The trade-off between the efficiency of an instrumental variable and its exogeneity is widely recognized but little understood. This paper specifies the terms of that trade-off by analyzing the asymptotic mean squared errors associated with the instrumental variables estimator when the instrument may not be perfectly exogenous. The analysis shows that even seemingly minor misspecifications can play havoc with statistical inferences based on "quasi-instrumental variable" estimators. Simple rules of thumb are derived by which intuition can be applied to choices among alternative estimators based on different instrumental variables, or between instrumental variable and ordinary least squares estimators. The theoretical analysis is applied to an example drawn from Jacobson's (1990) and Green and Krasno's (1990) work on congressional campaign spending and is bolstered by Monte Carlo simulations that, for the most part, reproduce the patterns of errors predicted by the asymptotic results.