Applied Econometrics Instrumental Variable Approach
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Transcript of Applied Econometrics Instrumental Variable Approach
Nguyen Ngoc AnhNguyen Ha Trang
Applied EconometricsInstrumental Variable Approach
DEPOCEN
Topics That Will Be Covered in this Workshop
Why use IV?– Discussion of endogeneity bias– Statistical motivation for IV
What is an IV?– Identification issues– Statistical properties of IV estimators
How is an IV model estimated?– Software and data examples– Diagnostics: IV relevance, IV exogeneity, Hausman
Review of the Linear Model (in metrix algebra) Population model: Y = α + βX + ε
– Assume that the true slope is positive, so β > 0 Sample model: Y = a + bX + e
– Least squares (LS) estimator of β:bLS = (X′X)–1X′Y = Cov(X,Y) / Var(X)
Under what conditions can we speak of bLS as a causal estimate of the effect of X on Y?
Review of the Linear Model Key assumption of the linear model:
– E(|x) = E( ) = 0 Cov(x, ) = E(x ) = 0 – E(X′e) = Cov(X,e) = E(e | X) = 0– Exogeneity assumption = X is uncorrelated with the
unobserved determinants of Y Important statistical property of the LS estimator
under exogeneity:
E(bLS) = β + Cov(X,e) / Var(X)plim(bLS) = β + Cov(X,e) / Var(X)
Second terms 0, so bLS unbiased and consistent
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Review of the Linear ModelWhen you regress Y on X, Y = β0 + β1X + ε and the OLS estimate of β1 can be described as
But since X and ε are correlated, bOLS does not estimate β1 but some other quantity that depends on the correlation of X and ε
Cov ,Cov ,Cov , Cov ,
Cov , Cov , Cov ,Cov , Cov ,
OLSX XY X
bX X X X
X X ε X ε XX X X X
0 1
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β β
ββ
ε
Endogeneity and the Evaluation Problem
When is the exogeneity assumption violated?– Measurement error → Attenuation bias– Instantaneous causation → Simultaneity bias– Omitted variables → Selection bias
Selection bias is the problem in observational research that undermines causal inference– Measurement error and instantaneous causation
can be posed as problems of omitted variables Potential outcome approach!!!!
When Is the Exogeneity Assumption Violated?
Omitted variable (W) that is correlated with both X and Y– Classic problem of omitted variables bias
Coefficient on X will absorb the indirect path through W, whose sign depends on Cov(X,W) and Cov(W,Y)
X Y
W
Things more complicated in applied settings because there are bound to be many W’s, not to mention that the “smearing” problem applies in this context also
Example #1: Police Hiring
Measurement error– Mobilization of sworn officers (M.E. in X) as well
as differential victim reporting or crime recording (M.E. in Y) may be correlated with police size
Instantaneous causation– More police might be hired during a crime wave
Omitted variables– Large departments may differ in fundamental
ways difficult to measure (e.g., urban, heterogeneous)
Example #2: Delinquent Peers
Measurement error– Highly delinquent youth probably overestimate the
delinquency of their peers (M.E. in X), and likely underestimate their own delinquency (M.E. in Y)
Instantaneous causation– If there is influence/imitation, then it is bidirectional
Omitted variables– High-risk youth probably select themselves into
delinquent peer groups (“birds of a feather”)
Regression EstimationIgnoring Omitted Variables
Suppose we estimate treatment effect model:Y = α + βX + ε
– Let’s assume without loss of generality that X is a binary “treatment” (= 1 if treated; = 0 if untreated)
Least squares estimator:bLS = Cov(X,Y) / Var(X) = E(Y | X = 1) – E(Y | X = 0)
– Simply the difference in means between “treated” units (X = 1) and “untreated” units (X = 0)
Estimating Treatment EffectsConsider treatment assignment (dummy variable) X and outcome
Y
Regress Y on X
Yi = β0 + β1Xi + εi
The estimate of β1 is just the difference between the mean Y for X = 1 (the treatment group) and the mean Y for X = 0 (the control group)
Thus the OLS estimate is
= β1 +
Y β β ε
Y β ε
1 0 1 1
0 0 0
1 0Y Y 1 0
Estimating Treatment Effects(With Random Assignment)
If the treatment is randomly assigned, then X is uncorrelated with ε (X is exogenous)
If X is uncorrelated with ε if and only if
But if , then the mean difference is
= β1 + = β1
This implies that standard methods (OLS) give an unbiased estimate of β1, which is the average treatment effect
That is, the treatment-control mean difference is an unbiased estimate of β1,
1 0
1 0
1 0Y Y 1 0
What goes wrong without randomization?
If we do not have randomization, there is no guarantee that X is uncorrelated with ε (X may be endogenous)
Thus the OLS estimate is still
= β1 +
If X is correlated with ε, then
Hence does not estimate β1, but some other quantity that depends on the correlation of X and ε
If X is correlated with ε, then standard methods give a biased estimate of β1
1 0Y Y 1 0
1 0Y Y
1 0
Omitted Variables in applied research
What variables of interest to us are surely endogenous?– Micro = Employment, education, marriage, military
service, fertility, conviction, family structure,....– Macro = Poverty, unemployment rate, collective
efficacy, immigrant concentration,.... Basically, EVERYTHING!
– (I’m sorry ....... But it suck)
Potential outcome framework
Traditional Strategies to Deal with Omitted Variables
Randomization (physical control) Covariate adjustment (statistical control)
– Control for potential W’s in a regression model– But...we have no idea how many W’s there are, so
model misspecification is still a real problem here
Quasi-Experimental Strategies to Deal with Omitted Variables
Difference in differences (fixed-effects model)– Requires panel data
Propensity score matching– Requires a lot of measured background variables
Similar to covariate adjustment, but only the treated and untreated cases which are “on support” are utilized
Instrumental variables estimation– Requires an exclusion restriction
Instrumental Variables Estimation Is a Viable Approach
An “instrumental variable” for X is one solution to the problem of omitted variables bias
Requirements for Z to be a valid instrument for X– Relevant = Correlated with X– Exogenous = Not correlated
with Y but through its correlation with X
Z
X Y
W
e
Important Point about Instrumental Variables Models I often hear...“A good instrument should not be
correlated with the dependent variable”– WRONG!!!
Z has to be correlated with Y, otherwise it is useless as an instrument– It can only be correlated with Y through X– (trong X có 2 phần, 1 phần dính với e một phần với
Y, muốn tận dụng phần dính với Y) A good instrument must not be correlated
with the unobserved determinants of Y
Important Point about Instrumental Variables Models
Not all of the available variation in X is used– Only that portion of X which is “explained” by
Z is used to explain Y
X Y
Z X = Endogenous variableY = Response variableZ = Instrumental variable
Important Point about Instrumental Variables Models
X Y
Z
Realistic scenario: Very little of X is explained by Z, or what is explained does not overlap much with Y
X YZ
Best-case scenario: A lot of X is explained by Z, and most of the overlap between X and Y is accounted for
Important Point about Instrumental Variables Models
The IV estimator is BIASED– In other words, E(bIV) ≠ β (finite-sample bias)– The appeal of IV derives from its consistency
“Consistency” is a way of saying that E(b) → β as N → ∞ So…IV studies often have very large samples
– But with endogeneity, E(bLS) ≠ β and plim(bLS) ≠ β anyway
Asymptotic behavior of IVplim(bIV) = β + Cov(Z,e) / Cov(Z,X)
– If Z is truly exogenous, then Cov(Z,e) = 0
Instrumental Variables Terminology
Three different models to be familiar with– First stage: X = α0 + α1Z + ω – Structural model: Y = β0 + β1X + ε – Reduced form: Y = δ0 + δ1Z + ξ
More on the Method of Two-Stage Least Squares (2SLS)
Step 1: X = a0 + a1Z1 + a2Z2 + + akZk + u – Obtain fitted values (X̃) from the first-stage model
Step 2: Y = b0 + b1X̃ + e – Substitute the fitted X̃ in place of the original X– Note: If done manually in two stages, the standard
errors are based on the wrong residual e = Y – b0 – b1X̃ when it should be e = Y – b0 – b1X
Best to just let the software do it for you
Some examples
Some examples
Including Control Variables in an IV/2SLS Model
Control variables (W’s) should be entered into the model at both stages– First stage: X = a0 + a1Z + a2W + u – Second stage: Y = b0 + b1X̃ + b2W + e
Control variables are considered “instruments,” they are just not “excluded instruments”– They serve as their own instrument
Functional Form Considerations with IV/2SLS
Binary endogenous regressor (X)– Consistency of second-stage estimates do not
hinge on getting first-stage functional form correct Binary response variable (Y)
– IV probit (or logit) is feasible but is technically unnecessary
In both cases, linear model is tractable, easily interpreted, and consistent– Although variance adjustment is well advised
Technical Conditions Required for Model Identification
Order condition = At least the same # of IV’s as endogenous X’s– Just-identified model: # IV’s = # X’s– Overidentified model: # IV’s > # X’s
Rank condition = At least one IV must be significant in the first-stage model– Number of linearly independent columns in a matrix
E(X | Z,W) cannot be perfectly correlated with E(X | W)
Instrumental Variables and Randomized Experiments
Imperfect compliance in randomized trials– Some individuals assigned to treatment group will
not receive Tx, and some assigned to control group will receive Tx
Assignment error; subject refusal; investigator discretion
– Some individuals who receive Tx will not change their behavior, and some who do not receive Tx will change their behavior
A problem in randomized job training studies and other social experiments (e.g., housing vouchers)
Durbin-Wu-Hausman (DWH) Test
Balances the consistency of IV against the efficiency of LS– H0: IV and LS both consistent, but LS is efficient– H1: Only IV is consistent
DWH test for a single endogenous regressor:DWH = (bIV – bLS) / √(s2
bIV – s2
bLS) ~ N(0,1)
– If |DWH| > 1.96, then X is endogenous and IV is the preferred estimator despite its inefficiency
Durbin-Wu-Hausman (DWH) Test
A roughly equivalent procedure for DWH:1. Estimate the first-stage model2. Include the first-stage residual in the structural
model along with the endogenous X3. Test for significance of the coefficient on residual
Note: Coefficient on endogenous X in this model is bIV (standard error is smaller, though)– First-stage residual is a “generated regressor”
Software Considerations
Basic model specification in Stataivreg y (x = z) w [weight = wtvar], options
y = dependent variablex = endogenous variablez = instrumental variablew = control variable(s)