Context: Identifying Contagion in Networks

• Ideally we would like to run a model in the form of \[ Y_i = f\left(X_i, Y_{-i}, G\right) \]

  where \(Y_{-i}\) is the behavior of individuals other than \(i\) and \(G\) is a graph.

• The problem: So far, traditional statistical models won’t work, why? because most of them rely on having IID. e.g

  1. In OLS \(\varepsilon_i\) must be IID, otherwise \(E[X\varepsilon]\neq 0\)
  2. In MLE (logit, probit, Survival analysis, etc.) must have IID in order to build the likelihood, otherwise will be optimizing a wrong log-likelihood.

  In both cases, contagion estimators will be biased.

Spatial Autocorrelation Models (SAR)

• Spatial Autocorrelation Models are mostly applied in the context of spatial statistics and econometrics.

• The core part of these model lies on specifying a grid and thus generating a network. Such network is then represented as an adjancency matrix with row sums equal to 1.

• The spatial autocorrelation model explicitly builds the autocorrelatation of a graph (a network) as follows

  \[ \begin{align} Y = & \alpha + \rho W Y + X\beta + \varepsilon,\quad\varepsilon\sim MVN(0,\sigma^2I_n) \\ \implies & Y = \left(I_n -\rho W\right)^{-1}(\alpha + X\beta + \varepsilon) \end{align} \]

  Where \(\rho\in[-1,1]\) and \(W=\{w_{ij}\}\), with \(\sum_j w_{ij} = 1\)

Spatial Autocorrelation Models (SAR) (cont.)

• This is more close than we might think, since the \(i\)-th element of \(Wy\) can be expressed as \(\frac{\sum_j a_{ij} y_j}{\sum_j a_{ij}}\), what we usually define as exposure in networks, where \(a_{ij}\) is an element of the \(\{0,1\}\)-adjacency matrix .

• Notice that \((I_n - \rho W)^{-1} = I_n - \rho W - \rho^2 W^2 - \dots\), hence there autocorrelation does consider effects over neighbors farther than 1 step away, which makes the specification of \(W\) no critical. (See LeSage, 2008)

• There are several ways of estimating this (besides of the one in the R package sna), CRAN has an especial task dedicated to it. (https://cran.r-project.org/web/views/Spatial.html)

• Neat, but this is not problem free… These models assume that \(W\) is exogenous, in other words, if there’s homophily you won’t be able to use it!

• This takes us to the next method

General overview

• Problem Homophily can be confounded with contagion

• Generically, if we observe homophily in a network, there is a set of variables \(Z_i\) that partly generated the graph \(G\).

• Furthermore, that same variable \(Z_i\) may have generated the behavior \(Y_i\), hence, if we project \(Y_i\) onto \(G\) (regression model or other model using exposure) we will see an association between the two of them.

• Such association will be spurious since comes throught \(Z_i\), and is not a direct effect of \(G\) over \(Y\). (See Shalizi and Thomas, 2011)

• Solution (according to Aral et al., 2009) Use propensity score matching, to rule out homophily-based contagion. First, lets take a look at the Rubin Causal Model.

Rubin Causal Model (1978)

• Potential outcomes \(Y_i(0), Y_i(1)\) are the possible states of the world, this is, the outcome of the variable \(Y\) for the treated and untreated version of \(i\). But we only observe one.

  \[ Y_i = Y_i(W_i) = Y_i(1)W_i + Y_i(0)(1-W_i) \]

• Assignment mechanism On what depends the treatment regime?

  1. Completely random (ideal case)
  2. Observable characteristics (we might do something)
  3. On the outcome (don’t even think about it)

  Here we see \(W_i\perp (Y_i(0),Y_i(1))\;|\;X_i\) (the second case), and refer to this as “Unconfounded assignment”.

• Stable Unit Treatment Value Assumption (SUTVA) “Joe’s change in blood pressure may not depend on whether or not Mary receives the drug” (wiki)

  “Only the level of the treatment applied to the specific individual is assumed to potentially affect outcomes for that particular individual” (Imbens and Wooldridge 2009, p.13)

Rosenbaum and Rubin (1983b)

Assumptions:

1. Unconfoundedness \(W_i \perp (Y_i(0), Y_i(1))\;|X_i\) “Beyond \(X_i\), there are no (unbserved) characteristics of the individual associated both with the potential outcomes and treatment.”

2. Overlap \(0 < pr(W_i=1|X_i=x) < 1,\;\forall x\) “The conditional distribution of \(X_i\) given \(W_i=0\) completely overlaps the conditional distribution of \(X_i\) given \(W_i=1\).” In other words, all estimated probabilities are strictly between 0 and 1.

   Unconfoundedness+Overlap = Strong Ignorability

Rosenbaum and Rubin (1983b) (cont.)

Usually, we would like to estimate the Average Treatment Effect (\(ATE\))

\[ \begin{align} ATE(x) &= E[Y_i(1) - Y_i(0)| X_i = x] \quad \mbox{(Which we can't estimate)}\\\\ & \mbox{From Unconfoundedness we have} \\ &= E[Y_i(1) | W_i = 1, X_i = x] - E[Y_i(0) | W_i = 0, X_i = x] \\\\ & \mbox{From overlap we get} \\ &= E[Y_i | W_i = 1, X_i = x] - E[Y_i | W_i = 0, X_i = x] \\ \end{align} \]

Which is something that we can estimate

Nearest Neighbor Matching Estimator

1. Roughly, for each individual \(i\), we find a set \(\mathcal{J}_M(i) \subset \{1,\dots,N\}\) of \(M\) nearest neighbors such that all \(j\in \mathcal{J}_M(i)\) minimize \(\|X_i - X_j\|\) and \(W_i \neq W_j\) (so we match on the opposite group).

   This \(\|X_i - X_j\|\) can be \(|\hat p(X_i) - \hat p(X_j)|\) as well, and further, with \(M=1\).

2. Then, for each individual \(i\) we generate the following

   \[ \begin{align} \hat Y_i(0) & = Y_i\times1\{W_i=0\} + \left(\frac{1}{M}\sum_{j\in\mathcal{J}_M(i)}Y_j\right)\times 1\{W_i = 1\} \\ \hat Y_i(1) & = Y_i\times1\{W_i=1\} + \left(\frac{1}{M}\sum_{j\in\mathcal{J}_M(i)}Y_j\right)\times 1\{W_i = 0\} \\ \end{align} \]

3. Then we estimate the \(ATE = \frac{1}{N}\sum_{i=1}^N\left(\hat Y_i(1) - \hat Y_i(0)\right)\)

Regression including propensity score

In a world where observations are statistically independent, Imbens and Wooldridge (2009) suggest estimating the following model

\[ \begin{align} &\min_{\alpha_0,\beta_0}\sum_{i:W_i=0}\frac{(Y_i - \alpha_0 - \beta_0'(X_i - \bar X))^2}{\hat p(X_i)}\\ \tag{20} \mbox{and} &\\ &\min_{\alpha_1,\beta_1}\sum_{i:W_i=1}\frac{(Y_i - \alpha_1 - \beta_1'(X_i - \bar X))^2}{1 - \hat p(X_i)} \end{align} \]

• Which can be done in the following steps:

  1. For the entire dataset, estimate propensity score \(\hat p(X_i)\),
  2. For each group, estimate parameters \(\alpha,\beta\).
  3. The ATE will be then \(\tau_{PATE} = \alpha_1 - \alpha_0\)

• They highlight the fact that “[E]ven if the logit model for the propensity score is misspecified, the binary response MLE […] still has a well-defined probaiblity limit” (Imbens and Wooldridge, 2009, p.39) Leading to consistent estimates.

Implementation

1. Probability of treatment They use logit model to estimate propensity scores

   \[ p_{lt}(X_{it}) = \Pr\{W_{ilt} = 1|X_{it}\} = \frac{\exp(\alpha_{lt} + \beta_{lt}X_{it})}{1 + \exp(\alpha_{lt} + \beta_{lt}X_{it})} \]

2. Removing outliers For each individual \(i\) in the set of treated they found a match \(j\) by solving the following problem \(j = \arg\min_{j: W_{ilt}\neq W_{jlt}} |p_{lt}(X_i) - p_{lt}(X_j) |\)

   subject to \(|p_{lt}(X_i) - p_{lt}(X_j) |\leq 2\sigma_d\), where \(\sigma_d = \sqrt{\hat V(|p_{lt}(X_i) - p_{lt}(X_j) |)}\)

3. Pseudo treatment effect Then, out of the entire set of matches, they counted the number of individuals who had adopted the behavior \(n_+\) and who had not \(n_-\) and computed the ratio.

One important point. The authors do not discuss another important assumption that takes place in the Rubin Causal Model. The Stable Unit Treatment Value Assumption (SUTVA) states that the treatment effects have no spill overs (Sekhon 2007). This can be assumed in the dynamic part of the model arguying that any spill overs happend with time lags and thus won’t affect the outcomes at the current time.

References

• Aral, S., Muchnik, L., & Sundararajan, A. (2009). Distinguishing influence-based contagion from homophily-driven diffusion in dynamic networks. Proceedings of the National Academy of Sciences of the United States of America, 106(51), 21544–21549. http://doi.org/10.1073/pnas.0908800106

• Imbens, G. W., & Wooldridge, J. M. (2009). Recent Developments in the Econometrics of Program Evaluation. Journal of Economic Literature, 47(1), 5–86. http://doi.org/10.1257/jel.47.1.5

• Sekhon, J. S. (2008). The Neyman-Rubin Model of Causal Inference and Estimation Via Matching Methods. The Oxford Handbook of Political Methodology. http://doi.org/10.1093/oxfordhb/9780199286546.003.0011

• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). Cambridge: MIT Press.

• LeSage, J. P. (2008). An Introduction to Spatial Econometrics. Revue D’économie Industrielle, 123(123), 19–44. http://doi.org/10.4000/rei.3887