We are interested in estimating the average e#ect of a binary treatment on a scalar outcome. If assignment to the treatment is independent of the potential outcomes given pretreatment variables, biases associated with simple treatment-control average comparisons can be removed by adjusting for di#erences in the pre-treatmentvariables. Rosenbaum and Rubin #1983, 1984# show that adjusting solely for di#erences between treated and control units in a scalar function of the pre-treatment variables, the propensity score, also removes the entire bias associated with di#erences in pre-treatment variables. Thus it is possible to obtain unbiased estimates of the treatment e#ect without conditioning on a possibly highdimensional vector of pre-treatment variables. Although adjusting for the propensity score removes all the bias, this can come at the expense of e#ciency. We show that weighting with the inverse of a nonparametric estimate of the propensity score, rather than the true propensity scor...

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ABSTRACT: I describe two new methods for estimating the optimal treatment regime (equivalently, protocol, plan or strategy) from very high dimesional observational and experimental data: (i) g-estimation of an optimal double-regime structural nested mean model (drSNMM) and (ii) g-estimation of a standard single regime SNMM combined with sequential dynamicprogramming (DP) regression. These methods are compared to certain regression methods found in the sequential decision and reinforcement learning literatures and to the regret modelling methods of Murphy (2003). I consider both Bayesian and frequentist inference. In particular, I propose a novel “Bayes-frequentist compromise ” that combines honest subjective non- or semiparametric Bayesian inference with good frequentist behavior, even in cases where the model is so large and the likelihood function so complex that standard (uncompromised) Bayes procedures have poor frequentist performance. 1

We address the problem that occurs when inferences about counterfactuals—predictions, ‘‘what-if’ ’ questions, and causal effects—are attempted far from the available data. The danger of these extreme counterfactuals is that substantive conclusions drawn from statistical models that fit the data well turn out to be based largely on speculation hidden in convenient modeling assumptions that few would be willing to defend. Yet existing statistical strategies provide few reliable means of identifying extreme counterfactuals. We offer a proof that inferences farther from the data allow more model dependence and then develop easyto-apply methods to evaluate how model dependent our answers would be to specified counterfactuals. These methods require neither sensitivity testing over specified classes of models nor evaluating any specific modeling assumptions. If an analysis fails the simple tests we offer, then we know that substantive results are sensitive to at least some modeling choices that are not based on empirical evidence. Free software that accompanies this article implements all the methods developed. 1

Inferences about counterfactuals are essential for prediction, answering ‘‘what if ’ ’ questions, and estimating causal effects. However, when the counterfactuals posed are too far from the data at hand, conclusions drawn from well-specified statistical analyses become based on speculation and convenient but indefensible model assumptions rather than empirical evidence. Unfortunately, standard statistical approaches assume the veracity of the model rather than revealing the degree of model-dependence, so this problem can be hard to detect. We develop easy-to-apply methods to evaluate counterfactuals that do not require sensitivity testing over specified classes of models. If an analysis fails the tests we offer, then we know that substantive results are sensitive to at least some modeling choices that are not based on empirical evidence. We use these methods to evaluate the extensive scholarly literatures on the effects of changes in the degree of democracy in a country (on any dependent variable) and separate analyses of the effects of UN peacebuilding efforts. We find evidence that many scholars are inadvertently drawing conclusions based more on modeling hypotheses than on evidence in the data. For some research questions, history contains insufficient information to be our guide. Free software that accompanies this paper implements all our suggestions. Social science is about making inferencesFusing facts we know to learn about facts we do not know. Some inferential targets (the facts we do not know) are factual, which means that they exist even if we do not know them. In early 2003, Saddam Hussein was obviously either alive or dead, but the world did not know which it was

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The Centre for Market and Public Organisation (CMPO) is a leading research centre, combining expertise in economics, geography and law. Our objective is to study the intersection between the public and private sectors of the economy, and in particular to understand the right way to organise and deliver public services. The Centre aims to develop research, contribute to the public debate and inform policy-making. CMPO, now an ESRC Research Centre was established in 1998 with two large

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tex June 8, 2000 probability 1: (1.2) for all > t and all t with t (t) = (). This assumption essentially says that the future does not determine the past. The observed data are linked to the counterfactual data by (1.3) Eq. (1.3) states that a subject's observed outcome history through end of follow-up'is equal to their counterfactual outcome history corresponding to the treatment they did indeed receive.' We assume - = (-, ) where is an outcome process of interest and. is the process of other recorded variables. Robins (1987) considers the sequential randomination assumption that for all l and 4, () II- () I z (-),[ (-) (1.4) where for any variable _Z () = {Z (u); u _> } is the history of that .variable from t onwards. We also refer to (1.4) as the assumption of no mmeasured confounders given prognostic factors L (), Because of measurability issues, (1.4) is not well-defined. If the A (t) process can only jump at discrete non-random times ,2,.-- and .the () process has left-hand limits, i.e., (-) = llm(u), (1.4) is formally, for each .f [. () I Z (t['),] (['),()] = .f [.(t) I Z (['),] ([')] (1.5) where f (. I ') is the conditional density of A (t) with respect to a dominating measure p (.). If A (t) is a marked point process that can jump in continuous time with CADLAG (continuous from the right with left-hand limits) step-function sample paths, then Eq. (1.4) is formally that ,x [IZ (-), (-),()] = ,x [ I Z(-),(?)] (Z.a) J-he 8, 2000 155 .r[ () I Z (-),] (-), t () - t (-),()] = .r [() I z (-),] (-),() - (-)] (1.b) Here, the intensity process ), (tl.) is _m Fr[A (t + fit) A (t-) I A (-), .I/at. Eq. (1.6a) says that given past treatment and confounder historyl the probability that the A process jumps at t does not depend on the future counterfact...