12.2 Instrumental Variable (IV) Method (

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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 ofZiis not assumed.

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

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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

X⁰X)₋1 1

√ X⁰u.

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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 ofui is not assumed.

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12 Instrumental Variable (

^{操作変数法}

)

12.1 Measurement Error (

^測定誤差

)

Errors in Variables

1. True regression model:

y= X˜β+u 2. Observed variable:

X = X˜ +V

V: is called themeasurement 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 inV are zeros.

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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 inXis uncorrelated with ˜Xin 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

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(b) uis not correlated withV. uis 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

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7. Example: The Case of Two Variables:

The regression model is given by:

y_t =α+βx˜_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

)

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Now we focus onβ.

βˆ is not consistent. because of:

plim( ˆβ)=β− σ²vβ σ²+σ²v

= β

1+σ²v/σ² < β

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12.2 Instrumental Variable (IV) Method (

^{操作変数法}

or IV

^法

)

Instrumental Variable (IV)

1. Consider the regression model:y= Xβ+uandu∼N(0, σ²I_n).

In the case of E(X⁰u), 0, OLSE ofβis inconsistent.

2. Proof:

βˆ =β+(1

nX⁰X)⁻¹1

nX⁰u −→ β+ M⁻_xx¹M_xu, where

1

nX⁰X −→ M_xx, 1

nX⁰u −→ M_xu ,0 3. Find theZwhich satisfies 1

nZ⁰u −→ M_zu= 0.

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MultiplyingZ⁰on both sides of the regression model: y=Xβ+u, Z⁰y=Z⁰Xβ+Z⁰u

Dividingnon both sides of the above equation, we take plim on both sides.

Then, we obtain the following:

plim (1

nZ⁰y )

= plim (1

nZ⁰X )

β+plim (1

nZ⁰u )

=plim (1

nZ⁰X )

β.

Accordingly, we obtain:

β= (

plim (1

nZ⁰X ))₋1

plim (1

nZ⁰y )

. Therefore, we consider the following estimator:

βIV =(Z⁰X)⁻¹Z⁰y,

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which is taken as an estimator ofβ.

=⇒ Instrumental Variable Method (操作変数法or IV法) 4. Assume the followings:

1

nZ⁰X −→ M_zx, 1

nZ⁰Z −→ M_zz, 1

nZ⁰u −→ 0 5. Asymptotic Distribution ofβIV:

βIV =(Z⁰X)⁻¹Z⁰y= (Z⁰X)⁻¹Z⁰(Xβ+u)= β+(Z⁰X)⁻¹Z⁰u, which is rewritten as:

√n(βIV −β)=(1

nZ⁰X)₋1( 1

√nZ⁰u)

Applying the Central Limit Theorem to( 1

√nZ⁰u)

, we have the following result:

√1

nZ⁰u −→ N(0, σ²M_zz).

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Therefore,

√n(βIV −β)=(1

nZ⁰X)₋1( 1

√nZ⁰u)

−→ N(0, σ²M_zx⁻¹MzzM_zx⁰ ⁻¹)

=⇒ Consistency and Asymptotic Normality 6. The variance ofβIV is given by:

V(βIV)= s²(Z⁰X)⁻¹Z⁰Z(X⁰Z)⁻¹, where

s² = (y−XβIV)⁰(y−XβIV) n−k .

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12.3 Two-Stage Least Squares Method (2

^{段階最小二乗法}

, 2SLS or TSLS)

1. Regression Model:

y= Xβ+u, u∼N(0, σ²I),

In the case of E(X⁰u), 0, OLSE is not consistent.

2. Find the variableZ which satisfies 1

nZ⁰u −→ M_zu=0.

3. UseZ = Xˆ for the instrumental variable.

Xˆ is the predicted value which regresses X on the other exogenous variables, sayW.

That is, consider the following regression model:

X= W B+V.

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EstimateBby OLS.

Then, we obtain the prediction:

Xˆ =WBˆ, where ˆB=(W⁰W)⁻¹W⁰X.

Or, equivalently,

Xˆ = W(W⁰W)⁻¹W⁰X. Xˆ is used for the instrumental variable ofX.

4. The IV method is rewritten as:

βIV =( ˆX⁰X)⁻¹Xˆ⁰y= (X⁰W(W⁰W)⁻¹W⁰X)⁻¹X⁰W(W⁰W)⁻¹W⁰y. Furthermore,βIV is written as follows:

β =β+ ⁰ ⁰ ⁻¹ ⁰ ⁻¹ ⁰ ⁰ ⁻¹ ⁰ .

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Therefore, we obtain the following expression:

√n(βIV −β)=((1

nX⁰W)(1

nW⁰W)₋1(1

nXW⁰)₀)₋1(1

nX⁰W)(1

nW⁰W)₋1( 1

√nW⁰u)

−→ N(

0, σ²(M_xwM_ww⁻¹M⁰_xw)⁻¹) .

5. Clearly, there is no correlation betweenW anduat least in the limit, i.e., plim(1

nW⁰u)

=0.

6. Remark:

Xˆ⁰X= X⁰W(W⁰W)⁻¹W⁰X = X⁰W(W⁰W)⁻¹W⁰W(W⁰W)⁻¹W⁰X= Xˆ⁰Xˆ. Therefore,

βIV =( ˆX⁰X)⁻¹Xˆ⁰y= ( ˆX⁰X)ˆ ⁻¹Xˆ⁰y,

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which implies the OLS estimator ofβin the regression model: y= Xˆβ+uand u∼ N(0, σ²I_n).

Example:

y_t =αx_t+βz_t+u_t, u_t ∼(0, σ²). Suppose that x_tis correlated withu_t butz_t is not correlated withu_t.

• 1st Step:

Estimate the following regression model:

x_t = γw_t +δz_t +· · ·+v_t, by OLS. =⇒ Obtain ˆx_tthrough OLS.

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• 2nd Step:

Estimate the following regression model:

y_t =αxˆ_t +βz_t+u_t, by OLS. =⇒ αivandβiv

Note as follows. Estimate the following regression model:

z_t =γ2w_t +δ2z_t+· · ·+v_2t, by OLS.

=⇒ γˆ2 = 0, ˆδ2 =1, and the other coeﬃcient estimates are zeros. i.e., ˆz_t =z_t.

Eviews Command:

tsls y x z @ w z ...