Assessing Studies Based on Multiple Regression :
Some pitfalls
(SW Chapter 9)
Is there a systematic way to assess regression studies?
Multiple regression has some key virtues:

It provides an estimate of the effect on Y of arbitrary changes X.

It resolves the problem of omitted variable bias, if an omitted variable can be measured and included.

It can handle nonlinear relations (effects that vary with the X’s)
Internal validity: the statistical inferences about causal effects are valid for the population being studied.
External validity: the statistical inferences can be generalized from the population and setting studied to other populations and settings, where the “setting” refers to the legal, policy, and physical environment and related salient features.
Threats to External Validity of Multiple Regression Studies
How far can we generalize class size results from California school districts?

Differences in populations

California in 2005?

Massachusetts in 2005?

Mexico in 2005?

California in 2009?

Differences in settings
Threats to Internal Validity
Internal validity: the statistical inferences about causal effects are valid for the population being studied.
Five threats to the internal validity of regression studies:

Omitted variable bias

Wrong functional form

Errorsinvariables bias

Sample selection bias

Simultaneous causality bias
All of these imply that E(u_{i}X_{1}_{i},…,X_{ki}) 0 – in which case OLS is biased and inconsistent.
1. Omitted variable bias
Omitted variable bias arises if an omitted variable is both:
(i) a determinant of Y and
(ii) correlated with at least one included regressor.
Potential solutions to omitted variable bias

If the variable can be measured, include it as an additional regressor in multiple regression;

Possibly, use panel data (Chapter 10) in which each entity (individual) is observed more than once;

If the variable cannot be measured, use instrumental variables regression (Chapter 12);

Run a randomized controlled experiment (Chapter 13).
2. Wrong functional form
3. Errorsinvariables bias

Data entry errors in administrative data

Recollection errors in surveys (when did you start your current job?)

Ambiguous questions problems (what was your income last year?)

Intentionally false response problems with surveys (What is the current value of your financial assets? How often do you drink and drive?)
In general, measurement error in a regressor results in “errorsinvariables” bias.
Illustration: suppose
Y_{i} = _{0} + _{1}X_{i} + u_{i}
is “correct” in the sense that the three least squares assumptions hold (in particular E(u_{i}X_{i}) = 0).
Let
X_{i} = unmeasured true value of X
= imprecisely measured version of X
Question: What will the bias for _{1} be if Y is regressed on ?
Potential solutions to errorsinvariables bias

Obtain better data.

Develop a specific model of the measurement error process.

Instrumental variables regression.
4. Sample selection bias
Units are often “selfselected”.
Kiær (1895) developed The representative method
Example: returns to education

What is the return to an additional year of education?

Empirical strategy:

Sampling scheme: simple random sample of employed college grads (employed, so we have wage data)

Data: earnings and years of education

Estimator: regress ln(earnings) on years_education

Ignore issues of omitted variable bias and measurement error – could there be sample selection bias?
Potential solutions to sample selection bias

Collect the sample in a way that avoids sample selection.

Returns to education example: sample college graduates, not workers (include the unemployed)

Randomized controlled experiment.

Construct a model of the sample selection problem and estimate that model (we won’t do this).
5. Simultaneous causality bias
So far we have assumed that X causes Y.
What if Y causes X, too?
Example: Class size effect

Low STR results in better test scores

But suppose districts with low test scores are given extra resources: as a result of a political process they also have low STR

What does this mean for a regression of TestScore on STR?
Simultaneous causality bias in equations
(a) Causal effect on Y of X: Y_{i} = _{0} + _{1}X_{i} + u_{i}
(b) Causal effect on X of Y: X_{i} = _{0} + _{1}Y_{i} + v_{i}

Large u_{i} means large Y_{i}, which implies large X_{i} (if _{1}>0)

Thus corr(X_{i},u_{i}) 0

Thus is biased and inconsistent.
Example: A district with particularly bad test scores given the STR (negative u_{i}) receives extra resources, thereby lowering its STR; so STR_{i} and u_{i} are correlated
Example: A key for distributing income from the state to Norwegian communes was based on linear regressions of communal expenditure on demographic and other “objective” variables characterizing the individual commune (the unit). The regression coefficients were used to construct the key. Why could we have a simultaneous causality bias here? What could be done?
Schweder, T. 1996. Rattsøutvalgets regresjoner. Sosialøkonomen, nr 4: 1518.
Potential solutions to simultaneous causality bias

Randomized controlled experiment. Because X_{i} is chosen at random by the experimenter, there is no feedback from the outcome variable to Y_{i} (assuming perfect compliance).

Develop and estimate a complete model of both directions of causality. This is the idea behind many large macro models (e.g. Federal Reserve BankUS). This is extremely difficult in practice.

Use instrumental variables regression to estimate the causal effect of interest (effect of X on Y, ignoring effect of Y on X).
Case study: Test Scores and Class Size
(SW Section 9.4)
Objective: Assess the threats to the internal and external validity of the empirical analysis of the California test score data.

External validity

Compare results for California and Massachusetts

Think hard…

Internal validity

Go through the list of five potential threats to internal validity and think hard…
The Massachusetts data set

220 elementary school districts

Test: 1998 MCAS test – fourth grade total (Math + English + Science)

Variables: STR, TestScore, PctEL, LunchPct, Income
Summary of Findings for Massachusetts

Coefficient on STR falls from –1.72 to –0.69 when control variables for student and district characteristics are included – an indication that the original estimate contained omitted variable bias.

The class size effect is statistically significant at the 1% significance level, after controlling for student and district characteristics

No statistical evidence on nonlinearities in the TestScore – STR relation

No statistical evidence of STR – PctEL interaction
How do the Mass and California results compare?
Comparison of estimated class size effects: CA vs. MA
Summary: Comparison of California and Massachusetts Regression Analyses

Class size effect falls in both CA, MA data when student and district control variables are added.

Class size effect is statistically significant in both CA, MA data.

Estimated effect of a 2student reduction in STR is quantitatively similar for CA, MA.

Neither data set shows evidence of STR – PctEL interaction.

Some evidence of STR nonlinearities in CA data, but not in MA data.
Threats to internal validity in the test score/class size example?
1. Omitted variable bias?
This analysis controls for:

district demographics (income)

some student characteristics (English speaking)
What is missing?

Additional student characteristics, for example native ability (but is this correlated with STR?)

Access to outside learning opportunities

Teacher quality (perhaps better teachers are attracted to schools with lower STR)
2. Wrong functional form?

We have tried quite a few different functional forms, in both the California and Mass. data

Nonlinear effects are modest

Plausibly, this is not a major threat at this point.
3. Errorsinvariables bias?

STR is a districtwide measure

Presumably there is some measurement error – students who take the test might not have experienced the measured STR for the district

Ideally we would like data on individual students, by grade level.
4. Selection bias?

Sample is all elementary public school districts (in California; in Mass.)

no reason that selection should be a problem.
5. Simultaneous Causality?

School funding equalization based on test scores could cause simultaneous causality.

This was not in place in California or Mass. during these samples, so simultaneous causality bias is arguably not important.
Do SW: 9.6, 9.11
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