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Local projections identify the same policy counterfactuals as empirical and structural models

Endong Wang

Abstract

We study policy counterfactuals that impose path restrictions on a policy instrument over a finite window. Under a sequential intervention design, we define two counterfactual objects, policy-peg impulse responses and policy-path effects, and we provide a novel local projection identification method. Under policy invariance and a linear moving average envelope, the local projection estimands coincide with the counterfactual outcomes implied by empirical vector autoregressions and linearized forward looking structural models, and the counterfactual outcomes are fully characterized by the relevant impulse responses. We also provide local projection identification of both counterfactual objects under an one-shot intervention design. In the empirical applications, we quantify the propagation of an oil-supply news shock under interest-rate pegs and study alternative liftoff paths during the post-pandemic tightening episode.

Local projections identify the same policy counterfactuals as empirical and structural models

Abstract

We study policy counterfactuals that impose path restrictions on a policy instrument over a finite window. Under a sequential intervention design, we define two counterfactual objects, policy-peg impulse responses and policy-path effects, and we provide a novel local projection identification method. Under policy invariance and a linear moving average envelope, the local projection estimands coincide with the counterfactual outcomes implied by empirical vector autoregressions and linearized forward looking structural models, and the counterfactual outcomes are fully characterized by the relevant impulse responses. We also provide local projection identification of both counterfactual objects under an one-shot intervention design. In the empirical applications, we quantify the propagation of an oil-supply news shock under interest-rate pegs and study alternative liftoff paths during the post-pandemic tightening episode.
Paper Structure (29 sections, 2 theorems, 45 equations, 3 figures)

This paper contains 29 sections, 2 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sequential policy counterfactuals
  3. Policy-peg impulse responses
  4. Policy-path effects
  5. Local projection identification
  6. Policy-peg impulse responses
  7. Policy-path responses
  8. Structural interpretation of LP counterfactuals
  9. An SVMA envelope
  10. From policy shocks to policy paths
  11. Counterfactual characterization
  12. Empirical SVAR-based counterfactuals
  13. A forward-looking structural counterfactuals
  14. Rule changes and the Lucas critique
  15. Local projections for one-shot policy counterfactuals
  16. ...and 14 more sections

Key Result

Theorem 4.1

Suppose Assumption ass:structural-env holds. Then $\beta_{h,\mathcal{Q}}=\beta_{h,\mathcal{Q}}^{LP}$ and $\boldsymbol\gamma_{h,\mathcal{Q}}= \boldsymbol\gamma_{h,\mathcal{Q}}^{LP}$.

Figures (3)

  • Figure 1: Policy-peg impulse responses to an oil-supply news shock under a sequential design. The figure reports impulse responses over monthly horizons $h=0,1,\ldots,48$ for, from left to right, WTI oil price, CPI inflation, industrial production, and the federal funds rate, denoted FFR. The first row reports responses to a 10 bps contractionary monetary policy shock. The second row reports responses to a one-standard-deviation oil-supply news shock, normalized so that the impact response of WTI is about 15%. Shaded bands in the first two rows are 68% confidence intervals computed from 5,000 block bootstrap replications with block size 24. Rows 3 to 5 report policy-peg counterfactual responses to the oil-supply news shock under peg windows $q=3$, $q=12$, and $q=24$ months. In these rows, the shaded region marks the intervention window during which the FFR is fixed at its baseline path. The black line reproduces the unrestricted oil-shock impulse response and the colored line reports the corresponding peg counterfactual. The required policy intervention for this peg, rounded to 5 bps increments, is $(-20,-5,0,5)$ over months $0$--$3$, $(0,0,5,0,0,5,0,0,-5)$ over months $4$--$12$, and $(-5,-5,0,0,-5,0,0,0,0,0,0,0)$ over months $13$--$24$.
  • Figure 2: Impulse responses to a 25-basis-point monetary policy shock. The figure reports impulse responses from a structural VAR identified with a high-frequency external instrument, the orthogonal monetary policy shock of bauer2023reassessment. The VAR includes the FFR, the unemployment rate, real GDP, CPI inflation, and the excess bond premium. The lag order is fifteen. The model is estimated over 1992:01--2020:02, 338 monthly observations. Shaded bands are 68% confidence intervals obtained from a block bootstrap with 5,000 replications and block length 24 months.
  • Figure 3: Hypothetical policy paths and implied outcome gaps in the post-pandemic era. The black line reports the historical series. The counterfactual policy path under the three-month intervention is shown in light ochre, and the counterfactual under the six-month intervention is shown in dark amber. The shaded band marks the intervention window beginning in November 2021. The left panel plots the three-month intervention (November 2021 to February 2022), the middle panel plots the six-month intervention (November 2021 to May 2022), and the right panel plots the corresponding deviations from the historical series implied by the two policy-path responses. For the three-month scenario, the required policy shock interventions are approximately $(10,10,40)$ bps, rounded to the nearest $5$ bps. For the six-month scenario, the required shocks are approximately $(10,10,40,25,30,20)$ bps.

Theorems & Definitions (4)

  • Theorem 4.1: Local projection identification of the sequential policy counterfactuals
  • Theorem 5.1: Local projection identification of the one-shot policy counterfactuals
  • proof
  • proof