Econometrics2011

CHAPTER 13. ENDOGENEITY

213

13.2 Reduced Form

The reduced form relationship between the variables or “regressors”xi and the instruments zi is found by linear projection. Let

= E zizi0 1 E zixi0

be the ` k matrix of coe¢ cients from a projection of xi

on zi; and de…ne

ui = xi 0zi

as the projection error. Then the reduced form linear relationship between xi and zi is

xi

= 0zi + ui:

(13.5)

In matrix notation, we can write (13.5) as

X = Z + U

(13.6)

where U is n k:

By construction,

E

(z

i

u0) = 0;

i

so (13.5) is a projection and can be estimated by OLS:

x

=

^

z + u^

^ =

z0z 1 z0x :

Substituting (13:6) into (13.2), we …nd

y = (Z + U) + e

=

Z + v;

(13.7)

where

=

(13.8)

and

v = U + e:

Observe that

E

(z

v

) =

E

z

u0

+

E

(z

e

) = 0:

i

i

i

i

i

i

Thus (13.7) is a projection equation and

may be estimated by OLS. This is

y

=

^

+ v^;

Z

Z0y

^ =

Z0Z 1

The equation (13.7) is the reduced form for y: (13.6) and (13.7) together are the reduced form

equations for the system
y	= Z + v
x	= Z + U:
		^
As we showed above, OLS yields the reduced-form estimates		;

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214

13.3Identi…cation

The structural parameter relates to ( ; ) through (13.8). The parameter is identi…ed,

meaning that it can be recovered from the reduced form, if
rank ( ) = k:	(13.9)
Assume that (13.9) holds. If ` = k; then = 1 : If `	> k; then for any W > 0; =
( 0W ) 1 0W :

If (13.9) is not satis…ed, then cannot be recovered from ( ; ) : Note that a necessary (although not su¢ cient) condition for (13.9) is ` k:

Since Z and X have the common variables X1; we can rewrite some of the expressions. Using (13.3) and (13.4) to make the matrix partitions Z = [Z1; Z2] and X = [Z1; X2] ; we can partitionas

		=		11	12
				21	22
				I	12
		=		0	22
(13.6) can be rewritten as
X1	=	Z1
X2	= Z1 12 + Z2 22 + U2:					(13.10)

is identi…ed if rank( ) = k; which is true if and only if rank( 22) = k2 (by the upper-diagonal structure of ): Thus the key to identi…cation of the model rests on the `2 k2 matrix 22 in (13.10).

13.4Estimation

The model can be written as

yi = x0i + ei

E(ziei) = 0

or

Egi ( ) = 0

gi ( ) = zi yi x0i :

This is a moment condition model. Appropriate estimators include GMM and EL. The estimators and distribution theory developed in those Chapter 8 and 9 directly apply. Recall that the GMM estimator, for given weight matrix W n; is

^ 0 0 1 0 0

= X ZW nZ X X ZW nZ y:

13.5Special Cases: IV and 2SLS

If the model is just-identi…ed, so that k = `; then the formula for GMM simpli…es. We …nd that

=	X0ZW nZ0X 1 X0ZW nZ0y
b		1 W			X0Z 1 X0ZW nZ0y
= Z0X			1
=	Z0X		n
		1 Z0y

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215

This estimator is often called the instrumental variables estimator (IV) of ; where Z is used as an instrument for X: Observe that the weight matrix W n has disappeared. In the just-identi…ed case, the weight matrix places no role. This is also the MME estimator of ; and the EL estimator. Another interpretation stems from the fact that since = 1 ; we can construct the Indirect Least Squares (ILS) estimator:

=

1

b

b

b

Z0X

1

1

Z

Z

1

Z

y

=

Z0Z 1

1 0

0

=

Z X

Z Z

Z

Z

Z

0

y

=

Z00X 1

Z00y

:

0

which again is the IV estimator.

Recall that the optimal weight matrix is an estimate of the inverse of =

E

ziz0e2

: In the

2

z

2

E(zizi0)

2

z

z

i i

j

) suggesting

special case that E ei

i

=

(homoskedasticity), then

=

/ E( i

i0

the weight matrix

W

= (Z

Z) 1 : Using this choice, the GMM estimator equals

n

b

0

1 Z0X 1

1 Z0y

2SLS = X0Z Z0Z

X0Z Z0Z

This is called the two-stage-least squares (2SLS) estimator. It was originally proposed by Theil (1953) and Basmann (1957), and is the classic estimator for linear equations with instruments. Under the homoskedasticity assumption, the 2SLS estimator is e¢ cient GMM, but otherwise it is ine¢ cient.

It is useful to observe that writing

=

1 Z0

P

=

Z Z0Z

X

=

P X

Z

then the 2SLS estimator is

c

b

= X0P X 1 X0P y

b

1

X0y:

=

X0X

The source of the “two-stage”name is since

it can be computed as follows

c c

c

First regress X on Z; vis., = (Z0Z) 1 (Z0X) and X = Z = P X:

c

b

c

c

c

c

b

1

0

Second, regress y on X;

vis., =

0

b

X X

X y:

It is useful to scrutinize

X: Recall, X

= [X1

; X2] and

Z

= [X1; Z2]: Then

the projection c

i

c

=

h

1c

2

[ c

X

=

X1

; X2

P X ; P X ]

=

[X1; P X2]

=

c

hX1; X2i;

since X1 lies in the span of X: Thus in the second stage, we regress y on X1

and X2: So only the

endogenous variables X2 are replaced by their …tted values:

c

X

= Z

+ Z

:

c2

1 b12

2 b22

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216

13.6Bekker Asymptotics

Bekker (1994) used an alternative asymptotic framework to analyze the …nite-sample bias in the 2SLS estimator. Here we present a simpli…ed version of one of his results. In our notation, the model is

y	=	X + e	(13.11)
X	=	Z + U	(13.12)
	= (e; U)
E( j Z)	=	0
E 0 j Z	=	S

As before, Z is n l so there are l instruments.

First, let’s analyze the approximate bias of OLS applied to (13.11). Using (13.12),

En1 X0e = E(xiei) = 0E(ziei) + E(uiei) = s21

and

1

E

X0X

= E xixi0

n

22

= 0Q

z0

z0

u0

u0

= 0

E

z

+

E

u

+ 0

E

z

+

E

u

i

i

i

i

i

i

i

i

+ S

where Q =

E

(ziz0) : Hence by a …rst-order approximation

i

E ^OLS

E

nX0X

E nX0e

1

1

1

1 s21

=

0Q + S22

(13.13)

which is zero only when s21 = 0 (when X is exogenous).

We now derive a similar result for the 2SLS estimator.

Let P = Z (Z0Z)

1

Z0.

^2SLS = X0P X 1 X0P y :

By the spectral decomposition of an idempotent matrix, P = H H0

where = diag (I

l

; 0) : Let Q = H0 S 1=2

which satis…es

Q0Q = In

and partition Q = (q0

Q0 )

E

1

2

where q1 is l 1: Hence

E

0P j Z

1

1

1

n

=

n

S1=20E Q0 Q j Z S1=2

=

S1=20E

q10 q1 S1=2

n

n

=

l

S1=20S1=2

n

=

S

where

l

=

:

n

Using (13.12) and this result,

n1 E X0P e = n1 E 0Z0e + n1 E U0P e = s21;

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ENDOGENEITY

217

and

= 0Q

S22:

1

1

X0P X = 0

z

z0

+ 0

(z

u

) +

u

z0

+

U0P U

nE

E

E

E

nE

i

i

i

i

i

i

+

Together

E ^2SLS

E

nX0P X

E nX0P e

1

1

1

1 s21:

= 0Q + S22

(13.14)

In general this is non-zero, except when s21 = 0 (when X is exogenous). It is also close to zero when = 0. Bekker (1994) pointed out that it also has the reverse implication –that when = l=n is large, the bias in the 2SLS estimator will be large. Indeed as ! 1; the expression in (13.14) approaches that in (13.13), indicating that the bias in 2SLS approaches that of OLS as the number of instruments increases.

Bekker (1994) showed further that under the alternative asymptotic approximation that is …xed as n ! 1 (so that the number of instruments goes to in…nity proportionately with sample

^

size) then the expression in (13.14) is the probability limit of 2SLS

13.7Identi…cation Failure

Recall the reduced form equation

X2 = Z1 12 + Z2 22 + U2:

The parameter fails to be identi…ed if 22 has de…cient rank. The consequences of identi…cation failure for inference are quite severe.

Take the simplest case where k = l = 1 (so there is no Z1): Then the model may be written as

yi	=	xi + ei
xi	=	zi + ui

and 22 = = E(zixi) =Ezi2: We see that is identi…ed if and only if 6= 0; which occurs when E(xizi) 6= 0. Thus identi…cation hinges on the existence of correlation between the excluded exogenous variable and the included endogenous variable.

Suppose this condition fails, so E(xizi) = 0: Then by the CLT

n

1

Xi

d

p

n

=1

ziei ! N1

N 0; E zi2ei2

(13.15)

n

n

1

Xi

1

X

d

p

n

=1

zixi =

p

n

ziui ! N2

N 0; E

zi2ui2

(13.16)

i=1

therefore

1

n

^

p

i=1 ziei

d N1

n

=

pn

P

Cauchy;

1

Pi=1 zixi

! N2

n

since the ratio of two normals is Cauchy. This is particularly nasty, as the Cauchy distribution does not have a …nite mean. This result carries over to more general settings, and was examined by Phillips (1989) and Choi and Phillips (1992).

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218

Suppose that identi…cation does not completely fail, but is weak. This occurs when 22 is full rank, but small. This can be handled in an asymptotic analysis by modeling it as local-to-zero, viz

22 = n 1=2C;

where C is a full rank matrix. The n 1=2 is picked because it provides just the right balancing to allow a rich distribution theory.

To see the consequences, once again take the simple case k = l = 1: Here, the instrument xi is

weak for zi if

= n 1=2c:

Then (13.15) is una¤ected, but (13.16) instead takes the form

n

n

n

1

X

1

Xi

zi2 +

1

X

p

zixi =

p

p

ziui

n

i=1

n

n

=1

i=1

1

n

1

n

Xi

X

= n

zi2c + pn

ziui

=1

i=1

d

! Qc + N2

therefore

N1

^

d

!

:

Qc + N2

^

As in the case of complete identi…cation failure, we …nd that is inconsistent for and the

^

asymptotic distribution of is non-normal. In addition, standard test statistics have non-standard distributions, meaning that inferences about parameters of interest can be misleading.

The distribution theory for this model was developed by Staiger and Stock (1997) and extended to nonlinear GMM estimation by Stock and Wright (2000). Further results on testing were obtained by Wang and Zivot (1998).

The bottom line is that it is highly desirable to avoid identi…cation failure. Once again, the equation to focus on is the reduced form

X2 = Z1 12 + Z2 22 + U2

and identi…cation requires rank( 22) = k2: If k2 = 1; this requires 22 =6 0; which is straightforward to assess using a hypothesis test on the reduced form. Therefore in the case of k2 = 1 (one RHS endogenous variable), one constructive recommendation is to explicitly estimate the reduced form equation for X2; construct the test of 22 = 0, and at a minimum check that the test rejects

H0 : 22 = 0.

When k2 > 1; 22 6= 0 is not su¢ cient for identi…cation. It is not even su¢ cient that each column of 22 is non-zero (each column corresponds to a distinct endogenous variable in Z2): So while a minimal check is to test that each columns of 22 is non-zero, this cannot be interpreted as de…nitive proof that 22 has full rank. Unfortunately, tests of de…cient rank are di¢ cult to implement. In any event, it appears reasonable to explicitly estimate and report the reduced form equations for Z2; and attempt to assess the likelihood that 22 has de…cient rank.

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219

Exercises

1. Consider the single equation model

yi = xi + ei;

^

where yi and zi are both real-valued (1 1). Let denote the IV estimator of using as an instrument a dummy variable di (takes only the values 0 and 1). Find a simple expression for the IV estimator in this context.

2. In the linear model

				yi	=	xi0 + ei
			E(ei	j xi)	=	0
suppose i2 = E ei2 j Xi			is known. Show that the GLS estimator of can be written as an
IV estimator	using some instrument z			: (Find an expression for z			:)
			i			i
3. Take the linear model
				y = X + e:
			^				^
Let the OLS estimator for be and the OLS residual be e^ = y							X .
Let the IV estimator for						~	~
		using some instrument Z be and the IV residual be e~ = y X .

If Z is indeed endogeneous, will IV “…t” better than OLS, in the sense that e~0e~ < e^0e^; at least in large samples?

4. The reduced form between the regressors xi and instruments zi takes the form

xi = 0zi + ui

or

X = Z + U

where Xi is k 1; zi is l 1; X is n k; Z is n l; U is n k; and is l k: The parameteris de…ned by the population moment condition

E ziui0 = 0 ^			1	(Z0X) :
Show that the method of moments estimator for is = (Z0Z)
5. In the structural model
y	=	X + e
X	=	Z + U

with l k; l k; we claim that is identi…ed (can be recovered from the reduced form) if rank( ) = k: Explain why this is true. That is, show that if rank( ) < k then cannot be identi…ed.

6. Take the linear model

yi = xi + ei

E(ei j xi) = 0:

where xi and are 1 1:

CHAPTER 13. ENDOGENEITY				220
(a) Show that E(xiei) = 0 and E x2ei		= 0: Is zi = (xi	x2)0	a valid instrumental variable
for estimation of ?	i		i

(b)De…ne the 2SLS estimator of ; using zi as an instrument for xi: How does this di¤er from OLS?

(c)Find the e¢ cient GMM estimator of based on the moment condition

E(zi (yi xi )) = 0:

Does this di¤er from 2SLS and/or OLS?

7.Suppose that price and quantity are determined by the intersection of the linear demand and supply curves

Demand	:	Q = a0 + a1P + a2Y + e1
Supply	:	Q = b0 + b1P + b2W + e2

where income (Y ) and wage (W ) are determined outside the market. In this model, are the parameters identi…ed?

8.The data …le card.dat is taken from David Card “Using Geographic Variation in College Proximity to Estimate the Return to Schooling”in Aspects of Labour Market Behavior (1995). There are 2215 observations with 29 variables, listed in card.pdf. We want to estimate a wage equation

log(W age) = 0 + 1Educ + 2Exper + 3Exper2 + 4South + 5Black + e

where Educ = Eduation (Years) Exper = Experience (Years), and South and Black are regional and racial dummy variables.

(a)Estimate the model by OLS. Report estimates and standard errors.

(b)Now treat Education as endogenous, and the remaining variables as exogenous. Estimate the model by 2SLS, using the instrument near4, a dummy indicating that the observation lives near a 4-year college. Report estimates and standard errors.

(c)Re-estimate by 2SLS (report estimates and standard errors) adding three additional instruments: near2 (a dummy indicating that the observation lives near a 2-year college), fatheduc (the education, in years, of the father) and motheduc (the education, in years, of the mother).

(d)Re-estimate the model by e¢ cient GMM. I suggest that you use the 2SLS estimates as the …rst-step to get the weight matrix, and then calculate the GMM estimator from this weight matrix without further iteration. Report the estimates and standard errors.

(e)Calculate and report the J statistic for overidenti…cation.

(f)Discuss your …ndings.

Chapter 14

Univariate Time Series

A time series yt is a process observed in sequence over time, t = 1; :::; T . To indicate the dependence on time, we adopt new notation, and use the subscript t to denote the individual observation, and T to denote the number of observations.

Because of the sequential nature of time series, we expect that yt and yt 1 are not independent, so classical assumptions are not valid.

We can separate time series into two categories: univariate (yt 2 R is scalar); and multivariate (yt 2 Rm is vector-valued). The primary model for univariate time series is autoregressions (ARs). The primary model for multivariate time series is vector autoregressions (VARs).

14.1Stationarity and Ergodicity

De…nition 14.1.1 fytg is covariance (weakly) stationary if

E(yt) =

is independent of t; and

cov (yt; yt k) = (k)

is independent of t for all k: (k) is called the autocovariance function.

(k) = (k)= (0) = corr(yt; yt k)

is the autocorrelation function.

De…nition 14.1.2 fytg is strictly stationary if the joint distribution of (yt; :::; yt k) is independent of t for all k:

De…nition 14.1.3 A stationary time series is ergodic if (k) ! 0 as k ! 1.

221

^(0)

CHAPTER 14. UNIVARIATE TIME SERIES

222

The following two theorems are essential to the analysis of stationary time series. There proofs are rather di¢ cult, however.

Theorem 14.1.1 If yt is strictly stationary and ergodic and xt = f(yt; yt 1; :::) is a random variable, then xt is strictly stationary and ergodic.

Theorem 14.1.2 (Ergodic Theorem). If yt is strictly stationary and ergodic and Ejytj < 1; then as T ! 1;

1	T	p
	Xt
T	=1 yt ! E(yt):

This allows us to consistently estimate parameters using time-series moments: The sample mean:

1 XT

^ = T t=1 yt

The sample autocovariance

1 XT

^(k) = T t=1 (yt ^) (yt k ^) :

The sample autocorrelation

^(k) = ^(k):

Theorem 14.1.3 If yt is strictly stationary and ergodic and Eyt2 < 1; then as T ! 1;

p

1. ^ ! E(yt);

p

2. ^(k) ! (k);

p

3. ^(k) ! (k):

Proof of Theorem 14.1.3. Part (1) is a direct consequence of the Ergodic theorem. For Part (2), note that

	1		T
			Xt	(yt ^) (yt k ^)
	^(k) = T =1
	1		T	1		T	1		T
			Xt			X			X
	= T			ytyt k T			yt^ T
			=1			t=1			yt k ^ + ^2:
									t=1