OTHER TOPICS IN THE SINGLE EQUATION LINEAR MODEL

MicroEconometría Avanzada

Ignacio Lobato (ITAM)

OTHER TOPICS IN THE SINGLE EQUATION LINEAR

MODEL

Estimated explanatory and instrumental variables

(see Wooldridge, chap 6)

Consider the linear model:

Y = β₀+Z0β+U, with E(U) =0,

where

Z=h(S,δ) with EkZk2 <∞,

andh is a known function, δ an unknown element of the parameter

space ∆,S is a d 1 vector of observable variables.

There is also a k 1 vector of functions gsuch that,

X=g(S,λ), EkXk2 <∞ andE(XU) =0,

for some functiong and an unknown parameter vector λ2Λ.

Estimated explanatory and instrumental variables

(see Wooldridge, chap 6)

Consider the linear model:

Y = β₀+Z0β+U, with E(U) =0,

where

Z=h(S,δ) with E_kZ_k2 <∞,

andh is a known function, δ an unknown element of the parameter

space ∆,S is a d 1 vector of observable variables.

There is also a k 1 vector of functions gsuch that,

X=g(S,λ), EkXk2 <∞ andE(XU) =0,

for some functiong and an unknown parameter vector λ2Λ.

Estimated explanatory and instrumental variables

(see Wooldridge, chap 6)

Consider the linear model:

Y = β₀+Z0β+U, with E(U) =0,

where

Z=h(S,δ) with EkZk2 <∞,

andh is a known function, δ an unknown element of the parameter space ∆,S is a d 1 vector of observable variables.

There is also a k 1 vector of functions gsuch that,

X=g(S,λ), EkXk2 <∞ andE(XU) =0,

for some functiong and an unknown parameter vector λ2Λ.

Estimated explanatory and instrumental variables

(see Wooldridge, chap 6)

Consider the linear model:

Y = β₀+Z0β+U, with E(U) =0,

where

Z=h(S,δ) with EkZk2 <∞,

andh is a known function, δ an unknown element of the parameter

space ∆,S is a d 1 vector of observable variables.

There is also a k 1 vector of functions gsuch that, X=g(S,λ), E_kX_k2 <∞ andE(XU) =0,

for some functiong and an unknown parameter vector λ2Λ.

Estimated explanatory and instrumental variables

(see Wooldridge, chap 6)

Consider the linear model:

Y = β₀+Z0β+U, with E(U) =0,

where

Z=h(S,δ) with EkZk2 <∞,

andh is a known function, δ an unknown element of the parameter

space ∆,S is a d 1 vector of observable variables.

There is also a k 1 vector of functions gsuch that,

X=g(S,λ), EkXk2 <∞ andE(XU) =0,

for some functiong and an unknown parameter vector λ2Λ.

Some remarks are in order:

1 Some elements _δ and _λare known, which means that some components ofZ andXare observable.

2 Some elements of ZandX are identical.

3 h andg may share the same or di¤erent variables in S.

Some remarks are in order:

1 Some elements _δ and _λare known, which means that some

components ofZ andXare observable.

2 Some elements of ZandX are identical.

3 h andg may share the same or di¤erent variables in S.

Some remarks are in order:

1 Some elements _δ and _λare known, which means that some

components ofZ andXare observable.

2 Some elements of ZandX are identical.

3 h andg may share the same or di¤erent variables in S.

EXAMPLE: Z= 0 B B B B @

Z(1)

δ1+Z(2)0δ2

X(1)

1 C C C C A, δ=

δ1 δ2 , X= 0 B @

X(1) X(2)

Φ Z(2) _X(2)_λ

1

λ2 λ3

1 C A, λ=

2 4 λ1 λ2 λ3 3 5,

with Φ the standard normal distribution.

EXAMPLE: Z= 0 B B B B @

Z(1)

δ1+Z(2)0δ2

X(1)

1 C C C C A, δ=

δ1 δ2 , X= 0 B @

X(1) X(2) Φ Z(2) _X(2)_λ

1

λ2 λ3 1 C A, λ=

2 4 λ1 λ2 λ3 3 5,

with Φ the standard normal distribution.

EXAMPLE: Z= 0 B B B B @

Z(1)

δ1+Z(2)0δ2

X(1)

1 C C C C A, δ=

δ1 δ2 , X= 0 B @

X(1) X(2) Φ Z(2) _X(2)_λ

1

λ2 λ3

1 C A, λ=

2 4 λ1 λ2 λ3 3 5,

with Φ the standard normal distribution.

We haveiid observations(Yi,Si),i =1, ...,n of(Y,S).

It is also assumed that there are estimators of δ andλ,δn and λn respectively, such that:

n1/2(δn δ) =Op(1) and n1/2(λn λ) =Op(1).

Assuming that the order and rank conditions are satis…ed and, hence,

βis identi…ed, the IV estimator is:

βIV

n = En

~

Z_nX~0_n En X~n~X0n

1

En ~Xn~Zn0

1

En ~Zn~X

0

n En ~XnX~

0

n

1

En ~XnY

with ~Zni =h(Si,δn),X~ni=g(Si,λn),and for any vectora,

a= 1

a .

We haveiid observations(Yi,Si),i =1, ...,n of(Y,S).

It is also assumed that there are estimators of δ andλ,δn and λn

respectively, such that:

n1/2(δn δ) =Op(1) and n1/2(λn λ) =Op(1).

Assuming that the order and rank conditions are satis…ed and, hence,

βis identi…ed, the IV estimator is:

βIV

n = En

~

Z_nX~0_n En X~n~X0n

1

En ~Xn~Zn0

1

En ~Zn~X

0

n En ~XnX~

0

n

1

En ~XnY

with ~Zni =h(Si,δn),X~ni=g(Si,λn),and for any vectora,

a= 1

a .

We haveiid observations(Yi,Si),i =1, ...,n of(Y,S).

It is also assumed that there are estimators of δ andλ,δn and λn respectively, such that:

n1/2(δn δ) =Op(1) and n1/2(λn λ) =Op(1).

Assuming that the order and rank conditions are satis…ed and, hence, βis identi…ed, the IV estimator is:

βIV

n = En

~

Z_nX~0_n En X~n~X0n

1

En ~Xn~Zn0

1

En ~Zn~X

0

n En ~XnX~

0

n

1

En ~XnY

with ~Zni =h(Si,δn),X~ni=g(Si,λn),and for any vectora,

a= 1

a .

We haveiid observations(Yi,Si),i =1, ...,n of(Y,S).

It is also assumed that there are estimators of δ andλ,δn and λn respectively, such that:

n1/2(δn δ) =Op(1) and n1/2(λn λ) =Op(1).

Assuming that the order and rank conditions are satis…ed and, hence,

βis identi…ed, the IV estimator is:

βIV

n = En

~

Z_nX~0_n En X~n~X0n

1

En ~Xn~Zn0

1

En ~Zn~X

0

n En ~XnX~

0

n

1

En ~XnY

with ~Zni =h(Si,δn),~Xni=g(Si,λn),and for any vectora,

a= 1 a .

ICONSISTENCY : Assume thatZ andXare di¤erentiable in a neighborhood of δ and λ,respectively,

and there exist functions h˙ :Rd !R+ _{andg˙ :R}d _!R+ _{such that:}

sup d₂∆krδ

h(S,d)_k h˙(S) with Ehh˙(S)2i<∞,

sup l_2Λkrλ

g(S,l)_k g˙ (S) withEhg˙ (S)2i< ∞,

with

rbf (b) =¯

∂

∂b0f (b) _b=b¯

.

There may be some rows of zeroes in _rδh(S,d) and_rλh(S,d).

ICONSISTENCY : Assume thatZ andXare di¤erentiable in a neighborhood of δ and λ,respectively,

and there exist functions h˙ :Rd _!R+ andg˙ :Rd _!R+ such that: sup

d₂∆krδ

h(S,d)_k h˙(S) with Ehh˙(S)2i<∞, sup

l2Λkrλg(S,l)k g˙(S) withE h

˙

g(S)2i<∞,

with

rbf (b) =¯

∂

∂b0f (b) _b=b¯

.

There may be some rows of zeroes in _rδh(S,d) and_rλh(S,d).

ICONSISTENCY : Assume thatZ andXare di¤erentiable in a neighborhood of δ and λ,respectively,

and there exist functions h˙ :Rd _!R+ andg˙ :Rd _!R+ such that:

sup d₂∆krδ

h(S,d)_k h˙(S) with Ehh˙(S)2i<∞,

sup

l2Λkrλg(S,l)k g˙(S) withE

h

˙

g(S)2i<∞,

with

rbf (b¯) =

∂

∂b0

f (b)

b=b¯

.

There may be some rows of zeroes in _rδh(S,d) and_rλh(S,d).

ICONSISTENCY : Assume thatZ andXare di¤erentiable in a neighborhood of δ and λ,respectively,

and there exist functions h˙ :Rd _!R+ andg˙ :Rd _!R+ such that:

sup d₂∆krδ

h(S,d)_k h˙(S) with Ehh˙(S)2i<∞,

sup

l2Λkrλg(S,l)k g˙(S) withE

h

˙

g(S)2i<∞,

with

rbf (b) =¯

∂

∂b0 f (b)

b=b¯

.

There may be some rows of zeroes in _rδh(S,d) and_rλh(S,d).

Notice that,

En ~ZnX~n0 = En ZX0 +En ~Zn Z X~n X

0

+En ~Zn Z X +En Z ~Xn X

0 ,

There exists δ¯(_njl): ¯δ_n(jl) δ kδn δk,and

¯

λ_n(jl) : λn(jl) λ kλn λksuch that,

En ~Zn Z X~n X

0

jl

= En rδh S,δ¯

(jl)

n (δn δ) (λn λ)0rλg S,λ¯

(jl)

n

0

jl

En h˙ (S)g˙ (S) kδn δk kλn λk = Op(1) Op n 1/2 Op n 1/2

= Op n 1 alll,j =1, ...,k,

provided that h˙(S) g˙ (S) satis…es aLLN.

There exists δ¯(_njl): ¯δ_n(jl) δ kδn δk,and ¯

λ_n(jl) : λn(jl) λ kλn λksuch that,

En ~Zn Z X~n X

0

jl

= En rδh S,δ¯ (jl)

n (δn δ) (λn λ)0rλg S,λ¯ (jl)

n

0

jl

En h˙ (S)g˙ (S) kδn δk kλn λk = Op(1) Op n 1/2 Op n 1/2

= Op n 1 alll,j =1, ...,k,

provided that h˙(S) g˙ (S) satis…es aLLN.

Also,

En ~Zn Z X En h˙(S)kXk kδn δk+Op n 1/2

= Op(1) Op n 1/2 +Op n 1/2 .

En Z ~Xn X

0

En(kZkg˙ (S)) kλn λk+Op n 1/2

= Op(1) Op n 1/2 +Op n 1/2 .

Therefore, after applying the LLN,

En ~Zn~X

0

n =E ZX0 +op(1).

Also,

En ~Zn Z X En h˙(S)kXk kδn δk+Op n 1/2

= Op(1) Op n 1/2 +Op n 1/2 .

En Z ~Xn X

0

En(kZkg˙ (S)) kλn λk+Op n 1/2

= Op(1) Op n 1/2 +Op n 1/2 .

Therefore, after applying the LLN,

En ~Zn~X

0

n =E ZX0 +op(1).

Likewise,

En ~XnX~

0

n = E XX0 +op(1)

En ~XnY = E(XY) +op(1)

Hence,

βIV_n =

h

E ZX0 E XX0 1E XZ0 +op(1)

i 1

h

E ZX0 E XX0 1E(XY) +op(1)

i

= β+op(1)

Likewise,

En ~XnX~

0

n = E XX0 +op(1)

En ~XnY = E(XY) +op(1)

Hence,

βIV_n = h

E ZX0 E XX0 1E XZ0 +op(1)

i 1

h

E ZX0 E XX0 1E(XY) +op(1)

i

= β+op(1)

IASYMPTOTIC NORMALITY :Assume that hand gare twice di¤erentiable in a neighborhood ofδ andλ,respectively,

and there exist functions h¨ :Rd _!R+ _{andg¨ :R}d _!R+ _{such that:}

sup d₂∆krδδ

h(S,d)_k h¨(S) with Ehh¨(S)2i<∞,

sup l₂Λkrλλ

g(S,l)_k g¨(S) with E

h

¨

g(S)2i<∞,

with

rbbf (b¯) =

∂

∂b∂b0f (b) _b=b¯

.

IASYMPTOTIC NORMALITY :Assume that hand gare twice di¤erentiable in a neighborhood ofδ andλ,respectively,

and there exist functions h¨ :Rd _!R+ _{andg¨ :R}d _!R+ _{such that:} sup

d₂∆krδδ

h(S,d)_k h¨(S) with Ehh¨(S)2i<∞, sup

l₂Λkrλλ

g(S,l)_k g¨(S) with E h

¨

g(S)2i<∞,

with

rbbf (b¯) =

∂

∂b∂b0f (b) _b=b¯

.

IASYMPTOTIC NORMALITY :Assume that hand gare twice di¤erentiable in a neighborhood ofδ andλ,respectively,

and there exist functions h¨ :Rd _!R+ _{andg¨ :R}d _!R+ _{such that:}

sup d₂∆krδδ

h(S,d)_k h¨(S) with Ehh¨(S)2i<∞,

sup l₂Λkrλλ

g(S,l)_k g¨(S) with E

h

¨

g(S)2i<∞,

with

rbbf (b¯) =

∂

∂b∂b0f (b) _b=b¯

.

Notice that:

n1/2 βIV

n β = En

~

Z_nX~0_n En X~n~X0n

1

En X~n~Z0n

1

En ~Zn~X

0

n En ~XnX~

0

n

1

n1/2En ~Xn Z ~Zn

0

β+n1/2En X~nU .

First,

n1/2En ~Xn Z ~Zn

0

β=

= n1/2En X Z ~Zn

0

β

| {z }

=_En(Xn1/2(δn δ)0rδh(S,δ)0)β+Op(n1/2k(δn δ)_k2)

+n1/2En ~Zn Z X~n X

0

β

| {z }

=Op(n 1/2)

=En Xn1/2(δn δ)0rδh(S,δ)0 β+Op n 1/2

Notice that:

n1/2 βIV

n β = En

~

Z_nX~0_n En X~n~X0n

1

En X~n~Z0n

1

En ~Zn~X

0

n En ~XnX~

0

n

1

n1/2En ~Xn Z ~Zn

0

β+n1/2En X~nU .

First,

n1/2En ~Xn Z ~Zn

0 β=

= n1/2En X Z ~Zn

0

β

| {z }

=_En(Xn1/2(δn δ)0rδh(S,δ)0)β+Op(n1/2k(δn δ)_k2)

+n1/2En ~Zn Z X~n X

0

β

| {z }

=Op(n 1/2)

=En Xn1/2(δn δ)0rδh(S,δ)0 β+Op n 1/2

Notice that:

n1/2 βIV

n β = En

~

Z_nX~0_n En X~n~X0n

1

En X~n~Z0n

1

En ~Zn~X

0

n En ~XnX~

0

n

1

n1/2En ~Xn Z ~Zn

0

β+n1/2En X~nU .

First,

n1/2En ~Xn Z ~Zn

0

β=

= n1/2En X Z ~Zn

0 β

| {z }

=_En(Xn1/2(δn δ)0rδh(S,δ)0)β+Op(n1/2k(δn δ)_k2)

+n1/2En ~Zn Z X~n X

0

β

| {z }

=Op(n 1/2)

=En Xn1/2(δn δ)0rδh(S,δ)0 β+Op n 1/2

Notice that:

n1/2 βIV

n β = En

~

Z_nX~0_n En X~n~X0n

1

En X~n~Z0n

1

En ~Zn~X

0

n En ~XnX~

0

n

1

n1/2En ~Xn Z ~Zn

0

β+n1/2En X~nU .

First,

n1/2En ~Xn Z ~Zn

0

β=

= n1/2En X Z ~Zn

0

β

| {z }

=_En(Xn1/2(δn δ)0rδh(S,δ)0)β+Op(n1/2k(δn δ)_k2)

+n1/2En ~Zn Z X~n X

0 β

| {z }

=Op(n 1/2)

=En Xn1/2(δn δ)0rδh(S,δ)0 β+Op n 1/2

Notice that:

n1/2 βIV

n β = En

~

Z_nX~0_n En X~n~X0n

1

En X~n~Z0n

1

En ~Zn~X

0

n En ~XnX~

0

n

1

n1/2En ~Xn Z ~Zn

0

β+n1/2En X~nU .

First,

n1/2En ~Xn Z ~Zn

0

β=

= n1/2En X Z ~Zn

0

β

| {z }

=_En(Xn1/2(δn δ)0rδh(S,δ)0)β+Op(n1/2k(δn δ)_k2)

+n1/2En ~Zn Z X~n X

0

β

| {z }

=Op(n 1/2)

=En Xn1/2(δn δ)0rδh(S,δ)0 β+Op n

1/2

Now, sincevec(ABC) = (C0 A)vec(B)and by the LLN,

En Xn1/2(δn δ)0rδh(S,δ)0 β

= En

h

β0rδh(S,δ) X

i

n1/2(δn δ)

=Eh β0rδh(S,δ) X i

n1/2(δn δ) +op(1)

Now, sincevec(ABC) = (C0 A)vec(B)and by the LLN,

En Xn1/2(δn δ)0rδh(S,δ)0 β

= En

h

β0rδh(S,δ) X i

n1/2(δn δ)

=Eh β0rδh(S,δ) X

i

n1/2(δn δ) +op(1)

REMARK: When the coe¢ cients βcorresponding to the estimated explanatory variables are zero,

En X~n Z ~Zn

0

β=0

and

Eh β0rδh(S,δ) X i

=0

Also,

n1/2En ~XnU = n1/2En(XU) +n1/2En ~Xn X U

= n1/2En(XU) +E Urλg(S,λ)0 n 1/2₍

λn λ)

+Op n1/2kλn λk2

REMARK: When the coe¢ cients βcorresponding to the estimated explanatory variables are zero,

En X~n Z ~Zn

0

β=0

and

Eh β0rδh(S,δ) X

i =0

Also,

n1/2En ~XnU = n1/2En(XU) +n1/2En ~Xn X U

= n1/2En(XU) +E Urλg(S,λ)0 n 1/2₍

λn λ)

+Op n1/2kλn λk2

REMARK: When the coe¢ cients βcorresponding to the estimated explanatory variables are zero,

En X~n Z ~Zn

0

β=0

and

Eh β0rδh(S,δ) X i

=0

Also,

n1/2En ~XnU = n1/2En(XU) +n1/2En ~Xn X U

= n1/2En(XU) +E Urλg(S,λ)0 n

1/2

(λn λ)

+Op n1/2kλn λk2

REMARK: In many instances, E U_rλg(S,λ) =0,e.g. g X( )_,

λ ,whereX( )are observable exogenous variables, and

E U_jX( ) _=0.

Therefore,

n1/2 βIV

n β =

h

E ZX0 E XX0 1E XZ0 +op(1)

i 1

h

E ZX0 E XX0 1+op(1)

i

n

n1/2En(XU) +E

h

β0rδh(S,δ) X i

n1/2(δn δ)

+E U_rλg(S,λ) n1/2(λn λ) o

+op(1)

REMARK: In many instances, E U_rλg(S,λ) =0,e.g.

g X( )_,

λ ,whereX( )are observable exogenous variables, and

E U_jX( ) _=0.

Therefore,

n1/2 βIV

n β =

h

E ZX0 E XX0 1E XZ0 +op(1)

i 1

h

E ZX0 E XX0 1+op(1)

i

n

n1/2En(XU) +E

h

β0rδh(S,δ) X

i

n1/2(δn δ)

+E U_rλg(S,λ) n

1/2_(λ

n λ)

o

+op(1)

Assuming that,

n1/2

0 @ En(

XU) (δn δ)

(λn λ)

1 A d

!N(0,G),

then,

n1/2 βIV

n β

d

!B 1E ZX0 E XX0 1Nk+1 0,M0GM

Assuming that,

n1/2

0 @ En(

XU) (δn δ) (λn λ)

1

A d

!N(0,G),

then,

n1/2 βIV

n β

d

!B 1E ZX0 E XX0 1Nk+1 0,M0GM

with

B=E ZX0 E XX0 1E XZ0 and M=

0 B @

Ik+1

Eh β0rδh(S,δ) X i₀

E U_rλg(S,λ) 0

1 C A.

Thus, βIV_n is CAN with

AsyVar βIV_n = 1

nB

1_CB 1_0,

and

C=E ZX0 E XX0 1M0GME XX0 1E XZ0 ,

which can be estimated from data.

Thus, βIV_n is CAN with

AsyVar βIV_n = 1

nB

1_CB 1_0,

and

C=E ZX0 E XX0 1M0GME XX0 1E XZ0 ,

which can be estimated from data.

Thus, βIV_n is CAN with

AsyVar βIV_n = 1

nB

1_CB 1_0,

and

C=E ZX0 E XX0 1M0GME XX0 1E XZ0 ,

which can be estimated from data.

REMARK: In many circumstances,δn andλn are estimators

computed from other sample than (Yi,Si),i =1, ...,n. Samples may

have di¤erent sample sizes, which introduces even more trouble. In practice, it seems reasonable, cheap and easy to use the bootstrap for approximating the distribution of the IV estimator.

REMARK: The discussion is perfectly valid in the OLS case, just apply the results with Z=X.

REMARK: If both

Eh β0rδh(S,δ) X i

= 0and E U_rλg(S,λ) =0,

M =

0 @

Ik+1 0 0

1 A

and, as usual,

C =E ZX0 E XX0 1E XX0U2 E XX0 1E XZ0 .

REMARK: In many circumstances,δn andλn are estimators

computed from other sample than (Yi,Si),i =1, ...,n. Samples may have di¤erent sample sizes, which introduces even more trouble. In practice, it seems reasonable, cheap and easy to use the bootstrap for approximating the distribution of the IV estimator.

REMARK: The discussion is perfectly valid in the OLS case, just apply the results with Z=X.

REMARK: If both

Eh β0rδh(S,δ) X i

= 0and E U_rλg(S,λ) =0,

M =

0 @

Ik+1 0 0

1 A

and, as usual,

C =E ZX0 E XX0 1E XX0U2 E XX0 1E XZ0 .

REMARK: In many circumstances,δn andλn are estimators

computed from other sample than (Yi,Si),i =1, ...,n. Samples may have di¤erent sample sizes, which introduces even more trouble. In practice, it seems reasonable, cheap and easy to use the bootstrap for approximating the distribution of the IV estimator.

REMARK: The discussion is perfectly valid in the OLS case, just apply the results with Z=X.

REMARK: If both

Eh β0rδh(S,δ) X

i

= 0and E U_rλg(S,λ) =0,

M =

0 @

Ik+1

0 0

1 A

and, as usual,

C =E ZX0 E XX0 1E XX0U2 E XX0 1E XZ0 .

REMARK: In many circumstances,δn andλn are estimators

computed from other sample than (Yi,Si),i =1, ...,n. Samples may have di¤erent sample sizes, which introduces even more trouble. In practice, it seems reasonable, cheap and easy to use the bootstrap for approximating the distribution of the IV estimator.

REMARK: The discussion is perfectly valid in the OLS case, just apply the results with Z=X.

REMARK: If both

Eh β0rδh(S,δ) X i

= 0and E U_rλg(S,λ) =0,

M =

0 @

Ik+1 0 0

1 A

and, as usual,

C =E ZX0 E XX0 1E XX0U2 E XX0 1E XZ0 .

IMPORTANT CONCLUSION:

If both, the coe…cients βcorreponding to the generated Xare zero

and _rλg(S,λ)andU are uncorrelated, then the asymptotic

distribution of βIV is identical to the asymptotic distribution of the

corresponding IV estimator with observable variables, i.e. with known

parameters.

The same conclusion is applicable to the OLS estimator, but now we require that _rλh(S,λ) andU are uncorrelated (Z=X).

Otherwise, when we have generated explanatory o instrumental variables, the asymptotic distribution changes.

IMPORTANT CONCLUSION:

If both, the coe…cients βcorreponding to the generated Xare zero and _rλg(S,λ)andU are uncorrelated, then the asymptotic distribution of βIV is identical to the asymptotic distribution of the corresponding IV estimator with observable variables, i.e. with known parameters.

The same conclusion is applicable to the OLS estimator, but now we require that _rλh(S,λ) andU are uncorrelated (Z=X).

Otherwise, when we have generated explanatory o instrumental variables, the asymptotic distribution changes.

IMPORTANT CONCLUSION:

If both, the coe…cients βcorreponding to the generated Xare zero

and _rλg(S,λ)andU are uncorrelated, then the asymptotic

distribution of βIV is identical to the asymptotic distribution of the

corresponding IV estimator with observable variables, i.e. with known

parameters.

The same conclusion is applicable to the OLS estimator, but now we require that _rλh(S,λ) andU are uncorrelated (Z=X).

Otherwise, when we have generated explanatory o instrumental variables, the asymptotic distribution changes.

IMPORTANT CONCLUSION:

If both, the coe…cients βcorreponding to the generated Xare zero

and _rλg(S,λ)andU are uncorrelated, then the asymptotic

distribution of βIV is identical to the asymptotic distribution of the

corresponding IV estimator with observable variables, i.e. with known

parameters.

The same conclusion is applicable to the OLS estimator, but now we require that _rλh(S,λ) andU are uncorrelated (Z=X).

Otherwise, when we have generated explanatory o instrumental variables, the asymptotic distribution changes.

Testing homoskedasticity

Consider the linear model:

Y =β₀+Z0β+U with E(U) =0.

Using OLS,we want to test the assumption,

H0 :E ZZ0U2 =E ZZ0 E U2 ,

versus

H1:E ZZ0U2 6=E ZZ0 E U2 .

Half of the restrictions are redundant in H0,which can alternatively

written as,

H0 :C ZjZl,U2 =0, j =1, ...,k, l =j, ...,k

Testing homoskedasticity

Consider the linear model:

Y =β₀+Z0β+U with E(U) =0.

Using OLS,we want to test the assumption,

H0 :E ZZ0U2 =E ZZ0 E U2 ,

versus

H1:E ZZ0U2 6=E ZZ0 E U2 .

Half of the restrictions are redundant in H0,which can alternatively

written as,

H0 :C ZjZl,U2 =0, j =1, ...,k, l =j, ...,k

Testing homoskedasticity

Consider the linear model:

Y =β₀+Z0β+U with E(U) =0.

Using OLS,we want to test the assumption,

H0 :E ZZ0U2 =E ZZ0 E U2 ,

versus

H1:E ZZ0U2 ₆=E ZZ0 E U2 .

Half of the restrictions are redundant in H0,which can alternatively

written as,

H0 :C ZjZl,U2 =0, j =1, ...,k, l =j, ...,k

Testing homoskedasticity

Consider the linear model:

Y =β₀+Z0β+U with E(U) =0.

Using OLS,we want to test the assumption,

H0 :E ZZ0U2 =E ZZ0 E U2 ,

versus

H1:E ZZ0U2 6=E ZZ0 E U2 .

Half of the restrictions are redundant inH0,which can alternatively

written as,

H0 :C ZjZl,U2 =0, j =1, ...,k, l =j, ...,k

Let Zbe the k(k+1)/2-vector with the di¤erent components of ZZ0 and the linear proyector,

L U2 1,Z =δ0+Z0δ.

Then,H0 andH1 can alternatively be written as,

H0:δ =0versus H1 :δ6=0.

A suitable estimator of δ is:

δn =Vn Z

1

Cn Z,Un2 ,

whereUni =Yi β_0n Z0_iβ_n are the OLS residuals.

Let Zbe the k(k+1)/2-vector with the di¤erent components of

ZZ0 and the linear proyector,

L U2 1,Z =δ0+Z0δ.

Then,H0 andH1 can alternatively be written as,

H0:δ =0versus H1 :δ6= 0.

A suitable estimator of δ is:

δn =Vn Z

1

Cn Z,Un2 ,

whereUni =Yi β_0n Z0_iβ_n are the OLS residuals.

Let Zbe the k(k+1)/2-vector with the di¤erent components of

ZZ0 and the linear proyector,

L U2 1,Z =δ0+Z0δ.

Then,H0 andH1 can alternatively be written as,

H0:δ =0versus H1 :δ6= 0.

A suitable estimator of δ is:

δn =Vn Z

1

Cn Z,Un2 ,

whereUni =Yi β_0n Z0_iβ_n are the OLS residuals.

Let Zbe the k(k+1)/2-vector with the di¤erent components of

ZZ0 and the linear proyector,

L U2 1,Z =δ0+Z0δ.

Then,H0 andH1 can alternatively be written as,

H0:δ =0versus H1 :δ6= 0.

A suitable estimator of δ is:

δn =Vn Z

1

Cn Z,Un2 ,

whereUni =Yi β_0n Z0_iβ_n are the OLS residuals.

Using standard arguments, under suitable moment conditions,

Cn Z,Un2 Cn Z,U2 =op n 1/2 ,

and

δn =δ+Vn Z

1

Cn Z,U2 δ0 Z0δ +op n 1/2 .

Therefore, by the CLT, δn is CAN with

AsyVar(δn) = 1

nV Z

1

DV Z 1,

with

D=E Z E Z Z E Z 0 U2 δ0 Z0δ

2

whose corresponding estimator is,

\

AsyVar(δn) = 1

nVn Z

1

DnVn Z

1

,

with

Dn=En Z En Z Z En Z 0 Un2 δ0n Z0δn

2

Using standard arguments, under suitable moment conditions,

Cn Z,Un2 Cn Z,U2 =op n 1/2 ,

and

δn =δ+Vn Z

1

Cn Z,U2 δ0 Z0δ +op n 1/2 .

Therefore, by the CLT, δn is CAN with

AsyVar(δn) = 1

nV Z

1

DV Z 1,

with

D=E Z E Z Z E Z 0 U2 δ0 Z0δ

2

whose corresponding estimator is,

\

AsyVar(δn) = 1

nVn Z

1

DnVn Z

1

,

with

Dn=En Z En Z Z En Z 0 Un2 δ0n Z0δn

2

Using standard arguments, under suitable moment conditions,

Cn Z,Un2 Cn Z,U2 =op n 1/2 ,

and

δn =δ+Vn Z

1

Cn Z,U2 δ0 Z0δ +op n 1/2 .

Therefore, by the CLT, δn is CAN with

AsyVar(δn) =

1

nV Z

1

DV Z 1,

with

D=E Z E Z Z E Z 0 U2 δ0 Z0δ

2

whose corresponding estimator is,

\

AsyVar(δn) = 1

nVn Z

1

DnVn Z

1

,

with

Dn=En Z En Z Z En Z 0 Un2 δ0n Z0δn

2

Using standard arguments, under suitable moment conditions,

Cn Z,Un2 Cn Z,U2 =op n 1/2 ,

and

δn =δ+Vn Z

1

Cn Z,U2 δ0 Z0δ +op n 1/2 .

Therefore, by the CLT, δn is CAN with

AsyVar(δn) = 1

nV Z

1

DV Z 1,

with

D=E Z E Z Z E Z 0 U2 δ0 Z0δ

2

whose corresponding estimator is,

\

AsyVar(δn) = 1

nVn Z

1

DnVn Z

1

,

with

Dn=En Z En Z Z En Z 0 Un2 δ0n Z0δn

2

Using standard arguments, under suitable moment conditions,

Cn Z,Un2 Cn Z,U2 =op n 1/2 ,

and

δn =δ+Vn Z

1

Cn Z,U2 δ0 Z0δ +op n 1/2 .

Therefore, by the CLT, δn is CAN with

AsyVar(δn) = 1

nV Z

1

DV Z 1,

with

D=E Z E Z Z E Z 0 U2 δ0 Z0δ

2

whose corresponding estimator is,

\

AsyVar(δn) =

1

nVn Z 1

DnVn Z

1

,

with

Dn=En Z En Z Z En Z 0 Un2 δ0n Z0δn

2

Using standard arguments, under suitable moment conditions,

Cn Z,Un2 Cn Z,U2 =op n 1/2 ,

and

δn =δ+Vn Z

1

Cn Z,U2 δ0 Z0δ +op n 1/2 .

Therefore, by the CLT, δn is CAN with

AsyVar(δn) = 1

nV Z

1

DV Z 1,

with

D=E Z E Z Z E Z 0 U2 δ0 Z0δ

2

whose corresponding estimator is,

\

AsyVar(δn) = 1

nVn Z

1

DnVn Z

1

,

with

Dn=En Z En Z Z En Z 0 Un2 δ0n Z0δn

2

We can test the null by means of the Wald test statistic:

Wn =δ0nAsyVar\ (δn) 1δn,

and under H0,

Wn d

!χ2k(k+1)

2 .

Alternatively, we could use the asymptotically equivalentLM statistic:

LMn =nRn2,

whereR_n2 is the coe¢ cient of determination in the model:

U_ni2 = δ0+Zi0δ+error i =1, ...,n.

We can test the null by means of the Wald test statistic:

Wn =δ0nAsyVar\ (δn) 1δn,

and under H0,

Wn d

!χ2k(k+1) 2

.

Alternatively, we could use the asymptotically equivalentLM statistic:

LMn =nRn2,

whereR_n2 is the coe¢ cient of determination in the model:

U_ni2 = δ0+Zi0δ+error i =1, ...,n.

We can test the null by means of the Wald test statistic:

Wn =δ0nAsyVar\ (δn) 1δn,

and under H0,

Wn d

!χ2k(k+1)

2 .

Alternatively, we could use the asymptotically equivalentLM statistic:

LMn =nRn2,

whereR_n2 is the coe¢ cient of determination in the model:

U_ni2 = δ0+Zi0δ+error i =1, ...,n.

We can test the null by means of the Wald test statistic:

Wn =δ0nAsyVar\ (δn) 1δn,

and under H0,

Wn d

!χ2k(k+1)

2 .

Alternatively, we could use the asymptotically equivalentLM statistic:

LMn =nRn2,

whereR_n2 is the coe¢ cient of determination in the model:

U_ni2 = δ0+Zi0δ+error i =1, ...,n.

In practice, many people prefer to test the necessary condition forH0 :

β0C U2,ZZ0 β | {z } =_C(U2_,L(Yj1,Z)2

)

= C U2,Z 0β | {z } =C(U2_,L(Yj1,Z))

=0.

IfRˆ_n2 is the coe¢ cient of determination in the model:

U_ni2 =γ₀+Ln(Yj1,Z=Zi)γ₁+Ln(Yj1,Z=Zi)2γ₂+error

for i =1, ...,n,

the test statistic is theLM test:

d

LMn =nRˆn2.

Under H0,

d

LMn d

!χ2₂.

However, tests based on dLMn are inconsistent in the direction of many alternatives.

In practice, many people prefer to test the necessary condition forH0 :

β0C U2,ZZ0 β

| {z }

=_C(U2_,L(Yj1,Z)2 )

= C U2,Z 0β

| {z }

=C(U2_,L(Yj1,Z)) =0.

IfRˆ_n2 is the coe¢ cient of determination in the model:

U_ni2 =γ₀+Ln(Yj1,Z=Zi)γ₁+Ln(Yj1,Z=Zi)2γ₂+error

for i =1, ...,n,

the test statistic is theLM test:

d

LMn =nRˆn2.

Under H0,

d

LMn d

!χ2₂.

However, tests based on dLMn are inconsistent in the direction of many alternatives.

In practice, many people prefer to test the necessary condition forH0 :

β0C U2,ZZ0 β

| {z }

=_C(U2_,L(Yj1,Z)2 )

= C U2,Z 0β

| {z }

=C(U2_,L(Yj1,Z)) =0.

IfRˆ_n2 is the coe¢ cient of determination in the model:

U_ni2 =γ₀+Ln(Yj1,Z=Zi)γ₁+Ln(Yj1,Z=Zi)2γ₂+error

for i =1, ...,n,

the test statistic is theLM test:

d

LMn =nRˆn2.

Under H0,

d

LMn d

!χ2₂.

However, tests based on dLMn are inconsistent in the direction of many alternatives.

In practice, many people prefer to test the necessary condition forH0 :

β0C U2,ZZ0 β

| {z }

=_C(U2_,L(Yj1,Z)2 )

= C U2,Z 0β

| {z }

=C(U2_,L(Yj1,Z)) =0.

IfRˆ_n2 is the coe¢ cient of determination in the model:

U_ni2 =γ₀+Ln(Yj1,Z=Zi)γ₁+Ln(Yj1,Z=Zi)2γ₂+error

for i =1, ...,n,

the test statistic is theLM test: d

LMn =nRˆn2.

Under H0,

d

LMn d

!χ2₂.

However, tests based on dLMn are inconsistent in the direction of many alternatives.

In practice, many people prefer to test the necessary condition forH0 :

β0C U2,ZZ0 β

| {z }

=_C(U2_,L(Yj1,Z)2 )

= C U2,Z 0β

| {z }

=C(U2_,L(Yj1,Z)) =0.

IfRˆ_n2 is the coe¢ cient of determination in the model:

U_ni2 =γ₀+Ln(Yj1,Z=Zi)γ₁+Ln(Yj1,Z=Zi)2γ₂+error

for i =1, ...,n,

the test statistic is theLM test:

d

LMn =nRˆn2.

Under H0,

d

LMn d

!χ2₂.

However, tests based on dLMn are inconsistent in the direction of many alternatives.

In practice, many people prefer to test the necessary condition forH0 :

β0C U2,ZZ0 β

| {z }

=_C(U2_,L(Yj1,Z)2 )

= C U2,Z 0β

| {z }

=C(U2_,L(Yj1,Z)) =0.

IfRˆ_n2 is the coe¢ cient of determination in the model:

U_ni2 =γ₀+Ln(Yj1,Z=Zi)γ₁+Ln(Yj1,Z=Zi)2γ₂+error

for i =1, ...,n,

the test statistic is theLM test:

d

LMn =nRˆn2.

Under H0,

d

LMn d

!χ2₂.

However, tests based on dLMn are inconsistent in the direction of

many alternatives.

Finite sample asymptotics

Suppose that we have a CAN estimator θn of θ0, θ0 scalar for

presentation purposes. When θn is CAN, we have, under suitable

regularity conditions, that a Berry-Essen bound is satis…ed, i.e. sup

x

Pr θn θ0

AsyVar(θn)1/2

x

!

Φ(x) Cn 1/2 eachn =1,2, ..,

for some suitable constantC,andΦ is the standard normal

distribution, i.e.,

Φ(w) =

Z w

∞φ(w¯)dw¯ with φ(w) = 1

(2π)1/2

exp 1

2w

2 _.

For instance, if Wi,i =1, ..,n are iid with mean µ and varianceσ2,

sup x

Pr n1/2(W¯n µ) x Φ(x) 33

4

EjW1 µj3 σ3

n 1/2,

Finite sample asymptotics

Suppose that we have a CAN estimator θn of θ0, θ0 scalar for

presentation purposes. When θn is CAN, we have, under suitable

regularity conditions, that a Berry-Essen bound is satis…ed, i.e.

sup x

Pr θn θ0

AsyVar(θn)1/2

x

!

Φ(x) Cn 1/2 eachn =1,2, ..,

for some suitable constantC,andΦ is the standard normal distribution, i.e.,

Φ(w) =

Z w

∞φ(w¯)dw¯ with φ(w) = 1

(2π)1/2 exp 1 2w

2 _.

For instance, if Wi,i =1, ..,n are iid with mean µ and varianceσ2,

sup x

Pr n1/2(W¯n µ) x Φ(x) 33

4

EjW1 µj3 σ3

n 1/2,

Finite sample asymptotics

Suppose that we have a CAN estimator θn of θ0, θ0 scalar for

presentation purposes. When θn is CAN, we have, under suitable

regularity conditions, that a Berry-Essen bound is satis…ed, i.e.

sup x

Pr θn θ0

AsyVar(θn)1/2

x

!

Φ(x) Cn 1/2 eachn =1,2, ..,

for some suitable constantC,andΦ is the standard normal

distribution, i.e.,

Φ(w) =

Z w

∞φ(w¯)dw¯ with φ(w) = 1

(2π)1/2

exp 1

2w

2 _.

For instance, ifWi,i =1, ..,n are iid with mean µand variance σ2, sup

x

Pr n1/2(W¯n µ) x Φ(x)

33 4

EjW1 µj3 σ3

n 1/2,

I(Isidro)Edgeworth Expansions: In many cases of practical

relevance, for a given sample size n,if θn is CAN, the distribution of

n1/2(θn θ0)can be expanded as a power series inn 1/2,known as

Edgeworth Expansion, for each n=1,2, ....

Pr θn θ0

AsyVar(θn)1/2

x

!

=Φ(x) + 1

n1/2p1(x)φ(x)

+1

np2(x)φ(x)...+

1

nj/2pj(x)φ(x) +...

where φ(x)is the standard normal density, and the p_j0s are

polynomials, of degree at most 3j 1,with coe¢ cients depending on

cumulants of θn θ0.

If we are able to estimate some pj,j =1,2, ..., we could improve the accuracy of the asymptotic approximation.

I(Isidro)Edgeworth Expansions: In many cases of practical

relevance, for a given sample size n,if θn is CAN, the distribution of

n1/2(θn θ0)can be expanded as a power series inn 1/2,known as

Edgeworth Expansion, for each n=1,2, ....

Pr θn θ0

AsyVar(θn)1/2

x

!

=Φ(x) + 1

n1/2p1(x)φ(x)

+1

np2(x)φ(x)...+

1

nj/2pj(x)φ(x) +...

where φ(x)is the standard normal density, and the p_j0s are

polynomials, of degree at most 3j 1,with coe¢ cients depending on

cumulants of θn θ0.

If we are able to estimate some pj,j =1,2, ..., we could improve the accuracy of the asymptotic approximation.

I(Isidro)Edgeworth Expansions: In many cases of practical

relevance, for a given sample size n,if θn is CAN, the distribution of

n1/2(θn θ0)can be expanded as a power series inn 1/2,known as

Edgeworth Expansion, for each n=1,2, ....

Pr θn θ0

AsyVar(θn)1/2

x

!

=Φ(x) + 1

n1/2p1(x)φ(x)

+1

np2(x)φ(x)...+

1

nj/2pj(x)φ(x) +...

whereφ(x)is the standard normal density, and the p_j0s are

polynomials, of degree at most 3j 1,with coe¢ cients depending on cumulants of θn θ0.

If we are able to estimate some pj,j =1,2, ..., we could improve the accuracy of the asymptotic approximation.

I(Isidro)Edgeworth Expansions: In many cases of practical

relevance, for a given sample size n,if θn is CAN, the distribution of

n1/2(θn θ0)can be expanded as a power series inn 1/2,known as

Edgeworth Expansion, for each n=1,2, ....

Pr θn θ0

AsyVar(θn)1/2

x

!

=Φ(x) + 1

n1/2p1(x)φ(x)

+1

np2(x)φ(x)...+

1

nj/2pj(x)φ(x) +...

whereφ(x)is the standard normal density, and the p_j0s are

polynomials, of degree at most 3j 1,with coe¢ cients depending on

cumulants of θn θ0.

If we are able to estimate some pj,j =1,2, ..., we could improve the

accuracy of the asymptotic approximation.

ICornish-Fisher Expansions: De…ne xα as,

Pr θn θ0

AsyVar(θn)1/2

xα

! =α,

the Cornish-Fisher expansion is the inversion of the Edgeworth expansion, and it has the form,

xα =zα+

1

n1/2p11(zα) +

1

np21(zα) +...+

1

nj/2pj1(zα) +...

where thep_j0₁s are polynomial derivable from thep_j0s,andzα solves Φ(zα) =α.

See, e.g., Hall (1992),The Bootstrap and Edgeworth Expansions,

Springer.

ICornish-Fisher Expansions: De…ne xα as,

Pr θn θ0

AsyVar(θn)1/2

xα !

=α,

the Cornish-Fisher expansion is the inversion of the Edgeworth expansion, and it has the form,

xα =zα+

1

n1/2p11(zα) +

1

np21(zα) +...+

1

nj/2pj1(zα) +...

where thep_j0₁s are polynomial derivable from thep_j0s,andzα solves Φ(zα) =α.

See, e.g., Hall (1992),The Bootstrap and Edgeworth Expansions,

Springer.

ICornish-Fisher Expansions: De…ne xα as,

Pr θn θ0

AsyVar(θn)1/2

xα !

=α,

the Cornish-Fisher expansion is the inversion of the Edgeworth expansion, and it has the form,

xα =zα+

1

n1/2p11(zα) +

1

np21(zα) +...+

1

nj/2pj1(zα) +...

where thep_j0₁s are polynomial derivable from thep_j0s,andzα solves

Φ(z_α) =α.

See, e.g., Hall (1992),The Bootstrap and Edgeworth Expansions,

Springer.

ICornish-Fisher Expansions: De…ne xα as,

Pr θn θ0

AsyVar(θn)1/2

xα !

=α,

the Cornish-Fisher expansion is the inversion of the Edgeworth expansion, and it has the form,

xα =zα+

1

n1/2p11(zα) +

1

np21(zα) +...+

1

nj/2pj1(zα) +...

where thep_j0₁s are polynomial derivable from thep_j0s,andzα solves Φ(zα) =α.

See, e.g., Hall (1992),The Bootstrap and Edgeworth Expansions,

Springer.

Bootstrap approximations

A bootstrap approximation often provides a more accurate approximation to the exact distribution of statistics than the asymptotic one.

Again, suppose that we have an estimator θn = θ(FWn)of

θ0 =θ(FW).

Let _Wn =fW1, ...,Wngbe a random sample of FW.Also, we have

an estimator ofFW,FˆWn say, which can be the sample distribution

function, FWn,a parametric estimator, or even a smooth estimator.

Let _Wn =_fW₁, ...,W_ngbe a random sample of FˆWn,e.g. if ˆ

FWn =FWn,Wn areiid observations from a distribution assigning a probability 1/n to each observation _fW1, ...,Wng.

Bootstrap approximations

A bootstrap approximation often provides a more accurate approximation to the exact distribution of statistics than the asymptotic one.

Again, suppose that we have an estimator θn = θ(FWn)of

θ0 =θ(FW).

Let _Wn =fW1, ...,Wngbe a random sample of FW.Also, we have

an estimator ofFW,FˆWn say, which can be the sample distribution

function, FWn,a parametric estimator, or even a smooth estimator.

Let _Wn =_fW₁, ...,W_ngbe a random sample of FˆWn,e.g. if ˆ

FWn =FWn,Wn areiid observations from a distribution assigning a probability 1/n to each observation _fW1, ...,Wng.

Bootstrap approximations

A bootstrap approximation often provides a more accurate approximation to the exact distribution of statistics than the asymptotic one.

Again, suppose that we have an estimator θn = θ(FWn)of

θ0 =θ(FW).

Let _Wn =fW1, ...,Wngbe a random sample of FW.Also, we have

an estimator ofFW,FˆWn say, which can be the sample distribution

function, FWn,a parametric estimator, or even a smooth estimator.

Let _Wn =_fW₁, ...,W_ngbe a random sample of FˆWn,e.g. if ˆ

FWn =FWn,Wn areiid observations from a distribution assigning a probability 1/n to each observation _fW1, ...,Wng.

Bootstrap approximations

A bootstrap approximation often provides a more accurate approximation to the exact distribution of statistics than the asymptotic one.

Again, suppose that we have an estimator θn = θ(FWn)of

θ0 =θ(FW).

Let _Wn =fW1, ...,Wngbe a random sample of FW.Also, we have

an estimator ofFW,FˆWn say, which can be the sample distribution

function, FWn,a parametric estimator, or even a smooth estimator.

Let _Wn =_fW₁, ...,W_ngbe a random sample of FˆWn,e.g. if ˆ

FWn =FWn,Wn areiid observations from a distribution assigning a

probability 1/n to each observation _fW1, ...,Wng.

We may be interested in estimating E[R(Wn,FW)],for a particular

functionR,which represent di¤erent distributional features of θn,say e.g.

1

R(_Wn,FW) = (θn θ(FW))!Bias(θn) =E(θn θ(FW))

2

R(_Wn,FW) = (θn E(θn))2 !V(θn) =E

h

(θn E(θn))2

i

3

R(Wn,FW) =1 θn θ(F_W)

\

AsyVar(θn) w

!Fθn θ(F_W)

\

AsyVar(θn)

(w) =E 1 _{θn θ(}F_W)

\

AsyVar(θn) w !

.

We may be interested in estimating E[R(Wn,FW)],for a particular

functionR,which represent di¤erent distributional features of θn,say e.g.

1

R(_Wn,FW) = (θn θ(FW))!Bias(θn) =E(θn θ(FW))

2

R(_Wn,FW) = (θn E(θn))2 !V(θn) =E

h

(θn E(θn))2

i

3

R(Wn,FW) =1 θn θ(F_W)

\

AsyVar(θn) w

!Fθn θ(F_W)

\

AsyVar(θn)

(w) =E 1 _{θn θ(}F_W)

\

AsyVar(θn) w !

.

We may be interested in estimating E[R(Wn,FW)],for a particular

functionR,which represent di¤erent distributional features of θn,say e.g.

1

R(_Wn,FW) = (θn θ(FW))!Bias(θn) =E(θn θ(FW))

2

R(_Wn,FW) = (θn E(θn))2 !V(θn) =E

h

(θn E(θn))2

i

3

R(Wn,FW) =1 θn θ(F_W)

\ AsyVar(θn) w

!Fθn θ(F_W)

\ AsyVar(θn)

(w) =E 1 _{θn θ(}F_W)

\ AsyVar(θn) w

! .

The bootstrap analog ofθn,θ_n,is the estimator computed with W_n,

For instance,θ_n =θ Fˆ_Wn ,whereFˆ_Wn(w) =n 1∑n_i=11_fW_i w_g is the empirical distribution based on _Wn =_fW₁, ...,W_ng,which are

iid variables with distribution FˆWn.

Then we have that the bootstrap analog ofE[R(_Wn,FW)]is, E [R(Wn,FWn)] =E[R(Wn,FWn)j Wn],

which is the expectation with respect to Fˆ_Wn.

The bootstrap analog ofθn,θ_n,is the estimator computed with W_n, For instance,θ_n =θ Fˆ_Wn ,whereFˆ_Wn(w) =n 1∑n_i=11_fW_i w_g is

the empirical distribution based on _Wn =_fW₁, ...,W_ng,which are

iid variables with distribution FˆWn.

Then we have that the bootstrap analog ofE[R(_Wn,FW)]is, E [R(Wn,FWn)] =E[R(Wn,FWn)j Wn],

which is the expectation with respect to Fˆ_Wn.

The bootstrap analog ofθn,θ_n,is the estimator computed with W_n,

For instance,θ_n =θ Fˆ_Wn ,whereFˆ_Wn(w) =n 1∑n_i=11_fW_i w_g is the empirical distribution based on _Wn =_fW₁, ...,W_ng,which are

iid variables with distribution FˆWn.

Then we have that the bootstrap analog ofE[R(_Wn,FW)]is,

E [R(Wn,FWn)] =E[R(Wn,FWn)j Wn],

which is the expectation with respect to Fˆ_Wn.

The bootstrap analog ofθn,θ_n,is the estimator computed with W_n,

For instance,θ_n =θ Fˆ_Wn ,whereFˆ_Wn(w) =n 1∑n_i=11_fW_i w_g is the empirical distribution based on _Wn =_fW₁, ...,W_ng,which are

iid variables with distribution FˆWn.

Then we have that the bootstrap analog ofE[R(_Wn,FW)]is, E [R(Wn,FWn)] =E[R(Wn,FWn)j Wn],

which is the expectation with respect to Fˆ_Wn.

Some approximations of distributional features of θn are:

1

d

Bias(θn) =E (θn) θ(FWn)

2

b

V(θn) =E

h

(θ_n E (θ_n))2

i

.

3 The bootstrap estimate of the quantil

Some approximations of distributional features of θn are:

1

d

Bias(θn) =E (θn) θ(FWn)

2

b

V(θn) =E

h

(θn E (θn))

2i

.

3 The bootstrap estimate of the quantil

Some approximations of distributional features of θn are:

1

d

Bias(θn) =E (θn) θ(FWn)

2

b

V(θn) =E

h

(θn E (θn))

2i

.

3 The bootstrap estimate of the quantil

Frequently,E (R(_Wn,FWn))is very expensive to compute,

but it can be approximated very accurately in practice generating B

resamples:

Wn(b) =

n

W₁(b), ...Wn(b)

o

, b =1, ...,B.

Then, we approximate E (R(Wn,FWn)), as accurately as desired, by

EB(R(Wn,FWn)) =

1

B

B

∑

b=1

R _Wn(b) ,FWn .

Frequently,E (R(_Wn,FWn))is very expensive to compute, but it can be approximated very accurately in practice generating B

resamples:

Wn(b) =

n

W₁(b), ...Wn(b)

o

, b =1, ...,B.

Then, we approximate E (R(Wn,FWn)), as accurately as desired, by

EB(R(Wn,FWn)) =

1

B

B

∑

b=1

R _Wn(b) ,FWn .

Frequently,E (R(_Wn,FWn))is very expensive to compute,

but it can be approximated very accurately in practice generating B

resamples:

Wn(b) =

n

W₁(b), ...Wn(b)

o

, b =1, ...,B.

Then, we approximate E (R(Wn,FWn)), as accurately as desired, by

EB(R(Wn,FWn)) =

1

B

B

∑

b=1

R _Wn(b) ,FWn .

Example. Suppose that the object of interest is the bias

R(_Wn,FW) = (θn θ(FW))!Bias(θn) =E(θn θ(FW))

we are going to estimate it by

EB(R(Wn,FWn)) =

1

B

B

∑

b=1

R _Wn(b) ,FWn

where we use as FˆWn the sample distribution function, FWn,(this is

also called nonparametric boostrap) This means doing this:

1. GenerateB independent random samples from FWn

2. For each of them estimate θ,call each estimate: θ_b,b=1, ...B

3. Then, the bootstrap estimate of the bias is just

1

B

B

∑

b=1

θ_b θn

Note that this is the analogue in the boostrap world of

Eθn θ

Example. Suppose that the object of interest is the bias

R(_Wn,FW) = (θn θ(FW))!Bias(θn) =E(θn θ(FW)) we are going to estimate it by

EB(R(Wn,FWn)) =

1

B

B

∑

b=1

R _Wn(b) ,FWn

where we use as FˆWn the sample distribution function, FWn,(this is

also called nonparametric boostrap) This means doing this:

1. GenerateB independent random samples from FWn

2. For each of them estimate θ,call each estimate: θ_b,b=1, ...B

3. Then, the bootstrap estimate of the bias is just

1

B

B

∑

b=1

θ_b θn

Note that this is the analogue in the boostrap world of

Eθn θ