Continuous-time Impulse Response Functions with functional approaches and mixed-frequency data

The impulse response function (IRF) characterizes how a given structural shock propagates over time through the economy. To measure the impact of a high-frequency shock on low-frequency macroeconomic aggregates, the usual approach relies on aggregating all data at the lowest frequency, potentially leading to a loss of information. This paper proposes a novel concept to measure this macroeconomic impact directly at high frequency, without requiring aggregation: the continuous-time IRF (CT-IRF). We express the response of the low-frequency target variable yt to the input signal as a convolution integral between the impulse-response and the input signal. Our approach is similar in spirit to local projections, with the key difference being that we can construct the entire IRF in a single step and handle mixed frequencies. The estimation problem is an ill-posed inverse problem, which we address using a penalized least squares estimator with a Sobolev-type penalty. We derive the convergence rate of our estimator as the sample size T grows and demonstrate its excellent finite-sample performance via a Monte Carlo study. Finally, we apply our method to estimate highfrequency IRFs of quarterly U.S. business investment to uncertainty shocks and of U.S. GDP growth rate to financial shocks.