Estimating DSGE models with zero interest rate policy

Once the policy rate is fixed at (or near) zero for an extended period, it stops varying as an observable, and the rational-expectations solution the researcher would otherwise fit — a single, time-invariant law of motion linking today’s state to yesterday’s state and expected shocks — no longer describes the data-generating process during that period, so a standard Kalman filter built around that one solution breaks down. The paper’s answer is to explicitly model a second regime, indexed by the current expected duration of the fixed-rate episode, and to estimate that expected duration itself as a parameter rather than assuming the researcher already knows how long agents expected the ZLB to last.

The expected duration of the fixed-rate regime, d^e_t, is treated as a discrete, time-varying free parameter representing how many periods agents currently expect the policy rate to remain fixed, and it is estimated jointly with the model’s structural parameters θ rather than calibrated or backed out from other shocks. Because the fixed-rate structural equations differ from the conventional-policy equations, the rational-expectations solution during the ZLB is not a single fixed matrix but a sequence of time-varying reduced-form matrices {C_t, Q_t, G_t} for t = 1,…,d, computed by backward recursion starting from the terminal (post-liftoff) conventional-policy solution; at each ZLB period, the matrices corresponding to the currently estimated expected duration are the ones that generate that period’s law of motion. The authors note this solution method is a special case of Kulish and Pagan (2017).

The federal funds rate is dropped from the set of observables during the ZLB (rather than treated as a zero-variance measurement), and the prior on expected duration is an equally weighted mixture of a uniform distribution and a survey-implied distribution built from the Federal Reserve Bank of New York’s Survey of Primary Dealers (available from 2011) and Blue Chip Financial Forecasts (March 2009-December 2010), with a maximum possible duration of 23 quarters. Estimation uses a two-block Metropolis-Hastings sampler: one block updates the (randomly blocked) sequence of integer-valued expected durations via an independent uniform proposal, and the other updates the structural parameters via a random-walk multivariate Student-t proposal centered on the pre-ZLB likelihood mode. The main chain runs 2 million draws (first half discarded as burn-in), with two additional chains used to check convergence via Brooks-Gelman-Rubin diagnostics, which the authors report show no evidence of non-convergence.

Estimated on a sample including the ZLB, the posterior mode implies a less aggressive monetary policy response to inflation (ψ₁ = 1.62 versus 2.04 in Smets-Wouters), a flatter Phillips curve (higher price-Calvo parameter, 0.91 versus 0.66), weaker consumption habits (0.44 versus 0.71), and, notably, much higher persistence of the risk-premium shock (ρ_b = 0.95 versus 0.22) and somewhat higher persistence of the monetary policy shock (ρ_m = 0.43 versus 0.15). The authors describe the overall set of structural estimates as “generally in line with previous findings,” while flagging the risk-premium persistence as the most economically consequential departure, since it is this shock’s persistence and size that later interact with the estimated expected-duration path.

The posterior mean of expected duration is short — around a year or less — through 2009, well below the roughly two-year prior mean and in line with survey data at the time, then rises sharply in 2011Q3 following the Federal Reserve’s first calendar-based forward guidance announcement (August 2011), with the posterior distribution shifting from mass concentrated below two years to mass concentrated above two years; the posterior mean subsequently peaks around 2012 and declines again with the 2013 “taper tantrum.” Throughout, the posterior standard deviation is considerably smaller than the prior’s, which the authors interpret as evidence that expected durations are reasonably well identified by the data, and there remains non-negligible posterior weight on durations of three to four years, consistent with the right-skewed distribution that Bauer and Rudebusch (2016) find using shadow-rate methods.

The model backs out two shadow rates — an unconditional shadow rate reflecting what the Taylor rule would prescribe from fundamentals alone, and a conditional shadow rate that also depends on the lagged observed rate and measures how binding the ZLB currently is — and finds the conditional shadow rate’s posterior mean is negative in every quarter since 2009, with the ZLB estimated to be binding with probability close to one in 17 of the 22 ZLB quarters. In the remaining quarters there is non-negligible (15-20 percent) posterior probability that the constraint was not binding, but even in the clearest case (2012Q1, with an 18.9 percent probability of a non-binding constraint) the average shadow rate over that range is only about 7 basis points above zero, which the authors describe as “quantitatively small” slack. The unconditional shadow rate is more negative than the conditional one throughout, reflecting the pull of fundamentals such as the large output gap and below-target inflation.

In a counterfactual that keeps the same estimated structural shocks but lets the (estimated) Taylor rule set negative interest rates instead of being constrained at zero, cumulative losses over the 22 ZLB quarters relative to the constrained path are large: 45.3 percent for output, 45.2 percent for consumption, and 97.9 percent for investment, with the 5-year yield roughly 1.5 percentage points lower in the unconstrained counterfactual. The corresponding difference in inflation is “much more modest,” a pattern the authors read as consistent with the recession being driven mainly by a risk-premium shock — in the Del Negro et al. (2015) interpretation, a shock that behaves like a hit to financial-sector net worth, producing a sharp real contraction alongside only a modest inflation decline. The authors summarize the overall conclusion as that the ZLB “placed a significant constraint on monetary policy, exacerbated the recession and delayed the recovery,” consistent with the independent finding of Gust et al. (2017).

Comparing conditional forecasts made as of 2011Q3 under a 4-quarter versus a 16-quarter expected duration, a longer expected duration brings forward higher near-term growth in output, consumption, investment, and hours (with the paths crossing around the time of eventual liftoff, since the longer-duration scenario implies lower long-term rates now but a stronger economy — and hence a faster subsequent rate rise — at liftoff), while expected duration has comparatively less effect on inflation. The reason the model does not produce the implausibly large responses to long expected durations documented as the “forward guidance puzzle” by Carlstrom et al. (2015) and Del Negro et al. (2012) is that, in the model’s Euler equation and investment (Tobin’s q) equation, the relevant real-rate term also embeds the risk-premium shock b̂_t; draws in which the expected duration is long also tend to carry a larger and more persistent negative risk-premium shock, which offsets the expansionary pull of the longer expected duration.

Constructing a generalized impulse response (following Koop et al. 1996) around the observed 2013Q2 fall in expected duration from roughly 10 to roughly 5 quarters — corresponding to the Federal Reserve’s surprise tapering announcement — the model implies an average cumulative increase in the federal funds rate path of 292 basis points and an immediate roughly 10-basis-point increase in the 5-year yield (in line with the observed market reaction), alongside negative median responses in output, consumption, and investment, a prolonged fall in hours, and only a “relatively small” fall in inflation. The authors are explicit about two limitations of the overall framework: it does not penalize estimated sequences of expected durations that would imply the ZLB should actually have been violated (an occasionally-binding-constraint check the model does not impose), and it assumes a single, common expected duration shared by all agents at each date rather than allowing for heterogeneous beliefs about the liftoff date.