Other estimation principles: Method of Moments …Other estimation principles: Method of Moments and Maximum Likelihood (1 of 2) Ragnar Nymoen University of Oslo 25 April 2013 1/21

Method of moments and instrumental variables estimation MM and estimation of simultaneous equations models

Other estimation principles: Method of Momentsand Maximum Likelihood (1 of 2)

Ragnar Nymoen

University of Oslo

25 April 2013

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References

I HGL, Ch 10.3-10.3.3, 10.3.5

I BN Kap 6.2, 6.3, 9.2.7

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Rational expectations models

Consistent estimation of a RE model IRE model from Lecture 17 (version with leading variable):

Yt = β0 + β1E (Xt+1 | It−1) + εt (1)

Xt = λXt−1 + εxt , − 1 < λ < 1 (2)

εt ∼ IID(0, σ2) (3)

εxt ∼ IID(0, σ2x ) (4)

Cov(εt , εxs) = 0 for all t and s (5)

From Lecture 17: OLS on

Yt = β0 + β1Xt+1 + ut (6)

gives an inconsistent estimate of β1, becauseut = εt − β1λεxt − β1εxt+1

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Consistent estimation of a RE model II

Assume however that we have an instrumental variable Zt withE (Zt) = µZ and the following properties:

1. Zt is correlated with Xt+1 in (6)

2. Zt is uncorrelated with ut in (1)

Consider

(Zt − Z̄ )Yt = β0(Zt − Z̄ ) + β1(Zt − Z̄ )Xt+1 + (Zt − Z̄ )ut

and take expectations on both sides of this equation:

E (Zt − Z̄ )Yt = β0E (Zt − Z̄ ) + β1E (Zt − Z̄ )Xt+1 + E (Zt − Z̄ )ut

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Consistent estimation of a RE model III

Bringing in the properties of Zt we obtain:

β1 =E [(Zt − Z̄ )Yt ]

E [(Zt − Z̄ )Xt+1]=

Cov(Yt ,Zt)

Cov(Xt+1, Zt)(7)

If we can find a Z variable that allows us to form empiricalmoments σ̂YZ and σ̂X+1Z that are consistent estimators of thetheoretical moments Cov(Yt ,Zt) and Cov(Xt+1, Zt) we wouldhave a consistent estimator of β1.

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Consistent estimation of a RE model IV

Let

β̂IV1 =

σ̂YZσ̂X+1Z

=∑t(Zt − Z̄ )Yt

∑t(Zt − Z̄ )Xt+1

=∑t(Zt − Z̄ ) [β0 + β1Xt+1 + ut ]

∑t(Zt − Z̄ )Xt+1

= β1 +∑t(Zt − Z̄ )ut

∑t(Zt − Z̄ )Xt+1

plim(β̂IV1 ) = β1 +

plim 1T ∑t(Zt − Z̄ )ut

plim 1T ∑t(Zt − Z̄ )Xt+1

= β1

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Consistent estimation of a RE model V

since

plim1

T ∑t

(Zt − Z̄ )ut = Cov(ut , Zt) = 0 by property 1

plim1

T ∑t

(Zt − Z̄ )Xt+1 = Cov(Xt+1, Zt) 6= 0 by property 2

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How to find a Z? I

I In the RE model (1)-(5) there is just one candidate: Xt−1.Based on the model specification

I Xt−1 is correlated with Xt+1 since λ 6= 0.I Xt−1 is uncorrelated with εxt , εxt+1, εxt+2,. . . and with εt+j

(j = 0,±1,±2, . . .).

I Therefore, Xt−1 is predetermined with respect to thedisturbance ut :

ut = εt − β1λεxt − β1εxt+1

in

Yt = β0 + β1Xt+1 + ut

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How to find a Z? II

I Can check that using Xt−1 as instrumental variable gives amoment estimator β̂IV

1 which is consistent:

β̂IV1 =

∑t(Xt−1 − X̄−1)Yt

∑t (Xt−1 − X̄−1)Xt+1

=∑t(Xt−1 − X̄−1)(β0 + β1Xt+1 + ut)

∑t (Xt−1 − X̄−1)Xt+1

= β1 +∑t(Xt−1 − X̄−1)ut

∑t (Xt−1 − X̄−1)Xt+1

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How to find a Z? III

I Since

plim∑t(Xt−1 − X̄−1)ut

∑t (Xt−1 − X̄−1)Xt+1= 0 (8)

from the assumptions made, we get the conclusion:

plim(β̂IV1 ) = β1 (9)

for the parameter β1 in the RE model (1)-(5).

I Question: Since it is Xt+1 that needs to be instrumented in(1), why not use Xt as the instrumental variable?

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How to find a Z? IV

I Question: If we replace (1) by the simpler equation

Yt = β1E (Xt | It−1) + εt (10)

and leave the rest of the model (1)-(5) unchanged. Whatwould the expression for methods of moment (instrumentalvariable) estimator be?

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Measurement-error models

Consistent estimation of measurement-error model I

I As we have seen: inconsistency of OLS in RE models is due tothe same econometric problem as the measurement-error bias.

I The solution is also the same: The method of moments givesa consistent estimator of β1( and β0) in

Yi = β0 + β1X ∗i + ε∗i , i = 1, 2, . . . , n (11)

X ∗i = Xi − ei (12)

ε∗i ∼ IID(0, σX ∗) (13)

ei ∼ IID(0, σ2e ) (14)

Cov(ε∗i , ej ) = 0 for all i and j (15)

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Measurement-error models

Consistent estimation of measurement-error model II

I provided that there is a instrumental variable Zi that we canuse to form the moments σ̂YZ and σ̂XZ that are consistentestimators of the theoretical moments Cov(Yi ,Zi ) andCov(X ∗i , Zi ).

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General remarks about Method of Moments

Generality of method of moments, MM I

I In the examples with RE and measurement errors, theeconometric model implies that the parameter of interest β1

can be expressed as

β1 =Cov(Z , Y )

Cov(Z , X )(16)

where we have dropped the subscript for random variablenumber t and i , to highlight that the parameter of interest β1

is expressed in terms of the theoretical moments.

I The MM principle: Replace the theoretical moments by theempirical moments that are consistent estimators of the twovariances.

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Generality of method of moments, MM II

I In general: if the parameter of interest is a function of ntheoretical moments. MM “requires” that we have n empiricalcounterparts that are consistent estimators of the n theoreticalmoments. (in our example: If we are interested in β0 as well,need to estimate E (Y ) and E (X ) by their empirical averages).

I It is not always that we “have” the n empirical moments. Inthis case the parameters of interest are not identified in theeconometric meaning. More about identification in E4160 andE4136.

I MM also works for other estimation problems than linear inparameters econometric equations. Seminar exercise 10.2 isabout applying MM to a non-linear statistical relationship.

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OLS as a MM estimator I

In the case with

Y = β0 + β1X + ε

E (ε) = 0,E (Y ) = µY , E (X ) = µX and Cov(X , ε) = 0We can form moments

µY = β0 + β1µX (17)

Cov(X , Y ) = β1Var(X ) (18)

(from (X − µX )Y = (X − µX )β0 + β1X (X − µX ) + (X − µX )ε ).

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OLS as a MM estimator II

I We get for β1:

β1 =Cov(X ,Y )

Var(X )=

Cov(X ,Y )

Cov(X , X )=

Cov(Z ,Y )

Cov(Z , X )

if and only if Z ≡ X

I When the assumptions of the regression model hold, X is aperfect instrument for itself and the MM principle leads to thesame estimator of β1 as the least-square principle does.

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Weak instruments

The importance of strong instruments I

I For the models were it is relevant to use MM, we have

plim(β̂OLS1 ) 6= β1 and plim(β̂IV

1 ) = β1

but

E (β̂OLS1 ) 6= β1 and E (β̂IV

1 ) 6= β1

since the MM/IV estimator is biased in finite samples.

I What about the variances of β̂IV1 ?

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Weak instruments

The importance of strong instruments III On page 411 in HGL (BN kap 9.2.7) we find an expression for

the case when Xi and Zi are deterministic (but it has anintuitive appeal):

Var(β̂IV1 ) = Var(β1 +

∑i (Zi − Z̄ )ui

∑i (Zi − Z̄ )Xi)

= σ2u

∑i (Zi − Z̄ )2

[∑i (Zi − Z̄ )Xi ]2= σ2

u

1

[∑i (Zi−Z̄ )Xi ]2

∑i (Zi−Z̄ )2

=σ2u

∑i (Xi − X̄ )21

[∑i (Zi−Z̄ )Xi ]2

∑i (Xi−X̄ )2 ∑i (Zi−Z̄ )2

= Var(β̂OLS1 )

1

r2XZ

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Weak instruments

The importance of strong instruments III

I This result shows that Var(β̂IV1 ) is larger than Var(β̂OLS

1 ),and that the difference can become large when r2XZ is small.

I The problem of low correlation between X and Z is known asthe problem of weak instruments

I One of the advantages of formulating a full model, such as inthe RE example, is that we can understand directly that a testof instrument-strength is to regress X on the set ofinstruments that is defined by the model. What is it?

I A significant t-ratio (or F-test) suggest that the instrumentsare relatively strong. More about this in E4160 and E4136.

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IV estimation of simultaneous equation models I

In class: Can we estimate b consistently in the model thatillustrated OLS simultaneous equations bias in Lecture 16?

Ct = a + b(GDPt) + εCt (19)

GDPt = Ct + It (20)

It = µI + εIt (21)

E (εCt | It) = 0, Var(εCt | It) = σ2C (22)

E (εIt) = 0, Var(εCt) = σ2I (23)

Kov(εCt , εIt) = 0 (24)

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Other estimation principles: Method of Moments …Other estimation principles: Method of Moments and Maximum Likelihood (1 of 2) Ragnar Nymoen University of Oslo 25 April 2013 1/21

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