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How Well Are State-Dependent Local Projections Capturing Nonlinearities?

Zhiheng You

Abstract

We evaluate how well state-dependent local projections recover true impulse responses in nonlinear environments. Using quadratic vector autoregressions as a laboratory, we show that linear local projections fail to capture any nonlinearities when shocks are symmetrically distributed. Popular state-dependent local projections specifications capture distinct aspects of nonlinearity: those interacting shocks with their signs capture higher-order effects, while those interacting shocks with lagged states capture state dependence. However, their gains over linear specifications are concentrated in tail shocks or tail states; and, for lag-based specifications, hinge on how well the chosen observable proxies the latent state. Our proposed specification-which augments the linear specification with a squared shock term and an interaction between the shock and lagged observables-best approximates the true responses across the entire joint distribution of shocks and states. An application to monetary policy reveals economically meaningful state dependence, whereas higher-order effects, though statistically significant, prove economically modest.

How Well Are State-Dependent Local Projections Capturing Nonlinearities?

Abstract

We evaluate how well state-dependent local projections recover true impulse responses in nonlinear environments. Using quadratic vector autoregressions as a laboratory, we show that linear local projections fail to capture any nonlinearities when shocks are symmetrically distributed. Popular state-dependent local projections specifications capture distinct aspects of nonlinearity: those interacting shocks with their signs capture higher-order effects, while those interacting shocks with lagged states capture state dependence. However, their gains over linear specifications are concentrated in tail shocks or tail states; and, for lag-based specifications, hinge on how well the chosen observable proxies the latent state. Our proposed specification-which augments the linear specification with a squared shock term and an interaction between the shock and lagged observables-best approximates the true responses across the entire joint distribution of shocks and states. An application to monetary policy reveals economically meaningful state dependence, whereas higher-order effects, though statistically significant, prove economically modest.
Paper Structure (39 sections, 18 theorems, 206 equations, 9 figures, 1 table)

This paper contains 39 sections, 18 theorems, 206 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Failure of Linear Local Projections
  3. Setup: QAR Model
  4. Why Linear Local Projections Fail
  5. Performance of State-Dependent Local Projections
  6. State-Dependent Local Projections
  7. A Distance Measure for Impulse Responses
  8. Monte Carlo Experiment
  9. Analytic Results
  10. Multivariate Extension: QVAR(1,1)
  11. Estimation and Inference
  12. Empirical Application
  13. Conclusion
  14. Derivation of Structural Functions
  15. QAR(1, 1)
  16. ...and 24 more sections

Key Result

Proposition 1

The CAR for the QAR(1,1) model is where $s$ is the realized value of state $s_{t-1}$, and the state-dependent loading$a_h$ and the higher-order coefficient$q_h$ are

Figures (9)

  • Figure 1: Comparing Impulse Responses: Varying $s$ and $y$
  • Figure 2: Comparing Impulse Responses: Varying $\delta$ and $S$
  • Figure 3: Conditional Distance by $s_{t-1}$ and $u_t$ Bins
  • Figure 4: Conditional MSE Given $s_{t-1}=s$
  • Figure 5: State Dependence of Impulse Responses
  • ...and 4 more figures

Theorems & Definitions (27)

  • Proposition 1
  • Remark 1: Properties of CAR
  • Remark 2: Alternative IRF Concepts for Nonlinear Models
  • Proposition 2
  • Remark 3
  • Proposition 3
  • Remark 4
  • Proposition 4
  • Remark 5
  • Remark 6
  • ...and 17 more