Marketing de redes sociales

Using discontinuities in assignment rules to identify causal effects dates back to Thistlethwaite and Campbell (1960) and has a long history in psychology (Cook, 2008). However the idea has received an explosion of attention in economics after Hahn et al. (2001) formalized RD design in the language common to program evalu- ation. Subsequent work has further clarified RD design’s underpinnings (Lee, 2008; Lee and Lemieux, 2009), developed the theory behind estimation (Porter, 2003; Sun, 2005; Lee and Card, 2008; Frandsen and Guiteras, 2010), provided practical guid- ance on bandwidth selection (Ludwig and Miller, 2007; Imbens and Kalyanaraman, 2008), and proposed tests of the underlying assumptions (McCrary, 2008). Imbens and Lemieux (2008) and Lee and Lemieux (2009) review this surge in RD design research, focusing on the standard scalar case.

In the standard RD setup, units have a continuous scalar covariate X and outcome

Y. The treatment mechanism can be classified as “sharp” or “fuzzy” depending on

whether the treatment is completely or partially determined by the covariate passing a cutoff (Trochim, 1984). In a sharp design, units receive a binary treatment W 2 {0, 1}

if and only if their covariate exceeds a cutoff c. That is,

W = 1{X c},

where 1 {·} is the indicator function.

Following Neyman (1923) and Rubin (1978), we can define causal effects in terms of potential outcomes. In the potential outcomes framework, Y (1) gives the outcome under treatment and Y (0) gives the outcome under the control. Potential outcomes

Y (1)and Y (0) are linked to the observed variables Y by

Y = Y (W ) = (1 W )_{· Y (0) + W · Y (1).}

We only ever observe the potential outcome associated with the treatment actually received. Average causal effects are then defined as averages over unit causal effects

Y (1) Y (0). Because we can only observe Y (1) or Y (0) but never both for a given

unit, causal inference entails comparisons across potentially dissimilar units.

Intuitively, sharp regression discontinuity design identifies the average causal effect of the treatment for units at the treatment boundary (X = c) by comparing units " above and below the treatment boundary as " goes to zero. Assuming the conditional regression functions of the potential outcomes E [Y (0) | X = x] and E [Y (1) | X = x] are continuous in x, units just above and below the discontinuity have the same average potential outcomes Y (0) and Y (1) but differ by their treatment status W and potential outcome actually observed Y = Y (W ). The average causal effect of the treatment for units at the treatment boundary is identified by taking the limits from above and below,

⌧SRD =E [Y (1) Y (0) | X = c] (2.1)

=E [Y | W = 1, X = c] E [Y | W = 0, X = c]

= lim

x#cE [Y | X = x] limx"c E [Y | X = x]

CHAPTER 2. BOUNDARY REGRESSION DISCONINUITY DESIGN 50

The practical problem becomes how to estimate the two limits limx#cE [Y | X = x]

and limx"cE [Y | X = x]. Given the boundary nature of the problem, Hahn et al.

(2001) propose estimating the limits by local linear regression. Porter (2003) proves that local linear regression is rate optimal for the regression discontinuity problem. With a rectangular kernel, local linear regression amounts to predicting both limits from a linear regression on a subset of observations around the discontinuity. Imbens and Kalyanaraman (2008) recommend local linear regression with an edge kernel and derive an optimal, data-dependent, bandwidth selection rule.

Fuzzy regression discontinuity design generalizes the RD setup to account for treatment rules that are discontinuous in the probability of treatment (Trochim, 1984; Hahn et al., 2001). That is, cases where

lim

x#c Pr(W = 1| X = x) > limx"cPr(W = 1| X = x)

but the probabilities are not necessarily one and zero. Many real assignment rules take this form due to noncompliance with treatment assignment or individual waivers to sharp treatment rules.

As pointed out by Hahn et al. (2001), fuzzy regression discontinuity design has a strong connection to instrumental variables estimation of treatment effects with unit-varying effects (Imbens and Angrist, 1994). When a fraction of units receive treatment on each side of the cutoff, the difference between average outcomes on either side of the discontinuity becomes an intent to treat effect—the average effect of the assignment or encouragement but not the treatment. The average outcomes on both sides of the discontinuity include a mixture of treated and untreated units due to noncompliance.

To see the connection to instrumental variables, assume that units receive an encouragement or assignment Z depending on whether they fall above or below the cutoff,

Z = 1_{X c_{} .}

For instance, schools may recommend students for promotion to the next grade if they score above c on an end-of-year exam but parents may choose otherwise. Given

imperfect compliance, encouragement Z may differ from treatment W . Let W (z) be the treatment the unit would receive given encouragement z. Under a monotonicity assumption that there are no defiers—no units with W (1) = 0 and W (0) = 1—the fuzzy RD estimate is

⌧FRD = limx#cE [Y | X = x] limx"cE [Y | X = x]

limx#cE [W | X = x] limx"cE [W | X = x] (2.2)

=E [Y (1) Y (0) | W (1) > W (0), X = c] ,

which takes the form a Wald estimate (Hahn et al., 2001).

The identified parameter has the standard local average treatment effect (LATE) interpretation of the average treatment effect for units that comply with the encour- agement Z and therefore have W (1) > W (0). The only difference compared to a standard encouragement design setup is that the estimate is local to the encour- agement cutoff c. The locality restriction arises because there is no overlap in the covariate distribution for units that receive and do not receive the encouragement and encouragement is only ignorable conditional on X. Under continuity, we can replace the exact conditioning with limits from above and below.

The practical problem is the same as for sharp RD but requires estimating the four limits in (2.2) rather than two in (2.1). Using a single bandwidth for all four con- ditional expectations, the local linear estimator can be implemented using weighted 2SLS with weights depending on the kernel (Imbens and Lemieux, 2008).