Introduction and Identification 

Todd Wagner

Econometrics with 

Observational Data
Goals for Course

 To enable researchers to conduct careful 

analyses with existing VA (and non-VA) 

datasets.

 We will 

– Describe econometric tools and their 

strengths and limitations

– Use examples to reinforce learning
Goals of Today’s Class

 Understanding causation with 

observational data

 Describe elements of an equation 

 Example of an equation 

 Assumptions of the classic linear model 
Terminology 

 Confusing terminology is a major barrier to 

interdisciplinary research

– Multivariable or multivariate

– Endogeneity or confounding

– Interaction or Moderation

– Right or Wrong

 Maciejewski ML, Weaver ML and Hebert PL. 

(2011) Med Care Res Rev 68 (2): 156-176
Polls 
Understanding Causation:

Randomized Clinical Trial

 RCTs are the gold-standard research design for 

assessing causality

 What is unique about a randomized trial?

The treatment / exposure is randomly assigned

 Benefits of randomization: 

Causal inferences
Randomization 





Random assignment distinguishes 

experimental and non-experimental design 

Random assignment should not be confused 

with random selection

– Selection can be important for generalizability 

(e.g., randomly-selected survey participants)

– Random assignment is required for 

understanding causation
Limitations of RCTs

 Generalizability to real life may be low

– Exclusion criteria may result in a select sample

 Hawthorne effect (both arms)

 RCTs are expensive and slow

 Can be unethical to randomize people to 

certain treatments or conditions

 Quasi-experimental design can fill an 

important role
Can Secondary Data Help us 

understand Causation?

Coffee may make high achievers slack off
Observational Data

 Widely available (especially in VA) 

 Permit quick data analysis at a low cost 

 May be realistic/ generalizable

 Key independent variable may not be 

exogenous – it may be endogenous
Endogeneity 

 A variable is said to be endogenous

when it is correlated with the error term 

(assumption 4 in the classic linear model) 

 If there exists a loop of causality between 

the independent and dependent variables 

of a model leads, then there is 

endogeneity 
Endogeneity 

 Endogeneity can come from:

– Measurement error

– Autoregression with autocorrelated errors

– Simultaneity

– Omitted variables

– Sample selection
Example of Endogeneity

 Qx: does greater use of PET screening 

decrease lung cancer mortality

 You observe that some facilities do a lot of 

PET screening while others do little

 You compare patients across facilities and find 

a negative correlation between PET screening 

intensity and mortality

 Why is PET screening intensity endogenous?
Econometrics v Statistics

 Often use different terms

 Cultural norms–if it seems endogenous, it 

probably is

 Underlying data generating model is 

economic. Rational actors concerned with

– Profit maximization

– Quantity maximization

– Time minimization
Elements of an Equation

Maciejewski ML, Diehr P, Smith MA, Hebert P. Common methodological terms in 

health services research and their synonyms. Med Care. Jun 2002;40(6):477-484.
Terms 

 Univariate– the statistical expression of 

one variable

 Bivariate– the expression of two 

variables 

 Multivariate– the expression of more than 

one variable (can be dependent or 

independent variables)
Dependent variable

Outcome measure Error Term

Intercept

Covariate, RHS variable,

Predictor, independent 

variable

Y X i =+ + ββ ε 0 1 i i

Note the similarity to the equation of a line (y=mx+B)
 “i” is an index. 

 

If we are analyzing people, then this 

typically refers to the person 

 There may be other indexes 

Y X i =+ + ββ ε 0 1 i i
DV

Two covariates

Error Term

Intercept

Y XZ i =+ + + ββ β ε 01 2 i ii
i 01 2 i i ij ij i

j

Y X Z BX =+ + + + ββ β ∑ ε

DV

j covariates

Error Term

Intercept

Different notation
Error term

 Error exists because

1. Other important variables might be omitted

2. Measurement error

3. Human indeterminacy

 Understand error structure and minimize 

error 

 Error can be additive or multiplicative

See Kennedy, P. A Guide to Econometrics
Example: is height associated with 

income?
 Y=income; X=height 

 Hypothesis: Height is not related to 

income (B1=0)

 If B1=0, then what is B0?

Y X i =+ + ββ ε 0 1 i i
Height and Income

0 50000 100000 150000 200000

Income

60 65 70 75

height

How do we want to describe the data?
Estimator

 A statistic that provides information on 

the parameter of interest (e.g., height) 

 Generated by applying a function to the 

data 

 Many common estimators

– Mean and median (univariate estimators)

– Ordinary least squares (OLS) (multivariate 

estimator)
Ordinary Least Squares (OLS)

0 50000 100000 150000 200000

60 65 70 75

height

Fitted values Income
Other estimators

 Least 

absolute 

deviations 

 Maximum 

likelihood 0 50000 100000 150000 200000

60 65 70 75

height

Fitted values Income
Choosing an Estimator

 Least squares

 Unbiasedness

 Efficiency (minimum variance)

 Asymptotic properties

 Maximum likelihood

 Goodness of fit

 We’ll talk more about identifying the “right” 

estimator throughout this course.
How is the OLS fit?

0 50000 100000 150000 200000

60 65 70 75

height

Fitted values Income
What about gender?

 How could gender affect the relationship 

between height and income? 

– Gender-specific intercept

– Interaction
Gender Indicator Variable

Gender Intercept

height

Y XZ i =+ + + ββ β ε 01 2 i ii
Gender-specific Indicator 00

0 50000 100000 150000 2000

60 65 70 75

height

B0 

B2 

B1 is the slope of the line
Interaction 

Interaction Term,

Effect modification,

Modifier

Note: the gender “main effect”

variable is still in the model

height gender 

Y X Z XZ i =+ + + + ββ β β ε 01 2 3 i i ii i
Gender Interaction

0 50000 100000 150000 200000

60 65 70 75

height

Interaction allows two 

groups to have different 

slopes 
Assumptions 

Classic Linear 

Regression (CLR)
Classic Linear Regression

 No “superestimator”

 CLR models are often used as the starting 

point for analyses 

 5 assumptions for the CLR 

 Variations in these assumption will guide 

your choice of estimator (and happiness 

of your reviewers)
Assumption 1

 The dependent variable can be calculated 

as a linear function of a specific set of 

independent variables, plus an error term

 For example,

Y X Z XZ i =+ + + + ββ β β ε 01 2 3 i i ii i
Violations to Assumption 1

 Omitted variables 

 Non-linearities

– Note: by transforming independent 

variables, a nonlinear function can be made 

from a linear function
Testing Assumption 1

 Theory-based transformations (e.g., CobbDouglas production)

 Empirically-based transformations

 Common sense

 Ramsey RESET test

 Pregibon Link test

Ramsey J. Tests for specification errors in classical linear least squares regression 

analysis. Journal of the Royal Statistical Society. 1969;Series B(31):350-371. 

Pregibon D. Logistic regression diagnostics. Annals of Statistics. 1981;9(4):705-724. 
Assumption 1 and Stepwise

 Statistical software allows for creating 

models in a “stepwise” fashion 

 Be careful when using it. 

– Little penalty for adding a nuisance variable

– BIG penalty for missing an important 

covariate
Bias if Gender is Ignored 0000

0 50000 100000 150000 20

60 65 70 75

height

Estimate 

without

gender 
Assumption 2 

 Expected value of the error term is 0

E(ui

)=0

 Violations lead to biased intercept 

 A concern when analyzing cost data 

(Smearing estimator when working with 

logged costs) 
Assumption 3

 IID– Independent and identically 

distributed error terms 

– Autocorrelation: Errors are uncorrelated 

with each other

– Homoskedasticity: Errors are identically 

distributed
Heteroskedasticity 
Violating Assumption 3

 Effects 

– OLS coefficients are unbiased

– OLS is inefficient

– Standard errors are biased

 Plotting is often very helpful

 Different statistical tests for heteroskedasticity

– GWHet--but statistical tests have limited power
Fixes for Assumption 3

 Transforming dependent variable may 

eliminate it

 Robust standard errors (Huber White or 

sandwich estimators)
Assumption 4

 Observations on independent variables 

are considered fixed in repeated samples

 E(xiui|x)=0 

 Violations

– Errors in variables

– Autoregression

– Simultaneity

Endogeneity 
Assumption 4: 

Errors in Variables

 Measurement error of dependent variable 

(DV) is maintained in error term.

 OLS assumes that covariates are 

measured without error. 

 Error in measuring covariates can be 

problematic
Common Violations

 Including a lagged dependent variable(s) as a 

covariate

 Contemporaneous correlation

– Hausman test (but very weak in small samples)

 Instrumental variables offer a potential 

solution
Assumption 5

 Observations > covariates

 No multicollinearity

 Solutions

– Remove perfectly collinear variables

– Increase sample size
Any Questions?
Statistical Software

 I frequently use SAS for data 

management

 I use Stata for my analyses

 Stattransfer
Regression References

 Kennedy A Guide to Econometrics

 Greene. Econometric Analysis. 

 Wooldridge. Econometric Analysis of 

Cross Section and Panel Data.

 Winship and Morgan (1999) The Estimation 

of Causal Effects from Observational Data 

Annual Review of Sociology, pp. 659-706.