In many predictive decision-making scenarios, such as credit scoring and academic testing, a decision-maker must construct a model that accounts for agents' incentives to ``game'' by changing their features to receive better decisions. Whereas the strategic classification literature has previously assumed that agents' outcomes are not causally dependent on their features (and thus strategic behavior is a form of lying), we join concurrent work in modeling agents' outcomes as a function of their changeable attributes. Our work introduces the realizable linear regression setting, and is the first to incorporate a crucial phenomenon: when agents act to change observable features, they may as a side effect perturb hidden features that causally affect their true outcomes. As our main contribution, we provide the efficient algorithms for optimizing three distinct decision-making objectives: accurately predicting agents' post-gaming outcomes (prediction risk minimization), incentivizing agents to improve these outcomes (agent outcome maximization), and estimating the coefficients of the true underlying model (parameter estimation). Our algorithms circumvent the hardness result of Miller et al. (2020) by allowing the decision maker to test a sequence of decision rules and observe agents' responses, in effect performing causal interventions by varying the chosen rule.