Nonlinear Econometrics for Finance Lecture 4

. Econometrics for Finance Lecture 4 1 / 35

Last class: some important ingredients

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Recall the criterion function:

QT(θ) = gT(θ)⊤ WT gT(θ).

1×1 1×N N×N N×1

Thus, for m = 1, …, d, the first derivative of the criterion function is:

∂QT (θ) ∂θ

∂QT (θ) 1 ∂g(Xt+1,θ) 1

⊤ ∂QT (θ)

T−1 T−1 ∂θ

= ··· where =2

g(X ,θ) Tt+1

∂QT (θ) ∂θd

Tt=1 N×N

and, for m, j = 1, …, d, the second derivative of the criterion function is:

∂2QT (θ) ⊤

∂2QT (θ) ∂θ1∂θ1 ∂2QT (θ)

∂2QT (θ) ∂θ1∂θ2 ∂2QT (θ)

∂2QT (θ) ∂θ1∂θd ···

∂ QT(θ) =2 1 ∂g(Xt+1,θ)

1222 ∂θ∂θ ··· ··· ··· ···

· · · · · ·

∂θm∂θj T t=1 ∂θm T t=1 ∂θj

T−1 ⊤T−1 +2 1∂g(Xt+1,θ) W 1g(X ,θ).

Tt+1 T t=1 ∂θm∂θj T t=1

· · · ∂2QT (θ) ∂θd∂θd

W 1 ∂g(Xt+1,θ)

. Econometrics for Finance Lecture 4

Last class: review of “consistency”

∂2QT (θ0) ∂θ∂θ⊤ ∂θ

→p 2Γ⊤0 WΓ0 →p 2Γ⊤0 W0 ∂g(Xt+1,θ0)

∂QT (θ0) p

Γ0 = E ∂θ⊤ .

. Econometrics for Finance Lecture 4

Last class: review of “asymptotic normality”

→p 2Γ⊤0 WΓ0 →d N(0,4Γ⊤0 WΦ0WΓ0)

→d N(0,(2Γ⊤0 WΓ0)−14Γ⊤0 WΦ0WΓ0(2Γ⊤0 WΓ0)−1) →d N(0,(Γ⊤0 WΓ0)−1Γ⊤0 WΦ0WΓ0(Γ⊤0 WΓ0)−1)

∂g(Xt+1,θ0) ⊤ ∂θ⊤ and Φ0 = E g(Xt+1, θ0)g(Xt+1, θ0) .

∂θ∂θ⊤ ∂θ

√ TθT−θ0 =−

√ ∂QT(θ0) T

. Econometrics for Finance Lecture 4

Last class: asymptotic variance of the GMM estimator

V(θT ) = 1 (Γ⊤0 W Γ0)−1Γ⊤0 W Φ0W Γ0(Γ⊤0 W Γ0)−1 T

1 The asymptotic variance of θT is what will allow us to compute standard errors and do inference.

2 Notice its “sandwich” form. We will return to it. (Intuitive, right? A well-defined variance-covariance matrix has to be symmetric.)

3 Notice that we are dividing by T . (Intuitive, right? The variance

of θT has to go to zero for the estimator to be consistent for θ0.) 4 We will see that in two cases the asymptotic variance simplifies.

. Econometrics for Finance Lecture 4 5 / 35

Important issues in GMM

Important issues in GMM

. Econometrics for Finance Lecture 4 6 / 35

Important issues in GMM

Two cases: N = d and N > d

1 The (exactly-identified) case N = d. The number of moment conditions N is the same as the number of parameters to estimate d. (We will see that the choice of WT is asymptotically irrelevant.)

2 The (over-identified) case N > d. (We will see that the choice of WT is important. We will choose WT optimally.)

. Econometrics for Finance Lecture 4 7 / 35

Important issues in GMM

First: the case N = d

If the number of moment conditions N is the same as the number

of parameters d ⇒ Exact identification of the model.

In such case, Γ0 becomes a (nonsingular) square matrix. So,

Γ0 WΓ0 Γ0 WΦ0WΓ0

⊤ −1 Γ0 WΓ0

−1 −1 ⊤−1 ⊤

= Γ0 W Γ0 Γ0 WΦ0WΓ0 (Γ0)

= Γ−1Φ Γ⊤−1 000

⊤ −1 −1 =Γ0Φ0Γ0 .

−1 −1 ⊤−1 W Γ0

Implication: when N = d, it does not really matter asymptotically what weight matrix we are using.

The weight matrix is not affecting the asymptotic variance of the

GMM estimator.

. Econometrics for Finance Lecture 4 8 / 35

Important issues in GMM

The case N = d: summary

If N = d, then √

d ⊤ −1 −1

→ N 0, Γ0 Φ0 Γ0

d ∂g(Xt+1, θ0) ⊤ ⊤−1

∂θ⊤ E g(Xt+1,θ0)g(Xt+1,θ0)

∂g(Xt+1, θ0) −1 E ∂θ⊤ ,

1⊤ −1 −1

. Econometrics for Finance Lecture 4 9 / 35

Important issues in GMM

Second: the case N > d

If the number of moment conditions is not the same as the number of parameters, then the optimal weight matrix matters (even asymptotically).

Important results: if W = Φ−1, then the asymptotic variance of the 0

GMM estimator is …

1⊤ −1 −1

1 … equal to T Γ0 Φ0 Γ0 (i.e., precisely the value that we

would obtain when N = d)

2 … the smallest variance possible.

We will prove (1) and, then, (2).

. Econometrics for Finance Lecture 4 10 / 35

Important issues in GMM

Second: the case N > d

If W = Φ−1, then the asymptotic variance of the GMM estimator is 0

1⊤ −1 −1 equal to T Γ0 Φ0 Γ0

Proof. Write

V(θ ) = 1 Γ⊤WΓ −1 Γ⊤WΦ WΓ Γ⊤WΓ −1

T 0000000 T

= 1 Γ⊤ Φ−1Γ −1 Γ⊤ Φ−1Φ Φ−1Γ Γ⊤ Φ−1Γ −1 T000 00000 000

1⊤ −1 −1 = T Γ0Φ0 Γ0 .

. Econometrics for Finance Lecture 4 11 / 35

Important issues in GMM

Second: the case N > d

If the asymptotic variance of the GMM estimator is equal to

1⊤ −1 −1 T Γ0 Φ0 Γ0

, then it is the smallest possible.

Proof. We will compare it to the general form of the GMM variance (which would result from the use of any W ), i.e.,

1 ⊤ −1 ⊤ ⊤ −1 T Γ0 WΓ0 Γ0 WΦ0WΓ0 Γ0 WΓ0

Ignoring T1 , we will then show that

Γ⊤ W Γ −1 Γ⊤ W Φ W Γ Γ⊤ W Γ −1 − Γ⊤ Φ−1Γ −1 ≥ 0.

0000000000

. Econometrics for Finance Lecture 4 12 / 35

Important issues in GMM

Second: the case N > d

Once more, we wish to show:

⊤ −1 ⊤ ⊤ −1 ⊤ −1 −1 Γ0WΓ0 Γ0WΦ0WΓ0 Γ0WΓ0 − Γ0Φ0 Γ0

Note that F−1 −G−1 ≥ 0 if G−F ≥ 0. Hence, we want to show that ⊤ −1 ⊤ ⊤ −1⊤

We can write

Γ0Φ0 Γ0− Γ0WΓ0 Γ0WΦ0WΓ0 Γ0WΓ0 ≥0.

⊤ −1/2 1/2 ⊤ −1 ⊤ 1/2 −1/2 ⊤ Γ0Φ0 IN− Φ0 WΓ0 Γ0WΦ0WΓ0 Γ0WΦ0 Φ0 Γ0=HPH ,

where H = Γ⊤ Φ−1/2 and P is idempotent and symmetric (i.e., a projection) 00

. Econometrics for Finance Lecture 4 13 / 35

Important issues in GMM

Second: the case N > d

Now, notice that

HPH⊤ = HPPH⊤ = HPP⊤H⊤ = (HP)(HP)⊤,

where the first equal sign derives from the idempotency of P and the second from the symmetry of P.

This quantity is necessarily positive semidefinite (≥ 0 in the language of matrices) since

Nonlinear Econometrics for Finance Lecture 4

z⊤(HP)(HP)⊤z ⊤

Important issues in GMM

The case N > d (with an optimal W ): summary

−1 If N > d and W = Φ0 = E

g(Xt+1,θ0)g(Xt+1,θ0)

d ⊤ −1 −1

→ N 0, Γ0 Φ0 Γ0

d ∂g(Xt+1, θ0) ⊤

∂g(Xt+1, θ0) −1 E ∂θ⊤ ,

⊤−1 ∂θ⊤ E g(Xt+1,θ0)g(Xt+1,θ0)

1⊤ −1 −1

. Econometrics for Finance Lecture 4 15 / 35

Important issues in GMM

Implementation

Estimation

If N = d, choosing the optimal weight matrix is not important, asymptotically. (It may still be a good idea to use it, but asymptotically any weight matrix would yield the same result.)

If N > d, we should employ the optimal weight matrix for estimation.

1 This matrix depends on θ0.

2 Idea: Estimate θ0 by using a first-stage estimate θ1 (computed T

with the identity matrix as the weight matrix).

3 Usethefirst-stageestimateθ1 tocomputeanestimateofthe

optimal weight matrix.

4 Second stage: Find θ2 using the estimated optimal weight matrix.

. Econometrics for Finance Lecture 4 16 / 35

Important issues in GMM

Implementation

The two-step procedure in detail

First stage: Estimate θ1 by T

arg min gT (θ)⊤IN gT (θ) , θ

where IN is an N × N identity matrix.

Then, estimate WT = Φ−1 using 0

T t=1 Second stage: Estimate θ2 by

WT = 1 g(Xt+1, θT1 )g(Xt+1, θT1 )⊤

arg min gT (θ)⊤WT gT (θ) . θ

. Econometrics for Finance Lecture 4

Important issues in GMM

In both cases (N = d and N > d with an optimal weight matrix), the asymptotic variance is

and we estimate it with

V(θ ) = 1 Γ⊤Φ−1Γ −1

1⊤ −1 −1

T0 V(θT ) = Γ0 Φ Γ0

1 T−1 ∂g(Xt+1,θT) 1 T−1

Γ0 = and Φ0 = g(Xt+1,θT)g(Xt+1,θT)⊤. T ∂θ⊤ T

. Econometrics for Finance Lecture 4 18 / 35

HAC estimation

Dependence in the observations: HAC estimation

. Econometrics for Finance Lecture 4 19 / 35

HAC estimation

HAC estimation

In deriving the asymptotic variance 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 ⇒ correlated with consumption levels at time t.

. Econometrics for Finance Lecture 4 20 / 35

HAC estimation

HAC estimation

With uncorrelated observations, we wrote ⊤

Φ0 = E g(Xt+1, θ0)g(Xt+1, θ0) . With correlated observations, we will write

(forward and backward) into account.

It is also completely analogous to variance estimation in the presence of correlated errors in the regression case.

⊤ E g(Xt+1, θ0)g(Xt+1+j , θ0) .

This is intuitive. The summation over j takes the correlation

. Econometrics for Finance Lecture 4 21 / 35

HAC estimation

HAC estimation

To estimate Φ0 we use the (truncated, to k terms) sample equivalent k k−|j| 1 T−j ⊤

ΦT = g(Xt+1,θT )g(Xt+1+j,θT ) (1)

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 for consistency of ΦT for Φ0 are k → 0 and k2 → ∞. TT

We use the first-stage estimates θ(1) to compute the HAC optimal weight T

matrix and implement the second-stage estimation. We then use the

second-stage estimates θ(2) to compute the HAC optimal weight matrix to T

evaluate the asymptotic variance of the GMM estimator.

. Econometrics for Finance Lecture 4 22 / 35

HAC estimation

Understanding HAC

Assume at is a stationary time series and E(at) = 0.

t12 t=1

V√=E at 2 2t=1

2E(a1 + a2 + a1a2 + a2a1)

1 2E(a2t ) + (E(atat+1) + E(atat−1)) . 2

. Econometrics for Finance Lecture 4

HAC estimation

Understanding HAC

Now, consider

t13 t=1

V√=E at 3 3t=1

= 3E((a1+a2+a3)(a1+a2+a3))

= 3E(a1 +a2 +a3 +a1a2 +a1a3 +a2a1

+ a2a3 + a3a1 + a3a2)

= 13(3E(a2t ) + 2 (E(atat+1) + E(atat−1)) + E(atat+2) + E(atat−2)).

. Econometrics for Finance Lecture 4

HAC estimation

Understanding HAC

The general version (for any k) is k

a t1 t=1

V √ = kk

[(kE(at ) + (k − 1) (E(atat+1) + E(atat−1))

(k − 2)(E(atat+2) + E(atat−2) + .. + E(a1ak) + E(aka1)] k k−|j|

k E(atat+j).

. Econometrics for Finance Lecture 4 25 / 35

Understanding HAC

1 We found:

2 When at = g(Xt+1, θ0), then

g(X ,θ0) t+1

HAC estimation

k k−|j| t=1

k j=−k k

E(atat+j). (2)

3 Notice that this is consistent with the way in which our estimator was written:

t=1 V √ = k

E(g(Xt+1,θ0)g(Xt+1+j,θ0) ). (3)

k T−j

ΦT = k−|j| 1 g(Xt+1,θT)g(Xt+1+j,θT)⊤ . (4) j=−k k T t=1

4 Naturally, as k → ∞, we have

k k−|j| t=1

k→∞ E(atat+j) →

k j=−k k

E(atat+j), (5)

which is the asymptotic expression of the variance (but infinite sums cannot be estimated and so we use Eq. (4)).

. Econometrics for Finance Lecture 4 26 / 35

Over-identifying restrictions

Testing for over-identifying restrictions

. Econometrics for Finance Lecture 4 27 / 35

Over-identifying restrictions

Hansen’s test

Once you have estimates, you can of course do inference on them. In the C-CAPM model, for example, is β = 0.95? Or, is γ = 3?

You can also evaluate whether the pricing errors are close to zero. Problem: are the moment conditions close to zero?

. Econometrics for Finance Lecture 4 28 / 35

Over-identifying restrictions

Hansen’s test

Implementation: The test statistic is

TQT (θT ) = TgT (θT )⊤Φ−1gT (θT ) Yd χ2N−d,

a Chi-Squared random variable with N − d degrees of freedom.

Non-mandatory proof

. Econometrics for Finance Lecture 4 29 / 35

Let us see GMM estimation in practice using Matlab …

. Econometrics for Finance Lecture 4 30 / 35

Hansen’s test: proof

We start with a Taylor’s expansion, stopped at the first order, around θ0:

1 T−1 1 T−1

g(Xt+1, θT ) = g(Xt+1, θ0)

1 T−1 ∂g(Xt+1,θ0)

and so we have

T t=1 ∂θ⊤ T−1 T−1

1a1√ √ g(Xt+1,θT)= √ g(Xt+1,θ0)+Γ0 T θT −θ0 ,

TT t=1 t=1

where“=a”signifies“asymptoticallyequivalentto…”.

. Econometrics for Finance Lecture 4 31 / 35

Now, recall from our proof of asymptotic normality for GMM, that

T θT −θ0 =−(Γ0Φ Γ0) Γ0Φ √

g(Xt+1,θ0).

a ⊤ −1 −1 ⊤ −1 1

Then, substituting in the main Taylor’s expansion,

T−1 T−1 1a1

g(Xt+1, θT ) = √ TT

g(Xt+1, θ0)

− Γ0(Γ⊤0 Φ−1Γ0)−1Γ⊤0 Φ−1 √ g(Xt+1, θ0)

1 T−1 0 0Tt=1

=a IN − Γ0(Γ⊤0 Φ−1Γ0)−1Γ⊤0 Φ−1 ΨT . 00

. Econometrics for Finance Lecture 4 32 / 35

Now, consider the following

−1/2 1 T−1

Φ √ g(Xt+1,θT) T T

a −1/21T−1

= Φ √ g(Xt+1, θT )

=a Φ−1/2 IN − Γ0(Γ⊤0 Φ−1Γ0)−1Γ⊤0 Φ−1 ΨT

=a IN − Φ−1/2Γ0(Γ⊤0 Φ−1Γ0)−1Γ⊤0 Φ−1/2 Φ−1/2ΨT 0000

=a PΦ−1/2ΨT, 0

where P is symmetric and idempotent.

. Econometrics for Finance Lecture 4 33 / 35

This, in turn, implies

=a Ψ⊤Φ−1/2P⊤PΦ−1/2Ψ T00T

=a Ψ⊤ Φ−1/2P Φ−1/2Ψ . T00T

Consider, now, the trace of P:

tr(IN − Φ−1/2Γ0(Γ⊤0 Φ−1Γ0)−1Γ⊤0 Φ−1/2)

= tr(IN ) − tr(Φ−1/2Γ0(Γ⊤0 Φ−1Γ0)−1Γ⊤0 Φ−1/2) 000

= N − tr((Γ⊤Φ−1Γ )−1(Γ⊤Φ−1Γ )) 000 000

= N−tr(Id)=N−d.

. Econometrics for Finance Lecture 4 34 / 35

Using Jordan’s decomposition:

= Ψ⊤T Φ−1/2P Φ−1/2ΨT 00

= 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 i=1

since the zis are independent standard normal random variables. (Chapter 0 contains a discussion of the properties used above.)

Back to test

T QT (θT )

. Econometrics for Finance Lecture 4 35 / 35

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