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Principles of Econometrics, 4th Edition Page 1Chapter 10: Random Regressors and
Moment-Based Estimation
Chapter 10Random Regressors and
Moment-Based Estimation
Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4th Edition Page 2Chapter 10: Random Regressors and
Moment-Based Estimation
10.1 Linear Regression with Random x’s10.2 Cases in Which x and e are Correlated10.3 Estimators Based on the Method of
Moments10.4 Specification Tests
Chapter Contents
Principles of Econometrics, 4th Edition Page 3Chapter 10: Random Regressors and
Moment-Based Estimation
We relax the assumption that variable x is not random
Principles of Econometrics, 4th Edition Page 4Chapter 10: Random Regressors and
Moment-Based Estimation
10.1 Linear Regression with Random x’s
Principles of Econometrics, 4th Edition Page 5Chapter 10: Random Regressors and
Moment-Based Estimation
Modified simple regression assumptions:
10.1Linear Regression with Random x’s
A10.1 yi = β1 + β2xi + ei correctly describes the relationship
between yi and xi in the population, where β1 and β2 are unknown
(fixed) parameters and ei is an unobservable random error term.
A10.2 The data pairs (xi, yi), i = 1, …, N, are obtained by random
sampling. That is, the data pairs are collected from the same
population, by a process in which each pair is independent of every
other pair. Such data are said to be independent and identically
distributed.
Principles of Econometrics, 4th Edition Page 6Chapter 10: Random Regressors and
Moment-Based Estimation
Modified simple regression assumptions (Continued):
10.1Linear Regression with Random x’s
A10.3 The expected value of the error term e, conditional on the value
of x, is zero.
If E(e|x) = 0, then we can show that it is also true that x and e are uncorrelated, and that cov(x, e) = 0. Explanatory variables that are not correlated with the error term are called exogenous variables.
Conversely, if x and e are correlated, then cov(x, e) ≠ 0 and we can show that E(e|x) ≠ 0. Explanatory variables that are
correlated with the error term are called endogenous variables.
A10.4 In the sample, x must take at least two different values.
Principles of Econometrics, 4th Edition Page 7Chapter 10: Random Regressors and
Moment-Based Estimation
Modified simple regression assumptions (Continued):
10.1Linear Regression with Random x’s
A10.5 var(e|x) = σ2. The variance of the error term, conditional on any x, is a constant σ2.
A10.6 The distribution of the error term is normal.
Principles of Econometrics, 4th Edition Page 8Chapter 10: Random Regressors and
Moment-Based Estimation
Assumption A10.2 states that both y and x are obtained by a sampling process, and thus are random– This is the only one new assumption on our list
10.1Linear Regression with Random x’s
Principles of Econometrics, 4th Edition Page 9Chapter 10: Random Regressors and
Moment-Based Estimation
The result that under the classical assumptions, and fixed x’s, the least squares estimator is the best linear unbiased estimator, is a finite sample, or a small sample– This means is that the result does not depend on
the size of the sample
10.1Linear Regression with Random x’s
10.1.1The Small Sample Properties of the
Least Squares Estimators
Principles of Econometrics, 4th Edition Page 10Chapter 10: Random Regressors and
Moment-Based Estimation
Under assumptions A10.1–A10.6:1. The least squares estimator is unbiased2. The least squares estimator is the best linear
unbiased estimator of the regression parameters, and the usual estimator of σ2 is unbiased
3. The distributions of the least squares estimators, conditional upon the x’s, are normal, and their variances are estimated in the usual way • The usual interval estimation and hypothesis
testing procedures are valid
10.1Linear Regression with Random x’s
10.1.1The Small Sample Properties of the
Least Squares Estimators
Principles of Econometrics, 4th Edition Page 11Chapter 10: Random Regressors and
Moment-Based Estimation
If x is random, as long as the data are obtained by random sampling and the other usual assumptions hold, no changes in our regression methods are required
10.1Linear Regression with Random x’s
10.1.1The Small Sample Properties of the
Least Squares Estimators
Principles of Econometrics, 4th Edition Page 12Chapter 10: Random Regressors and
Moment-Based Estimation
For the purposes of a ‘‘large sample’’ analysis of the least squares estimator, it is convenient to replace assumption A10.3 by:
A10.3* E(e) = 0 and cov(x, e) = 0
10.1Linear Regression with Random x’s
10.1.2Large Sample
Properties of the Least Squares
Estimators
Principles of Econometrics, 4th Edition Page 13Chapter 10: Random Regressors and
Moment-Based Estimation
Now we can say: – Under assumptions A10.1, A10.2, A10.3*, A10.4, and A10.5, the least
squares estimators:1. Are consistent.
– They converge in probability to the true parameter values as N→∞.
2. Have approximate normal distributions in large samples, whether the errors are normally distributed or not. – Our usual interval estimators and test statistics are valid, if the
sample is large.3. If assumption A10.3* is not true, and in particular if cov(x,e) ≠ 0
so that x and e are correlated, then the least squares estimators are inconsistent. – They do not converge to the true parameter values even in very
large samples. – None of our usual hypothesis testing or interval estimation
procedures are valid.
10.1Linear Regression with Random x’s
10.1.2Large Sample
Properties of the Least Squares
Estimators
Principles of Econometrics, 4th Edition Page 14Chapter 10: Random Regressors and
Moment-Based Estimation
10.1Linear Regression with Random x’s
10.1.3Why Least Squares
Estimation Fails
FIGURE 10.1 (a) Correlated x and e
Principles of Econometrics, 4th Edition Page 15Chapter 10: Random Regressors and
Moment-Based Estimation
10.1Linear Regression with Random x’s
10.1.3Why Least Squares
Estimation Fails
FIGURE 10.1 (b) Plot of data, true and fitted regression functions
Principles of Econometrics, 4th Edition Page 16Chapter 10: Random Regressors and
Moment-Based Estimation
The statistical consequences of correlation between x and e is that the least squares estimator is biased — and this bias will not disappear no matter how large the sample – Consequently the least squares estimator is
inconsistent when there is correlation between x and e
10.1Linear Regression with Random x’s
10.1.3Why Least Squares
Estimation Fails
Principles of Econometrics, 4th Edition Page 17Chapter 10: Random Regressors and
Moment-Based Estimation
10.2 Cases in Which x and e are
Correlated
Principles of Econometrics, 4th Edition Page 18Chapter 10: Random Regressors and
Moment-Based Estimation
When an explanatory variable and the error term are correlated, the explanatory variable is said to be endogenous – This term comes from simultaneous equations
models• It means ‘‘determined within the system’’
– Using this terminology when an explanatory variable is correlated with the regression error, one is said to have an ‘‘endogeneity problem’’
10.2Cases in Which x
and e are Correlated
Principles of Econometrics, 4th Edition Page 19Chapter 10: Random Regressors and
Moment-Based Estimation
The errors-in-variables problem occurs when an explanatory variable is measured with error – If we measure an explanatory variable with
error, then it is correlated with the error term, and the least squares estimator is inconsistent
10.2Cases in Which x
and e are Correlated
10.2.1Measurement Error
Principles of Econometrics, 4th Edition Page 20Chapter 10: Random Regressors and
Moment-Based Estimation
Let y = annual savings and x* = the permanent annual income of a person – A simple regression model is:
– Current income is a measure of permanent income, but it does not measure permanent income exactly.• It is sometimes called a proxy variable • To capture this feature, specify that:
10.2Cases in Which x
and e are Correlated
10.2.1Measurement Error
*1 2i i iy x v Eq. 10.1
*i i ix x u Eq. 10.2
Principles of Econometrics, 4th Edition Page 21Chapter 10: Random Regressors and
Moment-Based Estimation
Substituting:
10.2Cases in Which x
and e are Correlated
10.2.1Measurement Error
*1 2
1 2
1 2 2
1 2
iy x v
x u v
x v u
x e
Eq. 10.3
Principles of Econometrics, 4th Edition Page 22Chapter 10: Random Regressors and
Moment-Based Estimation
In order to estimate Eq. 10.3 by least squares, we must determine whether or not x is uncorrelated with the random disturbance e – The covariance between these two random
variables, using the fact that E(e) = 0, is:
10.2Cases in Which x
and e are Correlated
10.2.1Measurement Error
*2
2 22 2
cov ,
0u
x e E xe E x u v u
E u
Eq. 10.4
Principles of Econometrics, 4th Edition Page 23Chapter 10: Random Regressors and
Moment-Based Estimation
The least squares estimator b2 is an inconsistent estimator of β2 because of the correlation between the explanatory variable and the error term– Consequently, b2 does not converge to β2 in
large samples– In large or small samples b2 is not
approximately normal with mean β2 and variance
10.2Cases in Which x
and e are Correlated
10.2.1Measurement Error
22var b x x
Principles of Econometrics, 4th Edition Page 24Chapter 10: Random Regressors and
Moment-Based Estimation
Another situation in which an explanatory variable is correlated with the regression error term arises in simultaneous equations models– Suppose we write:
10.2Cases in Which x
and e are Correlated
10.2.2Simultaneous
Equations Bias
1 2Q P e Eq. 10.5
Principles of Econometrics, 4th Edition Page 25Chapter 10: Random Regressors and
Moment-Based Estimation
There is a feedback relationship between P and Q– Because of this, which results because price and
quantity are jointly, or simultaneously, determined, we can show that cov(P, e) ≠ 0
– The resulting bias (and inconsistency) is called the simultaneous equations bias
10.2Cases in Which x
and e are Correlated
10.2.2Simultaneous
Equations Bias
Principles of Econometrics, 4th Edition Page 26Chapter 10: Random Regressors and
Moment-Based Estimation
When an omitted variable is correlated with an included explanatory variable, then the regression error will be correlated with the explanatory variable, making it endogenous
10.2Cases in Which x
and e are Correlated
10.2.3Omitted Variables
Principles of Econometrics, 4th Edition Page 27Chapter 10: Random Regressors and
Moment-Based Estimation
Consider a log-linear regression model explaining observed hourly wage:
–What else affects wages? What have we omitted?
10.2Cases in Which x
and e are Correlated
10.2.3Omitted Variables
21 2 3 4ln β β β βWAGE EDUC EXPER EXPER e Eq. 10.6
Principles of Econometrics, 4th Edition Page 28Chapter 10: Random Regressors and
Moment-Based Estimation
We might expect cov(EDUC, e) ≠ 0– If this is true, then we can expect that the least
squares estimator of the returns to another year of education will be positively biased,
E(b2) > β2, and inconsistent• The bias will not disappear even in very
large samples
10.2Cases in Which x
and e are Correlated
10.2.3Omitted Variables
Principles of Econometrics, 4th Edition Page 29Chapter 10: Random Regressors and
Moment-Based Estimation
Estimating our wage equation, we have:
–We estimate that an additional year of education increases wages approximately 10.75%, holding everything else constant • If ability has a positive effect on wages, then
this estimate is overstated, as the contribution of ability is attributed to the education variable
10.2Cases in Which x
and e are Correlated
10.2.4Least Squares
Estimation of a Wage Equation
2ln 0.5220 0.1075 0.0416 0.0008
se 0.1986 0.0141 0.0132 0.0004
WAGE EDUC EXPER EXPER
Principles of Econometrics, 4th Edition Page 30Chapter 10: Random Regressors and
Moment-Based Estimation
10.3 Estimators Based on the Method of
Moments
Principles of Econometrics, 4th Edition Page 31Chapter 10: Random Regressors and
Moment-Based Estimation
When all the usual assumptions of the linear model hold, the method of moments leads to the least squares estimator – If x is random and correlated with the error
term, the method of moments leads to an alternative, called instrumental variables estimation, or two-stage least squares estimation, that will work in large samples
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 32Chapter 10: Random Regressors and
Moment-Based Estimation
The kth moment of a random variable Y is the expected value of the random variable raised to the kth power:
– The kth population moment in Eq. 10.7 can be estimated consistently using the sample (of size N) analog:
10.3.1Method of Moments
Estimation of a Population Mean
and Variance
10.3Estimators Based on
the Method of Moments
th moment of kkE Y k Y Eq. 10.7
thˆ sample moment of k kk iE Y k Y y N Eq. 10.8
Principles of Econometrics, 4th Edition Page 33Chapter 10: Random Regressors and
Moment-Based Estimation
The method of moments estimation procedure equates m population moments to m sample moments to estimate m unknown parameters– Example:
10.3.1Method of Moments
Estimation of a Population Mean
and Variance
10.3Estimators Based on
the Method of Moments
Eq. 10.9 22 2 2var Y E Y E Y
Principles of Econometrics, 4th Edition Page 34Chapter 10: Random Regressors and
Moment-Based Estimation
The first two population and sample moments of Y are:
10.3.1Method of Moments
Estimation of a Population Mean
and Variance
10.3Estimators Based on
the Method of Moments
Eq. 10.10
1
2 22 2
Population Moments Sample Momentsˆ
ˆi
i
E Y y N
E Y y N
Principles of Econometrics, 4th Edition Page 35Chapter 10: Random Regressors and
Moment-Based Estimation
Solve for the unknown mean and variance parameters:
and
10.3.1Method of Moments
Estimation of a Population Mean
and Variance
10.3Estimators Based on
the Method of Moments
Eq. 10.11 ˆ iy N y
22 2 22 2 2
2ˆ ˆ ii i y yy y Nyy
N N N
Eq. 10.12
Principles of Econometrics, 4th Edition Page 36Chapter 10: Random Regressors and
Moment-Based Estimation
In the linear regression model y = β1 + β2x + e, we usually assume:
– If x is fixed, or random but not correlated with e, then:
10.3.2Method of Moments
Estimation in the Simple Linear
Regression Model
10.3Estimators Based on
the Method of Moments
Eq. 10.13
Eq. 10.14
1 20 0i i iE e E y x
1 20 0E xe E x y x
Principles of Econometrics, 4th Edition Page 37Chapter 10: Random Regressors and
Moment-Based Estimation
We have two equations in two unknowns:
10.3.2Method of Moments
Estimation in the Simple Linear
Regression Model
10.3Estimators Based on
the Method of Moments
Eq. 10.15
1 2
1 2
1 0
1 0
i i
i i i
y b b xN
x y b b xN
Principles of Econometrics, 4th Edition Page 38Chapter 10: Random Regressors and
Moment-Based Estimation
These are equivalent to the least squares normal equations and their solution is:
– Under "nice" assumptions, the method of moments principle of estimation leads us to the same estimators for the simple linear regression model as the least squares principle
10.3.2Method of Moments
Estimation in the Simple Linear
Regression Model
10.3Estimators Based on
the Method of Moments
Eq. 10.16
2 2
1 2
i i
i
x x y yb
x x
b y b x
Principles of Econometrics, 4th Edition Page 39Chapter 10: Random Regressors and
Moment-Based Estimation
Suppose that there is another variable, z, such that:1. z does not have a direct effect on y, and thus it
does not belong on the right-hand side of the model as an explanatory variable
2. z is not correlated with the regression error term e• Variables with this property are said to be
exogenous3. z is strongly [or at least not weakly] correlated
with x, the endogenous explanatory variableA variable z with these properties is called an instrumental variable
10.3.3Instrumental
Variables Estimation in the Simple Linear Regression Model
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 40Chapter 10: Random Regressors and
Moment-Based Estimation
If such a variable z exists, then it can be used to form the moment condition:
– Use Eqs. 10.13 and 10.16, the sample moment conditions are:
10.3.3Instrumental
Variables Estimation in the Simple Linear Regression Model
10.3Estimators Based on
the Method of Moments
1 20 0E ze E z y x Eq. 10.16
1 2
1 2
1 ˆ ˆ 0
1 ˆ ˆ 0
i i
i i i
y xN
z y xN
Eq. 10.17
Principles of Econometrics, 4th Edition Page 41Chapter 10: Random Regressors and
Moment-Based Estimation
Solving these equations leads us to method of moments estimators, which are usually called the instrumental variable (IV) estimators:
10.3.3Instrumental
Variables Estimation in the Simple Linear Regression Model
10.3Estimators Based on
the Method of Moments
Eq. 10.18
2
1 2
ˆ
ˆ ˆ
i ii i i i
i i i i i i
z z y yN z y z yN z x z x z z x x
y x
Principles of Econometrics, 4th Edition Page 42Chapter 10: Random Regressors and
Moment-Based Estimation
These new estimators have the following properties:– They are consistent, if z is exogenous, with
E(ze) = 0 – In large samples the instrumental variable
estimators have approximate normal distributions • In the simple regression model:
10.3.3Instrumental
Variables Estimation in the Simple Linear Regression Model
10.3Estimators Based on
the Method of Moments
Eq. 10.19
2
2 2 22ˆ ~ ,
zx i
Nr x x
Principles of Econometrics, 4th Edition Page 43Chapter 10: Random Regressors and
Moment-Based Estimation
These new estimators have the following properties (Continued):– The error variance is estimated using the
estimator:
10.3.3Instrumental
Variables Estimation in the Simple Linear Regression Model
10.3Estimators Based on
the Method of Moments
2
1 22ˆ ˆ
ˆ2
i i
IV
y x
N
Principles of Econometrics, 4th Edition Page 44Chapter 10: Random Regressors and
Moment-Based Estimation
Note that we can write the variance of the instrumental variables estimator of β2 as:
– Because the variance of the instrumental variables estimator will always be larger than the variance of the least squares estimator, and thus it is said to be less efficient
10.3.3aThe Importance of
Using Strong Instruments
10.3Estimators Based on
the Method of Moments
22
2 2 22
varˆvarzxzx i
brr x x
2 1zxr
Principles of Econometrics, 4th Edition Page 45Chapter 10: Random Regressors and
Moment-Based Estimation
To extend our analysis to a more general setting, consider the multiple regression model:
– Let xK be an endogenous variable correlated with the error term
– The first K - 1 variables are exogenous variables that are uncorrelated with the error term e - they are ‘‘included’’ instruments
10.3.4Instrumental
Variables Estimation in the Multiple
Regression Model
10.3Estimators Based on
the Method of Moments
1 2 2β β βK Ky x x e
Principles of Econometrics, 4th Edition Page 46Chapter 10: Random Regressors and
Moment-Based Estimation
We can estimate this equation in two steps with a least squares estimation in each step
10.3.4Instrumental
Variables Estimation in the Multiple
Regression Model
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 47Chapter 10: Random Regressors and
Moment-Based Estimation
The first stage regression has the endogenous variable xK on the left-hand side, and all exogenous and instrumental variables on the right-hand side– The first stage regression is:
– The least squares fitted value is:
10.3.4Instrumental
Variables Estimation in the Multiple
Regression Model
10.3Estimators Based on
the Method of Moments
1 2 2 1 1 1 1K K K L L Kx x x z z v Eq. 10.20
1 2 2 1 1 1 1ˆ ˆˆ ˆ ˆˆK K K L Lx x x z z Eq. 10.21
Principles of Econometrics, 4th Edition Page 48Chapter 10: Random Regressors and
Moment-Based Estimation
The second stage regression is based on the original specification:
– The least squares estimators from this equation are the instrumental variables (IV) estimators
– Because they can be obtained by two least squares regressions, they are also popularly known as the two-stage least squares (2SLS) estimators • We will refer to them as IV or 2SLS or
IV/2SLS estimators
10.3.4Instrumental
Variables Estimation in the Multiple
Regression Model
10.3Estimators Based on
the Method of Moments
Eq. 10.22 *1 2 2 ˆβ β βK Ky x x e
Principles of Econometrics, 4th Edition Page 49Chapter 10: Random Regressors and
Moment-Based Estimation
The IV/2SLS estimator of the error variance is based on the residuals from the original model:
10.3.4Instrumental
Variables Estimation in the Multiple
Regression Model
10.3Estimators Based on
the Method of Moments
Eq. 10.23 2
1 2 22ˆ ˆ ˆβ β β
σi i K Ki
IV
y x x
N K
Principles of Econometrics, 4th Edition Page 50Chapter 10: Random Regressors and
Moment-Based Estimation
In the simple regression, if x is endogenous and we have L instruments:
– The two sample moment conditions are:
10.3.4aUsing Surplus Instruments in
Simple Regression
10.3Estimators Based on
the Method of Moments
1 1 1ˆ ˆˆˆ L Lx z z
1 2
1 2
1 ˆ ˆβ β 0
1 ˆ ˆˆ β β 0
i i
i i i
y xN
x y xN
Principles of Econometrics, 4th Edition Page 51Chapter 10: Random Regressors and
Moment-Based Estimation
Solving using the fact that , we get:
10.3.4aUsing Surplus Instruments in
Simple Regression
10.3Estimators Based on
the Method of Moments
2
1 2
ˆ ˆ ˆβ
ˆˆ ˆ
ˆ ˆβ β
i i i i
i ii i
x x y y x x y yx x x xx x x x
y x
x x
Principles of Econometrics, 4th Edition Page 52Chapter 10: Random Regressors and
Moment-Based Estimation
Sometimes we have more instrumental variables at our disposal than are necessary – Suppose we have L = 2 instruments, z1 and z2
– Then we have:
10.3.4bSurplus Moment
Conditions
10.3Estimators Based on
the Method of Moments
2 2 1 2β β 0E z e E z y x
Principles of Econometrics, 4th Edition Page 53Chapter 10: Random Regressors and
Moment-Based Estimation
We have three sample moment conditions:
10.3.4bSurplus Moment
Conditions
10.3Estimators Based on
the Method of Moments
1 2
1 1 2 2
2 1 2 3
1 ˆ ˆ ˆβ β 0
1 ˆ ˆ ˆβ β 0
1 ˆ ˆ ˆβ β 0
i i i
i i i
i i i
y x mN
z y x mN
z y x mN
Principles of Econometrics, 4th Edition Page 54Chapter 10: Random Regressors and
Moment-Based Estimation
The first stage regression is a key tool in assessing whether an instrument is ‘‘strong’’ or ‘‘weak’’ in the multiple regression setting
10.3.5Assessing
Instrument Strength Using the First Stage
Model
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 55Chapter 10: Random Regressors and
Moment-Based Estimation
Suppose the first stage regression equation is:
– The key to assessing the strength of the instrumental variable z1 is the strength of its relationship to xK after controlling for the effects of all the other exogenous variables
10.3.5aOne Instrumental
Variable
10.3Estimators Based on
the Method of Moments
1 2 2 1 1 1 1K K K Kx x x z v Eq. 10.24
Principles of Econometrics, 4th Edition Page 56Chapter 10: Random Regressors and
Moment-Based Estimation
Suppose the first stage regression equation is:
–We require that at least one of the instruments be strong
10.3.5bMore Than One
Instrumental Variable
10.3Estimators Based on
the Method of Moments
Eq. 10.25 1 2 2 1 1 1 1K K K L L Kx x x z z v
Principles of Econometrics, 4th Edition Page 57Chapter 10: Random Regressors and
Moment-Based Estimation
Consider the model with an instrumental variable MOTHEREDUC:
10.3.6Instrumental
Variables Estimation of the Wage
Equation
10.3Estimators Based on
the Method of Moments
29.7751 0.0489 0.0013 0.2677 se 0.4249 0.0417 0.0012 0.0311EDUC EXPER EXPER MOTHEREDUC
Eq. 10.26
Principles of Econometrics, 4th Edition Page 58Chapter 10: Random Regressors and
Moment-Based Estimation
To implement instrumental variables estimation using the two-stage least squares approach, we obtain the predicted values of education from the first stage equation and insert it into the log-linear wage equation to replace EDUC – Then estimate the resulting equation by least
squares
10.3.6Instrumental
Variables Estimation of the Wage
Equation
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 59Chapter 10: Random Regressors and
Moment-Based Estimation
The instrumental variables estimates of the log-linear wage equation are:
10.3.6Instrumental
Variables Estimation of the Wage
Equation
10.3Estimators Based on
the Method of Moments
2ln 0.1982 0.0493 0.0449 0.0009
se 0.4729 0.0374 0.0136 0.0004
WAGE EDUC EXPER EXPER
Principles of Econometrics, 4th Edition Page 60Chapter 10: Random Regressors and
Moment-Based Estimation
Using FATHEREDUC, the first stage equation is:
10.3.6Instrumental
Variables Estimation of the Wage
Equation
10.3Estimators Based on
the Method of Moments
21 2 3 1 2γ γ γ θ θEDUC EXPER EXPER MOTHEREDUC FATHEREDUC v
Principles of Econometrics, 4th Edition Page 61Chapter 10: Random Regressors and
Moment-Based Estimation
10.3.6Instrumental
Variables Estimation of the Wage
Equation
10.3Estimators Based on
the Method of Moments
Table 10.1 First-Stage Equation
Principles of Econometrics, 4th Edition Page 62Chapter 10: Random Regressors and
Moment-Based Estimation
The IV/2SLS estimates are:
10.3.6Instrumental
Variables Estimation of the Wage
Equation
10.3Estimators Based on
the Method of Moments
2ln 0.0481 0.0614 0.0442 0.0009
se 0.4003 0.0314 0.0134 0.0004
WAGE EDUC EXPER EXPER Eq. 10.27
Principles of Econometrics, 4th Edition Page 63Chapter 10: Random Regressors and
Moment-Based Estimation
In a multiple regression model, the coefficients are the effect of a unit change in an explanatory, independent, variable on the expected outcome, holding all other things constant– In calculus terminology, the coefficients are
partial derivatives
10.3.7Partial Correlation
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 64Chapter 10: Random Regressors and
Moment-Based Estimation
We can net out or partial out the effects of explanatory variables– Regression coefficients can be thought of
measuring the effect of one variable on another after removing, or partialling out, the effects of all other variables
– The sample correlation between two residuals is called the partial correlation coefficient
10.3.7Partial Correlation
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 65Chapter 10: Random Regressors and
Moment-Based Estimation
The multiple regression model, including all K variables, is:
10.3.8Instrumental
Variables Estimation in a General Model
10.3Estimators Based on
the Method of Moments
1 2 2 1 1
G exogenous variables B endogenous variables
G G G G K Ky x x x x e Eq. 10.28
Principles of Econometrics, 4th Edition Page 66Chapter 10: Random Regressors and
Moment-Based Estimation
Think of G = Good explanatory variables, B = Bad explanatory variables and L = Lucky instrumental variables– It is a necessary condition for IV estimation that L ≥ B– If L = B then there are just enough instrumental variables
to carry out IV estimation • The model parameters are said to just identified or
exactly identified in this case• The term identified is used to indicate that the model
parameters can be consistently estimated – If L > B then we have more instruments than are
necessary for IV estimation, and the model is said to be overidentified
10.3.8Instrumental
Variables Estimation in a General Model
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 67Chapter 10: Random Regressors and
Moment-Based Estimation
Consider the B first-stage equations:
The predicted values are:
In the second stage of estimation we apply least squares to:
10.3.8Instrumental
Variables Estimation in a General Model
10.3Estimators Based on
the Method of Moments
1 2 2 1 1 ,
1, ,G j j j Gj G j Lj L jx x x z z v
j B
Eq. 10.29
1 2 2 1 1ˆ ˆˆ ˆ ˆˆ ,
1, ,G j j j Gj G j Lj Lx x x z z
j B
*1 2 2 1 1ˆ ˆG G G G K Ky x x x x e Eq. 10.30
Principles of Econometrics, 4th Edition Page 68Chapter 10: Random Regressors and
Moment-Based Estimation
Consider the model with B = 2:
– The first-stage equations are:
10.3.8aAssessing
Instrument Strength in a General Model
10.3Estimators Based on
the Method of Moments
Eq. 10.31 1 2 2 1 1 1 1G G G G G Gy x x x x e
1 11 21 2 1 11 1 21 2 1
2 12 22 2 2 12 1 22 2 2
γ γ γ θ θγ γ γ θ θ
G G G
G G G
x x x z z vx x x z z v
Principles of Econometrics, 4th Edition Page 69Chapter 10: Random Regressors and
Moment-Based Estimation
When testing the null hypothesis H0: βk = c, use of the test statistic is valid in large samples – It is common, but not universal, practice to use
critical values, and p-values, based on the distribution rather than the more strictly appropriate N(0,1) distribution
– The reason is that tests based on the t-distribution tend to work better in samples of data that are not large
10.3.8bHypothesis Testing with Instrumental
Variables Estimates
10.3Estimators Based on
the Method of Moments
ˆ ˆsek kt c
Principles of Econometrics, 4th Edition Page 70Chapter 10: Random Regressors and
Moment-Based Estimation
When testing a joint hypothesis, such as H0: β2 = c2, β3 = c3, the test may be based on the chi-square distribution with the number of degrees of freedom equal to the number of hypotheses (J) being tested – The test itself may be called a “Wald” test, or a
likelihood ratio (LR) test, or a Lagrange multiplier (LM) test
– These testing procedures are all asymptotically equivalent
10.3.8bHypothesis Testing with Instrumental
Variables Estimates
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 71Chapter 10: Random Regressors and
Moment-Based Estimation
Unfortunately R2 can be negative when based on IV estimates– Therefore the use of measures like R2 outside
the context of the least squares estimation should be avoided
10.3.8cGoodness-of-Fit
with Instrumental Variables Estimates
10.3Estimators Based on
the Method of Moments
Principles of Econometrics, 4th Edition Page 72Chapter 10: Random Regressors and
Moment-Based Estimation
10.4 Specification Tests
Principles of Econometrics, 4th Edition Page 73Chapter 10: Random Regressors and
Moment-Based Estimation
1. Can we test for whether x is correlated with the error term?
– This might give us a guide of when to use least squares and when to use IV estimators
2. Can we test if our instrument is valid, and uncorrelated with the regression error, as required?
10.4Specification Tests
Principles of Econometrics, 4th Edition Page 74Chapter 10: Random Regressors and
Moment-Based Estimation
The null hypothesis is H0: cov(x, e) = 0 against the alternative H1: cov(x, e) ≠ 0
10.4Specification Tests
10.4.1The Hausman Test for Endogeneity
Principles of Econometrics, 4th Edition Page 75Chapter 10: Random Regressors and
Moment-Based Estimation
If null hypothesis is true, both the least squares estimator and the instrumental variables estimator are consistent
– Naturally if the null hypothesis is true, use the more efficient estimator, which is the least squares estimator
If the null hypothesis is false, the least squares estimator is not consistent, and the instrumental variables estimator is consistent
– If the null hypothesis is not true, use the instrumental variables estimator, which is consistent
10.4Specification Tests
10.4.1The Hausman Test for Endogeneity
Principles of Econometrics, 4th Edition Page 76Chapter 10: Random Regressors and
Moment-Based Estimation
There are several forms of the test, usually called the Hausman test
10.4Specification Tests
10.4.1The Hausman Test for Endogeneity
Principles of Econometrics, 4th Edition Page 77Chapter 10: Random Regressors and
Moment-Based Estimation
Consider the model:
– Let z1 and z2 be instrumental variables for x.
1. Estimate the model by least squares, and obtain the residuals .
• If there are more than one explanatory variables that are being tested for endogeneity, repeat this estimation for each one, using all available instrumental variables in each regression
10.4Specification Tests
10.4.1The Hausman Test for Endogeneity 1 2y x e
1 1 1 2 2x z z v
1 1 1 2 2ˆ ˆˆv x z z
Principles of Econometrics, 4th Edition Page 78Chapter 10: Random Regressors and
Moment-Based Estimation
Consider the model (Continued):
2. Include the residuals computed in step 1 as an explanatory variable in the original regression,
– Estimate this "artificial regression" by least squares, and employ the usual t-test for the hypothesis of significance
10.4Specification Tests
10.4.1The Hausman Test for Endogeneity 1 2y x e
1 2 ˆy x v e
0
1
: 0 no correlation between and : 0 correlation between and
H x eH x e
Principles of Econometrics, 4th Edition Page 79Chapter 10: Random Regressors and
Moment-Based Estimation
Consider the model (Continued):
3. If more than one variable is being tested for endogeneity, the test will be an F-test of joint significance of the coefficients on the included residuals
10.4Specification Tests
10.4.1The Hausman Test for Endogeneity 1 2y x e
Principles of Econometrics, 4th Edition Page 80Chapter 10: Random Regressors and
Moment-Based Estimation
A test of the validity of the surplus moment conditions is:
1. Compute the IV estimates using all available instruments, including the G variables x1=1, x2, …, xG that are presumed to be exogenous, and the L instruments
2. Obtain the residuals
3. Regress on all the available instruments described in step 1
10.4Specification Tests
10.4.2Testing Instrument
Validity
ˆk
1 2 2ˆˆ .ˆ ˆ
K Ke y x x
e
Principles of Econometrics, 4th Edition Page 81Chapter 10: Random Regressors and
Moment-Based Estimation
A test of the validity of the surplus moment conditions is (Continued):
4. Compute NR2 from this regression, where N is the sample size and R2 is the usual goodness-of-fit measure
5. If all of the surplus moment conditions are valid, then
• If the value of the test statistic exceeds the 100(1−α)-percentile from the distribution, then we conclude that at least one of the surplus moment conditions restrictions is not valid
10.4Specification Tests
10.4.2Testing Instrument
Validity
2 2( ) .~ L BNR
2( )L B
Principles of Econometrics, 4th Edition Page 82Chapter 10: Random Regressors and
Moment-Based Estimation
10.4Specification Tests
10.4.3Specification Tests
for the Wage Equation
Table 10.2 Hausman Test Auxiliary Regression
Principles of Econometrics, 4th Edition Page 83Chapter 10: Random Regressors and
Moment-Based Estimation
Key Words
Principles of Econometrics, 4th Edition Page 84Chapter 10: Random Regressors and
Moment-Based Estimation
asymptotic propertiesconditional expectationendogenous variableserrors-in-variablesexogenous variablesfinite sample propertiesfirst stage regressionHausman test
Keywords
instrumental variableinstrumental variable estimatorjust identified equationslarge sample propertiesover identified equationspopulation momentsrandom sampling
reduced form equationsample momentssimultaneous equations biastest of surplus moment conditionstwo-stage least squares estimationweak instruments
Principles of Econometrics, 4th Edition Page 85Chapter 10: Random Regressors and
Moment-Based Estimation
Appendices
Principles of Econometrics, 4th Edition Page 86Chapter 10: Random Regressors and
Moment-Based Estimation
We can use the conditional pdf to compute the conditional mean of Y given X:
Similarly we can define the conditional variance of Y given X:
10AConditional and
Iterated Expectations
Eq. 10A.1
10A.1Conditional
Expectations
| | |y y
E Y X x yP Y y X x yf y x
2var | | |
yY X x y E Y X x f y x
Principles of Econometrics, 4th Edition Page 87Chapter 10: Random Regressors and
Moment-Based Estimation
The law of iterated expectations says that the expected value of the conditional expectation of Y given X:
10AConditional and
Iterated Expectations
Eq. 10A.2
10A.2Iterated
Expectations
|XE Y E E Y X
Principles of Econometrics, 4th Edition Page 88Chapter 10: Random Regressors and
Moment-Based Estimation
We can now show:
10AConditional and
Iterated Expectations
10A.2Iterated
Expectations
,
|
| [by changing order of summation]
|
|
y y x
y x
x y
x
X
E Y yf y y f x y
y f y x f x
yf y x f x
E Y X x f x
E E Y X
Principles of Econometrics, 4th Edition Page 89Chapter 10: Random Regressors and
Moment-Based Estimation
Two other results can be shown to be true:
10AConditional and
Iterated Expectations
10A.2Iterated
Expectations
|XE XY E XE Y X
cov , |X XX Y E X E Y X
Eq. 10A.3
Eq. 10A.4
Principles of Econometrics, 4th Edition Page 90Chapter 10: Random Regressors and
Moment-Based Estimation
The following can be shown to hold:
10AConditional and
Iterated Expectations
10A.3Regression
Model Application
Eq. 10A.5
Eq. 10A.6
| 0 0i x i i xE e E E e x E
| 0 0i i x i i i x iE x e E x E e x E x
cov , | 0 0i i x i x i i x i xx e E x E e x E x Eq. 10A.7
Principles of Econometrics, 4th Edition Page 91Chapter 10: Random Regressors and
Moment-Based Estimation
If E(e|x) = 0 it follows that E(e) = 0, E(xe) = 0, and cov(x, e) = 0– However, if E(e|x) ≠ 0 then cov(x, e) ≠ 0
10AConditional and
Iterated Expectations
10A.3Regression
Model Application
Principles of Econometrics, 4th Edition Page 92Chapter 10: Random Regressors and
Moment-Based Estimation
This is an algebraic proof that the least squares estimator is not consistent when cov(x, e) ≠ 0 – The regression model is y = β1 + β2x + e. – Under Eq. A10.3, E(e) = 0, so that
E(y) = β1 + β2E(x)
10BThe
Inconsistency of the Least Squares
Estimator
Principles of Econometrics, 4th Edition Page 93Chapter 10: Random Regressors and
Moment-Based Estimation
Subtract this expectation from the original equation:
–Multiply both sides by x – E(x):
– Take expected values of both sides:
or
10BThe
Inconsistency of the Least Squares
Estimator
2i i i i iy E y x E x e
22i i i i i i i i ix E x y E y x E x x E x e
22i i i i i i i i iE x E x y E y E x E x E x E x e
2cov , var cov ,x y x x e
Principles of Econometrics, 4th Edition Page 94Chapter 10: Random Regressors and
Moment-Based Estimation
Solve for β2:
– If we assume cov(x, e) = 0, then:
– The least squares estimator can be expressed as:
10BThe
Inconsistency of the Least Squares
Estimator
2
cov , cov ,var var
x y x ex x
Eq. 10B.1
2
cov ,var
x yx
Eq. 10B.2
2 2 2
/ 1 cov( , )var( )/ 1
i i i i
i i
x x y y x x y y N x ybxx x x x N
Eq. 10B.3
Principles of Econometrics, 4th Edition Page 95Chapter 10: Random Regressors and
Moment-Based Estimation
The sample variance and covariance converge to the true variance and covariance as the sample size N increases, so that the least squares estimator converges to β2
– If cov(x, e) = 0, then:
– If cov(x, e) ≠ 0, then:
10BThe
Inconsistency of the Least Squares
Estimator
2 2
cov( , ) cov( , )var( )var( )
x y x ybxx
2
cov , cov ,var var
x y x ex x
Principles of Econometrics, 4th Edition Page 96Chapter 10: Random Regressors and
Moment-Based Estimation
The least squares estimator now converges to:
10BThe
Inconsistency of the Least Squares
Estimator
2 2 2
cov , cov ,
var varx y x e
bx x
Eq. 10B.4
Principles of Econometrics, 4th Edition Page 97Chapter 10: Random Regressors and
Moment-Based Estimation
The IV estimator can be expressed as:
– For large samples:
10CThe Consistency
of the IV Estimator
Eq. 10C.1
2
1 cov ,ˆ1 cov ,
i i
i i
z z y y N z yz z x x N z x
2
cov ,ˆcov ,
z yz x
Eq. 10C.2
Principles of Econometrics, 4th Edition Page 98Chapter 10: Random Regressors and
Moment-Based Estimation
Following the steps from Appendix 10B, we get:
If cov(x, e) = 0, then:
10CThe Consistency
of the IV Estimator
Eq. 10C.3
Eq. 10C.4
2
cov , cov ,cov , cov ,
z y z ez x z x
2 2
cov ,ˆcov ,
z yz x
Principles of Econometrics, 4th Edition Page 99Chapter 10: Random Regressors and
Moment-Based Estimation
Start with the simple regression model:
We can describe the relationship between an instrumental variable z, which must be correlated with x but uncorrelated with e, as:
10DThe Logic of the Hausman Test
Eq. 10D.1
Eq. 10D.2
1 2y x e
0 1x z v
Principles of Econometrics, 4th Edition Page 100Chapter 10: Random Regressors and
Moment-Based Estimation
We can divide x into two parts, a systematic part and a random part, as:
– Substituting:
10DThe Logic of the Hausman Test
Eq. 10D.3
Eq. 10D.4
x E x v
1 2 1 2
1 2 2
y x e E x v eE x v e
Principles of Econometrics, 4th Edition Page 101Chapter 10: Random Regressors and
Moment-Based Estimation
An estimated analog of Eq. 10D.3 is:
Substitute Eq. 10D.5 into the original Eq. 10D.1:
10DThe Logic of the Hausman Test
Eq. 10D.5
Eq. 10D.6
ˆ ˆx x v
1 2 1 2
1 2 2
ˆ ˆˆ ˆ
y x e x v ex v e
Principles of Econometrics, 4th Edition Page 102Chapter 10: Random Regressors and
Moment-Based Estimation
To reduce confusion, write:
– If we omit
10DThe Logic of the Hausman Test
Eq. 10D.7
Eq. 10D.8
1 2 ˆ ˆy x v e
1 2 ˆy x e
v
Principles of Econometrics, 4th Edition Page 103Chapter 10: Random Regressors and
Moment-Based Estimation
Carrying out the test is made simpler by playing a trick on Eq. 10D.7:
10DThe Logic of the Hausman Test
Eq. 10D.9
1 2 2 2
1 2 2
1 2
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ
y x v e v v
x v v e
x v e
Principles of Econometrics, 4th Edition Page 104Chapter 10: Random Regressors and
Moment-Based Estimation
Using canonical correlations there is a solution to the problem of identifying weak instruments when an equation has more than one endogenous variable– Canonical correlations are a generalization of
the usual concept of a correlation between two variables and attempt to describe the association between two sets of variables
10ETesting for Weak
Instruments
Principles of Econometrics, 4th Edition Page 105Chapter 10: Random Regressors and
Moment-Based Estimation
A test for weak identification, the situation that arises when the instruments are correlated with the endogenous regressors but only weakly, is based on the Cragg-Donald F-test statistic
10ETesting for Weak
Instruments
10E.1A Test for Weak Identification
2 2Cragg-Donald 1B BN G B L r r Eq. 10E.1
Principles of Econometrics, 4th Edition Page 106Chapter 10: Random Regressors and
Moment-Based Estimation
Two particular consequences of weak instruments:– Relative Bias: In the presence estimator can
become large– Rejection Rate (Test Size): When estimating a
model with endogenous regressors, testing hypotheses about the coefficients of the endogenous variables is frequently of interest
10ETesting for Weak
Instruments
10E.1A Test for Weak Identification
Principles of Econometrics, 4th Edition Page 107Chapter 10: Random Regressors and
Moment-Based Estimation
10ETesting for Weak
Instruments
10E.1A Test for Weak Identification
Table 10E.1 Critical Values for the Weak Instrument Test Basedon IV Test Size (5% level of significance)
Principles of Econometrics, 4th Edition Page 108Chapter 10: Random Regressors and
Moment-Based Estimation
10ETesting for Weak
Instruments
10E.1A Test for Weak Identification
Table 10E.2 Critical Values for the Weak Instrument Test Basedon IV Relative Bias (5% level of significance)
Principles of Econometrics, 4th Edition Page 109Chapter 10: Random Regressors and
Moment-Based Estimation
Consider the following HOURS supply equation specification:
where
10ETesting for Weak
Instruments
10E.2Examples of
Testing for Weak Identification
1 2 3 4 5β β β β 6 βHOURS MTR EDUC KIDSL NWIFEINC e Eq. 10E.2
1000NWIFEINC FAMINC WAGE HOURS
Principles of Econometrics, 4th Edition Page 110Chapter 10: Random Regressors and
Moment-Based Estimation
Weak IV Example 1: Endogenous: MTR; Instrument: EXPER– The estimated first-stage equation for MTR is
Model (1) of Table 10E.3– The estimated coefficient of MTR in the
estimated HOURS supply equation in Model (1) of Table 10E.4 is negative and significant at the 5% level
10ETesting for Weak
Instruments
10E.2Examples of
Testing for Weak Identification
Principles of Econometrics, 4th Edition Page 111Chapter 10: Random Regressors and
Moment-Based Estimation
10ETesting for Weak
Instruments
10E.2Examples of
Testing for Weak Identification
Table 10E.3 First-stage Equations
Principles of Econometrics, 4th Edition Page 112Chapter 10: Random Regressors and
Moment-Based Estimation
10ETesting for Weak
Instruments
10E.2Examples of
Testing for Weak Identification
Table 10E.4 IV Estimation of Hours Equation
Principles of Econometrics, 4th Edition Page 113Chapter 10: Random Regressors and
Moment-Based Estimation
Weak IV Example 2: Endogenous: MTR; Instruments: EXPER, EXPER2, LARGECITY– The first-stage equation estimates are reported
in Model (2) of Table 10E.3– The estimated coefficient of MTR in the
estimated HOURS supply equation in Model (2) of Table 10E.4 is negative and significant at the 5% level, although the magnitudes of all the coefficients are smaller in absolute value for this estimation than the model in Model (1)
10ETesting for Weak
Instruments
10E.2Examples of
Testing for Weak Identification
Principles of Econometrics, 4th Edition Page 114Chapter 10: Random Regressors and
Moment-Based Estimation
Weak IV Example 3 Endogenous: MTR, EDUC; Instruments: MOTHEREDUC, FATHEREDUC– The first-stage equations for MTR and EDUC
are Model (3) and Model (4) of Table 10E.3– The estimates of the HOURS supply equation,
Model (3) of Table 10E.4, shows parameter estimates that are wildly different from those in Model (1) and Model (2), and the very small t-statistic values imply very large standard errors, another consequence for instrumental variables estimation in the presence of weak instruments
10ETesting for Weak
Instruments
10E.2Examples of
Testing for Weak Identification
Principles of Econometrics, 4th Edition Page 115Chapter 10: Random Regressors and
Moment-Based Estimation
If instrumental variables are ‘‘weak,’’ then the instrumental variables, or two-stage least squares, estimator is unreliable When there is a single endogenous variable, the first-stage F-test of the joint significance of the external instruments is an indicator of instrument strengthIf there is more than one endogenous variable on the right-hand side of an equation, then the F-test statistics from the first stage equations do not provide reliable information about instrument strength
10ETesting for Weak
Instruments
10E.3Testing for Weak
Identification: Conclusions
Principles of Econometrics, 4th Edition Page 116Chapter 10: Random Regressors and
Moment-Based Estimation
We do two types of simulations1. We generate a sample of artificial data and
give numerical illustrations of the estimators and tests
2. We carry out a Monte Carlo simulation to illustrate the repeated sampling properties of the least squares and IV/2SLS estimators under various conditions
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 117Chapter 10: Random Regressors and
Moment-Based Estimation
We create an artificial sample of y values by adding e to the systematic portion of the regression– The least squares estimates are
10F.1Illustrations
Using Simulated Data
ˆ 0.9789 1.7034
se 0.088 0.090LSy x
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 118Chapter 10: Random Regressors and
Moment-Based Estimation
The IV estimates using z1 are:
The IV estimates using z2 are:
10F.1Illustrations
Using Simulated Data
1_ˆ 1.1011 1.1924
se 0.109 0.195IV zy x
2_ˆ 1.3451 0.1724
se 0.256 0.797IV zy x
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 119Chapter 10: Random Regressors and
Moment-Based Estimation
If we use instrumental variables estimation with the invalid instrument, we get:
10F.1Illustrations
Using Simulated Data
3_ˆ 0.9640 1.7657
se 0.095 0.172IV zy x
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 120Chapter 10: Random Regressors and
Moment-Based Estimation
The outcome of two-stage least squares estimation using the two instruments z1 and z2 where we first obtain the first-stage regression of x on the two instruments z1 and z2:
– The instrumental variables estimates are:
10F.1Illustrations
Using Simulated Data
1 2ˆ 0.1947 0.5700 0.2068
se 0.079 0.089 0.077
x z z Eq. 10F.1
1 2_ ,ˆ 1.1376 1.0399
se 0.116 0.194IV z zy x
Eq. 10F.2
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 121Chapter 10: Random Regressors and
Moment-Based Estimation
To implement the Hausman test, estimate the first-stage equation shown in Eq. 10F.1 using the instruments z1 and z2 – Compute the residuals:
– Include the residuals as an extra variable in the regression equation and apply least squares:
10F.1.1The Hausman
Test
1 2ˆ ˆ 0.1947 0.5700 0.2068v x x x z z
ˆ ˆ1.1376 1.0399 0.9957se 0.080 0.133 0.163
y x v
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 122Chapter 10: Random Regressors and
Moment-Based Estimation
If we consider using just z1 as an instrument, the estimated first-stage equation is:
If we use just z2 as an instrument, the estimated first-stage equation is:
10F.1.2Test for Weak Instruments
1ˆ 0.2196 0.5711
t 6.24
x z
2ˆ 0.2140 0.2090
t 2.28
x z
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 123Chapter 10: Random Regressors and
Moment-Based Estimation
If we use z1, z2, and z3 as instruments, there are two surplus moment conditions. – The IV estimates using these three instruments
are:
– Obtaining the residuals and regressing them on the instruments yields:
• The R2 from this regression is 0.1311 and NR2 = 13.11
10F.1.3Testing the Validity of Surplus
Instruments
1 2 3_ , ,ˆ 1.0626 1.3535IV z z zy x
1 2 3ˆ 0.0207 0.1033 0.2355 0.1798e z z z
10FMonte Carlo Simulation
Principles of Econometrics, 4th Edition Page 124Chapter 10: Random Regressors and
Moment-Based Estimation
10F.2The Repeated
Sampling Properties of
IV/2SLS
Table 10F.1 Monte Carlo Simulation Results10F
Monte Carlo Simulation
Principles of Econometrics, 4th Edition Page 125Chapter 10: Random Regressors and
Moment-Based Estimation
In the case in which ρ = 0.8 and π = 0.1, the mean square error for the least squares estimator is:
The IV estimator it is
10000 22 21
β 10000 0.6062mmb
10F.2The Repeated
Sampling Properties of
IV/2SLS
21000022 21
β β 10000 1.0088mm
10FMonte Carlo Simulation