Pages from ludvigson methodslecture 2

EZW Recursive PreferencesUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

CFL: semiparametric approach to estimate EZW modelwithout:

Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models

Need to proxy Rw,t+1 with observable returns.

Loglinearizing the model.

Parametric restrictions on law of motion or joint dist. of Ct

and Ri,t, or on value of key preference parameters.

Obtain estimates of β, RRA θ, EIS ρ−1

Evaluate EZW model’s ability to fit asset return datarelative to competing model specifications.

Investigate implications for Rw,t+1 and return to humanwealth.

Semiparametric approach is sieve minimum distance

(SMD) procedure.

First order conditions for optimal consumption choice:

)−ρ

ρ−θ

Ri,t+1 − 1

= 0 (8)

)−ρ

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

[(1 − β) + βRt

)1−ρ] 1

1−ρ

into (8):

)−ρ

Vt+1Ct+1

Ct+1Ct

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

)−ρ

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

[(1 − β) + βRt

)1−ρ] 1

1−ρ

into (8):

)−ρ

Vt+1Ct+1

Ct+1Ct

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

N test asset returns, {Ri,t+1}Ni=1. (9) is a x-sect asset pricing model.

)−ρ

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

[(1 − β) + βRt

)1−ρ] 1

1−ρ

into (8):

)−ρ

Vt+1Ct+1

Ct+1Ct

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

Moment restrictions (9) form the basis of empirical investigation.

)−ρ

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

[(1 − β) + βRt

)1−ρ] 1

1−ρ

into (8):

)−ρ

Vt+1Ct+1

Ct+1Ct

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

Moment restrictions (9) form the basis of empirical investigation.

Empirical model is semiparametric: δ ≡ (β, θ, ρ)′ denote finitedimensional parameter vector; Vt/Ct unknown function.

Assume VtCt

an unknown function F: R2 → R of form

(Vt−1

Ct−1,

Ct−1

Assume VtCt

(Vt−1

Ct−1,

Ct−1

Assume {Ct/Ct−1 : t = 1, ...} is strictly stationary ergodic;and F(·) is such that the process{Vt/Ct : t = 1, ...} isasymptotically stationary ergodic.

Assume VtCt

(Vt−1

Ct−1,

Ct−1

Justified if, for example,

Assume VtCt

(Vt−1

Ct−1,

Ct−1

∆ log(Ct+1) is (possibly nonlinear) function of a hiddenfirst-order Markov process xt.

Under general assumptions, information in xt issummarized by Vt−1/Ct−1 and Ct/Ct−1.

Assume VtCt

(Vt−1

Ct−1,

Ct−1

With a nonlinear Markov process for xt, F(·) can displaynonmonotonicities in both arguments.

Assume VtCt

(Vt−1

Ct−1,

Ct−1

Note: Markov assumption only a motivation for argumentsof F(·). Econometric methodology itself leaves LOM for∆ ln Ct unspecified.

Ft denotes agents information set at time t.

zt+1 contains all observations at t + 1 and

γi(zt+1, δ, F) ≡ β

)−ρ

,Ct+1Ct+1

Vt−1Ct−1

, CtCt−1

)}1−ρ− (1 − β)

]} 11−ρ

ρ−θ

Ri,t+1 − 1

γi(zt+1, δ, F) ≡ β

)−ρ

,Ct+1Ct+1

Vt−1Ct−1

, CtCt−1

)}1−ρ− (1 − β)

]} 11−ρ

ρ−θ

Ri,t+1 − 1

δo ≡ (βo, θo, ρo)′, Fo ≡ Fo(zt, δo) denote true parameters thatuniquely solve the conditional moment restrictions (Eulerequations):

E {γi(zt+1, δo, Fo (·, δo))|Ft} = 0 i = 1, ..., N, (10)

γi(zt+1, δ, F) ≡ β

)−ρ

,Ct+1Ct+1

Vt−1Ct−1

, CtCt−1

)}1−ρ− (1 − β)

]} 11−ρ

ρ−θ

Ri,t+1 − 1

δo ≡ (βo, θo, ρo)′, Fo ≡ Fo(zt, δo) denote true parameters thatuniquely solve the conditional moment restrictions (Eulerequations):

E {γi(zt+1, δo, Fo (·, δo))|Ft} = 0 i = 1, ..., N, (10)

Let wt ⊆ Ft. Equation (10) ⇒E {γi(zt+1, δo, Fo (·, δo))|wt} = 0. i = 1, ..., N.

Intuition behind minimum distance procedure:

Theory ⇒mt ≡ E {γi(zt+1, δo, Fo (·, δo))|wt} = 0. i = 1, ..., N.

Since mt = 0, mt must have zero variance, mean.

Thus can find params by minimizing variance or quadraticnorm: min E[(mt)2]. Don’t observe mt ⇒ need estimate mt.

Since (11) is cond. mean, must hold for each observation, t.

Obs > params, need way to weight each obs; using samplemean is one way: min ET[(mt)2].

Minimum distance procedure useful for distribution-freeestimation involving conditional moments.

Minimum distance procedure useful for distribution-freeestimation involving conditional moments: min ET[(mt)2].

Contrast with GMM, used for unconditional moments:E[f (xt, α)] = 0.

With GMM take sample counterpart to population mean:gT = ∑

Tt=1 f (xt, α) = 0.

Then choose parameters α to min g′TWgT.

Tt=1 f (xt, α) = 0.

Then choose parameters α to min g′TWgT.

With GMM we average and then square.

With SMD, we square and then average.

True parameters δo and Fo (·, δo) solve:

minδ∈D

infF∈V

E[m(wt, δ, F)′m(wt, δ, F)

where m(wt, δ, F) = E{γ(zt+1, δ, F)|wt}

γ(zt+1, δ, F) = (γ1(zt+1, δ, F), ..., γN(zt+1, δ, F))′

minδ∈D

infF∈V

For any candidate δ ≡ (β, θ, ρ)′ ∈ D, defineV∗ ≡ F∗ (zt, δ) ≡ F∗ (·, δ) as:

F∗ (·, δ) = arg infF∈V

minδ∈D

infF∈V

For any candidate δ ≡ (β, θ, ρ)′ ∈ D, defineV∗ ≡ F∗ (zt, δ) ≡ F∗ (·, δ) as:

F∗ (·, δ) = arg infF∈V

It is clear that Fo (zt, δo) = F∗ (zt, δo)

EZW Recursive Preferences: Two-Step ProcedureUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

First step: For any candidate δ ∈ D, an initial estimate ofF∗ (·, δ) obtained using SMD that consists of two parts:(Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07).

1 Replace the conditional expectation with a consistent,nonparametric estimator (specified later).

2 Approximate the unknown function F by a sequence offinite dimensional unknown parameters (sieves) FKT

Approximation error decreases as KT increases with T.

2 Approximate the unknown function F by a sequence offinite dimensional unknown parameters (sieves) FKT

Approximation error decreases as KT increases with T.

Second step: estimates of δo is obtained by solving asample minimum distance problem such as GMM.

EZW Recursive Preferences: First Step SMD Est of F∗Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

Approximate VtCt

Vt−1

Ct−1, Ct

Ct−1; δ)

with a bivariate sieve:

(Vt−1

Ct−1,

Ct−1; δ

)≈ FKT

(·, δ) = a0(δ)+KT

∑j=1

aj(δ)Bj

(Vt−1

Ct−1,

Ct−1

Sieve coefficients {a0, a1, ..., aKT} depend on δ

Basis functions {Bj(·, ·) : j = 1, ..., KT} have knownfunctional forms independent of δ

Initial value for VtCt

at time t = 0, denoted V0C0

, taken as aunknown scalar parameter to be estimated.

Given V0C0

j=1and data on consumption

Ct−1

t=1, use FKT

to generate a sequence{

Recall m(wt, δo, F∗(·, δo)) ≡ E {γ(zt+1, δo, F∗ (·, δo))|wt} = 0.

First-step SMD estimate F (·) for F∗ (·) based on

F (·, δ) = arg minFKT

∑t=1

m(wt, δ, FKT(·, δ))′m(wt, δ, FKT

(·, δ)),

m(wt, δ, FKT(·, δ)) any nonpara. estimator of m.

Do this for a three dimensional grid of values ofδ = (β, θ, ρ)′ .

Example of nonparametric estimator of m:

p0j(wt), j = 1, 2, ..., JT

dw → R be instruments.

pJT (·) ≡ (p01 (·) , ..., p0JT(·))′

Define T × JT matrix P ≡(pJT (w1) , ..., pJT (wT)

)′. Then:

m(w, δ, F) =

∑t=1

γ(zt+1, δ, F)pJT(wt)′(P′P)−1

)pJT (w)

m(·) a sieve LS estimator of m(w, δ, F).

Procedure equivalent to regressing each γi on instrumentsand taking fitted values as estimate of conditional mean.

EZW Preferences: First Step SMD Est of F∗Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

Attractive feature of this estimator of F∗: implemented as GMM

FT (·, δ) = arg minFT∈VT

[gT(δ,FT ; yT)

]′{IN⊗

(P′P)−1}

︸︷︷︸W

[gT(δ,FT; yT)

where yT =(z′T+1, ...z′2, w′

T, ...w′1

)′denotes vector of all obs and

gT(δ,FT ; yT) =1

∑t=1

γ(zt+1, δ,FT)⊗pJT (wt) (13)

[gT(δ,FT ; yT)

]′{IN⊗

(P′P)−1}

︸︷︷︸W

[gT(δ,FT; yT)

where yT =(z′T+1, ...z′2, w′

T, ...w′1

gT(δ,FT ; yT) =1

∑t=1

Weighting gives greater weight to moments more highlycorrelated with instruments pJT(·).

[gT(δ,FT ; yT)

]′{IN⊗

(P′P)−1}

︸︷︷︸W

[gT(δ,FT; yT)

where yT =(z′T+1, ...z′2, w′

T, ...w′1

gT(δ,FT ; yT) =1

∑t=1

Weighting gives greater weight to moments more highlycorrelated with instruments pJT(·).

Weighting can be understood intuitively by noting that variationin conditional mean m(wt, δ, F) is what identifies F∗(·, δ).

EZW Preferences: Second Step GMM Est of δoUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

Under correct specification, δo satisfies :

E {γi(zt+1, δo, F∗ (·, δo)) ⊗ xt} = 0, i = 1, ..., N.

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.

E {γi(zt+1, δo, F∗ (·, δo)) ⊗ xt} = 0, i = 1, ..., N.

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.

Regardless the model is correctly or incorrectly specified,estimate δ by minimizing GMM objective:

δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

E {γi(zt+1, δo, F∗ (·, δo)) ⊗ xt} = 0, i = 1, ..., N.

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.

δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

Examples: W = I, W = G−1T .

E {γi(zt+1, δo, F∗ (·, δo)) ⊗ xt} = 0, i = 1, ..., N.

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.

δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

F (·, δ) not held fixed in this step: depends on δ!

E {γi(zt+1, δo, F∗ (·, δo)) ⊗ xt} = 0, i = 1, ..., N.

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.

δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

Estimator F (·, δ) obtained using min. dist over a grid of values δ.

E {γi(zt+1, δo, F∗ (·, δo)) ⊗ xt} = 0, i = 1, ..., N.

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.

δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

Estimator F (·, δ) obtained using min. dist over a grid of values δ.

Choose the δ and corresponding F (·, δ) that minimizes GMMcriterion.

EZW Recursive Preferences: Two Step EstimationUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

Why estimate in two steps? All params could be estimatedin one step by minimizing the SMD criterion.

Less desirable for asset pricing:

1 Want estimates of RRA and EIS to reflect values required tomatch unconditional risk premia. Not possible using SMDwhich emphasizes conditional moments.

2 SMD procedure effectively changes set of test assets–linearcombinations of original portfolio returns. But we may beinterested in explaining original returns!

3 Linear combinations may imply implausible long and shortpositions, do not necessarily deliver a large spread inunconditional mean returns.

Procedure allows for model misspecification:

Euler equation need not hold with equality.

As before, compare models by relative magnitude ofmisspecification, rather than...

...asking whether each model individually fits dataperfectly (given sampling error).

Use W = G−1 in second step, compute HJ distance.

Test whether HJ distances of competing models arestatistically different (White reality check–Chen andLudvigson ’09).

EZW Preferences With Restricted Dynamics:Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07

Bansal, Gallant, Tauchen ’07: SMM estimation of LRRmodel: Bansal & Yaron ’04.

Structural estimation of EZW utility, restricting to specificlaw of motion for cash flows (“long-run risk”LRR).

Cash flow dynamics in BGT version of LRR model:

∆ct+1 = µc + xc,t + σtεc,t+1

∆dt+1 = µd + φx xc,t︸︷︷︸LR risk

+ φsst + σεdσtεd,t+1

xc,t = φxc,t−1 + σεx σεxc,t

σ2t = σ2 + ν(σ2

t−1 − σ2) + σwwt

st = (µd − µc) + dt − ct

εc,t+1, εd,t+1, εxc,t, wt ∼ N.i.i.d (0, 1)

SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,Tauchen ’97, Tauchen ’97.

Outline of SMM steps

1 Solve the model over grid of values of deep parameters:ρd = (β, θ, ρ, φ, φx, µc, µd, σ, σǫd

, σǫxν, φs, σw)′

2 For each value of ρd on the grid, combine solutions withlong simulation of length N of model.

3 Simulation: Monte Carlo draws from the Normaldistribution for primitive shocks.

4 Form obs eqn for simulated and historical data, e.g.,yt = (dt − ct, ct − ct−1, pt − dt, rd,t)

5 Choose value ρd that most closely “matches” momentsbetween dist of simulated and historical data ( “match”made precise below.)

Let {yt}Nt=1 denote simulated data (in obs eqn).

Let {yt}Tt=1 denote historical data on same variables.

Auxiliary model of hist. data: e.g., VAR, with densityf (yt|yt−L, ...yt−1, α), good LOM for data–f -model.

Score function of f -model:

sf (yt|yt−L, ...yt−1, α) =∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

Score function of f -model:

sf (yt|yt−L, ...yt−1, α) =∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

QMLE estimator of auxiliary model on historical data

α = arg maxα

LT(α, {yt}Tt=1)

LT(α, {yt}Tt=1) =

∑t=L+1

ln f (yt|yt−L, ..., yt−1, α)

First-order-condition:

∂αLT(α, {yt}T

t=1) = 0 or,1

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.

∂αLT(α, {yt}T

t=1) = 0 or,1

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.

Idea: since above, good estimator for ρd is one that sets

∑t=L+1

sf (yt(ρd)|yt−L(ρd), ..., yt−1(ρd), α) ≈ 0.

∂αLT(α, {yt}T

t=1) = 0 or,1

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.

∑t=L+1

If dim(α) >dim(ρd), use GMM:

mT(ρd, α)︸︷︷︸dim(α)×1

∑t=L+1

sf (yt(ρd)|yt−L(ρd), ..., yt−1(ρd), α)

∂αLT(α, {yt}T

t=1) = 0 or,1

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.

∑t=L+1

If dim(α) >dim(ρd), use GMM:

mT(ρd, α)︸︷︷︸dim(α)×1

∑t=L+1

sf (yt(ρd)|yt−L(ρd), ..., yt−1(ρd), α)

The GMM estimator is

ρd = arg minρd

{mT(ρd, α)′I−1mT(ρd, α)

GMM: ρd = arg minρd

{mT(ρd, α)′I−1mT(ρd, α)}.

I−1 is inv. of var. of score, data determined from f -model

∑t=1

∂αln[f (yt|yt−L, ..., yt−1, α)]

∑t=1

∂αln[f (yt|yt−L, ..., yt−1, α)]

Sims {y}Nt=1 follow stationary dens. p(yt−L, ..., yt|ρd). Note:

no closed-form for p(·|ρd).

∑t=1

∂αln[f (yt|yt−L, ..., yt−1, α)]

Intuition: mT(ρd, α)as→ m(ρd, α) as N → ∞, where

m(ρd, α) =∫

· · ·∫

s(yt−L, ..., yt, α)p(yt−L, ..., yt|ρd)dyt−L · · ·dyt

∑t=1

∂αln[f (yt|yt−L, ..., yt−1, α)]

m(ρd, α) =∫

· · ·∫

⇒ use Monte Carlo compute expect. of s(·) under p(·|ρd).

m(ρd, α) =∫

· · ·∫

m(ρd, α) =∫

· · ·∫

If f = p above is mean of scores of likelihood. Should bezero, given f.o.c for MLE estimator.

m(ρd, α) =∫

· · ·∫

Thus, if data do follow the structural model p(·|ρd), thenm(ρo

d, αo) = 0, forms basis of a specification test.

m(ρd, α) =∫

· · ·∫

Thus, if data do follow the structural model p(·|ρd), thenm(ρo

d, αo) = 0, forms basis of a specification test.

Summary: solve model for many values of ρd, store longsimulations of model each time, do one-time estimation ofauxiliary f -model. Choose ρd to minimize GMM criterionabove.

Advantages of using score functions as moments:

1 Computational: one-time estimation of structural model;useful if f -model is nonlinear.

2 If f -model good description of data, under null, MLEefficiency is obtained.

If dim(α) >dim(ρd), score-based SMM is consistent,asymptotically normal, assuming:

That the auxiliary model is rich enough to identifynon-linear structural model. Sufficient conditions foridentification unknown.

Big issue: are these the economically interesting moments?Regards both choice of moments, and weighting function.

Consumption-Based Asset Pricing: Final Thoughts

Little work linking financial markets to macroeconomicrisks, given by primitives in the IMRS over consumption.

No model that relates returns to other returns can explainasset prices in terms of primitive economic shocks. Suchmodels of SDF only describe asset prices.

So far many consumption-based models have beenevaluated using calibration exercises ⇒

A crucial next step in evaluating consumption-basedmodels is structural econometric estimation. But...

...models are imperfect and will never fit data infallibly.

Argue here for need to move away from testing if modelsare true, towards comparison of models based on magnitudeof misspecification.

Example: scaled consumption models.

Rather than ask whether scaled models are true...

...ask whether allowing for state-dependence of SDF onconsumption growth reduces misspecification over theanalogous non-state-dependent model.

Macroeconomic data, unlike financial measured with error.

⇒ Can’t expect such models to perform as well as financialfactor models of SDF.

True systematic risk factors are macroeconomic in nature;asset prices derived endogenously from these.

Financial factors could represent projection of true Mt onportfolios (i.e., mimicking portfolios).

In which case, they will always perform at least as well, orbetter than, mismeasured macro factors from true Mt ⇒

Not sensible to run horse races between financial factormodels and macro models.

Goal: not to find better factors, but rather to explainfinancial factors from deeper economic models.

Pages from ludvigson methodslecture 2

Documents

Transcript of Pages from ludvigson methodslecture 2

Pages from 20100505_PF_2010_studio4

Pages From 362

Pages from 20100505_PF_2010_vs_makeitwork

Pages from S24h20150212-

Pages From c2TR

Pages from SARHAN_Transcripts

Ludvigson methodslecture1

Pages from iogear_KVM_GCS632U

Lettau, Ludvigson 2002. Time Varying Risk Premia Coct of Cap Alter Impli Q Theory

Pages From Brochure_INLGESE

Pages from 20100505_PF_2010_studio1

Pages From Apics_cscp

Pages From Phaser_3100MFPServiceManual2

Pages from thebestfromthetanningblog2013

Pages From MDDbook

Pages From 2010_amrfangary

Extracted pages from 4Extracted pages from 248610400-NATO-Logistics-Handbook-2012.pdf

Sydney C. Ludvigson Serena Ng y Columbia University New ...sn2294/papers/handbook-big.pdfSydney C. Ludvigson New York University and NBER Serena Ng y Columbia University July 20, 2009

Pages from Boilers_and_ThermicFluidHeaters.pdf

Pages from tialo_uspenski_pechat