For more details see also: Empirical Work in The Social Sciences; Mixed, Fixed, and Random Effects Models; Linear Models;

*Interaction Models; Instrumental Variables; Instrumental Variables and Selection Bias; Difference-in-Difference Estimators & Propensity Score Matching in Higher Education Research*

**Ordinary Least Squares(OLS):**

y = b

_{0}+ b_{1}x_{ }+e
y = b

_{0}+ b_{1}x_{1}+ b_{2}x_{2}+e*2-variable case***y = bx + e , b = (x’x)**

^{-1}x’y*vectorized multivariable case*

Ordinary Least Squares(OLS) provides the best linear
approximation to the population conditional expectation function f(x) = E(y|x),
even if the CEF is non-linear. OLS does
not hinge on linearity as an empirical tool for assessing the essential
features of causal relationships. The least squares regression equation is causal to the extent that the population CEF is causal.

**Interaction Models:**y = b

_{0}+ b

_{1}x

_{ }+ b

_{2}z +b

_{3}xz + e

The relationship between x and y is conditional on z. Ex: If
z is binary, the the marginal effect of x on y can be expressed as follows:

∂y/∂x = b

_{1}+b_{3}z for z= 1
∂y/∂x = b

_{1}for z= 0**Selection Bias:**

C

_{i}= choice/selection/treatment
Y

_{0i}= baseline potential outcome
Y

_{1i}= potential treatment outcome
E[Y

_{i}|c_{i}=1] - E[Y_{i}|c_{i}=0] =E[Y_{1i}-Y_{0i}|c_{i}=1] +{ E[Y_{0i}|c_{i}=1] - E[Y_{0i}|c_{i}=0]}*Observed effect = treatment effect on the treated + {selection bias}*

*If the potential outcomes ‘Y*

_{0i}’ for those that are treated non-randomly or self selected (c

_{i}=1) differ from potential outcomes ‘Y

_{0i}’ from those that are not treated or don’t self select then the term { E[Y

_{0i}|c

_{i}=1] - E[Y

_{0i}|c

_{i}=0]} could have a positive or negative value, creating selection bias. When we calculate the observed difference between treated and untreated groups E[Y

_{i}|c

_{i}=1] - E[Y

_{i}|c

_{i}=0]

*selection bias*becomes confounded with the actual treatment effect E[Y

_{1i}-Y

_{0i}|c

_{i}=1] .

**Conditional Independence Assumption (CIA):**

E[Y

_{i}|x_{i},c_{i}=1]- E[Y_{i}|x_{i},c_{i}=0]= E[Y_{1i}-Y_{0i}|x_{i}]
With

*conditional on ‘x ‘*comparisons, selection bias disappears.**Matching Estimators:**make comparisons across groups with similar ‘matched’ covariate values.

E[Y

_{1i}-Y_{0i}|c_{i}=1] Î£ Î´_{x}P(x_{i}=x|c_{i}=1)
The matching estimator calculates
an average of the difference between groups( Î´

_{x})weighted by the distribution of covariates P(x_{i}=x|c_{i}=1) .**Regression Estimators:**OLS provides a matching estimator based on a variance weighted average of treatment effects: b = (x’x)

^{-1}x’y

**or Cov(x,y)/Var(x).**

Propensity score matching works
similar to covariate matching except comparisons are made based on scores vs.
specific covariate values. OLS with proper controls (the same covariates used
in matching, or the same x’s used to generate the propensity scores)provides a
robust matching estimator that gives very similar results to explicit matching
estimators and propensity score matching estimators.

**Fixed Effects and Heterogeneity:**

Mixed Models: y = bx
+ zÎ± + e ; bx = fixed effects
(FE); zÎ± +e = random effects (RE)

Heterogeneity: unobserved
individual effects

FE: capture individual effects by
shifting the regression equation with a dummy variable ‘d’

y = bx + d Î± + e

RE: assumes that individual
effects are randomly distributed across individuals, modeled as a random
intercepts model the RE model is a special case of the general mixed model

y = bx + Î± + e

**Instrumental Variables:**

Suppose you are trying to assess
the treatment effect of ‘s’, but there is some omitted factor A that is not
accounted for, or there is some component ‘A’ related to ‘s’ that is not
observable or measurable. We say that there is measurement error in ‘s’.

y = b

_{0}+ b_{1}s + b_{2}A +e*full regression*
y = b

_{0}+ b_{1}s + e*short regression given observable data on ‘s’ omitting ‘A’; e = b*_{2}A +e
To the extent that ‘A’ is
correlated with‘s’ we have omitted variable bias in our estimate of b

_{1}, the treatment effect. IV techniques attempt to estimate b_{1}using only the ‘quasi-experimental’ proportion of variation related to ‘s’ but unrelated to A. If we could observe ‘A’ we would just include it in the regression, or if we could properly measure ‘s’ we would not require IVs.
IV estimation requires that we
find some variable ‘z’ correlated with ‘s’ but uncorrelated with the
measurement error; E(z,e)=0. The only
relationship between ‘z ‘and the outcome ‘y ‘should be through ‘s’, so we have z→s→y and

*E(y|z) = b E(s|z) + E(e|z)*; given E(e|z) = 0 b

*=*

_{IV}*E(y|z)/E(s|z)*=(z’s)

^{-1}z’y

IV estimates are 2-stage
regression estimates:

1
s* = bz

2
y =bs*→b

_{IV}**Difference in Difference:**

DD estimators assume that in
absence of treatment the difference between control and treatment groups would
be constant or ‘fixed’ over time. DD estimators are a special type of fixed
effects estimator.

(A-B) :Differences in groups
pre-treatment represent the ‘normal’ difference between groups.

(A’-B) = total post treatment effect
= normal effect (A-B) + treatment effect (A’-A)

DD estimates compare the
difference in group averages for ‘y’ pre-treatment to the difference in group
averages post treatment. The larger the difference post treatment the larger
the treatment effect.

This can also be represented in
the regression context with interactions where t = time indicating pre and post
treatment and x is an indicator for treatment and control groups. At t= 1 there
are no treatments so those terms = 0.
The parameter b

_{3}is our difference in difference estimator.
y = b

_{0}+ b_{1}x_{ }+ b_{2}t+b_{3}xt + e**References:**
Greene, Econometric Analysis, 5th
Edition

Understanding Interaction Models: Improving Empirical Analyses. Thomas Bramber, William Roberts Clark, Matt Golder. Political Analysis (2006) 14:63-82

Elements of Econometrics. Jan Kmenta. Macmillan (1971)

Angrist and Pischke, Mostly Harmless Econometrics, 2009

Using Instrumental Variables to Account for Selection Effects in Research on First-Year Programs

Gary R. Pike, Michele J. Hansen and Ching-Hui Lin

Research in Higher Education

Volume 52, Number 2, 194-214, DOI: 10.1007/s11162-010-9188-x

Program Evaluation and the

Difference-in-Difference Estimator

Course Notes

Education Policy and Program Evaluation

Vanderbilt University

October 4, 2008

Difference in Difference Models, Course Notes

ECON 47950: Methods for Inferring Causal Relationships in Economics

William N. Evans

University of Notre Dame

Spring 2008

Understanding Interaction Models: Improving Empirical Analyses. Thomas Bramber, William Roberts Clark, Matt Golder. Political Analysis (2006) 14:63-82

Elements of Econometrics. Jan Kmenta. Macmillan (1971)

Angrist and Pischke, Mostly Harmless Econometrics, 2009

Using Instrumental Variables to Account for Selection Effects in Research on First-Year Programs

Gary R. Pike, Michele J. Hansen and Ching-Hui Lin

Research in Higher Education

Volume 52, Number 2, 194-214, DOI: 10.1007/s11162-010-9188-x

Program Evaluation and the

Difference-in-Difference Estimator

Course Notes

Education Policy and Program Evaluation

Vanderbilt University

October 4, 2008

Difference in Difference Models, Course Notes

ECON 47950: Methods for Inferring Causal Relationships in Economics

William N. Evans

University of Notre Dame

Spring 2008