Broadening the Scope of Bootstrap in Complex Problems

Transcripción

Broadening the Scope of Bootstrap in Complex Problems
Víctor Aguirre Torres
ITAM y CIMAT
Seminario Dirección General de Investigación Económica, Banco de
México
9 de Marzo de 2007
Joint work with Manuel Domínguez, ITAM
Víctor Aguirre Torres ITAM y CIMAT
9 de Marzo de 2007 Joint work with Manuel D
Broadening
Seminario Dirección
the Scope General
of Bootstrap
de Investigación Económica, Banco de México () / 16
Outline
Unconditional Bootstrap Pivots (UBP’s)
For large n and using UBP’s, it is possible to draw statistically
meaningful inferences (size and power) with a relatively small number
of bootstrap replications, say B 50.
Examples: procedures that require smoothing or computer intensive
methods.
Increasing B produces marginal improvement.
Broadening
the Scope General
of Bootstrap
Set Up
Data Zn = fZ1 , Z2 , ..., Zn g
DGP FZ1 Z2 ...Zn .
Operator of interest
Tn ! d T
Limiting distribution unknown.
Known but large sample approximation unreliable.
Known but unable to compute quantiles.
Broadening
the Scope General
of Bootstrap
Bootstrap Approximation
Bootstrap the DGP and obtain Zn .
Compute Tn .
Justi…cation:
FT n jZn ! FT in probability, or
Tn !d T in probability.
With basically the same regularity conditions for Tn to converge to T .
Broadening
the Scope General
of Bootstrap
Main Theorem
Let Zn = fZ1 , Z2 , ..., Zn g be a collection of rv´ s, and let Sn (Zn ) be
such that FSn ! FS .
n
o
Let Hn = H1 , H2 , ..., Hm (n ) be a collection of rv´ s, m (n) non
decreasing integer valued, and let Rn (Zn , Hn ) be such that
FR n jZn ! FR in probability, where FR is not stochastic.
Then FSn R n ! FS FR
Example: Tn =
p
θn
n b
θ and Tn =
p
θn
n b
b
θn
Broadening
the Scope General
of Bootstrap
Consequence on Bootstrap
If Tn !d T and Tn,b !d T in probability for b = 1, 2, ..., B. Then,
for …xed B, as n increases to in…nity we have that
Tn , Tn,1 , ..., Tn,B !d T B ,
where T B is a rv with 1 + B iid coordinates.
Tn , Tn,b and Tn,b 0 are unconditionally asymptotically independent and
equally distributed.
Broadening
the Scope General
of Bootstrap
Example: Quantile Regression
Data: (Xi , Yi )
i = 1, ..., n.
Model: Yi = β1 + β2 Xi + Ui
fU unknown.
Robust estimation:
b
β = arg min ∑ qp (Yi
qp (u ) = u pIfu
0g
(1
β1
p )Ifu <0 g
β2 Xi )
0<p<1
Non di¤erentiable objective function.
Broadening
the Scope General
of Bootstrap
AN in Quantile Regression
Take for example p = 1/2, L1 estimator.
Under regularity conditions, Koenker and Basset (1978, 1982):
p
where
n b
β2
β2 !d N (0, σ2 )
σ2 = fU2 (FU 1 (1/2))V (X )/4
Inference on requires smoothing to estimate fU and then σ.
Inference depends on the choice of smoothing constant.
Broadening
the Scope General
of Bootstrap
Usual Application of Bootstrap, Quantile Regression
Bootstrap resample: (Xi , Yi )
i = 1, ..., n.
Under the same regularity conditions for AN, Hahn (1995)
p
n b
β2
b
β2 !d N (0, σ2 )
Using percentiles of b
β2b directly is equivalent to AN.
t-bootstrap requires computation of
p
zb =
n b
β2
bb
σ
b
β2
, b = 1, ..., B
This demands smoothing for each b = 1, ..., B to estimate b
fU and a
large B (above 1000).
Broadening
the Scope General
of Bootstrap
Alternative: UBP
Consider
p
β2
n b
β2 ,
p
β2,1
n b
b
β2 , ...,
p
β2,B
n b
From the main theorem the limit is i.i.d. N (0, σ2 ). Then:
tB =
(bβ2 β2 )
1
b
B ∑b =1 ( β2,b
B
b
β2 )
2
!d t (B )
t
b
β2
.
UBP
No smoothing required.
No consistency of denominator to estimate σ required.
Test of hypothesis. Asymptotic size , global power of one.
Broadening
the Scope General
of Bootstrap
MC exercise, signi…cance level.
tB quantile regression, (10,000 MC replications)
B=5
300
250
Relative Error
200
t* ; α=0.1
t* ; α=0.05
t* ; α=0.01
N ; α=0.1
N ; α=0.05
N ; α=0.01
150
100
50
0
0
200
400
600
800
1000
1200
-50
Sample size
relative error= αnα α
Broadening
the Scope General
of Bootstrap
Local Power for B Fixed
Local alternatives H1n : θ = θ 0 +
cσ
p
.
n
Power for B and n …nite:
π B ,n (c ) = Pc (jtB j > tB ,1
α/2 )
Limit power, B …xed:
π B (c ) = limn !∞ π B ,n (c )
π B (c ) = P (jt (B, c )j > tB ,1
α/2 )
Limit power, B ! ∞
π (c ) = P (jN (c, 1)j > z1
α/2 )
Broadening
the Scope General
of Bootstrap
Local Power Curves as a Function of B
1
0.9
Power of Unconditional Pivot
0.8
0.7
0.6
5
10
20
50
Limit
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
Standardized Difference
Broadening
the Scope General
of Bootstrap
Joint Tests
Tn !d Np (0, Σ)
Then, for …xed B and large n
Hn,B = (Tn )T
h
1
B
∑Bb=1
Tn,b
Tn,b
T
i
1
Tn !d TB2 ,p
Hotelling’s T-square,
Bq
α
B q +1 F (q, B
T2
q + 1)
UBP
Broadening
the Scope General
of Bootstrap
Another Example
Construct an asymptotic pivot by combining
Tn , Tn,1 , ..., Tn,B
For example, if Tn !d κχ2 (ν) then
Hn,B =
F
Tn
(∑Bb=1 T n,b )/B
!d F (ν, νB )
UBP
Broadening
the Scope General
of Bootstrap
Summary
For very complex procedures and large n bootstrap could be applied
with drastic reduction in B.
For certain procedures that require smoothing, for large n bootstrap
could be applied avoiding the use of smoothing constant.
Choice of B is nonstochastic. Properties of inferential procedure
remain known.
Convenient determination of B considering size and power.
No need of imposing extra regularity conditions on the DGP or on the
asymptotic properties of the operator.
Broadening
the Scope General
of Bootstrap

Broadening the Scope of Bootstrap in Complex Problems

Transcripción

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