Would it be reasonable to attempt to use a propensity score as an instrumental variable? Would a propensity score be valid as an instrumental variable in a quasi experimental context?
I've seen papers that explore the question from the opposite direction: can an instrumental variable be used in the calculation of propensity scores. Thanks.
 A: No.
Let's say we have treatment $T$, outcome $Y$, and some other variables $X$. The propensity score is then defined as
$$p(x) = P(T = 1|X = x)$$
This is first and foremost a purely statistical construct which has no direct connection to causality. Causality comes into the picture if we assume "ignorability":
$$P(Y_{t}|T, X) = P(Y_{t}|X)$$
where $Y_{t}$ is the potential outcome of $Y$ when $T$ is set to $t$. This holds for example when the back-door criterion holds on the causal graph you assume (i.e., $X$ blocks all back-door paths from $T$ to $Y$). 
Rubin and Rosenbaum argued that it will then hold that 
$$P(Y_{t}|T, p(X)) = P(Y_{t}|p(X))$$
which makes estimation easier since one does not need to deal with the potentially high-dimensional $X$, but simply $p(x)$, which is a single number.
For instrumental variable analysis, you are looking for a $X$ such that at the very least it holds that ("instrument independence")
$$P(Y_{t}|X) = P(Y_{t})$$
(usually, you assume more, but let's ignore this). However, typically, $X$ that fulfill the "ignorability" assumption above contains variables that influence both $T$ and $Y$. Influencing $Y$ would violate instrument independence. Since the propensity score is simply a function of these $X$, they also will violate instrument independence.
Literature:
Pearl, Judea: Understanding propensity scores. In: Causality, 2nd ed. CUP.
A: 3 questions need to pay attention when using propensity score (PS) as covariate in the regression model. In the practice, PS is got from logistic regression. For example: the linear model. 
1: $Y=\beta_0 + \beta_1 T + \beta_2P +\epsilon$ where $T$ is treatment and $P$ is PS. Because $P=\frac {\exp(X\beta)}{1+\exp(X\beta)}$, we have  $Y=\beta_0 + \beta_1 T + \beta_2\frac {\exp(X\beta)}{1+\exp(X\beta)} +\epsilon$. The question is: Why the response variable $Y$ has this strange relation with covariate $X$? Why we need this limitation?
2: Should we consider the variance of $P$, because $P$ is estimated from data, not observed true value?
3: If $X$ has high dimensional and/or co-linearity problem when the $X$ is used in linear model directly, $X$ has high dimensional and/or co-linearity problem in the logistic regression when you try to get PS. Is it possible to resolve this problem in logistic regression?  
