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Estimate a fiscal multiplier from an instrumented shock

A multiplier is dollars of output per dollar of spending. It is not an impulse response, and the difference is where most of the applied literature's confusion lives. tsecon.lp_multiplier implements the Ramey-Zubairy (2018) one-step integral estimator: at each horizon it regresses the cumulated outcome on the cumulated impulse, instrumented by the contemporaneous shock. Both sides accumulate over the same window, so the coefficient is a ratio of dollars to dollars.

The recipe

import numpy as np, tsecon

rng = np.random.default_rng(11)                    # 80 years of quarterly data
n = 320
news = rng.standard_normal(n)                      # the exogenous shock (e.g. military news)
u = np.zeros(n)
g = np.zeros(n)
for t in range(1, n):
    u[t] = 0.5 * u[t - 1] + rng.standard_normal()  # business-cycle noise
    g[t] = 0.7 * g[t - 1] + news[t] + 0.8 * u[t]   # spending also REACTS to the cycle
y = 1.2 * g + u                                    # true multiplier = 1.2

m = tsecon.lp_multiplier(y, g, news, horizons=8, n_lag_controls=4)
for h in (0, 2, 4, 6, 8):
    print(f"h={h}   multiplier {m['multiplier'][h]:+.3f}   se {m['se'][h]:.3f}"
          f"   first-stage F {m['first_stage_f'][h]:7.1f}")

h=0   multiplier +1.274   se 0.054   first-stage F   454.2
h=2   multiplier +1.267   se 0.055   first-stage F   120.2
h=4   multiplier +1.255   se 0.050   first-stage F    76.5
h=6   multiplier +1.224   se 0.061   first-stage F    47.1
h=8   multiplier +1.203   se 0.079   first-stage F    24.6

Reading it

multiplier[h] is the cumulative multiplier through horizon h: 1.20 at h=8 recovers the true 1.2 built into the simulation. se[h] is the kernel-HAC standard error of that single 2SLS coefficient — inference on the multiplier itself, not a delta-method ratio of two impulse responses and not one leg's standard error relabelled. Build a band the usual way: multiplier ± 1.96 · se.

first_stage_f is not optional reading. It falls from 454 at h=0 to 25 at h=8 — instruments always weaken as the accumulation window grows, and a multiplier estimated off a weak first stage is a number with a confidence interval you cannot trust. The conventional F > 10 rule of thumb is a floor, not a target. nobs_per_h tells you how much sample each horizon actually used.

Why not just run OLS on the cumulated pair?

Because spending responds to the cycle, so cumulated g is correlated with the cumulated error:

cy = np.cumsum(y)[4:]                              # what OLS on the same cumulated pair says
cg = np.cumsum(g)[4:]
naive = tsecon.ols(cy, np.column_stack([np.ones(cy.size), cg]), se_type="hac")
print(f"OLS on cumulated y and g : {naive['params'][1]:+.3f}   (true 1.2)")
print(f"LP-IV at h=4             : {m['multiplier'][4]:+.3f}")

OLS on cumulated y and g : +1.375   (true 1.2)
LP-IV at h=4             : +1.255

OLS overstates the multiplier by 15% here because it attributes to spending the part of output that was driving spending. The instrument is what buys you the causal reading; the accumulation is what makes it a multiplier.

The trap this function exists to avoid

The other route people take is tsecon.lp_iv(..., cumulative=True) and then dividing the peak outcome response by the peak impulse response. That ratio is not a multiplier: the two peaks generally occur at different horizons, and the ratio of two estimated responses has no standard error you can read off either regression. Ramey and Zubairy's point is that the one-step estimator gets both the estimand and the inference right. Use lp_multiplier when you want dollars per dollar; use lp_iv when you want a response path.

Gotchas

  • n_lag_controls controls both series. It adds that many lags of the outcome and the impulse; too few and the shock is not exogenous conditional on the controls.
  • The instrument must be contemporaneous with the impulse, and exogenous. Feeding a raw endogenous variable gives you a correlation with an IV-shaped standard error.
  • Units matter. If your data are in logs, convert to a dollar scale (the usual sample mean of Y/G rescaling) before calling this the multiplier.

See also