12.1 Measurement Error (

7. Some Formulas:

Let X_nand Y_nbe the random variables which satisfy plim X_n =c and plim Y_n= d. Then,

(a) plim (X_n+Y_n)=c+d (b) plim X_nY_n =cd

(c) plim X_n/Y_n =c/d for d ,0

(d) plim g(X_n)=g(c) for a function g(·)

=⇒ Slutsky’s Theorem (スルツキー定理)

8. Central Limit Theorem (中心極限定理)

Univariate Case: X₁, X₂,· · ·, X_n are mutually independently and identically distributed as Xi ∼(µ, σ²).

Then,

X−E(X)

√ V(X)

= X−µ σ/√

n −→ N(0,1), which implies

√n(X−µ)= 1

√n

∑n i=1

(X_i−µ) −→ N(0, σ²).

Multivariate Case: X₁, X₂,· · ·, X_nare mutually independently and identically distributed as X_i ∼(µ, Σ).

Then,

√1 n

∑n i=1

(X_i−µ) −→ N(0,Σ) 9. Central Limit Theorem (Generalization)

X₁, X₂, · · ·, X_n are mutually independently and identically distributed as X_i ∼ (µ, Σi).

Then,

√1 n

∑n i=1

(X_i−µ) −→ N(0,Σ), where

Σ = lim

n→∞



1 n

∑n i=1

Σi



.

10. Definition: Let ˆθnbe a consistent estimator ofθ. Suppose that √

n(ˆθn−θ) converges to N(0,Σ) in distribution.

Then, we say that ˆθnhas an asymptotic distribution (漸近分布): N(θ,Σ/n).

11. X₁,X₂,· · ·,X_n are random variables with density function f (x;θ).

Let ˆθnbe a maximum likelihood estimator ofθ.

Then, under some regularity conditions. ˆθn is a consistent estimator ofθand the asymptotic distribution of √

n(ˆθ−θ) is given by: N

0,lim (I(θ)

n )₋1

. 12. Regularity Conditions:

(a) The domain of X_i does not depend onθ.

(b) There exists at least third-order derivative of f (x;θ) with respect toθ, and their derivatives are finite.

13. Thus, MLE is

(i) consistent，

(ii) asymptotically normal，and (iii) asymptotically efficient.

11 Consistency and Asymptotic Normality of OLSE

Regression model: y= Xβ+u, u∼ (0, σ²I_n).

Consistency:

1. Let ˆβn = (X⁰X)⁻¹X⁰y be the OLS with sample size n.

Consistency: As n is large, ˆβnconverges toβ. 2. Assume the stationarity assumption for X, i.e.,

1

nX⁰X −→ M_xx. Then, we have the following result:

1

nX⁰u −→ 0.

Proof:

According to Chebyshev’s inequality, for g(Z)≥0, P(g(Z)≥k) ≤ E(g(Z))

k , where k is a positive constant.

Set g(Z)=Z⁰Z, and Z = 1 nX⁰u.

Apply Chebyshev’s inequality.

E( (1

nX⁰u)⁰1 nX⁰u)

= 1 n²E(

u⁰XX⁰u)

= 1 n²E(

tr(u⁰XX⁰u))

= 1 n²E(

tr(XX⁰uu⁰))

= 1 n²tr(

XX⁰E(uu⁰))

= σ²

n²tr(XX⁰)= σ²

n²tr(X⁰X)= σ² n tr(1

nX⁰X). Therefore,

P( (1

nX⁰u)⁰1

nX⁰u≥ k)

≤ σ² nktr(1

nX⁰X)−→0×tr(Mxx)=0.

Note that from the assumption, 1

nX⁰X −→ Mxx. Therefore, we have:

(1 nX⁰u)⁰1

nX⁰u−→0, which implies:

1

nX⁰u−→0, because (1

nX⁰u)⁰1

nX⁰u indicates a quadratic form.

3. Note that 1

nX⁰X −→ Mxx results in (1

nX⁰X)⁻¹ −→ M⁻_xx¹.

=⇒Slutsky’s Theorem

(*) Slutsky’s Theorem g(ˆθ)−→g(θ), when ˆθ−→θ.

4. OLS is given by:

βˆn=β+(X⁰X)⁻¹X⁰u= β+(1

nX⁰X)⁻¹(1 nX⁰u). Therefore,

βˆn−→β+M⁻_xx¹×0=β Thus, OLSE is a consitent estimator.

Asymptotic Normality:

1. Asymptotic Normality of OLSE

√n( ˆβn−β) −→ N(0.σ²M⁻¹_xx), when n −→ ∞.

2. Central Limit Theorem: Greenberg and Webster (1983)

Z₁, Z₂,· · ·, Z_nare mutually indelendently distributed with meanµand variance Σi.

Then, we have the following result:

√1 n

∑n i=1

(Z_i−µ) −→ N(0,Σ), where

Σ = lim

n→∞



1 n

∑n i=1

Σi



. The distribution of Ziis not assumed.

3. Define Z_i = x_i⁰u_i. Then,Σi =Var(Z_i)=σ²x⁰_ix_i.

4. Σis defined as:

Σ = lim

n→∞



1 n

∑n i=1

σ²x⁰_ix_i



= σ²lim

n→∞

(1 nX⁰X

)

=σ²M_xx,

where

X =









x₁ x2

...

x_n









5. Applying Central Limit Theorem (Greenberg and Webster (1983), we obtain the following:

√1 n

∑n i=1

x⁰_iui = 1

√nX⁰u−→ N(0, σ²Mxx). On the other hand, from ˆβn =β+(X⁰X)⁻¹X⁰u, we can rewrite as:

√n( ˆβ−β)=(1

nX⁰X)₋1 1

√nX⁰u.

Var((1

nX⁰X)₋₁ 1

√nX⁰u )

=E((1

nX⁰X)₋₁ 1

√nX⁰u((1

nX⁰X)₋₁ 1

√nX⁰u)₀)

=(1

nX⁰X)₋₁(1

nX⁰E(uu⁰)X)(1

nX⁰X)₋₁

=σ²(1

nX⁰X)₋1

−→ σ²M⁻_xx¹.

Therefore,

√n( ˆβ−β) −→ N(0, σ²M_xx⁻¹)

=⇒Asymptotic normality (漸近的正規性) of OLSE The distribution of uiis not assumed.

12 Instrumental Variable ( ^{操作変数法} )

12.1 Measurement Error (

)

Errors in Variables

1. True regression model:

y= X˜β+u 2. Observed variable:

X = X˜ +V

V: is called the measurement error (測定誤差or観測誤差).

3. For the elements which do not include measurement errors in X, the corre- sponding elements in V are zeros.

4. Regression using observed variable:

y= Xβ+(u−Vβ) OLS ofβis:

βˆ =(X⁰X)⁻¹X⁰y=β+(X⁰X)⁻¹X⁰(u−Vβ)

5. Assumptions:

(a) The measurement error in X is uncorrelated with ˜X in the limit. i.e., plim(1

nX˜⁰V)

=0. Therefore, we obtain the following:

plim(1 nX⁰X)

=plim(1 nX˜⁰X˜)

+plim(1 nV⁰V)

= Σ + Ω

(b) u is not correlated with V.

u is not correlated with ˜X.

That is,

plim(1 nV⁰u)

=0, plim(1 nX˜⁰u)

=0.

6. OLSE ofβis:

βˆ =β+(X⁰X)⁻¹X⁰(u−Vβ)=β+(X⁰X)⁻¹( ˜X+V)⁰(u−Vβ). Therefore, we obtain the following:

plim ˆβ=β−(Σ + Ω)⁻¹Ωβ

7. Example: The Case of Two Variables:

The regression model is given by:

y_t =α+β˜x_t+u_t, x_t = ˜x_t+v_t. Under the above model,

Σ =plim(1 nX˜⁰X˜)

= plim





1 1

n

∑

˜xi

1 n

∑

˜x_i 1 n

∑

˜x²_i



=

(1 µ

µ µ²+σ² )

, whereµandσ²represent the mean and variance of ˜x_i.

Ω =plim(1 nV⁰V)

=plim

(0 0

0 1

n

∑ v²_i

)

=

(0 0

0 σ²v

) .

Therefore, plim

(αˆ βˆ )

= (α

β )

−

((1 µ

µ µ²+σ² )

+

(0 0

0 σ²_v

))−1(0 0 0 σ²_v

) (α β )

= (α

β )

− 1 σ²+σ²v

(−µσ²_vβ σ²vβ

)

Now we focus onβ.

βˆ is not consistent. because of:

plim( ˆβ)=β− σ²vβ σ²+σ²v

= β

1+σ²v/σ² < β