Omitted Variable Bias

1. Omitted Variable Bias Methods of Economic Investigation Lecture 7

2. Today’s Lecture • Review Regression Framework • Linking last term to this term • Understanding conditional expectations • Covariates Added and Removed • Why is this “Controlling” • What if you leave something out?

3. Review: This Course So Far • Do we know what the effect of X is on y? • Defining concepts of “treatment” and “control” • Experimental settings: Explaining the usefulness of Random Assignment • Non-Experimental settings: variation is `as if’ random • In Theory, all you need to do if you have random assignment is take a difference in means • Go backwards a bit…

4. General Goal in Causal Inference • E(Y| T=1) – E(Y | T=0) or E(Y| T=1, X) – E(Y | T=0, X) • So we need to know how to estimate a conditional expectation function • We’ll worry about if can be causally interpreted later—right now let’s just if we can even show how we could estimate it

5. What is a conditional expectation • The conditional expectation function (CEF) is the population average of Y with X held fixed • CEF =E[Yi | Xi] • It is a function of X and because X is a random variable, the CEF is itself random • We have been doing the special case where the CEF only takes 2 values • E[Yi | Ti=1] and E[Yi | Ti=0] • In practice, could be continuous but we’ll come back to that

6. Defining the CEF • For a continuous Y with a density fy(. | X=xi) • Let’s ground this in reality a bit: • The expectation is a concept because we rarely observe the entire population • Generally we’re going to use data to make inferences about the sample CEF • Last term you proved under what conditions and assumptions this sample CEF converges to the population CEF

7. How is Y related to the CEF • The observed outcome Y can be decomposed into the bit that is explained by X (the CEF) and the unexplained bit ε • Then last term you learned that • CEF is the “best” (in a MMSE sense) summary of the relationship between Y and X • Can also decompose the variance usefully into explained variance and unexplained variance

8. How is OLS related to this? • If the true CEF is linear, it’s the sample estimate • Example joint normality • Saturated model (different parameter for every possible combination of values for X) • Best Linear Predictor (BLUE) • Best we can relative to other linear estimates • Common justification—local linear approximation • Minimizes Mean Squared Error • Implies it is the best linear approximation to CEF

9. This should be familiar • You are probably used to thinking OLS as • Turns out that is the thing is the parameter we need to estimate the CEF • And the sample analog that we estimate will converge to the true β (under appropriate conditions

10. How do we establish Causality? • Conditional Independence Assumption • In words: on average, outcomes would be the same in the treatment and control groups • In Math: Let Y1 be potential outcome if T=1 and Y0 be potential outcome of T=0 • Can write Y = Y0 + (Y1 – Y0)*T • Then CIA: {Y0i,Y1i} independent of Ti(or Ti | Xi) Or E[Yi| T=1 ] – E[Yi| T=0] = E[Y1i – Y0i] • Intuition: since we never observe both Y0 andY1 for any one individual, we want to think on average, the two groups are the same

11. Selection Bias • The Reason we need CIA: Selection bias • In words: different people choose to be in the treatment or control group. The underlying differences in these people would generate differences in outcomes regardless of the treatment • E(Y| T=1) – E(Y1 | T=0) =E(Y- Y0 |Ti=1) +{E(Y0 | T=1) – E(Y0 | T=0)}

12. What is generating the bias? • If it’s selection on observables • we can condition on X—estimate the CEF • Convenient that the regression can partial this out: let’s see how • Simple example: 2 variables, T and X, each of which are either zero or one. So, the CEF can have 4 values • E[Y| T=0, X=0]=α • E[Y| T=1, X=0]= α+β • E[Y| T=0, X=1]= α+γ • E[Y| T=1, X=1]= α+β+γ+δ

13. Extracting β from the estimated CEF • Imagine we estimate each CEF’ • Write a single equation for the CEF E[Y | T, X] = α + βT +γX+δ(T*X) What is β = E[Y| T=1, X=0]- E[Y| T=0, X=0] • Why is X fixed? So we can get β out regression • Generalize this to a continuous X, and we just hold it at some fixed point (e.g. it’s mean) Main effect of T Main effect of X Interaction

14. What if T didn’t vary with X? • If X has some fixed association with Y independent of T, then the CEF only has 2 values: • E[Y| T=0, X=0]=α • E[Y| T=1, X=0]= α+β • E[Y| T=0, X=1]= α • E[Y| T=1, X=1]= α+β [for ease we can set γ and δ to zero…] • In that case—we’re ok and can exclude X. But we can increase precision because we get 2 estimates of the CEF if we include X

15. What T and X are correlated? • Then we’ve got a problem • E[Y| T=0, X=0]=α • E[Y| T=1, X=0]= α+β • E[Y| T=0, X=1]= α+γ • E[Y| T=1, X=1]= α+β+γ+δ • If we don’t have X: E[Y | T=1] has two potential values and we cannot say: E[Y | T=1] –E[Y | T=0 ≠ E[Y1 - Y0 | T=1] • Another way to say this is we’ve got “selection on unobservables”

16. Omitted Variable Bias Formula • Simply from before, suppose δ=0 so our true CEF is: E[Y | T, X] = α + βT +γX (REMEMBER: This means Y = E[Y | T, X] +e) • We estimate

17. What is happening when we estimate? • What this is doing is given observed Y’s for the various values of T it is choosing the α and β that minimize the mean square error • intuitively, the bias happens because the values of Y’s are changing not only with respect to T but also the unobserved X • In more formal terms we get: Should look familiar True β (this is what we want) Relationship between X and Y

18. What is relationship between X and T • Pretend we could observe X…then w could think about estimating • Well, we know, since this is OLS and assuming no OVB in this equation that • So our omitted variable bias formula can be re-written as:

19. Signing your bias • Generally, the concern with OVB is that it biases your estimate upwards. • The sign of the bias will be as follows:

20. What did we learn today • Conditional Expectation Functions (CEFs) are how we determine “causal effects” • Regressions allow the “best” way to estimate these CEFs • Selection on observables is not something that will bias our estimates • Selection on unobservables is a problem

21. Next Class • Review of OVB • Error Correction Models • Fixed Effects • Random Effects (interaction terms)