Two Misconceptions

The following two statements about selection on observables are false:

1. Under selection on observables, if I know the value of someone’s covariate vector \(X\), then learning her treatment status \(D\) provides no additional information about the average value of her observed outcome \(Y\).
2. Selection on observables requires the treatment \(D\) and potential outcomes \((Y_0,Y_1)\) to be conditionally independent given covariates \(X\).

If you’ve studied treatment effects, pause for a moment and see if you can figure out what’s wrong with each of them before reading further.

The First Misconception

The first statement:

Under selection on observables, if I know the value of someone’s covariate vector \(X\), then learning her treatment status \(D\) provides no additional information about the average value of her observed outcome \(Y\).

is a verbal description of the following conditional mean independence condition: \[ \mathbb{E}[Y|X,D] = \mathbb{E}[Y|X]. \] So what’s wrong with this equality? The potential outcomes \((Y_0, Y_1)\) and the observed outcome \(Y\) are related according to \[ Y = Y_0 + D (Y_1 - Y_0). \] Taking conditional expectations of both sides and using the selection on observables assumption \[ \begin{aligned} \mathbb{E}[Y|X,D] &= \mathbb{E}[Y_0|X,D] + D \mathbb{E}[Y_1 - Y_0|D,X]\\ &= \mathbb{E}[Y_0|X] + D \mathbb{E}[Y_1 - Y_0|X]. \end{aligned} \] In contrast, conditioning on \(X\) alone gives \[ \begin{aligned} \mathbb{E}[Y|X] &= \mathbb{E}[Y_0|X] + \mathbb{E}[D(Y_1 - Y_0)|X]\\ &= \mathbb{E}[Y_0|X] + \mathbb{E}_{D|X}[D\mathbb{E}(Y_1 - Y_0|D,X)]\\ &= \mathbb{E}[Y_0|X] + \mathbb{E}_{D|X}[D\mathbb{E}(Y_1 - Y_0|X)]\\ &= \mathbb{E}[Y_0|X] + \mathbb{E}(D|X) \cdot \mathbb{E}(Y_1 - Y_0|X) \end{aligned} \] by iterated expectations and the selection on observables assumption, since \(\mathbb{E}(Y_1 - Y_0|X)\) is a measurable function of \(X\). Subtracting these expressions, we find that \[ \mathbb{E}(Y|X,D) - \mathbb{E}(Y|X) = \left[ D - \mathbb{E}(D|X) \right] \cdot \mathbb{E}(Y_1 - Y_0|X) \] so that \(\mathbb{E}(Y|X,D) = \mathbb{E}(Y|X)\) if and only if the RHS equals zero.

So how could the RHS equal zero? One way is if \(D = \mathbb{E}(D|X)\). Since \(D\) is a binary random variable, this would require \(\mathbb{E}(D|X)\) to be a binary random variable as well. But notice that \(\mathbb{E}(D|X) = \mathbb{P}(D=1|X)\) is simply the propensity score \(p(X)\). Because \(X\) is a random variable, so is \(p(X)\). But \(p(X)\) cannot take on the values zero or one. If it did, this would violate the overlap assumption: \(0 < p(X) < 1\).

So we can’t have \(D = \mathbb{E}(D|X)\), but what about \(\mathbb{E}(Y_1 - Y_0|X)=0\)? Since \((Y_1 - Y_0)\) is the treatment effect of \(D\), it follows that \(\mathbb{E}(Y_1 - Y_0|X)\) is the conditional average treatment effect \(\text{ATE}(X)\) given \(X\). It’s not a contradiction for \(\text{ATE}(X)\) to equal zero, but think about what it would mean: it would require that the average treatment effect for a person with covariates \((X = x)\) is exactly zero regardless of \(x\). Moreover, by iterated expectations it would imply that \[ \text{ATE} = \mathbb{E}(Y_1 - Y_0) = \mathbb{E}_X[\mathbb{E}(Y_1 - Y_0| X)] = \mathbb{E}[\text{ATE}(X)] = 0 \] so the average treatment effect would also be zero. Again, this is not a contradiction but it would definitely be odd to assume that the treatment effect is zero before you even try to estimate it!

To summarize: the first statement above cannot be an implication of selection on observables because it would either require a violation of the overlap assumption, or imply that there is no treatment effect whatsoever. To correct the statement, we simply need to change the last three words:

Under selection on observables, if I know the value of someone’s covariate vector \(X\), then learning her treatment status \(D\) provides no additional information about the average values of her potential outcomes \((Y_0, Y_1)\).

This is a correct verbal statement of the mean exclusion restriction \(\mathbb{E}(Y_0|D,X) = \mathbb{E}(Y_0|X)\) and \(\mathbb{E}(Y_1|D,X) = \mathbb{E}(Y_1|X)\).

The Second Misconception

And this leads nicely to the second misconception:

Selection on observables requires the treatment \(D\) and potential outcomes \((Y_0,Y_1)\) to be conditionally independent given covariates \(X\).

To see why this is false, consider an example in which \[ \begin{aligned} Y &= (1 - D) \cdot (\alpha_0 + X'\beta_0 + U_0) + D \cdot (\alpha_1 + X' \beta_1 + U_1)\\ U_0|(D,X) &\sim \text{Normal}(0,1 - D/2)\\ U_1|(D,X) &\sim \text{Normal}(0,1 + D). \end{aligned} \] Notice that the distributions of \(U_0\) and \(U_1\) given \((D,X)\) depend on \(D\). Now, by iterated expectations, \[ \begin{aligned} \mathbb{E}(U_0|X) &= \mathbb{E}_{(D|X)}[\mathbb{E}(U_0|D,X)] = 0\\ \mathbb{E}(U_0) &= \mathbb{E}_{X}[\mathbb{E}(U_0|X)] = 0 \end{aligned} \] and similarly \(\mathbb{E}(U_1|X) = \mathbb{E}(U_1)=0\). Substituting \(D=0\) and \(D=1\), we can calculate the potential outcomes and average treatment effect as follows \[ \begin{aligned} Y_0 &= \alpha_0 + X'\beta_0 + U_0 \\ Y_1 &= \alpha_1 + X'\beta_1 + U_1 \\ \text{ATE} &= \mathbb{E}(Y_1 - Y_0) = (\alpha_1 - \alpha_0) + \mathbb{E}[X'](\beta_1 - \beta_0). \end{aligned} \] It follows that \(D\) is not conditionally independent of \((Y_0, Y_1)\) given \(X\). In particular, the variance of the potential outcomes depends on \(D\) even after conditioning on \(X\): \[ \begin{aligned} \text{Var}(Y_0|X,D) &= \text{Var}(U_0|X,D) = 1 - D/2\\ \text{Var}(Y_1|X,D) &= \text{Var}(U_1|X,D) = 1 + D. \end{aligned} \] In spite of this, the selection on observables assumption still holds: \[ \begin{aligned} \mathbb{E}(Y_0|D,X) &= \alpha_0 + X'\beta_0 + \mathbb{E}(U_0|D,X) = \alpha_0 + X'\beta_0\\ \mathbb{E}(Y_0|X) &= \alpha_0 + X'\beta_0 + \mathbb{E}(U_0|X) = \alpha_0 + X'\beta_0\\ \end{aligned} \] and similarly \(\mathbb{E}(Y_1|D,X) = \mathbb{E}(Y_1|X) = \alpha_1 + X'\beta_0\). While this example is admittedly a bit peculiar, the point is more general: because the average treatment effect is an expectation, identifying it only requires assumptions about conditional means.² The second statement is even easier to correct than the first: we need only add a single word:

Selection on observables requires the treatment \(D\) and potential outcomes \((Y_0,Y_1)\) to be conditionally mean independent given covariates \(X\).

Conditional independence implies conditional mean independence, but the converse is false.

Epilogue

So what’s the moral here? First, it’s crucial to distinguish between the observed outcome \(Y\) and the potential outcomes \((Y_0, Y_1)\). Second, the various notions of “unrelatedness” between random variables—independence, conditional mean independence, and uncorrelatedness—can be confusing. Be sure to pay attention to exactly which condition is used and why. In a future post, I’ll have more to say about the relationships between these notions.