Essays on weak identification in high-dimensional models with applications in macroeconomics

This thesis consists of four self-contained chapters. Chapter 2 (co-authored with Prof. Sophocles Mavroeidis and Prof. Anders Kock) considers hypothesis testing in instrumental variable (IV) regression models with few included exogenous covariates but many instruments--possibly more than the number of observations. We look for a method of inference that controls asymptotic size when there is heteroskedasticity and the instruments may be arbitrarily weak. We show that a ridge-regularised version of the jackknifed Anderson Rubin (1949, henceforth AR) test achieves this objective. This test weakens the assumptions needed for recently proposed jackknifed AR tests, and extends their scope to situations in which there are more instruments than observations. Monte-Carlo simulations indicate that our method has favourable finite-sample size and power properties compared to recently proposed alternative approaches in the literature. An empirical application on the elasticity of substitution between immigrants and natives in the US illustrates the usefulness of the proposed method for practitioners. Chapter 3 considers limited-information inference on New Keynesian Phillips Curves (NKPCs) in the presence of weak and high-dimensional IVs. Beyond the efficiency concerns previously raised in the literature, I show by simulation that ad-hoc selection procedures can lead to substantial biases in post-selection inference. I propose a Sup Score test that remains valid under dependent data, arbitrarily weak identification, and a number of IVs that increases exponentially with the sample size. Conducting inference on a standard NKPC with 361 IVs and 179 observations, I find substantially wider confidence sets than those commonly found. Chapter 4 (co-authored with Dr. Gerrit Koester and Dr. Christiane Nickel) addresses the endogeneity of slack in Phillips Curves in reduced-form contexts. Endogeneity of the labour market slack in reduced-form Phillips Curves is usually addressed either by including proxies for omitted supply shocks, or by using instrumental variables. Using the Kiviet (2022) Kinky Least Squares estimator, we find evidence that supply-shock proxies should not be omitted from PCs, and that many popular instrumental variables seem to be invalid. We estimate a standard backward-looking wage Phillips Curve by Kinky Least Squares and find that unless a large negative correlation between the slack variable and the error term is assumed, the coefficient of the slack variable is significantly negative. Chapter 5 provides a weak-identification robust IV method for identifying the structural covariance matrix in structural vector autoregressions (SVARs) where data for the reduced-form VAR are available over a longer horizon than for the IV. I apply this method to analyse the effect of US monetary shocks on real and financial variables in the US and other countries. While the effects on US real and financial variables are qualitatively similar to those reported previously in the literature, I find little evidence for US monetary policy spillovers in countries other than the US, in line with the theoretical predictions of the Mundellian Trilemma.