Part 2A: Basic Econometrics [ 1/75] Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business
Part 2A: Basic Econometrics [ 1/75]
Econometric Analysis of Panel Data
William Greene
Department of Economics
Stern School of Business
Part 2A: Basic Econometrics [ 2/75]
Endogeneity y = X+ε, Definition: E[ε|x]≠0 Why not?
Omitted variables Unobserved heterogeneity (equivalent to omitted
variables) Measurement error on the RHS (equivalent to omitted
variables) Structural aspects of the model Endogenous sampling and attrition Simultaneity (?)
Part 2A: Basic Econometrics [ 3/75]
Instrumental Variable Estimation One “problem” variable – the “last” one yit = 1x1it + 2x2it + … + KxKit + εit E[εit|xKit] ≠ 0. (0 for all others) There exists a variable zit such that
E[xKit| x1it, x2it,…, xK-1,it,zit] = g(x1it, x2it,…, xK-1,it,zit)In the presence of the other variables, zit “explains” xit
E[εit| x1it, x2it,…, xK-1,it,zit] = 0 In the presence of the other variables, zit and εit are
uncorrelated. A projection interpretation: In the projection XKt =θ1x1it,+ θ2x2it + … + θk-1xK-1,it + θK zit, θK ≠ 0.
Part 2A: Basic Econometrics [ 4/75]
The First IV Study: Natural Experiment(Snow, J., On the Mode of Communication of Cholera, 1855)
http://www.ph.ucla.edu/epi/snow/snowbook3.html
London Cholera epidemic, ca 1853-4 Cholera = f(Water Purity,u)+ε.
‘Causal’ effect of water purity on cholera? Purity=f(cholera prone environment (poor,
garbage in streets, rodents, etc.). Regression does not work.
Two London water companies
Lambeth Southwark
Main sewage discharge
Paul Grootendorst: A Review of Instrumental Variables Estimation of Treatment Effects…http://individual.utoronto.ca/grootendorst/pdf/IV_Paper_Sept6_2007.pdf
River Thames
Part 2A: Basic Econometrics [ 5/75]
IV Estimation
Cholera=f(Purity,u)+ε Z = water company Cov(Cholera,Z)=δCov(Purity,Z) Z is randomly mixed in the population
(two full sets of pipes) and uncorrelated with behavioral unobservables, u)
Cholera=α+δPurity+u+ε Purity = Mean+random variation+λu Cov(Cholera,Z)= δCov(Purity,Z)
Part 2A: Basic Econometrics [ 6/75]
Cornwell and Rupert DataCornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file are
EXP = work experienceWKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA = 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by union contractED = years of educationLWAGE = log of wage = dependent variable in regressions
These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data were downloaded from the website for Baltagi's text.
Part 2A: Basic Econometrics [ 7/75]
Part 2A: Basic Econometrics [ 8/75]
Specification: Quadratic Effect of Experience
Part 2A: Basic Econometrics [ 9/75]
The Effect of Education on LWAGE
1 2 3 4 ... ε
What is ε? ,... + everything elAbil seity, Motivation
Ability, Motivation = f( , , , ,...)
LWAGE EDUC EXP
EDUC GENDER SMSA SOUTH
2EXP
Part 2A: Basic Econometrics [ 10/75]
What Influences LWAGE?
1 2
3 4
Ability, Motivation
Ability, Motivat
( , ,...)
...
ε( )
Increased is associated with increases in
ion
Ability
Ability, Motivati( , ,on
LWAGE EDUC X
EXP
EDUC X
2EXP
2
...) and ε( )
What looks like an effect due to increase in may
be an increase in . The estimate of picks up
the effect of and the hidden effect of .
Ability, Motivation
Ability
Ability
EDUC
EDUC
Part 2A: Basic Econometrics [ 11/75]
An Exogenous Influence
1 2
3 4
( , , ,...)
...
ε( )
Increased is asso
Abili
ciate
ty, Motivation
Ability, Motivation
Ability, Motiva
d with increases in
( , , ,ti .n .o
LWAGE EDU Z
Z
Z
C X
EXP
EDUC X
2EXP
2
.) and not ε( )
An effect due to the effect of an increase on will
only be an increase in . The estimate of picks up
the effect of only.
Ability, Motiv
ation
EDUC
EDUC
ED
Z
Z
UC
is an Instrumental Variable
Part 2A: Basic Econometrics [ 12/75]
Instrumental Variables
Structure LWAGE (ED,EXP,EXPSQ,WKS,OCC,
SOUTH,SMSA,UNION) ED (MS, FEM)
Reduced Form: LWAGE[ ED (MS, FEM), EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION ]
Part 2A: Basic Econometrics [ 13/75]
Two Stage Least Squares Strategy
Reduced Form: LWAGE[ ED (MS, FEM,X), EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION ]
Strategy (1) Purge ED of the influence of everything but MS, FEM
(and the other variables). Predict ED using all exogenous information in the sample (X and Z).
(2) Regress LWAGE on this prediction of ED and everything else.
Standard errors must be adjusted for the predicted ED
Part 2A: Basic Econometrics [ 14/75]
OLS
Part 2A: Basic Econometrics [ 15/75]
The weird results for the coefficient on ED happened because the instruments, MS and FEM are dummy variables. There is not enough variation in these variables.
Part 2A: Basic Econometrics [ 16/75]
Source of Endogeneity
LWAGE = f(ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) +
ED = f(MS,FEM, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) + u
Part 2A: Basic Econometrics [ 17/75]
Remove the Endogeneity LWAGE = f(ED,
EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) + u +
LWAGE = f(ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) + u +
Strategy Estimate u Add u to the equation. ED is uncorrelated with
when u is in the equation.
Part 2A: Basic Econometrics [ 18/75]
Auxiliary Regression for ED to Obtain Residuals
Part 2A: Basic Econometrics [ 19/75]
OLS with Residual (Control Function) Added
2SLS
Part 2A: Basic Econometrics [ 20/75]
A Warning About Control Functions
Sum of squares is not computed correctly because U is in the regression.A general result. Control function estimators usually require a fix to the estimated covariance matrix for the estimator.
Part 2A: Basic Econometrics [ 21/75]
On Sat, May 3, 2014 at 4:48 PM, … wrote:
Dear Professor Greene,
I am giving an Econometrics course in Brazil and we are using your great textbook. I got a question which I think only you can help me. In our last class, I did a formal proof that var(beta_hat_OLS) is lower or equal than var(beta_hat_2SLS), under homoscedasticity. We know this assertive is also valid under heteroscedasticity, but a graduate student asked me the proof (which is my problem). Do you know where can I find it?
Part 2A: Basic Econometrics [ 22/75]
Part 2A: Basic Econometrics [ 23/75]
Part 2A: Basic Econometrics [ 24/75]
Part 2A: Basic Econometrics [ 25/75]
The General Problem1 2
1 1
2 2
2
1 2
2
Cov( , ) , K variables
Cov( , ) , K variables
is
OLS regression of y on ( , ) cannot estimate ( , )
consistently. Some other estimator is needed.
Additional structure:
endogenous
y X X
X 0
X 0
X
X X
X
1 2
= + where Cov( , ) but Cov( , )= .
An estimator based on ( , , ) may
be able to estimate ( , ) consistently.
instrumental variable
Z V V 0 Z 0
(IV X) X Z
Part 2A: Basic Econometrics [ 26/75]
Instrumental Variables Framework: y = X + , K variables in X. There exists a set of K variables, Z such that
plim(Z’X/n) 0 but plim(Z’/n) = 0
The variables in Z are called instrumental variables.
An alternative (to least squares) estimator of is
bIV = (Z’X)-1Z’y
We consider the following: Why use this estimator? What are its properties compared to least
squares? We will also examine an important application
Part 2A: Basic Econometrics [ 27/75]
IV Estimators
* Consistent
bIV = (Z’X)-1Z’y
= (Z’X/n)-1 (Z’X/n)β+ (Z’X/n)-1Z’ε/n = β+ (Z’X/n)-1Z’ε/n β* Asymptotically normal (same approach to proof as for
OLS)* Inefficient – to be shown.* By construction, the IV estimator is consistent. We have
an estimator that is consistent when least squares is not.
Part 2A: Basic Econometrics [ 28/75]
IV Estimation
Why use an IV estimator? Suppose that X and are not uncorrelated. Then least squares is neither unbiased nor consistent.
Recall the proof of consistency of least squares:
b = + (X’X/n)-1(X’/n).
Plim b = requires plim(X’/n) = 0. If this does not hold, the estimator is inconsistent.
Part 2A: Basic Econometrics [ 29/75]
A Popular MisconceptionIf only one variable in X is correlated with , the other
coefficients are consistently estimated. False.
The problem is “smeared” over the other coefficients.
1
111
21-1
1
1
1
Suppose only the first variable is correlated with
0Under the assumptions, plim( /n) = . Then
...
.
q0 q
plim = plim( /n)... .... q
times
K
ε
X'ε
b- β X'X
-1 the first column of Q
Part 2A: Basic Econometrics [ 30/75]
Consistency and Asymptotic Normality of the IV Estimator
-1
-1
-1
ˆ ( )
( )
ˆplim plim( N) N
ˆAsy.Var[ Asy.Var[ N]
( N) [plim N] (1/
-1ZX
-1 -1ZX XZ
-1 -1ZX XZ
β= Z'X Z'y
=β + Z'X Z'ε
(β - β )= Z'X/ (Z'ε/ )
= Q 0
β - β]= Q Z'ε/ Q
= 1/ Q Z'ΩZ/ Q =
i,t it it-1 -1
i
N)
(Assuming "well behaved" data)
ˆN ( N) N = ( N) N N
Invoke Slutsky and Lindberg-Feller for N to assert asymptotic normality.
ˆEstimate Asy.Var[ ] with
-1 -1ZX ZZ XZQ Ω Q
z(β - β )= Z'X/ Z'X/ w
w
β2
,t it it -1 -1ˆ(y )
[ ] [ ][ ]N or (N-K)
x βZ'X Z'Z XZ
Part 2A: Basic Econometrics [ 31/75]
Asymptotic Covariance Matrix of bIV
1IV
1 -1IV IV
2 1 -1IV IV
( ) '
( )( ) ' ( ) ' ' ( )
E[( )( ) '| ] ( ) ' ( )
b Z'X Z
b b Z'X Z Z X'Z
b b X,Z Z'X Z Z X'Z
Part 2A: Basic Econometrics [ 32/75]
Asymptotic EfficiencyAsymptotic efficiency of the IV estimator. The
variance is larger than that of LS. (A large sample type of Gauss-Markov result is at work.)
(1) It’s a moot point. LS is inconsistent.(2) Mean squared error is uncertain:
MSE[estimator|β]=Variance + square of bias.
IV may be better or worse. Depends on the data
Part 2A: Basic Econometrics [ 33/75]
Two Stage Least Squares
How to use an “excess” of instrumental variables(1) X is K variables. Some (at least one) of the K variables in X are correlated with ε.(2) Z is M > K variables. Some of the variables in Z are also in X, some are not. None of the variables in Z are correlated with ε.(3) Which K variables to use to compute Z’X and
Z’y?
Part 2A: Basic Econometrics [ 34/75]
Choosing the Instruments Choose K randomly? Choose the included Xs and the remainder
randomly? Use all of them? How? A theorem: (Brundy and Jorgenson, ca. 1972)
There is a most efficient way to construct the IV estimator from this subset: (1) For each column (variable) in X, compute the
predictions of that variable using all the columns of Z. (2) Linearly regress y on these K predictions.
This is two stage least squares
Part 2A: Basic Econometrics [ 35/75]
Algebraic Equivalence
Two stage least squares is equivalent to (1) each variable in X that is also in Z is
replaced by itself. (2) Variables in X that are not in Z are replaced
by predictions of that X using All other variables in X that are not correlated with ε All the variables in Z that are not in X.
Part 2A: Basic Econometrics [ 36/75]
The weird results for the coefficient on ED happened because the instruments, MS and FEM are dummy variables. There is not enough variation in these variables.
Part 2A: Basic Econometrics [ 37/75]
2SLS Algebra
1
1
ˆ
ˆ ˆ ˆ( )
But, = ( ) and ( ) is idempotent.
ˆ ˆ ( )( ) ( ) so
ˆ ˆ( ) = a real IV estimator by the definition.
ˆNote, plim( /n) =
-1
2SLS
-1Z Z
Z Z Z
2SLS
X Z(Z'Z) Z'X
b X'X X'y
Z(Z'Z) Z'X I -M X I -M
X'X= X' I -M I -M X= X' I -M X
b X'X X'y
X' 0
-1
ˆ since columns of are linear combinations
of the columns of , all of which are uncorrelated with
( ) ] ( )
2SLS Z Z
X
Z
b X' I -M X X' I -M y
Part 2A: Basic Econometrics [ 38/75]
Asymptotic Covariance Matrix for 2SLS
2 1 -1IV IV
2 1 -12SLS 2SLS
General Result for Instrumental Variable Estimation
E[( )( ) '| ] ( ) ' ( )
ˆSpecialize for 2SLS, using = ( )
ˆ ˆ ˆ ˆE[( )( ) '| ] ( ) ' ( )
Z
b b X,Z Z'X Z Z X'Z
Z X= I -M X
b b X,Z X'X X X X'X
2 1 -1
2 1
ˆ ˆ ˆ ˆ ˆ ˆ ( ) ' ( )
ˆ ˆ ( )
X'X X X X'X
X'X
Part 2A: Basic Econometrics [ 39/75]
2SLS Has Larger Variance than LS
2 -1
2 -1
A comparison to OLS
ˆ ˆAsy.Var[2SLS]= ( ' )
Neglecting the inconsistency,
Asy.Var[LS] = ( ' )
(This is the variance of LS around its mean, not )
Asy.Var[2SLS] Asy.Var[LS] in the matrix sense.
Com
X X
X X
β
-1 -1 2
2 2Z Z
pare inverses:
ˆ ˆ{Asy.Var[LS]} - {Asy.Var[2SLS]} (1/ )[ ' ' ]
(1/ )[ ' '( ) ]=(1/ )[ ' ]
This matrix is nonnegative definite. (Not positive definite
as it might have some rows and columns
X X- X X
X X- X I M X X M X
which are zero.)
Implication for "precision" of 2SLS.
The problem of "Weak Instruments"
Part 2A: Basic Econometrics [ 40/75]
Estimating σ2
2
2 n1i 1 in
Estimating the asymptotic covariance matrix -
a caution about estimating .
ˆSince the regression is computed by regressing y on ,
one might use
ˆ (y )ˆ
This is i
2sls
x
x'b
2 n1i 1 in
nconsistent. Use
(y )ˆ
(Degrees of freedom correction is optional. Conventional,
but not necessary.)
2slsx'b
Part 2A: Basic Econometrics [ 41/75]
Robust estimation of VC
-1 2 -1i,t it it
Counterpart to the White estimator allows heteroscedasticity
ˆˆ ˆ ˆ ˆˆ ˆEst.Asy.Var[ ]=( ) (y ) ( ) it itX'X x β x x X'X
“Actual” X
“Predicted” X
Part 2A: Basic Econometrics [ 42/75]
2SLS vs. Robust Standard Errors+--------------------------------------------------+| Robust Standard Errors |+---------+--------------+----------------+--------+|Variable | Coefficient | Standard Error |b/St.Er.|+---------+--------------+----------------+--------+ B_1 45.4842872 4.02597121 11.298 B_2 .05354484 .01264923 4.233 B_3 -.00169664 .00029006 -5.849 B_4 .01294854 .05757179 .225 B_5 .38537223 .07065602 5.454 B_6 .36777247 .06472185 5.682 B_7 .95530115 .08681261 11.000 +--------------------------------------------------+| 2SLS Standard Errors |+---------+--------------+----------------+--------+|Variable | Coefficient | Standard Error |b/St.Er.|+---------+--------------+----------------+--------+ B_1 45.4842872 .36908158 123.236 B_2 .05354484 .03139904 1.705 B_3 -.00169664 .00069138 -2.454 B_4 .01294854 .16266435 .080 B_5 .38537223 .17645815 2.184 B_6 .36777247 .17284574 2.128 B_7 .95530115 .20846241 4.583
Part 2A: Basic Econometrics [ 43/75]
Endogeneity Test? (Hausman)
Exogenous Endogenous
OLS Consistent, Efficient Inconsistent 2SLS Consistent, Inefficient Consistent
Base a test on d = b2SLS - bOLS
Use a Wald statistic, d’[Var(d)]-1d
What to use for the variance matrix? Hausman: V2SLS - VOLS
Part 2A: Basic Econometrics [ 44/75]
Hausman Test
Part 2A: Basic Econometrics [ 45/75]
Hausman Test: One at a Time?
Part 2A: Basic Econometrics [ 46/75]
Endogeneity Test: Wu
Considerable complication in Hausman test (text, pp. 234-237)
Simplification: Wu test. Regress y on X and estimated for the
endogenous part of X. Then use an ordinary Wald test.
X̂
Part 2A: Basic Econometrics [ 47/75]
Wu Test
Part 2A: Basic Econometrics [ 48/75]
Regression Based Endogeneity Test
it it it
An easy t test. (Wooldridge 2010, p. 127)
y q
= a set of M instruments.
Write = +
Can be estimated by ordinary least squares.
Endogeneity concerns correlation between v and .
ˆAdd v
itx δ
Z
q Zπ v
it it it it
= q - to the equation and use OLSˆ
ˆy q v + { error}
Simple t test on whether equals 0.
ˆEven easier, algebraically identical, (Wu, 1973), add
to the equation and do the same tes
it
z
x δ
q
t.
Part 2A: Basic Econometrics [ 49/75]
Testing Endogeneity of WKS(1) Regress WKS on 1,EXP,EXPSQ,OCC,SOUTH,SMSA,MS. U=residual, WKSHAT=prediction(2) Regress LWAGE on 1,EXP,EXPSQ,OCC,SOUTH,SMSA,WKS, U or WKSHAT+---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Constant -9.97734299 .75652186 -13.188 .0000 EXP .01833440 .00259373 7.069 .0000 19.8537815 EXPSQ -.799491D-04 .603484D-04 -1.325 .1852 514.405042 OCC -.28885529 .01222533 -23.628 .0000 .51116447 SOUTH -.26279891 .01439561 -18.255 .0000 .29027611 SMSA .03616514 .01369743 2.640 .0083 .65378151 WKS .35314170 .01638709 21.550 .0000 46.8115246 U -.34960141 .01642842 -21.280 .0000 -.341879D-14+---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|+---------+--------------+----------------+--------+---------+----------+ Constant -9.97734299 .75652186 -13.188 .0000 EXP .01833440 .00259373 7.069 .0000 19.8537815 EXPSQ -.799491D-04 .603484D-04 -1.325 .1852 514.405042 OCC -.28885529 .01222533 -23.628 .0000 .51116447 SOUTH -.26279891 .01439561 -18.255 .0000 .29027611 SMSA .03616514 .01369743 2.640 .0083 .65378151 WKS .00354028 .00116459 3.040 .0024 46.8115246 WKSHAT .34960141 .01642842 21.280 .0000 46.8115246
Part 2A: Basic Econometrics [ 50/75]
General Test for Endogeneity
2
Extending the Wu test
There are M K instruments .
ˆTo carry out the test, compute column
by column with regressions on .
Compute by OLS the extended model
ˆ
Use
1 1 2 2
2
1 1 2 2 2
y= X β +X β +ε
Z
X
Z
y= X β +X β + X + error
0 an F test to test H : 0
Part 2A: Basic Econometrics [ 51/75]
Alternative to Hausman’s Formula?
H test requires the difference between an efficient and an inefficient estimator.
Any way to compare any two competing estimators even if neither is efficient?
Bootstrap? (Maybe)
Part 2A: Basic Econometrics [ 52/75]
Part 2A: Basic Econometrics [ 53/75]
Weak Instruments Symptom: The relevance condition, plim Z’X/n not zero,
is close to being violated. Detection:
Standard F test in the regression of xk on Z. F < 10 suggests a problem.
F statistic based on 2SLS – see text p. 351. Remedy:
Not much – most of the discussion is about the condition, not what to do about it.
Use LIML? Requires a normality assumption. Probably not too restrictive.
Part 2A: Basic Econometrics [ 54/75]
Weak Instruments (cont.)
-1ˆplim = + [Cov( )] Cov( )
If Cov( ) is "small" but nonzero, small
Cov( ) may hugely magnify the effect.
IV is not only inefficient, may be very badly
biased by "weak" instruments.
Solutions
β β Z,X Z,ε
Z,ε
Z,X
? Can one "test" for weak instruments?
Part 2A: Basic Econometrics [ 55/75]
Weak Instruments
-1 -1ZX XZ
-1XZ
Which is better?
LS is inconsistent, but probably has smaller variance
LS may be more precise
IV is consistent, but probably has larger variance
ˆAsy.Var[ ] =
may be lZZβ Q Ω Q
Q
ZX
arge. (Compared to what?)
Strange results with "small"
IV estimator tends to resemble OLS (bias) (not a
function of sample size).
Contradictory result. Suppose z is perfectly correlated
Q
with x. IV MUST be the same as OLS.
Part 2A: Basic Econometrics [ 56/75]
A study of moral hazardRiphahn, Wambach, Million: “Incentive Effects in the Demand for Healthcare”Journal of Applied Econometrics, 2003
Did the presence of the ADDON insurance influence the demand for health care – doctor visits and hospital visits?
For a simple example, we examine the PUBLIC insurance (89%) instead of ADDON insurance (2%).
Part 2A: Basic Econometrics [ 57/75]
Application: Health Care Panel Data
German Health Care Usage Data, 7,293 Individuals, Varying Numbers of PeriodsVariables in the file areData downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,293 individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice. This is a large data set. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). Note, the variable NUMOBS below tells how many observations there are for each person. This variable is repeated in each row of the data for the person. (Downloaded from the JAE Archive) DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years MARRIED = marital status EDUC = years of education
Part 2A: Basic Econometrics [ 58/75]
Evidence of Moral Hazard?
Part 2A: Basic Econometrics [ 59/75]
Regression Study
Part 2A: Basic Econometrics [ 60/75]
Endogenous Dummy Variable
Doctor Visits = f(Age, Educ, Health, Presence of Insurance, Other unobservables)
Insurance = f(Expected Doctor Visits, Other unobservables)
Part 2A: Basic Econometrics [ 61/75]
Approaches (Parametric) Control Function: Build a
structural model for the two variables (Heckman)
(Semiparametric) Instrumental Variable: Create an instrumental variable for the dummy variable (Barnow/Cain/ Goldberger, Angrist, Current generation of researchers)
(?) Propensity Score Matching (Heckman et al., Becker/Ichino, Many recent researchers)
Part 2A: Basic Econometrics [ 62/75]
Heckman’s Control Function Approach Y = xβ + δT + E[ε|T] + {ε - E[ε|T]} λ = E[ε|T] , computed from a model for whether T = 0 or 1
Magnitude = 11.1200 is nonsensical in this context.
Part 2A: Basic Econometrics [ 63/75]
Instrumental Variable ApproachConstruct a prediction for T using only the exogenous informationUse 2SLS using this instrumental variable.
Magnitude = 23.9012 is also nonsensical in this context.
Part 2A: Basic Econometrics [ 64/75]
Propensity Score Matching Create a model for T that produces probabilities for T=1: “Propensity Scores” Find people with the same propensity score – some with T=1, some with T=0 Compare number of doctor visits of those with T=1 to those with T=0.
Part 2A: Basic Econometrics [ 65/75]
Treatment Effect
Earnings and Education: Effect of an additional year of schooling
Estimating Average and Local Average Treatment Effects of Education when Compulsory Schooling Laws Really Matter Philip Oreopoulos AER, 96,1, 2006, 152-175
Part 2A: Basic Econometrics [ 66/75]
Treatment Effects and Natural Experiments
Part 2A: Basic Econometrics [ 67/75]
How do panel data fit into this? We can use the usual models.
We can use far more elaborate models We can study effects through time
Observations are surely correlated. The same individual is observed more than once Unobserved heterogeneity that appears in the
disturbance in a cross section remains persistent across observations (on the same ‘unit’).
Procedures must be adjusted. Dynamic effects are likely to be present.
Part 2A: Basic Econometrics [ 68/75]
Appendix: Structure and
Regression
Part 2A: Basic Econometrics [ 69/75]
Least Squares Revisited
-1 -1
-1
-1 -1
( ) ( )
plim plim( N) plim( N)
plim ( ) ( )= ( N) ( N)
[ N]
-1XX
-1 -1XX XX
-1 -1 -1XX XX XX
b= X'X X'y=β + X'X X'ε
b=β + X'X/ X'ε/ = β +Q γ
b- b = β + X'X X'ε - β +Q γ X'X/ X'ε/ - Q γ
Asy.Var[b - Q γ]= Q Asy.Var X'ε/ Q
i,t it it-1
-1
(1 N) [plim N] (1 N)
(Assuming "well behaved" data)
plim ) ( N) NN
= ( N) N ( plim )
Invoke Slutsky and Lind
-1 -1 -1 -1XX XX XX XX XX
-1XX
= / Q X'ΩX/ Q = / Q Ω Q
xN(b - b = X'X/ Q γ
X'X/ w - w
berg-Feller for ( plim ) to assert asymptotic
normality. is also asymptotically normally distributed, but inconsistent.
N w - w
b
Part 2A: Basic Econometrics [ 70/75]
Inference with IV Estimators
(1) Wald Statistics:
ˆ ˆ ˆ( ) ( )
(E.g., the usual 't-statistics')
(2) A type of F statistic:
ˆ ˆCompute SSUA=( )'( ) without restrictions
ˆ ˆˆ ˆCompute SSR=( )'( ) wit
-1
u u
R R
Rβ- q ' {Est.Asy.Var[β]} Rβ- q
y Xβ y Xβ
y Xβ y Xβ
h restrictions
ˆ ˆˆ ˆCompute SSU=( )'( ) without restrictions
(SSR SSU) / JF = ~F[J ,N K]
SSUA/(N-K)
U Uy Xβ y Xβ
Part 2A: Basic Econometrics [ 71/75]
Comparing OLS and IV
-1XX
-1 -1XX XX
XX
plim = +
Asy.Var[ ] =
Precision = Mean squared error[b| ]
= Asy.Var[b] + [plim( - )][plim( - )]
=
XX
Least squares b
b β Q
b Q Ω Q
β
b β b β
Q -1 -1 -1 -1XX XX XX
-1 -1ZX XZ
-1 -1ZX XZ
ˆ
ˆ plim =
ˆ Asy.Var[ ] =
ˆ Precision = Mean squared error[ | ]
=
XX
ZZ
ZZ
Ω Q Q Q
Instrumental variables β
β β
β Q Ω Q
β β
Q Ω Q
Part 2A: Basic Econometrics [ 72/75]
Testing for Endogeneity(?)
it it it
-1
A test for endogeneity? Consider one variable:
y q ==> = +
q may be endogenous. = a set of M instruments.
(1) OLS: = ( ) Inconsistent if Cov[q, ] 0.
itx δ y Xβ ε
Z
b X'X X'y
-1
2 -1 2,2SLS ,OLS
Consistent and efficient if Cov[q, ] = 0.
ˆ ˆ ˆ ˆ(2) 2SLS: ( ) Always consistent.
Inefficient if Cov[q, ] = 0.
ˆ ˆ ˆHausman test? [ ]'{ ( ) (ˆ ˆ
β = X'X X'y
β - b X'X
-1 1
2
-1 -1 12,2SLS
ˆ) } [ ]'
(a) Need to use the same estimator of .
1 ˆ ˆˆ ˆ[ ]' {( ) ( ) } [ ]'ˆ
(b) Even with this fix, the resulting matrix is singular.
X'X β - b
β - b X'X X'X β - b
Part 2A: Basic Econometrics [ 73/75]
Structure vs. Regression
Reduced Form vs. Stuctural Model Simultaneous equations origin
Q(d) = a0 + a1P + a2I + e(d) (demand)Q(s) = b0 + b1P + b2R + e(s) (supply)Q(.) = Q(d) = Q(s)What is the effect of a change in I on Q(.)?(Not a regression)
Reduced form: Q = c0 + c1I + c2R + v.(Regression)
Modern concepts of structure vs. regression: The search for causal effects.
Part 2A: Basic Econometrics [ 74/75]
Implications The structure is the theory The regression is the conditional mean There is always a conditional mean
It may not equal the structure It may be linear in the same variables What is the implication for least squares
estimation? LS estimates regressions LS does not necessarily estimate structures Structures may not be estimable – they may not be
identified.
Part 2A: Basic Econometrics [ 75/75]
Structure and Regression
Simultaneity? What if E[ε|x]≠0 y=x+ε, x=δy+u. Cov[x, ε]≠0
x is not the regression? What is the regression?
Reduced form: Assume ε and u are uncorrelated. y = [/(1- δ)]u + [1/(1- δ)]ε x= [1/(1- δ)]u + [δ /(1- δ)]ε Cov[x,y]/Var[x] =
The regression is y = x + v, where E[v|x]=0
2 2 2 2 2
2 2 2 2
[ ] /[ ]
(1 )(1/ ) where w= /[ ]u u
u uw w
Part 2A: Basic Econometrics [ 76/75]
Structure vs. RegressionSupply = a + b*Price + c*Capacity
Demand = A + B*Price + C*Income