GMM, HAC estimators, & Standard Errors for Business Cycle … · GMM, HAC estimators, & Standard...
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GMM, HAC estimators,& Standard Errors for Business Cycle
Statistics
Wouter J. Den HaanLondon School of Economics
c© Wouter J. Den Haan
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GMM HAC estimators Examples
Overview
• Generic GMM problem• Estimation• Heteroskedastic and Autocorrelation Consistent (HAC)estimators to calcuate optimal weighting matrix and standarderrors
• Simple applications• OLS with correct standard errors• IV with multiple instruments• standard errors for business cycle statistics
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GMM HAC estimators Examples
GMM problem
Underlying true model:
E [h(xt; θ)] = 0p
• θ : m× 1 vector with parameters• x : n× 1 vector of observables• h(·) : p× 1 vector-valued function with p ≥ m• 0p : p× 1 vector with zeros
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GMM HAC estimators Examples
Examples
OLS:qt = a+ bpt + ut
E[
ututpt
]=
[00
]
• IV:qt = a+ bpt + ut
E[
ututzt
]=
[00
]
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GMM HAC estimators Examples
ExamplesDSGE:
c−γt = Et
[β (1+ rt+1) c−γ
t+1
]ut+1 = β (1+ rt+1) c−γ
t+1 − c−γt
Et [ut+1f (zt)] = 0p =⇒ E [ut+1f (zt)] = 0p
where
• zt is a vector with variables in information set in period t and• f (·) is a measurable function
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GMM HAC estimators Examples
GMM estimation
• Let
g(θ; YT) =T
∑t=1
h (xt; θ) /T,
where YT contains data
• Idea of estimation: choose θ such that g(θ; YT) is as small aspossible
• Throughout the slides, remember that g(θ; YT) is a mean
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GMM HAC estimators Examples
GMM estimation
θ̂T = arg minθ∈Θ
g(θ; YT)′WTg(θ; YT)
• whereWT : p× p weighting matrix
• No weighting matrix needed if p = m
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GMM HAC estimators Examples
Asymptotic standard errors
√T(
θ̂T − θ0
)−→ N(0, V)
V =(DWD′
)−1 DWΣ0W′D′(DWD′
)−1
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GMM HAC estimators Examples
Asymptotic standard errors
W = plimT−→∞
WT
D′ = plimT−→∞
∂g (θ; YT)
∂θ′
∣∣∣∣θ=θ0
θ0 : true θ
Σ0 =∞
∑j=−∞
E[h (xt; θ0) h
(xt−j; θ0
)′]
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GMM HAC estimators Examples
Terms showing up in V
• D measures how sensitive g is to changes in θ0.
• less precise estimates of θ0 if g is not very sensitive to changesin θ.
• Σ0 is the variance-covariance matrix of√
Tg(θ0; YT) asT −→ ∞.• if the mean underlying the estimation is more volatile =⇒estimates of θ0 less precise
• obtaining an estimate for Σ0 is often the tricky bit (more onthis below)
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GMM HAC estimators Examples
Two cases when formula for V simplifies
1 p = m (no overidentifying restrictions)
√T(
θ̂T − θ0
)−→ N(0, V)
V =(
DΣ−10 D′
)−1
2 using optimal weighting matrix, i.e., the matrix W thatminimizes V.
Woptimal = Σ−10√
T(
θ̂T − θ0
)−→ N(0, V)
V =(
DΣ−10 D′
)−1
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GMM HAC estimators Examples
Example 1
• YT = {xt}Tt=1 and we want to estimate the mean µ
h (xt; µ) = xt − µ
g (µ; YT) =∑T
t=1 (xt − µ)
T
µ̂T =∑T
t=1 xt
T
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GMM HAC estimators Examples
Example 1•
D = 1
• Σ̂T equals the variance of√
T(
∑Tt=1 (xt − µ)
)/T, which
equals variance of√
T(
∑Tt=1 xt
)/T
• Variance of ∑Tt=1 xt equals variance of x1 + covariance of
x1&x2 + · · ·+ covariance of x1&xT+ covariance of x1&x2 +variance x2+ etc.
• IF xt serially uncorrelated, then variance of√
T(
∑Tt=1 xt
)/T
equals variance of xt
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GMM HAC estimators Examples
Example 2• x1,t and x2,t have the same mean
• for simplicity assume that x1,t and x2,t are not correlated witheach other and are not serially correlated
• YT = {x1,t, x2,t}Tt=1
h (xt; µ) =
[x1,t − µx2,t − µ
]& D =
[1 1
]Σ0 =
[σ2
x10
0 σ2x2
]
W =
[1/σ2
x10
0 1/σ2x2
]
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GMM HAC estimators Examples
Estimating variance-covariance matrix
• g(θ; YT) is the mean of h(xt; θ)
• variance of g(θ; YT) is easy to calculate if h (xt; θ) is seriallyuncorrelated
• but in general it is diffi cult
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GMM HAC estimators Examples
Estimate variance of a mean
• The (limit of the) variance-covariance of√
Tg(θ; YT) equals
Σ0 =∞
∑j=−∞
E[h (xt; θ0) h
(xt−j; θ0
)′]This is the spectral density of h (xt; θ0) at frequency 0
• Since Σ0 is a variance-covariance matrix, it should be positivesemi-definite (PSD), that is, z′Σ0z should be non-negative forany non-zero column vector z
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GMM HAC estimators Examples
HAC estimators
1 Kernel-based (truncated, Newey-West, Andrews)
2 Parametric (Den Haan & Levin)
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GMM HAC estimators Examples
Estimate variance of a mean
Σ̂T =J
∑j=−J
κ(j; J)∑
min{T,T+j}t=max{1,j+1}
[h (xt; θ0) h
(xt−j; θ0
)′]T
where κ(·; J) is the kernel with bandwidth parameter J.
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GMM HAC estimators Examples
Truncated kernel
• Truncatedκ(j; J) =
{1 if |j| ≤ J0 o.w.
• disadvantages• potential bias because autocorrelation at j > J is ignored• answer not necessarily positive semi-definite
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GMM HAC estimators Examples
Newey-West (Bartlett) kernel
κ(j; J) = 1− jJ+ 1
for |j| ≤ J
• Newey-West kernel always gives PSD
• disadvantages• potential bias because autocorrelation at j > J is ignored• bias because k (j; J) < 1 for included terms
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GMM HAC estimators Examples
Choice of bandwidth parameter
• a high (low) J leads to• less (more) bias• more (less) sampling variability
• J should be higher if h (·; θ0) is more persistent
• Joptimal −→ ∞ as T −→ ∞ but at a slower rate
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GMM HAC estimators Examples
Choice of bandwidth parameter
• There are papers that give expressions for Joptimal,but adding a constant to these expressions does not affectoptimality (that is, the analysis only gives an optimal rate).
• Thus, in a finite sample you simply have to experiment andhope your answer is robust
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GMM HAC estimators Examples
Cost of imposing PSD
• Suppose h (xt; θ0) is an MA(1)=⇒ E
[h (xt; θ0) h
(xt−j; θ0
)]= 0 for |j| > 1
• If J = 1 then you have a biased estimate since k(1; 1) = 1/2• If J > 1 then you must include estimates of expectations thatwe know are zero
• Suppose h(xt; θ0) has a persistent and not persistentcomponent=⇒ you must use the same J for both components
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GMM HAC estimators Examples
VARHAC
Idea:
• Estimate a flexible time-series process (VAR) for ht, that is,
h (xt; θ0) =J
∑j=1
Ajh(xt−j; θ0
)+ εt
• Use implied spectrum at frequency zero as the estimate for Σ0
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GMM HAC estimators Examples
VARHAC
h (xt; θ0) =J
∑j=1
Ajh(xt−j; θ0
)+ εt
Then the implied spectrum at frequency zero, i.e., Σ0, equals
Σ̂T =
[Ip −
J
∑j=1
Aj
]−1
Σ̂ε,T
[Ip −
J
∑j=1
A′j
]−1
with
Σ̂ε,T =∑T
j=J+1 εtε′t
T
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GMM HAC estimators Examples
VARHAC
• Estimate is PSD by construction (PSD is obtained withoutimposing additional bias)
• Only bias is due to lag length potentially being too short• You can use standard model selection criteria (AIC, BIC) todetermine lag length
• You could estimate a VARMA• VARHAC also gives a consistent estimate for nonlinearprocesses since Σ0 only depends on second-order processeswhich can be captured with VAR
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GMM HAC estimators Examples
Back to GMM
• Exactly identified case:• estimate θ̂T and calculate VT
• Over-identified case:• You need WT to estimate θ̂T• Woptimal
T depends on Σ0, which depends on θ0
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GMM HAC estimators Examples
Back to GMM
• What to do in practice?• obtain (consistent) estimate of θ0 with simple WT (identityalthough scaling is typically a good idea)
• use this estimate to calculate WoptimalT
• use WoptimalT to get a more effi cient estimate of θ0
• calculate variance of θ̂T using(
D̂TΣ̂−1T D̂′T
)−1
• you could iterate on this
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GMM HAC estimators Examples
Examples
• OLS with heteroskedastic and serially correlated errors
• IV with multiple instruments
• Business cycle statistics
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GMM HAC estimators Examples
Key lesson of today’s lecture
If you can write the estimation problem as
E [h(xt; θ] = 0p
then you can use GMM and we know how to calculate standarderrors
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GMM HAC estimators Examples
OLS
qt = θpt + ut
h (xt) = utpt
= qtpt − θp2t
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GMM HAC estimators Examples
OLS & GMM
√T(
θ̂T − θ0
)−→ N(0, V)
V =(
DΣ−10 D′
)−1
D = E(
p2t
)
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GMM HAC estimators Examples
OLS & GMM
V =
((E(
p2t
))Σ−1
0
(E(
p2t
))′)−1
• If E[utut+τ] = 0 for τ 6= 0, then
Σ−10 = E
[(utpt)
2]and
V =
((E(
p2t
))E[(utpt)
2] (E(
p2t
))′)−1
• If errors are homoskedastic, then
E[(utpt)
2]= E
[u2
t
]E[p2
t
]V =
(E(
p2t
))−1E[u2
t
]
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GMM HAC estimators Examples
OLS & GMMSuppose that
qt = θpt + ut
ut = εtpt
εt i.i.d, Et [ptεt] = 0, and εt homoskedastic
• Then you should do GLSqt
pt= θ + εt
• GMM does not by itself do the transformation, i.e., it does notdo GLS, but it does give you standard errors that correct forheteroskedasticity
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GMM HAC estimators Examples
IV
Suppose that
qt = θpt + ut,E [utz1,t] = 0 and E [utz2,t] = 0
• With GMM you can find the optimal weighting of the twoinstruments
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GMM HAC estimators Examples
Business cycles statistics
1
ψ =standard deviation (ct)
standard deviation (yt)
2
ρ = correlation (ct, yt)
Statistics are easy to estimate, but what is the standard error?
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GMM HAC estimators Examples
Business cycles statistics
GMM problem for ψ
h (xt; θ0) =
ct − µcyt − µy(
yt − µy
)2ψ2 − (ct − µc)
2
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GMM HAC estimators Examples
Business cycles statistics
corresponding D matrix
D =
−1 0 00 −1 0
2E [(ct − µc)] 2E[(
yt − µy
)ψ2]
2E[(
yt − µy
)2ψ
]
=
−1 0 00 −1 0
0 0 2E[(
yt − µy
)2ψ
]
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GMM HAC estimators Examples
Steps to follow1 Use standard estimates for µc, µy, and ψ
2 Obtain estimate for D
3 Obtain estimate for Σ0 using
h(
xt; θ̂T
)=
ct − µ̂c,Tyt − µ̂y,T(
yt − µ̂y,T
)2ψ̂
2 −(
ct − µ̂c,T
)2
4 Obtain estimate for V using(D̂TΣ̂−1
T D̂′T)−1
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GMM HAC estimators Examples
Business cycles statistics
GMM problem for ρ
h(x′t; θ
)=
ct − µcyt − µy(
yt − µy
)2ψ2 − (ct − µc)
2(yt − µy
)2ψρ− (ct − µc)
(yt − µy
)
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GMM HAC estimators Examples
Business cycles statistics
corresponding D matrix
D =
−1 0 0 00 −1 0 0
0 0 2E[(
yt − µy
)2ψ
]0
0 0 E[(
yt − µy
)2ρ
]E[(
yt − µy
)2ψ
]
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GMM HAC estimators Examples
References
• Den Haan, W.J., and A. Levin, 1997, A practioner’s guide to robustcovariance matrix estimation
• a detailed survey of all the ways to estimate Σ0 with moredetailed information