# Econometrics: Models with Endogenous Explanatory Variables

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Econometrics:Models with Endogenous Explanatory Variables

Burcu Eke

UC3M

Endogeneity

I Given the following linear regression model:

Y = 0 + 1X1 + 2X2 + . . .+ kXk +

If E [|X1, X2, . . . Xk] = 0 Xj , then we say that we haveexplanatory exogenous variablesIf, for some reason such as omission of relevant variables,measurement errors, simultaneity, etc., Xj is correlatedwith , we say that Xj is an endogenous explanatoryvariable.

2

Endogeneity

I OLS estimators of the model parameters are invalid(inconsistent, etc.) under the existence of endogenousexplanatory variables.

I In this topic, we will study how to obtain consistentestimators of the model parameters in the presence ofendogenous explanatory variables using instrumentalvariables and applying the two-stage least squaresestimation (2SLS).

3

Endogeneity Example 1: Measurement Error inthe Explanatory Variables

I Recall Y = 0 + 1X + , where the classical assumptions

are satisfied, hence:E [|X] = 0 E [Y |X] = L(Y |X) = 0 + 1X. Underthis case,

E[] = 0 and C(X, ) = 0

0 = E[Y ] 1E[X] and 1 =C(Y,X)

V (X)

4

Endogeneity: Example 1

I However, X has measurement error. We observe X suchthat X = X + v1, where v1 is the measurement error

I Substituting X = X v1, we get

Y = 0 + 1X + 1v1 u

I Then, C(X,u) 6= 0 X is endogenous.

5

Endogeneity Example 2: Omission ofExplanatory Variables

I Recall the case of omitting a relevant variable

I Let Y = 0 + 1X1 + u, where u = + 2X2 and 2 6= 0.Then this model is misspecified by omitting a relevantvariable

I In general, C(X1, u) 6= 0 X1 is endogenous.

6

Endogeneity Examples

1. Ability is not observed in a wage equation such as:log(wage) = 0 + 1educ + 2ability + .

Since the ability is not observable, we are left with thefollowing simple regression model:log(wage) = 0 + 1educ + u, where the ability isincluded in the error term uIf we estimate this model by OLS, we will obtain a biasedand inconsistent estimator of 1 if C(educ, ability) 6= 0.

2. Effect of smoking on wages (ignoring the level of education)

3. Effect of smoking on Cancer (ignoring the physical healthstate)

7

Endogeneity Example 3: Simultaneity

I It is quite common that the realizations of distict variablesare economically related.

I This causes the equation for the dependent variable to be apart of a system of simultaneous equations:

Some of the variables on the right side of the equation ofinterest appear as dependent variables in other equations,and vice versa.

8

Endogeneity Example 3a: Market EquilibriumModel

I Consider the following system:

Y1 = 1Y2 + 2X1 + u1 (Demand)

Y2 = 3Y1 + 4X2 + 5X3 + u2 (Supply)

I The endogenous variables are Y1 =quantity, Y2 =price aredetermined by

the exogenous variables, X1 =rent, X2 =salary, andX3=interest rateand by the disturbances: u1 =demand shock, andu2 =supply shock

I The variables Y1 and Y2, both of which appeared on theright hand side of the supply and demand equations, arenot orthogonal to their respective shocks:

E[Y1|Y2, X1] = 1Y2 + 2X1 + E[u1|Y2, X1] 6=0

9

Endogeneity Example 3b: Production Function

I If a company is a profit maximizer, or a cost minimizer

The quantities of the inputs are simultaneously determinedwith the level of output

The disturbance, that is, the realization of the technologicshocks, is in general correlated with the quantity of theinputs.

10

Instrumental Variables

I Instrumental variables (IV) approach allows us to getconsistent estimators of the population parameters whenthe OLS estimators are inconsistent (in situations such asomitting a relevant variable, measurement errors, andsimultaneity)

11

Instrumental Variables

I In general, we have to use the model: Y = 0 + 1X + ,

where C(X, ) 6= 0 0 and 1 are not the same as theparameters of the linear projection, L(Y |X)

The OLS estimators (0 and 1) of the linear projectionof Y on X are inconsistent estimators of 0 and 1:(xi = Xi X), (yi = Yi Y )

12

Instrumental Variables

p limn

1 =

p limn

(1

n

i

xiyi

)

p limn

(1

n

i

x2i

)

=

p limn

(1

n

i

xi(1xi + i)

)

p limn

(1

n

i

x2i

)

= 1 +

p limn

(1

n

i

xii

)

p limn

(1

n

i

x2i

) = 1 + C(X, )V (X)

6= 1

13

Instrumental Variables: Definition

I In the model:Y = 0 + 1X + (1)

, where C(X, ) 6= 0, in order to get consistent estimatorsof 0 and 1, we need additional information in theform of additional variables.

I Suppose we have a new variable, Z, which is called theinstrumental variable, with the following properties:

1. It is uncorrelated with the error term: (a) C(Z, ) = 0;2. It is correlated with the endogenous variable X: (b)

C(Z,X) 6= 0.

14

Instrumental Variables: IV Estimation in theSimple Model

I Using Z as an instrument, we can obtain consistentestimates 0 and 1.

I From (1) we get: C(Z, Y ) = 1C(Z,X) + C(Z, ), which,given (a), implies that:

1 =C(Z, Y )

C(Z,X)

0 = E[Y ] 1E[X] = E[Y ]C(Z, Y )

C(Z,X)E[X]

15

Instrumental Variables: IV Estimation in theSimple Model

I Assuming we have a random sample of size n of thepopulation, and by replacing population moments with thesample values in the expression above, (principle ofanalogy), we get the Instrumental Variable Estimator (IV):

1 =SY ZSXZ

=

i ziyii zixi

0 = Y 1X

I yi = Yi Y , xi = Xi X, and zi = Zi Z

16

Instrumental Variables: Properties of IVEstimators in the Simple Model

I Provided that (a) and (b) hold, the IV estimator will be aconsistent estimator:

p limn

1 =

p limn

(1

n

i

ziyi

)

p limn

(1

n

i

z2i

)

=

p limn

(1

n

i

zi(1zi + i)

)

p limn

(1

n

i

z2i

)

= = 1 +C(Z, )

V (Z)= 1

17

Instrumental Variables: Properties of IVEstimators in the Simple Model, (a)

I Any instrument or instrumental variable must meet thetwo properties: (a) and (b). Regarding to this:

The condition (a): C(Z, ) = 0, can not be confirmed. Sowe must depend our choice of Z on economic behavior orsome theory we must be very careful when choosing Z.

18

Instrumental Variables: Properties of IVEstimators in the Simple Model, (b)

I The condition (b): C(Z,X) = 0 can be tested using thesample data.

The simplest way is to consider the linear projection of Xon Z: X = 0 + 1Z + v,then, estimate this by OLS and perform the following test:H0 : 1 = 0 vs H1 : 1 6= 0

I Note: If Z = X, we obtain the OLS estimation. That is,when X is exogenous, can be used as its own instrument,and the IV estimator is then identical to the OLSestimator.

19

Instrumental Variables: The Variance of the IVEstimator

I In general, the IV estimator will have a larger variancethan the OLS estimator.

I To see this, lets drive the estimated variance of the IVestimator 1, S

21

:

S21 V

(1

)=2S2ZnS2ZX

=2

nr2ZXS2X

where rZX =SZXSZSX

, is the sample correlation coefficient

between X and Z, which measures the degree of linearrelationship between X and Z in the sample.

20

Instrumental Variables: The Variance of the IVEstimator

I Recall the variance of the OLS estimator of 1, 1:

S21

=2

nS2X

where 2 =1

n

i

2i

I If X is really exogenous, then, the OLS estimators areconsistent, and p lim

n2 = p lim

n2 = 2 =

Since 0 < |rZX | < 1, this implies that S21 > S21

, and the

difference will be bigger, the lower the |rZX | is.

21

Instrumental Variables: The Variance of the IVEstimator

I Therefore, if X is exogenous, using IV estimator insteadof the OLS estimator is costly, in terms of efficiency.

I The lower the correlation between Z and X, the greaterthe difference between the IV variance and the OLSvariance, in favor of the OLS variance.

In the case where both 1 and 1 are consistent, thenasymptotically the IV estimator variance and that of theOLS estimator have the following relationship:

p lim(S21/S2

1

)= 1/2ZX

22

Instrumental Variables: The Variance of the IVEstimator

I This implies, when n

I If ZX = 1% = 0.01, then V (1) = 10000V (1), and

therefore,V (1) = 100

V (1)

I If ZX = 10% = 0.1, then V (1) = 100V (1), and

therefore,V (1) = 10

V (1)

I Even with a relatively high correlation, ZX = 50% = 0.5,

then V (1) = 4V (1), and therefore,V (1) = 2

V (1)

23

Instrumental Variables: The Variance of the IVEstimator

I BUT if X is endogenous, the comparison between OLSand the IV variances has no meaning because the OLSestimator is inconsistent.

24

Instrumental Variables: Inferences with the IVEstimator

I Consider the simple model: Y = 0 + 1X +

I Assume the conditional homoscedasticity assumptionholds: V (|Z) = 2 = V ()

I then, it can be shown that:

1 1S1

asyN(0, 1)

25

Instrumental Variables: Inferences with the IVEstimator

I S1 is the standard error of the IV estimato

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