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I'm trying to articulate the differences between identification in regression models and assumptions with the potential outcomes framework. In particular what (if anything) does SUTVA add beyond an assumption like mean independence. This is how I've been thinking about it:

For a regression, we have data $(Y_i,X_i)$ where $X_i$ is binary. In the model $Y_i = b_0 + b_1 X_i + u_i$, we need $\mathbb{E}[u_i|x_i]=0$ for $\hat{b}_{1,OLS} \to b_1$.

In the potential outcomes world we invoke SUTVA ("no interference" and "no hidden versions of the treatment"—thanks for the formal definitions here) and call $(Y_i(0),Y_i(1))$ the potential outcomes, where $Y_i = (1-X_i)Y_i(0) + X_iY_i(1)$. If we assume independence ($(Y_i(1),Y_i(0))\perp X_i$), then I believe $\bar{Y}_i X_i - \bar{Y}_i (1-X_i) \to b_1$.

Then how is $\mathbb{E}[u_i|x_i]=0$ related to the SUTVA assumption? At first blush, it seems like the independence assumption should be the one related to the mean independence assumption, which would mean SUTVA is buying us something else. Is it buying something more? or is independence of potential outcomes weaker than mean independence of errors (without SUTVA)? Or should I think of SUTVA as clarifying the interpretation of the target parameter more than identification?

Imagine we prescribe medication to an (random) treatment group ($X_i=1$) and control group ($X_i=0$). If these groups share medicine, we would expect $\bar{Y}_i X_i = \bar{Y}_i(1-X_i)$ in essence because everyone is "treated" (albeit with a half dose). This is often given as an example for why we need SUTVA (this was a helpful discussion for me), but my intuition says that in the analogous regression model, if there is sharing we would expect $u_i$ (the unobserved determinants of the outcome) to be larger for $X_i=0$ group and smaller for $X_i=1$ group, violating $\mathbb{E}[u_i|x_i]=0$ (or stated another way, we have omitted variables: "I gave my medicine to my friends" and "I got medicine from my friends").

So yes, OLS is biased, but is it because the assumptions aren't strong enough or because they aren't met? (In either case there is also a little bit of semantics about whether $X$ is being assigned to receive the medicine or actually taking it... in an IV setting this seems like it would be an encouragement design with no first stage...)

I've also tried connecting the dots by using the potential outcomes in the regression equation. $$ \begin{align*} b_0 + b_1 X_i + u_i &= Y_i \\ &= (1-X_i)Y_i(0) + X_iY_i(1) \\ &=Y_i(0) + b_i X_i \\ &=\mu_{Y0} + e_{i0} + (\mu_{b} + e_{i1}) X_i \\ &=\mu_{Y0} + \mu_b X_i + (e_{i0}+e_{i1} X_i) \end{align*} $$ where the $\mu$s are population averages for and $e$s are deviations from untreated outcomes and treatment effects. Looking at this makes me think that if SUTVA is true then independence of potential outcomes implies mean independence of the errors. But I'm still feeling confused about how to think about the relationship if STUVA isn't true.

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A violation of SUTVA implies that the potential outcomes for n individual exist not only for the possible values of their treatment but also for the possible values of the treatment of other individuals. For simplicity, assume our population consists of two people: $1$ and $2$. For unit $1$, we have four potential outcomes: $Y^{00}$, $Y^{01}$, $Y^{10}$, and $Y^{11}$, corresponding to $Y^{X_1 X_2}$. That means a structural model for the observed outcome for unit $i$ ($i\ne j$) might be the following: $$ Y_i = \beta_0 + \beta_1 X_i + \beta_2 X_j + \beta_3 X_i X_j + \varepsilon_i $$ So, for unit $1$, $$ Y_1 = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2 + \varepsilon_1 $$ and for unit $2$, $$ Y_2 = \beta_0 + \beta_1 X_2 + \beta_2 X_1 + \beta_3 X_2 X_1 + \varepsilon_2 $$ When is OLS unbiased for $\beta_1$? My understanding is that when $E[X_i(\beta_2 X_j + \beta_3 X_i X_j + \varepsilon_i)]=0$, an OLS regression of $Y_i$ on $X_i$ will be unbiased for $\beta_1$. In an observational study with no confounding but where unit $1$'s treatment status is related to unit $2$'s treatment status, that assumption will not be met. Before that, though, you need to define the treatment effect, which may not correspond to any parameter in the structural model, in which case OLS alone is not enough to estimate the treatment effect.

With more units, the structural model for the outcome expands, with terms for each unit and combination of units. You may make simplifying assumptions, like that an individual's outcome only depends on the treatment status of those near them, or is only additive in other units' treatment statuses, or depends only on the proportion of treated units and not their identity. regardless, it will be possible to write down a structural model for that outcome. You can then use standard econometric methods to determine whether OLS will be sufficient to recover the structural parameter of interest. It's another story whether that parameter corresponds to a meaningful causal effect.

In general, to think about this issue, it's worth taking the following steps:

  1. Write down a structural model for $Y_i$. Under SUTV violations, this will involve the treatment status of other units.
  2. Define the causal effect interest. This should be in terms of potential outcomes.
  3. Write the causal effect as a function of the structural model parameters.
  4. Determine whether OLS would be sufficient for consistent estimation of the structural parameters using standard methods.
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  • $\begingroup$ I think the most helpful piece here was that the consistent estimation of a parameter from the structural model via OLS is a slightly different question from whether that structural parameter is the treatment effect of interest. Is it right to say then that SUTVA answers the second issue by guaranteeing that there is a well defined treatment effect where as the independence argument is about the first? $\endgroup$ Mar 23, 2021 at 18:13
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    $\begingroup$ Sort of. My understanding is that OLS can still give unbiased parameter estimates even with correlated errors. SUTVA and iid are two completely separate and unrelated issues. You can have SUTVA without iid: students in the same class have correlated errors but the effect of an intervention on each student doesn't depend on which other student get it. You can have iid without SUTVA: errors are independent among students but the outcome for each student depends on which of their friends get the intervention. iid considers the stochastic part of the model and SUTVA the structural. $\endgroup$
    – Noah
    Mar 24, 2021 at 3:34
  • $\begingroup$ Sure! That's definitely the case, and a great example. I'm realizing too late that I said independence when I really meant unconfoundedness (from the regression perspective) or that treatment status is independent of potential outcomes (from the experimental perspective). Sorry does that make more sense? $\endgroup$ Mar 24, 2021 at 15:45
  • $\begingroup$ Again, sort of, because SUTVA and unconfoundedness are also unrelated. SUTVA is a matter of potential outcomes and unconfoundedness is a matter of treatment assignment. But because potential outcomes occur before treatment is assigned, you can talk about SUTVA without actually assigning treatment (i.e., by writing the potential outcome functions, which don't involve the treatment). SUTVA is about how many different potential outcome functions there are, and unconfoundedness is about whether the error in those functions is also a cause of treatment. $\endgroup$
    – Noah
    Mar 24, 2021 at 16:05

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