Empirical Methods for Corporate Finance. Regression Discontinuity Design

Transcription

1 Empirical Methods for Corporate Finance Regression Discontinuity Design

2 Basic Idea of RDD Observations (e.g. firms, individuals, ) are treated based on cutoff rules that are known ex ante For instance, for some variable x an observation is treated if x x This type of cutoffs creates a discontinuity We focus on how this treatment influences the outcome variable of interest There exist many natural cutoffs Accounting variables exceeding some thresholds cause covenant violation Market capitalization exceeding some thresholds cause index inclusion Progressive income tax (step functions) 4/22/2015 Empircal Corporate Finance 2

3 Basic Idea of RDD Assignment to treatment is NOT random (unlike DiD) Assignment is based on value of x Assignment to treatment (and control) is not random, but whether individual observation is treated is assumed to be random We assume that observations cannot perfectly manipulate the value of their x Hence whether an observation's x is located just above or below the cutoff is assumed to be random This is the source of identification in RDD 4/22/2015 Empircal Corporate Finance 3

4 Notations The variable x is the forcing variable x is the threshold y(0) is the outcome when no treatment y(1) is the outcome when treatment Sharp RDD Assignment only depends on the forcing variable x (treatment=1 if x x ) Fuzzy RDD x x increases the probability of treatment, but other factors (other than x) determine whether an observation is treated or not 4/22/2015 Empircal Corporate Finance 4

5 Sharp RDD Assumption 1 Assignment to treatment occurs through a known and measured deterministic decision rule: 1 0 Treatment occurs if the value of the forcing variable meets or exceeds the threshold x Weak inequality is unimportant (why?) Need x on both sides of the threshold value (why?) 4/22/2015 Empircal Corporate Finance 5

6 Sharp RDD Source: Roberts and Whited (2010) 4/22/2015 Empircal Corporate Finance 6

7 Thoughts Given the treatment assignment, one could identify treatment effect using: NO! d is correlated with x. So if x affects y, there is no proper identification If x is uncorrelated with u, you do NOT need RDD (as OLS is fine) Or we could control for x and estimate: NO! Does NOT make use of random assignment that happens AROUND the threshold 4/22/2015 Empircal Corporate Finance 7

8 Sharp RDD Assumption 2 Local continuity Potential outcomes y(0) and y(1), conditional on the forcing variable x, are continuous at the threshold x Equivalently: (u x) is continuous in x at x In other words, the outcome variable y would be smooth around the threshold if there were no treatment 4/22/2015 Empircal Corporate Finance 8

9 Local Continuity The only reason for different outcome around the threshold is the treatment: Source: Roberts and Whited (2010) =0 4/22/2015 Empircal Corporate Finance 9

10 Local identification Comparison of outcomes just above and just below the threshold identifies the treatment effect (β) Creates a Bias versus Noise tradeoff Small number of observation around the threshold so the estimate will be noisy A wider range on each side of the threshold reduces noise, but introduce bias as observations further away from the threshold might vary for other reasons (such as the direct effect of x on y) Two ways to do estimation: Use all data, and control for the effect of x on y Only use data in a small window around the threshold 4/22/2015 Empircal Corporate Finance 10

11 Bias versus Noise y Close to threshold few points x If I compare average y on both side, I will estimate a discontinuity but there is none 4/22/2015 Empircal Corporate Finance 11

12 RDD with all data Estimation of two separate regressions: Estimate first equation using only data below x and second equation with data above x f() and g() can be any continuous function of x i x. They control for any effect of x on y Subtracting x i from x means that the estimated intercepts will provide the value of the regression functions at the threshold (and not zero) Treatment effect is: 4/22/2015 Empircal Corporate Finance 12

13 Combine both sides Estimation can be done in one step: So that The interaction term makes sure that the functional forms may be different How do we choose f() and g()? Usually polynomials Example: quadratic polynomial: True order is not known, so check robustness 4/22/2015 Empircal Corporate Finance 13

14 RDD with data around threshold only Do as before but focus on a smaller window around x E.g. Local linear regression (Imbens and Lemieux (2008)) where h is the window width (or bandwidth) which is subjective Lower need to have high order polynomial in smaller windows (why?) 4/22/2015 Empircal Corporate Finance 14

15 Use a Graph to get intuition Divide x into bins, and make sure that no bin contains the threshold x as an interior point Calculate average y in each bin Plot these average (mid point in each bin) Estimate RDD and plot predicted value of y Choice of bandwidth is subjective (bias vs noise tradeoff_ 4/22/2015 Empircal Corporate Finance 15

16 Graph Clear discontinuity in estimation and plot Dash line would be the predicted value of linear RDD. BUT no effect (cubic) Source: Roberts and Whited (2010) 4/22/2015 Empircal Corporate Finance 16

17 Fuzzy RDD Assumption 1 Assignment to treatment occurs in a stochastic manner where the probability of treatment has a know discontinuity at x Instead of a 0 1 step function, the treatment probability as a function of x can contain jump at the cutoff that is less than one Treatment is not purely driven by x E.g. FICO score >620 increases likelihood of loan being securitized, but other factor matter as well 4/22/2015 Empircal Corporate Finance 17

18 Fuzzy RDD Prob. of treatment increases with x Some untreated for x>x and some treated for x<x Source: Roberts and Whited (2010) 4/22/2015 Empircal Corporate Finance 18

19 Sharp vs Fuzzy With sharp RDD, we compare average y just above and below x With fuzzy RDD, the average change in y around the thresholds understate the treatment effect This comparison assumes that all observations are treated, but this is not true. If all observations are treated, change in y would be even larger We have to rescale based on change in probability as x>x just increase the probability of treatment 4/22/2015 Empircal Corporate Finance 19

20 Estimation of fuzzy RDD Idea: use x>x as instrumental variable for treatment! d i = 1 if treated, 0 otherwise New threshold indicator T i 1 0 E.g. d i = 1 if loan is securitized, T i =1 if FICO >620, which increases the probability that the loan is securitized 2SLS: with T i as IV for d i 4/22/2015 Empircal Corporate Finance 20

21 Estimation of fuzzy RDD Needed assumption for IV? T affects the probability of treatment (relevance) but is unrelated to y conditional on d and control f() Satisfied under earlier assumptions Again, f() is typically a polynomial function Same practical issues as with Sharp RDD Show robustness to different specifications and graphs Should also plot graph of d on x (prob. of treatment ) 4/22/2015 Empircal Corporate Finance 21

22 Internal Validity? Is there room for manipulation? Is the threshold chosen because of some pre existing discontinuity in x? or lack of comparability on both sides of the threshold? Can firms manipulate their x around the threshold? Does this matter? Yes, manipulation can cause violation of the local continuity assumption The outcome variable could display a jump around the threshold even in the absence of treatment because of the manipulation Look for bunching around the threshold (although imperfect) 4/22/2015 Empircal Corporate Finance 22

23 Internal validity? RDD assumes observation near the threshold, but on opposite sides, are comparable We need to check this! Graphs (or RDD) of other variables that could be related to y do not display jump at the threshold This does NOT proves the validity of RDD (again there is no way to prove causality) Add controls in RDD estimations (should not change the magnitude ) 4/22/2015 Empircal Corporate Finance 23

24 External validity? Identification in RDD only comes from observations close to the threshold Effect of treatment might be different for observations further away from the threshold Be careful about broad statements for these observations In fuzzy RDD, the treatment effect is estimated only using compliers The estimates only capture the effect of those where the discontinuity is what pushed them into treatment Same as with IV, extrapolation might be difficult 4/22/2015 Empircal Corporate Finance 24

25 Summary RDD relies on treatment assignment that is NOT random, but where the process follows some known cutoff rules Two types of RDD: sharp and fuzzy Sharp RDD: treatment is deterministic and only depends on x Fuzzy RDD: treatment is stochastic and the probability of treatment has discontinuity at x Fuzzy RDD is really an IV Many internal checks are needed 4/22/2015 Empircal Corporate Finance 25

26 Prevalence? RDD Source: Bowen, Fresard, and Taillard (2014) 4/22/

27 Application 1: Malenko and Shen (2015) Malenko and Shen, 2015, The role of proxy advisory firms: evidence from a regression discontinuity design Question: What is the effect of voting recommendations made by proxy advisory firms? Do they have an effect on voting outcomes? What channel? Motivation: Increasing role of proxy advisors (e.g. ISS). Critics mention onesize fits all approach, and potential conflicts of interest. Empirical Strategy: RDD using thresholds that generate deeper analysis by ISS of the companies compensation practices (below cutoff) Results: Recommendations matters! Negative recommendation by ISS on a say on pay proposal cause 25% lower voting support This operates through a certification effect (investors seek protection) 4/22/2015 Empircal Corporate Finance 27

28 Location in the field Governance Real decisions Financing Valuation Institutional framework: laws, regulations, taxes, markets, Proxy advisors 4/22/

29 Identification problem? A positive correlation between shareholder support (votes) and ISS recommendation does not imply causal effect. Why? Omitted variables could affect both ISS recommendations and shareholders vote 4/22/2015 Empircal Corporate Finance 29

30 Basic idea behind RDD Guidelines used by ISS on say on pay proposals Identifies underperforming firms as those with one and three year TSR (total shareholder returns) below median of other four digit GICS group TSR calculated on the last day of the quarter closest to fiscal year end ISS uses Compustat to compute TSR Only Russell 3000 firms (to define other four digit GICS group) In depth qualitative review of compensation practice of underperforming firms before giving a say on pay recommendation Probability of negative recommendation increases discontinuously below the cutoff 4/22/2015 Empircal Corporate Finance 30

31 Fuzzy RDD Instrument a negative recommendation with an dummy variable if a firm falls below the cutoff Assumption: relation between shareholder votes and MaxTSR would be smooth in the absence of differential recommendation around the cutoff Plausible? Yes as the rule is rather specific They provide three good validity tests (later ) 4/22/2015 Empircal Corporate Finance 31

32 Fuzzy RDD 2SLS specification: Coefficient of interest Local linear regressions with a 5% bandwidth: 4/22/2015 Empircal Corporate Finance 32

33 Data Votes and recommendations from ISS Voting Analytics Focus on say on pay proposals Defined Vote as the fraction of votes in favor of the proposal Recommendation is either positive or negative Use Compustat and CRSP to compute TSR Focus on 2010 and 2011 (when cutoff rules was used) 1,932 firms and 2,020 proposals 4/22/2015 Empircal Corporate Finance 33

34 Summary Statistic 4/22/2015 Empircal Corporate Finance 34

35 Real Discontinuities? Bins of 1%... (how many firms and proposal per bin?) 4/22/2015 Empircal Corporate Finance 35

36 First stage: Prob. of negative reco. ~0.15 Bandwidth of 5% 403 observations 4/22/2015 Empircal Corporate Finance 36

37 First stage: Prob. of negative reco. 4/22/2015 Empircal Corporate Finance 37

38 Second stage: effect on votes ~0.04 4/22/2015 Empircal Corporate Finance 38

39 Second stage: effect on votes 25% lower! Why different from graph? The effect is large (maybe too large ) 4/22/2015 Empircal Corporate Finance 39

40 Robustness 4/22/2015 Empircal Corporate Finance 40

41 Validity Tests Manipulation of the forcing variable is unlikely here (why?) Difficult to push TSR Cutoff is defined as a function of other firms TSR Density seems continuous No evidence of bunching 4/22/2015 Empircal Corporate Finance 41

42 Validity Tests Discontinuity in other variables? NO! 4/22/2015 Empircal Corporate Finance 42

43 Validity Tests Alternative Samples where threshold does NOT exist 4/22/2015 Empircal Corporate Finance 43

44 Validity Test 4/22/2015 Empircal Corporate Finance 44