We develop a test for instrument strength based on the bias of two-stage least squares (2SLS) that (1) generalizes Stock and Yogo’s and Sanderson and Windmeijer’s tests to be robust to heteroskedasticity and autocorrelation, and (2) extends Montiel Olea and Pflueger’s robust test for models with a single endogenous regressor to multiple endogenous regressors. Our test can be based either on an absolute bias criterion or on the 2SLS bias relative to a worst-case benchmark. We also develop extensions to test whether weak instruments cause bias in individual 2SLS coefficients. In simulations, our test controls size and is powerful, and we provide efficient code packages for its practical implementation. We demonstrate our testing procedures in the context of the estimation of state-dependent fiscal multipliers, following recent leading estimates.

We show that the financial accelerator may be very large in a liquidity trap. We study a sticky price model with real estate and a financial friction specified as a collateral constraint. Expectations can lead the economy to a self-fulfilling liquidity trap equilibrium where the lower bound on the nominal interest rate binds. We model these equilibria as stochastic sunspots. As in the Great Depression, a liquidity trap entails house price depreciation and potentially large output losses. Higher leverage implies much larger output losses but at the same time rules out the existence of short-lived liquidity traps.

Based on a narrative classification of all significant postwar changes in R&D appropriations for five major federal agencies, we find that an increase in nondefense R&D appropriations leads to increases in various measures of innovative activity and higher business-sector productivity in the long run. We structurally estimate the production function elasticity of nondefense government R&D capital using the SP-IV methodology of Lewis and Mertens (2024) and obtain implied returns of 140 to 210 percent over the postwar period. The estimates indicate that government-funded R&D accounts for one-fifth of business-sector TFP growth since WWII and imply substantial underfunding of nondefense R&D.