Using the Propensity Score in Regressions for Causal Effects

$$ \begin{aligned} \ \end{aligned} $$

Introduction

• This chapter discusses two simple methods to use the propensity score:

  • the propensity score as a covariate in regressions
  • running regressions weighted by the inverse of the propensity score

• Reasons:

  • they are easy to implement, which involve only standard statistical software packages for regressions;
  • their properties are comparable to many more complex methods;
  • they can be easily extended to allow for flexible statistical models including machine learning algorithms.

Outline

• Regressions with the propensity score as a covariate
  • Theorem 14.1
  • Proposition 14.1

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• Regressions weighted by the inverse of the propensity score
  • Average causal effect
    • Theorem 14.2
  • Average causal effect on the treated units
    • Table 14.1
    • Proposition 14.2
    • Theorem 14.3

Regressions with the propensity score as a covariate

$$ \text { Theorem 11.1 If } Z \perp\!\!\!\perp\{Y(1), Y(0)\} \mid X \text {, then } {\color{red}Z \perp\!\!\!\perp\{Y(1), Y(0)\} \mid e(X)} \text {. } $$

• By Theorem 11.1, if unconfoundedness holds conditioning on $X$, then it also holds conditioning on $e(X)$: $\color{red}{Z \perp\!\!\!\perp\{Y(1), Y(0)\} \mid e(X) }.$
• Analogous to (10.5), $\tau$ is also nonparametrically identified by $$ \tau=E[E\{Y \mid Z=1, e(X)\}-E\{Y \mid Z=0, e(X)\}], $$
• $\Rightarrow$ The simplest regression specification is the OLS fit of $Y$ on $\{1, Z, e(X)\}$, with the coefficient of $Z$ as an estimator, denoted by $\tau_e$: $$ \arg \min _{a, b, c} E\{Y-a-b Z-c e(X)\}^2 $$
• $\tau_e$ defined as the coefficient of $Z$.
• It is consistent for $\tau$ if
  • have a correct propensity score model
  • the outcome model is indeed linear in $Z$ and $e(X)$
• $\tau_e$ estimates $\tau_{\mathrm{O}}$ if we have a correct propensity score model even if the outcome model is completely misspecified

Regressions with the propensity score as a covariate

Theorem 14.1 If $Z \perp\!\!\!\perp\{Y(1), Y(0)\} \mid X$, then the coefficient of $Z$ in the OLS fit of $Y$ on $\{1, Z, e(X)\}$ equals $$ \tau_e=\tau_{\mathrm{O}}=\frac{E\left\{h_{\mathrm{O}}(X) \tau(X)\right\}}{E\left\{h_{\mathrm{O}}(X)\right\}}, $$ recalling that $h_{\mathrm{O}}(X)=e(X)\{1-e(X)\}$ and $\tau(X)=E\{Y(1)-Y(0) \mid X\}$.

$$ \begin{aligned} \ \ \end{aligned} $$

Corollary 14.1 If $Z \perp\!\!\!\perp\{Y(1), Y(0)\} \mid X$, then

• the coefficient of $Z-e(X)$ in the OLS fit of $Y$ on $Z-e(X)$ or $\{1, Z-e(X)\}$ equals $\tau_{\mathrm{O}}$;
• the coefficient of $Z$ in the OLS fit of $Y$ on $\{1, Z, e(X), X\}$ equals $\tau_{\mathrm{O}}$.

Regressions with the propensity score as a covariate

• An unusual feature of Theorem 14.1 is that the overlap condition ($0 < e(x) < 1$) is not needed any more.
• Even if some units have propensity score $e(X)$ equaling 0 or 1, their associate weight $e(X)\{1-e(X)\}$ is zero so that they do not contribute anything to the final parameter $\tau_O$.

Frisch–Waugh–Lovell Theorem

The Frisch–Waugh–Lovell (FWL) theorem reduces multivariate OLS to univariate OLS and therefore facilitate the understanding and calculation of the OLS coefficients.

Theorem A2.2 (sample FWL) With data $\left(Y, X_1, X_2, \ldots, X_p\right)$ containing column vectors, the coefficient of $X_1$ equals the coefficient of $\tilde{X}_1$ in the OLS fit of $Y$ or $\tilde{Y}$ on $\tilde{X}_1$, where

• $\tilde{Y}$ is the residual vector from the OLS fit of $Y$ on $\left(X_2, \ldots, X_p\right)$
• $\tilde{X}_1$ is the residual from the OLS fit of $X_1$ on $\left(X_2, \ldots, X_p\right)$.

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Based on the FWL theorem, we can obtain $\tau_e$ in two steps:

• first, we obtain the residual $\tilde{Z}$ from the OLS fit of $Z$ on ${1, e(X)}$;
• then, we obtain $\tau_e$ from the OLS fit of $Y$ on $\tilde{Z}$.

Proof of Theorem 14.1

The coefficient of $e(X)$ in the OLS fit of $Z$ on $\{1, e(X)\}$ is $$ \begin{aligned} \frac{\operatorname{cov}\{Z, e(X)\}}{\operatorname{var}\{e(X)\}} & =\frac{E[\operatorname{cov}\{Z, e(X) \mid X\}]+\operatorname{cov}\{E(Z \mid X), e(X)\}}{\operatorname{var}\{e(X)\}} \\ &=\frac{0+\operatorname{var}\{e(X)\}}{\operatorname{var}\{e(X)\}}=1, \end{aligned} $$

• the intercept is $E(Z)-E\{e(X)\}=0$
• the residual is $\tilde{Z}=Z-e(X)$ (This makes sense since $Z-e(X)$ is uncorrelated with any function of $X$).

Therefore, we can obtain $\tau_e$ from the univariate OLS fit of $Y$ on $Z-e(X)$ : $$\small{\tau_e=\frac{\operatorname{cov}\{Z-e(X), Y\}}{\operatorname{var}\{Z-e(X)\}}}$$ The denominator simplifies to $$ \begin{aligned} \operatorname{var}\{Z-e(X)\} & =E\{Z-e(X)\}^2 =e(X)+e(X)^2-2 e(X)^2=h_{\mathrm{O}}(X) \end{aligned} $$

Proof of Theorem 14.1

The numerator simplifies to $$ \begin{aligned} & \operatorname{cov}\{Z-e(X), Y\} \\ = & E[\{Z-e(X)\} Y] \\ = & E[\{Z-e(X)\} Z Y(1)]+E[\{Z-e(X)\}(1-Z) Y(0)] \\ & \quad \quad \quad{\color{red}(\text { since } Y=Z Y(1)+(1-Z) Y(0))} \\ = & E[\{Z-Z e(X)\} Y(1)]-E[e(X)(1-Z) Y(0)] \\ = & E[Z\{1-e(X)\} Y(1)]-E[e(X)(1-Z) Y(0)] \\ = & E\left[e(X)\{1-e(X)\} \mu_1(X)\right]-E\left[e(X)\{1-e(X)\} \mu_0(X)\right] \\ & \quad \quad \quad\text { {\color{red}(tower property and ignorability)} } \\ = & E\left\{h_{\mathrm{O}}(X) \tau(X)\right\} . \end{aligned} $$

• From the proof of Theorem 14.1, we can simply run the OLS of $Y$ on the centered treatment $\tilde{Z} = Z - e(X)$.
• Moreover, we can also include $X$ in the OLS fit which may improve efficiency in finite sample.

Comments of Theorem 14.1 and Corollary 14.1

• Theorem 14.1 motivates a two-step estimator for $\tau_{\mathrm{O}}$:
  • first, fit a propensity score model to obtain $\hat{e}\left(X_i\right)$;
  • second, run OLS of $Y_i$ on $\left(1, X_i, \hat{e}\left(X_i\right)\right)$ to obtain the coefficient of $Z_i$.
• Corollary 14.1 motivates another two-step estimator for $\tau_{\mathrm{O}}$:
  • first, fit a propensity score model to obtain $\hat{e}\left(X_i\right)$;
  • second, run OLS of $Y_i$ on $Z_i-\hat{e}\left(X_i\right)$ to obtain the coefficient of $Z_i$.

Remark: OLS is convenient for obtaining point estimators, the corresponding standard errors are incorrect due to the uncertainty in the first step estimation of the propensity score. We can use the bootstrap to approximate the standard errors.

Regressions weighted by the inverse of the propensity score

We first re-examine the Hajek estimator of $\tau$ : $$ \hat{\tau}^{\text {hajek }}=\frac{\sum_{i=1}^n \frac{Z_i Y_i}{\hat{e}\left(X_i\right)}}{\sum_{i=1}^n \frac{Z_i}{\hat{e}\left(X_i\right)}}-\frac{\sum_{i=1}^n \frac{\left(1-Z_i\right) Y_i}{1-\hat{e}\left(X_i\right)}}{\sum_{i=1}^n \frac{1-Z_i}{1-\hat{e}\left(X_i\right)}}, $$

• which equals the difference between the weighted means of the outcomes in the treatment and control groups.
• Numerically, it is identical to the coefficient of $Z_i$ in the following weighted least squares (WLS) of $Y_i$ on $\left(1, Z_i\right)$.

Proposition 14.1 $\hat{\tau}^{\text {hajek }}$ equals $\hat{\beta}$ from the following $WLS$ :

$$ (\hat{\alpha}, \hat{\beta})=\arg \min_{\alpha, \beta} \sum_{i=1}^n w_i\left(Y_i-\alpha-\beta Z_i\right)^2 $$

with weights $$ w_i=\frac{Z_i}{\hat{e}\left(X_i\right)}+\frac{1-Z_i}{1-\hat{e}\left(X_i\right)}= \begin{cases}\frac{1}{\hat{e}\left(X_i\right)} & \text { if } Z_i=1 \\ \frac{1}{1-\hat{e}\left(X_i\right)} & \text { if } Z_i=0\end{cases} $$

Average causal effect

• By Proposition 14.1, it is convenient to obtain $\hat{\tau}^{\text {hajek }}$ based on WLS.
• However, due to the uncertainty in the estimated propensity score, the standard error reported by WLS is incorrect for the true standard error $\Rightarrow$ bootstrap

• Why does the WLS give a consistent estimator for $\tau$ ?

• Recall that in the CRE with a constant propensity score, we can simply use the coefficient of $Z_i$ in the OLS fit of $Y_i$ on $(1, Z_i)$ to estimate $\tau$.
• In observational studies, units have different probabilities of receiving the treatment and control, respectively.
• If we weight the treated units by $1 / e(X_i)$ and the control units by $1 /\{1-e(X_i)\}$, then they can represent the whole population and we effectively have a pseudo randomized experiment.
• Consequently, the difference between the weighted means are consistent for $\tau$.
• The numerical equivalence of $\hat{\tau}^{\text {hajek }}$ and WLS is not only a fun numerical fact itself but also useful for motivation more complex estimator with covariate adjustment.

An extension to a more complex estimator

• In the CRE, we can use the coefficient of $Z_i$ in the OLS fit of $Y_i$ on $(1, Z_i, X_i, Z_i X_i)$ to estimate $\tau$, where the covariates are centered with $\bar{X}=0$.
• This is Lin (2013)’s estimator which uses covariates to improve efficiency.
• A natural extension to observational studies is to estimate $\tau$ using the coefficient of $Z_i$ in the WLS fit of $Y_i$ on $\left(1, Z_i, X_i, Z_i X_i\right)$ with weights defined in (14.1).
• If the linear models $$ E(Y \mid Z=1, X)=\beta_{10}+\beta_{1 x}^{\top} X, \quad E(Y \mid Z=0, X)=\beta_{00}+\beta_{0 x}^{\top} X, $$ are correctly specified, then both OLS and WLS give consistent estimators for the coefficients and the estimators of the coefficient of $Z$ is consistent for $\tau$.
• More interestingly, the estimator of the coefficient of $Z$ based on WLS is also consistent for $\tau$ if the propensity score model is correct and the outcome model is incorrect. $\Rightarrow$ the estimator based on WLS is doubly robust.

A doubly robust estimator

• Let $\hat{e}(X_i)$ be the fitted propensity score and $(\mu_1(X_i, \hat{\beta}_1), \mu_0(X_i, \hat{\beta}_0))$ be the fitted values of the outcome means based on the WLS.
• The outcome regression estimator is

$$ \hat{\tau}_{\mathrm{wls}}^{\mathrm{reg}}= \frac{1}{n}\sum_{i=1}^n\mu_1\left(X_i, \hat{\beta}_1\right)-\frac{1}{n} \sum_{i=1}^n \mu_0\left(X_i, \hat{\beta}_0\right) $$

• The doubly robust estimator for $\tau$ is

$$ \hat{\tau}_{\mathrm{wls}}^{\mathrm{dr}}=\hat{\tau}_{\mathrm{wls}}^{\mathrm{reg}}+\frac{1}{n} \sum_{i=1}^n \frac{Z_i\left\{Y_i-\mu_1\left(X_i, \hat{\beta}_1\right)\right\}}{\hat{e}\left(X_i\right)}-\frac{1}{n} \sum_{i=1}^n \frac{\left(1-Z_i\right)\left\{Y_i-\mu_0\left(X_i, \hat{\beta}_0\right)\right\}}{1-\hat{e}\left(X_i\right)} . $$

• An interesting result is that this doubly robust estimator equals the outcome regression estimator, which reduces to the coefficient of $Z_i$ in the WLS fit of $Y_i$ on $\left(1, Z_i, X_i, Z_i X_i\right)$ if we use weights (14.1).

Theorem 14.2

If $\bar{X}=0$ and $(\mu_1(X_i, \hat{\beta}_1), \mu_0(X_i, \hat{\beta}_0))=(\hat{\beta}_{10}+\hat{\beta}_{1 x}^{\top} X_i, \hat{\beta}_{00}+\hat{\beta}_{0 x}^{\top} X_i)$ based on the WLS fit of $Y_i$ on $\left(1, Z_i, X_i, Z_i X_i\right)$ with weights (14.1), then

$$ \hat{\tau}_{\mathrm{wls}}^{\mathrm{dr}}=\hat{\tau}_{\mathrm{wls}}^{\mathrm{reg}}=\hat{\beta}_{10}-\hat{\beta}_{00}, $$

which is the coefficient of $Z_i$ in the WLS fit.

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• Freedman and Berk (2008) showed that when the outcome model is correct, the WLS estimator is worse than the OLS estimator
• When the errors have variance proportional to the inverse of the propensity scores, the WLS estimator will be more effcient than the OLS estimator.
• The estimated standard error based on the WLS fit is not consistent for the true standard error because it ignores the uncertainty in the estimated propensity score.
• This can be easily fixed by using the bootstrap to approximate the variance of the WLS estimator.
• Nevertheless, they found that “weighting may help under some circumstances” because when the outcome model is incorrect, the WLS estimator is still consistent if the propensity score model is correct.

Average causal effect on the treated units

Proposition 14.2 $\hat{\tau}_{\mathrm{T}}^{\text {hajek }}$ is numerically identical to $\hat{\beta}$ in the following WLS: $$ (\hat{\alpha}, \hat{\beta})=\arg \min_{\alpha, \beta} \sum_{i=1}^n w_{\mathrm{T} i}\left(Y_i-\alpha-\beta Z_i\right)^2 $$ with weights $$ w_{\mathrm{T} i}=Z_i+\left(1-Z_i\right) \hat{o}\left(X_i\right)= \begin{cases}1 & \text { if } Z_i=1 \\ \hat{o}\left(X_i\right) & \text { if } Z_i=0\end{cases} $$ where $\hat{o}\left(X_i\right)=\hat{e}\left(X_i\right) /\{1-\hat{e}\left(X_i\right)\}$

Regression estimators

Average causal effect on the treated units

• If we center covariates with $\hat{X}(1)=0$, then we can estimate $\tau_{\mathrm{T}}$ using the coefficient of $Z_i$ in the WLS fit of $Y_i$ on $\left(1, Z_i, X_i, Z_i X_i\right)$ with weights defined in (14.2). Similarly, this estimator equals the regression estimator

$$ \hat{\tau}_{\mathrm{T}, \mathrm{wls}}^{\mathrm{reg}}=\hat{\bar{Y}}(1)-\frac{1}{n_1} \sum_{i=1}^n Z_i \mu_0\left(X_i, \hat{\beta}_0\right) $$

which also equals the doubly robust estimator

$$ \hat{\tau}_{\mathrm{T}, \mathrm{wls}}^{\mathrm{dr}}=\hat{\tau}_{\mathrm{T}, \mathrm{wls}}^{\mathrm{reg}}-\frac{1}{n_1} \sum_{i=1}^n \hat{o}\left(X_i\right)\left(1-Z_i\right)\left\{Y_i-\mu_0\left(X_i, \hat{\beta}_0\right)\right\}. $$

Theorem 14.3 If $\hat{\bar{X}}(1)=0$ and $\mu_0(X_i, \hat{\beta}_0)=\hat{\beta}_{00}+\hat{\beta}_{0x}^{\top} X_i$ based on the WLS fit of $Y_i$ on $(1, Z_i, X_i, Z_i X_i)$ with weights (14.2), then

$$ \hat{\tau}_{\mathrm{T}, \mathrm{wls}}^{\mathrm{dr}}=\hat{\tau}_{\mathrm{T}, \mathrm{wls}}^{\mathrm{reg}}=\hat{\beta}_{10}-\hat{\beta}_{00}, $$

which is the coefficient of $Z_i$ in the $W L S$ fit.