Causal: Week 11

Transcript

1. S William Lowe Hertie School th November
2. S Sensitivity to → Collider bias: nope → Mediator-outcome confounding: week → Confounding... → Exclusion violation restrictions? [link] (Conley et al., ; van Kippersluis & Rietveld, ) e state of the art → Omitted variable bias in coe cients → Omitted variable bias in R → Plots, plots, plots
3. U D X F Z Y Problem: → We know about X but we wonder if there’s an unmeasured Z too Cinelli and Hazlett ( ) example: → What is the e ect of direct harm D due to government-organized or perpetrated violence on Y peace preferences in Darfur - ? → X demographics including age, occupation, household size, voting experience → F being female → Z? village centrality, asset types, village accessibility... See Hazlett ( ) for study details
4. E O peace index (Y): → min; , max: , → mean: . , sd: . , → : %, : % directlyharmed (D): → : %, : % female (F): → : %, : % Model 1 (Intercept) 1.082 (0.315) ∗∗∗ directlyharmed 0.097 (0.023) ∗∗∗ age −0.002 (0.001) ∗ farmer dar −0.040 (0.029) herder dar 0.014 (0.032) pastvoted −0.048 (0.024) ∗ hhsize darfur 0.001 (0.002) female −0.232 (0.024) ∗∗∗ R2 0.512 Num. obs. 1276 ∗∗∗ p < 0.001; ∗∗ p < 0.01; ∗ p < 0.05
5. O What we want to estimate Y = Dˆ τ + X ˆ β + Z ˆ γ + ˆ є What we estimate Y = Dˆ τres + X ˆ βres + ˆ єres e di erence (bias) in estimates bias = ˆ τres − ˆ τ
6. O What we want to estimate Y = Dˆ τ + X ˆ β + Z ˆ γ + ˆ є What we estimate Y = Dˆ τres + X ˆ βres + ˆ єres e di erence (bias) in estimates bias = ˆ τres − ˆ τ What is ˆ τres as a function of ˆ τ?
7. O Cinelli and Hazlett use A B to mean the residuals from a regression of A on B → what’s le of A a er the ‘e ect’ of B has been removed Reminder: the regression coe cient of B predicting A in a bivariate regression is βB = cov(B, A) var(B) and from the Frisch-Lowell-Waugh theorem, the regression coe cient of B predicting A controlling for X is βB = cov(B X , A X ) var(B X)
8. O en ˆ τres = cov(D X , Y X ) var(D X) = cov(D X , D X + Z X ) var(D X) = ˆ τ + ˆ γ cov(D X , Z X ) var(D X) = ˆ τ + ˆ γ ˆ δ So bias = ˆ γ ˆ δ D X F Z Y δ τ γ → δ is a measure of ‘imbalance’ → γ is a measure of the (not necessarily causal) ‘impact’ of Z on Y
9. B the only way in which Z’s relationship to D enters the bias is captured by its ‘linear im- balance’, parameterized by ˆ δ. In other words, the linear regression of Z on D and X need not re ect the correct expected value of Z—rather it serves to capture the aspects of the relationship between Z and D that a ects the bias. (Cinelli & Hazlett, )
10. Plotting ˆ τres − ˆ γ ˆ δ
11. L → Hard to interpret if Z is not binary → What about multiple Z? → How much ‘robustness’ is enough? Compared to what? → Can get sensitivity for more than the coe cient’s point estimate? Suggestion: Switch from coe cients to (partial) R s Reminder: R is symmetrical in bivariate regressions: RZ∼D = var( ˆ Z) var(Z) = − var(Z D ) var(Z) = cor(Z, ˆ Z) = cor(Z, D) and so is partial R RZ∼D X = − var(Z D,X ) var(Z X) = cor(Z X , D X ) = RD∼Z X
12. A e relevant quantities are now: → How much variance in Y is explained by Z controlling for everything else: R Y∼Z X,D → How much variance in D is explained by Z controlling for everything else: R D∼Z X ere can be lots of Zs working in concert → eir scales are irrelevant now we work in variance explained, a.k.a. ‘explanatory power’ D X F Z Y R D∼Z X R Y∼Z X,D
13. S Functions of these two quantities: e robustness value RVq = RY∼Z X,D = RD∼Z X describes how strong Z has to be to reduce the ˆ τ by a factor of q → e.g. RV is the strength of confounding that would reduce it to zero Comparative robustness measures, e.g. relative bounds → the maximum e ect that a Z not more than k times as strong as, say ‘Female’ would have on ˆ τ → ese results are upper bounds on the e ects of multiple Zs
14. G Interpretation: → For multivariate Z contours are upper bounds → RVq is a summary of the diagonal of this plot → ˆ τ when Z is k times as powerful as ‘Female’ are shown as diamonds
15. T → Unmeasured confounders with equal e ect on D and Y would have to explain . % to reduce ˆ τ = . to zero → or . % to make ˆ τ not statistically distinguishable from zero (at the % level) → if confounders explained % of the residual variance of the Y, they would need to explain at least . % of the residual variance of the D to reduce ˆ τ to zero → the footnote shows the two relevant quantities for a Z like ‘Female’. Notice that % < . %, so even in the worst case scenario above, ˆ τ would not be reduced to zero
16. A : Sensitivity to misspeci cation comes for free!
17. S What thresholds are reasonable for sensitivity testing? → Terrible question: this question makes (almost) no sense
18. S What thresholds are reasonable for sensitivity testing? → Terrible question: this question makes (almost) no sense So what do I do with all this information then?
19. R Cinelli, C. & Hazlett, C. ( ). ‘Making sense of sensitivity: Extending omitted variable bias’. Journal of the Royal Statistical Society: Series B (Statistical Methodology), ( ), – . Conley, T. G., Hansen, C. B. & Rossi, P. E. ( ). ‘Plausibly exogenous’. e Review of Economics and Statistics, ( ), – . Hazlett, C. ( ). ‘Angry or weary? how violence impacts attitudes toward peace among darfurian refugees’. Journal of Con ict Resolution, ( ), – . van Kippersluis, H. & Rietveld, C. A. ( ). ‘Beyond plausibly exogenous’. e Econometrics Journal, ( ), – .