0) ∂θ
∂gT (θ0)⊤
∂θ⊤ WT gT (θ0),
where ∂gT (θ0) will, also, be shown to converge in probability to a finite number and
g(Xt+1, θ0) → E(g(Xt+1, θ0),
gT(θ0)= T1 g(Xt+1,θ0)→p 0
In other words, θT is a consistent estimator of θ.
• Let us now turn to asymptotic normality. After standardizing by
∂2QT (θ0)−1√
∂QT (θ0) ∂θ
θ T →p θ 0 .
T , we have
• Let us focus on term (a) first. What is √T ∂QT (θ0)? ∂θ
√ ∂QT (θ0) √ ∂gT (θ0)⊤
T ∂θ = T2 ∂θ⊤ WTgT(θ0)
Nonlinear econometrics for finance
1 T−1 ∂g(X ,θ ) 1 T−1
0 WT √ g(Xt+1,θ0) T t=1
N×N
d×N after transposing
where Φ0 = E
by the WLLN.
Yd 2Γ⊤0 WN(0,Φ0),
Yd N(0,4Γ⊤0 WΦ0WΓ0),
g(Xt+1, θ0)g(Xt+1, θ0) , because
1T−1∂g(X ,θ) p ∂g(X ,θ) t+10→E t+10=Γ0,
T t=1 ∂θ⊤ ∂θ⊤
• Note: the (weak) convergence result in Eq. (7) derives from an application of the T−1
CLT to √1 g(Xt+1,θ0) (see Chapter 0). In fact, T t=1
g(Xt+1, θ0) − E(g(Xt+1, θ0) = √ g(Xt+1,θ0)
T t=1 Yd N(0,Φ0),
where Φ0 = E g(Xt+1, θ0)g(Xt+1, θ0)⊤, if the data has no dependence structure.
Chapter 1 Nonlinear econometrics for finance • Now, we need to consider the additional term (b), namely ∂2QT (θ0). Write
∂2QT (θ0) ∂θ∂θ′
1 ∂g(Xt+1, θ0) 1 ∂g(Xt+1, θ0)
= 2T ∂θ t=1
WT T ∂θ m t=1
N×1 N×1
1×N after transposing
∂θ ∂θ WT T g(Xt+1,θ0) m j t=1
T−1 T−1 1 ∂2g(Xt+1,θ0) 1
1×N after transposing
for m,j = 1,...,d.
• Notice, again, that the last term T1 g(Xt+1,θ0) converges to zero by the WLLN
t=1 (since E(g(Xt+1, θ0) = 0). Hence,
∂2QT (θ0) p ∂θ∂θ′ → 2E
∂g(Xt+1, θ0)⊤ ∂θ⊤
∂g(Xt+1, θ0)
WE ∂θ⊤ (7)
• At this juncture, we can put everything together and obtain, by Slutsky’s theorem:
∂2QT (θ0)−1 √ ∂QT (θ0)
∂g(Xt+1,θ0) Γ0=E ∂θ⊤ ,
∂θ∂θ YN 0, Γ0WΓ0
−1⊤ ⊤ Γ0WΦ0WΓ0 Γ0WΓ0
Chapter 1 Nonlinear econometrics for finance
⊤ Φ0 = E g(Xt+1, θ0)g(Xt+1, θ0) .
We conclude that θT is asymptotically normal.
• In sum: the GMM estimator is consistent (θT →p θ). Also, the estimator θT con- √
verges to θ0 at speed T. Its precision can be quantified by evaluating the asymp- totic variance-covariance matrix.
2.2 Important issues in GMM 2.2.1 The case N = d
• If the number of moment conditions N is the same as the number of parameters (the case of exact identification of the model), then Γ0 becomes a (nonsingular) square matrix. So,
Γ⊤WΓ −1 Γ⊤WΦ WΓ Γ⊤WΓ −1 0000000
= Γ−1W−1 Γ⊤−1 Γ⊤WΦ W (Γ )(Γ )−1 W−1 Γ⊤−1 0000000
= Γ−1Φ Γ⊤−1 000
⊤ −1 −1 =Γ0Φ0Γ0 .
• For a review of the properties we used, see Chapter 0.
• Implication: when N = d, it does not matter - asymptotically - what weight matrix we are using. The weight matrix is not affecting the asymptotic distribution of the parameter estimates.
• This is a fairly standard “textbook” expression of the asymptotic variance matrix. The matrix has a typical “sandwich” form.
• So,ifN=d,then
Nonlinear econometrics for finance
d ⊤−1−1
YN 0, Γ0Φ0 Γ0
∂g(Xt+1, θ0)⊤
E g(Xt+1, θ0)g(Xt+1, θ0)⊤
∂g(Xt+1, θ0)−1
2.2.2 The case N > d: optimal weighting matrix
• If the number of moment conditions is not the same as the number of parameters, then the optimal weight matrix matters (even asymptotically).
• Important result: if W = Φ−1, then the asymptotic matrix of the GMM estimator 0
⊤ −1 −1 is minimal and equal to Γ0 Φ0 Γ0
obtain when N = d).
• LetusshowthatifW =Φ0 ,thentheasymptoticmatrixisequalto Γ0Φ0 Γ0 .
(which is precisely the value that we would
Γ⊤WΓ −1 Γ⊤WΦ WΓ Γ⊤WΓ −1 0000000
= Γ⊤ Φ−1Γ −1 Γ⊤ Φ−1Φ Φ−1Γ Γ⊤ Φ−1Γ −1 000 00000 000
⊤ −1 −1 =Γ0Φ0Γ0 .
• Importantly, it holds that
⊤ −1 ⊤ ⊤ −1
Γ0WΓ0 Γ0WΦ0WΓ0 Γ0WΓ0
thereby implying that the resulting asymptotic variance is the smallest possible.
• Note that F −1 − G−1 ≥ 0 if G − F ≥ 0. 12
Chapter 1 Nonlinear econometrics for finance • Hence, we want to show that
Γ⊤Φ−1Γ −Γ⊤WΓΓ⊤WΦWΓ−1Γ⊤WΓ≥0. 0000000000
⊤ −1/2 1/2 ⊤ −1 ⊤ 1/2 −1/2 Γ0 Φ0 I − Φ0 WΓ0 Γ0 WΦ0WΓ0 Γ0 WΦ0 Φ0
where H = Γ⊤ Φ−1/2 and P is a symmetric and idempotent matrix, i.e., a projection
matrix. • Thus,
HPPH⊤ = HPP⊤H⊤ = (HP)(HP)⊤.
• This is necessarily positive semidefinite (≥ 0 in the language of matrices) since
z⊤(HP)(HP)⊤z ≥ 0 ∀z ∈ Rd. • To recap, the optimal weight matrix is
−1 ⊤−1 W = Φ0 = E g(Xt+1, θ0)g(Xt+1, θ0)
and the asymptotic variance of the GMM estimator is:
∂g(Xt+1, θ0) ∂θ⊤
⊤ ∂g(Xt+1, θ0)
E g(Xt+1,θ0)g(Xt+1,θ0) E ∂θ⊤ Again, the limiting variance as a “sandwich” form.
Chapter 1 Nonlinear econometrics for finance
Implementation
• If N > d, one should employ the optimal weight matrix. However, this matrix depends on θ0. We estimate θ0 by using a preliminary estimate θ1 .
• First stage: Implement
where IN is an N × N identity matrix.
• Estimate W = Φ−1 using
• Second stage: Implement the following:
2.2.3 Inference
θT1 = arg min gT (θ)⊤IN gT (θ) , θ
WT = 1 g(Xt+1, θT1 )g(Xt+1, θT1 )⊤
θT2 = arg min gT (θ)⊤WT gT (θ) .
• GMM inference on the parameter values requires estimation of the asymptotic vari- ance matrix.
• What we want is an estimate of
∂g(Xt+1, θ0) ∂θ⊤
⊤ ∂g(Xt+1, θ0)
E g(Xt+1,θ0)g(Xt+1,θ0) E ∂θ⊤ 14
Chapter 1 Nonlinear econometrics for finance • As always, replace the expectations with empirical moments. Thus,
T . T ∂θ⊤
asyV(θ )
∂g(Xt+1,θ2) T
T−1 −1 1 ⊤
T−1 −1 1 ∂g(Xt+1,θ2)
g(Xt+1, θT2 )g(Xt+1, θT2 )
• By the WLLN and the convergence in probability of θT to θ0, asyV(θT ) → asyV(θ0).
2.2.4 Dependence in the observations: HAC estimation
• In deriving the asymptotic distribution of θT we used the fact that the observations X are uncorrelated. This is clearly restrictive in finance.
• In the C-CAPM case, for example, Xt+1 includes consumption levels at time t + 1. It is not believable that these are uncorrelated with consumption levels at time t.
• What changes if observations are correlated? Only Φ0!
• We write
rather than
⊤ E g(Xt+1, θ0)g(Xt+1−j , θ0) ,
E g(Xt+1, θ0)g(Xt+1, θ0) .
• This is, of course, completely analogous to variance estimation in the presence of
correlated errors in the regression case.
• Estimation:
k T−1 kT
ΦT = k−|j| 1 g(Xt+1,θT)g(Xt+1−j,θT)⊤ (8) j =−k t=1
Chapter 1 Nonlinear econometrics for finance
• The estimator is a weighted sum of auto-covariances computed using Bartlett-type weights. We will see that these weights derive naturally from the problem.
• The usual requirements are k → 0 and k2 → ∞. TT
• We use θ(1) to implement the secon-stage estimation problem and θ(2) to compute TT
the asymptotic variance of the GMM estimator.
Aside to understand HAC estimation: Assume at is a stationary time-series such
that E(at) = 0. Consider 2
at 122 t=1
V √2 = 2E at t=1
Now, consider
V √3 = 3E at t=1
= 2E(a1 + a2 + a1a2 + a2a1)
= 21 2E(a2t ) + (E(atat+1) + E(atat−1)) . at 132
More generally,
= 3E((a1+a2+a3)(a1+a2+a3))
= 3E(a1 +a2 +a3 +a1a2 +a1a3 +a2a1 +a2a3 +a3a1 +a3a2)
= 31(3E(a2t ) + 2 (E(atat+1) + E(atat−1)) + E(atat+2) + E(atat−2)).
Nonlinear econometrics for finance
at 1 t=1 2
V √k = k[(kE(at)+(k−1)(E(atat+1)+E(atat−1))
+(k − 2)(E(atat+2) + E(atat−2) +… + E(a1ak) + E(aka1)]
E(atat+j) → E(atat+j). j=−∞
k E(atat+j).
This is, however, completely consistent with the way in which the estimator was written, c.f., Eq. (8).
Naturally, as k → ∞, then k
at k k−|j| t=1
2.2.5 Hansen’s test of overidentifying restrictions
• Once you have estimates, you can of course do inference on them. In the C-CAPM model with CRRA utility, for example, is β = 0.95? Or, is α = 3?
• You can also evaluate whether the pricing errors are close to zero.
• Problem: are the moment conditions close to zero?
• Implementation:
TQT (θT ) = TgT (θT )⊤Φ−1gT (θT ) Yd χ2N−d.
Chapter 1 Nonlinear econometrics for finance • Proof. By Taylor expansion, stopped at the first order, around θ0:
T−1 T−1 1a1√
√ g(Xt+1,θT)= √ g(Xt+1,θ0)+Γ0 T θT −θ0 , TT
1T−1 1T−1 1T−1∂g(X ,θ) g(X,θ)=g(X,θ)+ t+10 θ−θ
t+1T t+10 ⊤ T0 TT T∂θ
t=1 t=1 t=1
t=1 t=1 where“=a”signifies“asymptoticallyequivalentto…”.
• Now, recall
• Now, write
−1/2 1 T−1
Φ √ g(Xt+1,θT) T T
√ a ⊤−1 −1⊤−11
T θT −θ0 =−(Γ0Φ0 Γ0) Γ0Φ0 √
g(Xt+1,θ0).
√ g(X ,θ )=a I −Γ (Γ⊤Φ−1Γ )−1Γ⊤Φ−1Ψ .
t+1 T N 0 0 0 0 0 0 T
a −1/21T−1
= Φ0 √ g(Xt+1, θT )
=a Φ−1/2 I − Γ (Γ⊤Φ−1Γ )−1Γ⊤Φ−1 Ψ
=a I − Φ−1/2Γ (Γ⊤Φ−1Γ )−1Γ⊤Φ−1/2 Φ−1/2Ψ N0000000 0T
=a PΦ−1/2Ψ , 0T
where P is symmetric and idempotent. 18
Chapter 1 • So,
Nonlinear econometrics for finance
• Consider, now, the trace of P :
tr(I − Φ−1/2Γ (Γ⊤Φ−1Γ )−1Γ⊤Φ−1/2)
=a Ψ⊤ Φ−1/2P ⊤ P Φ−1/2Ψ
T00T =a Ψ⊤Φ−1/2PΦ−1/2Ψ .
= tr(I ) − tr(Φ−1/2Γ (Γ⊤Φ−1Γ )−1Γ⊤Φ−1/2)
N0000000 = N − tr((Γ⊤Φ−1Γ )−1(Γ⊤Φ−1Γ ))
= N−tr(Id)=N−d. • Thus, by Jordan’s decomposition:
T Q (θ ) = Ψ⊤ Φ−1/2P Φ−1/2Ψ TTT00T
= Ψ⊤ Φ−1/2QΛQ⊤ Φ−1/2Ψ , T00T
where Λ is a matrix of ones and zeros with N − d ones. Finally,
N−d TQT(θT)=zi2Yd χ2N−d
since the zi⊤ s are independent normal (0, 1).
• Chapter 0 contains a discussion of the properties used above.
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