BIS Working Papers
No 715
A time series model of interest rates with the effective lower bound
by Benjamin K. Johannsen and Elmar Mertens
Monetary and Economic Department
April 2018
JEL classification: C32, C34, C53, E43, E47
Keywords: Shadow Rate, Effective Lower Bound, Trend Real Rate, Monetary Policy Shocks, Bayesian Time Series
This publication is available on the BIS website (www.bis.org).
© Bank for International Settlements 2017. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.
ISSN 1020-0959 (print)
ISSN 1682-7678 (online)
A Time Series Model of Interest Rates With the Effective Lower Bound
Benjamin K. Johannsen Federal Reserve Board
Elmar Mertens Bank for International Settlements
Abstract
Modeling nominal interest rates requires their effective lower bound (ELB) to be taken into account. We propose a flexible time series approach that includes a “shadow rate” — a notional rate identical to the actual nominal rate except when the ELB binds. We apply this approach to a trend-cycle decomposition of interest rates and macroeconomic variables that generates competitive interest-rate forecasts. Our estimates of the real-rate trend have edged down somewhat in recent decades, but not significantly so. We identify monetary policy shocks from shadow-rate surprises and find that they were particularly effective at stimulating economic activity during the ELB period.
JEL Classification Numbers: C32, C34, C53, E43, E47
Key words: Shadow Rate, Effective Lower Bound, Trend Real Rate, Monetary Policy Shocks, Bayesian Time Series
1. Introduction
This paper models nominal interest rates, along with other macroeconomic data, using a flexible time series model that explicitly incorporates the effective lower bound (ELB) on nominal interest rates. We employ a modeling device that we refer to as a “shadow rate” — the nominal interest rate that would prevail in the absence of the ELB — which is conceptually similar to shadow rates studied in the dynamic term-structure literature exemplified by Wu and Xia (2016), among others. Our time series approach allows us to estimate the relationship between interest rates and macroeconomic data in a flexible way and, similar to the approach taken in Diebold and Li (2006), we do not impose rigid no-arbitrage restrictions across the term-structure of interest rates.
We use our approach to estimate a trend-cycle model of U.S. data on interest rates, economic activity, and inflation over a sample that includes the recent spell at the ELB. Since the global financial crisis of 2008, real interest rates have been historically low, prompting some — for example, Summers (2014) and Rachel and Smith (2015) — to argue that the long-run level of the real interest rate has fallen. However, trend-cycle decompositions hinge on estimates of the relative size of shocks to trend and cycle, which has likely changed between the Great Moderation period that began in the 1980s and the last recession and its aftermath. We embed estimation of trend real rates in a stochastic volatility model whose estimates see recent declines in real rates as largely cyclical in nature. As a result, our estimates of the trend real rate are less variable than those reported, for example, by Laubach and Williams (2003, 2015) having edged down only modestly in recent decades. Similar to Laubach and Williams and others, we find considerable uncertainty around estimates of the real rate in the long run. In light of the wide uncertainty bands surrounding the trend estimates, the modest downward drift in our real-rate trend estimates is not significant.
Explicitly modeling the ELB has large effects on inference about out-of-sample expected interest rates over the past several years. Our estimated shadow rates are less than the ELB by definition, and our model delivers predicted paths for future short-term interest rates that include extended periods at the ELB. We compare interest-rate forecasts from our model with forecasts from the Survey of Professional Forecasters (SPF) and find that our model performs better than the SPF at longer horizons. In addition, we compare interest-rate forecasts from our model to forecasts from the shadow-rate term-structure model of Wu and Xia (2016), which imposes no-arbitrage crossequation restrictions, and the random-walk model. In both cases, we find our model’s forecasts are competitive.
At the ELB, the shadow rate is an unobserved state variable that matters for forecasting future outcomes in the policy rate and other variables. When the policy rate is above the ELB, the shadow rate is equal to the policy rate and thus also observed. In either case, innovations in the shadow rate can be interpreted as reflecting changes in monetary policy — be they implemented in the form of conventional variations in the policy rate when the ELB is not binding, or through unconventional tools (such as asset purchases or forward guidance) if otherwise. In this spirit, we estimate impulse responses to monetary policy shocks, which are identified by imposing short- run restrictions on shadow-rate surprises. The use of shadow-rate surprises for the identification of policy shocks at the ELB has already been pioneered by Wu and Xia (2016); a novelty of our approach is to estimate the impulse responses jointly with the shadow rate itself. Moreover, our stochastic-volatility model generates impulse responses that are time-varying: Our policy shocks are identified from conventional short-run restrictions imposed on the covariance structure of the model’s reduced-form forecast errors. As the relative importance of shocks to trends and cycles changes, so does the covariance structure of the model’s reduced-form forecast errors as well as the persistence of their effects.
We find that monetary policy shocks identified from shadow rate innovations affect yield spreads more markedly when the ELB has been binding, consistent with the notion that shadow rates capture the effects of unconventional policies. Importantly, our results suggest that additional monetary accommodation during the depths of the most-recent recession would have provided more stimulus than would monetary accommodation provided at other times. Sims (1980) argued that agnostic time series models should inform economic models. In this spirit, our model offers evidence about differences in the economy’s responses to policy shocks at or away from the ELB while imposing only minimal theoretical restrictions.
The way we incorporate the ELB in the estimation of our model can be extended to a broad class of time series models. With short-term nominal interest rates at or near their ELBs in many parts of the world, time series models that include interest rates but ignore the ELB — like a standard vector autoregression (VAR) — have been unable to adequately address the data. Moreover, reduced-form explorations of the empirical relationship between short- and longer-term interest rates — such as Campbell and Shiller (1991) — have often ignored the truncation in the distribution of future short-term interest rates. Our modeling approach overcomes these shortcomings in a wide class of otherwise conditionally linear Gaussian state-space models. Examples include the VARs studied in Sims (1980) and the models with time-varying parameters studied in Primiceri (2005) and Cogley and Sargent (2005).
Following work by Black (1995), the no-arbitrage dynamic term-structure models studied, among others, by Kim and Singleton (2011), Krippner (2015), and Wu and Xia (2016) identify shadow rates by imposing no-arbitrage cross-equation restrictions. In the context of a correctly specified model, cross-equation restrictions resulting from no-arbitrage conditions should lead to improved model estimates. However, the class of models for which no-arbitrage restrictions are implemented in the aforementioned papers also preclude certain model features, such as stochastic model parameters, that might otherwise be important. Time-varying parameters can easily be incorporated in our flexible time series approach. In addition, our shadow-rate estimates do not only reflect information embedded in longer-term yield data but also condition on direct readings about business cycle conditions embedded in macro variables such as inflation or the output- and unemployment-rate gaps (as measured by the Congressional Budget Office, “CBO”).
The papers by Iwata and Wu (2006), Nakajima (2011), Chan and Strachan (2014) also estimate time series models that incorporate the ELB and are closely related to ours. In all of these studies, lagged observed interest rates (rather than shadow rates) are explanatory variables in the dynamic system. We instead allow lagged shadow rates to be explanatory variables. In doing so we are able to more closely align our approach with the no-arbitrage term-structure literature, and, in addition, connect the concept of the shadow rate with the level of the short-term rate that would prevail in the absence of the ELB because we allow it to have the same persistence and co-variance properties as short-term interest rates. Nevertheless, our approach is flexible enough to include both shadow rates and observed rates as lagged explanatory variables.
The paper proceeds as follows. Section 2 describes our shadow rate approach, the model we estimate, and our estimation procedure. Section 3 presents time series estimates of the shadow rate as well as the expected real interest rate in the long run. Section 4 describes forecast from our model and compares our model’s forecasting performance to other benchmarks. Section 5 describes the time-varying impulse responses generated by our model. Section 6 concludes.
2. A Shadow-Rate Model for Interest Rates at the ELB
In this section we describe our time series model, which explicitly includes the ELB. The model includes inflation (π_{t}), nominal interest rates of maturity three months (i_{t}), two (y_{t}^{2}), five (y_{t}), and 10 years (y_{t}^{10}), as well as a cyclical measure of real activity (c_{t}) (henceforth referred to as “business cycle measure”). We use CBO output gap estimates as the business cycle measure; the supplementary appendix reports similar results based on CBO unemployment-rate gap estimates.
Our data set includes a short-term interest rate, the three-month Treasury rate, which is constrained by the ELB. We model the short-term interest rate as the observation of a censored variable. In particular, we assume that the nominal interest rate is the maximum of the ELB and a shadow rate (s_{t}) so that
The ELB might arise because of an arbitrage between bonds and cash, although the world has seen negative short-term nominal interest rates in a number of countries. It also might be thought of as a level below which monetary authorities are unwilling to push short-term interest rates. For our purposes, it is taken as an exogenous known constant (which could be made time-varying). In our empirical application, the ELB on nominal rates of all maturities is assumed to bind at 25 basis points (the rate of interest paid during that period by the Federal Reserve on excess reserves). We proceed by modeling the shadow rate, in conjunction with the other variables in the model, using standard time series methods, and account for the ELB when conditioning the posterior distribution of our model on observed interest rate data.
In principle, a separate version of equation (1) applies for nominal interest rates of different maturities; but, in estimating our model, the distinction between longer-term yields and their shadow rates would be moot because the ELB has not been binding for the other yields in the data.
We assume that all of our variables, except for the business cycle measure, can be decomposed into trend and gap components. Measured as deviation from an infinite-horizon forecast, the defining feature of the gap component, X_{t}, is that it has a zero ergodic mean. Considering our business cycle measures, we assume that c_{t} satisfies the definition of a gap component, as expressed by X_{t} in (2).^{ }That is, for any variable x_{t} Є {π_{t}, s_{t}, y_{t}^{2}, y_{t}^{5}, y_{t}^{10}} we write
The trend components xt are similar in spirit to the trend concept of Beveridge and Nelson (1981); however, by treating the trends as unobserved components we allow for the conditional expectations, E_{t}(-) in (2), to reflect a possibly wider information set than what is known to an econometrician at time t (Mertens, 2016). Defining the trend components as infinite-horizon expectations implies that changes in xt follow martingale-difference processes resulting in unit root dynamics for the trend components.
For example, U.S. inflation dynamics are well captured by such a trend-cycle decomposition when trend shocks have time-varying volatility; see Stock and Watson (2007), Cogley and Sargent (2015) or Mertens (2016). So, for the trend component of inflation, we write
Imposing a long-run Fisher equation, we assume that the shadow-rate trend can be decomposed into the inflation trend and a real-rate trend, denoted, r_{t}
To capture a connection between the short-term interest rate and the other interest rates, we assume that the other interest rates in our model (yf, y_{t}^{5}, y_{t}) share a common trend with the shadow rate, adjusted for average term-premiums denoted (p^{2}, p^{5}, p^{10})
By assuming that yield spreads are stationary, we impose the same cointegrating relationship on nominal yields that has also been used by Campbell and Shiller (1987) and King and Kurmann (2002) or more recently by Bauer and Rudebusch (2017), Del Negro et al. (2017), and Benati (2017). In sum, there are two stochastic trends in our model: π_{t} and f_{t}.^{10} While we model shocks to trend inflation with stochastic volatility, our baseline specification assumes homoscedastic shocks to the trend real rate. Prior evidence suggests that time-varying volatility in shocks to trend inflation serves well to capture changes in the anchoring of public perceptions of long-term inflation expectations and the credibility of policymaker’s inflation goals in U.S. postwar data (Stock and Watson, 2007). By contrast, the variability of the trend real rate is more likely to reflect slow- moving changes in long-term growth expectations, demographic trends and other secular drivers (Rachel and Smith, 2015; Gagnon et al., 2016; Eggertsson et al., 2017). Furthermore, as discussed by Laubach and Williams (2003), trend variability probably accounts for only a small share of the variability in real rates, which cautions us against fitting a stochastic volatility process for changes in this trend. As a robustness check, we estimate an alternative version of our model where shocks to the real-rate trend are also subject to stochastic volatility. As discussed in Section 3.3, this model variant fits the data worse — as measured by the marginal data density — than our baseline specification.
The evolution of the gap components are described by a VAR in our model. In order to give the model the flexibility to capture changes in the size of the business cycle over time, shocks to
the gap VAR are assumed to have stochastic volatility. That is, we assume that
where A(L) is a polynomial in the lag operator that has all roots outside the unit circle, B is a
—1/2
unit-lower-triangular matrix, e_{t} is a multi-variate standard normal random variable, and £_{t} is a diagonal matrix of stochastic volatilities; the vector collects the diagonal elements of £_{t} = £_{t} ( ^{£}_{t} 1 . The volatilities follow stationary AR(1) processes with mutually correlated shocks
where £ is a vector of mean log-variances, £ is a diagonal matrix of lag coefficients that are all inside the unit circle, and is the variance-covariance matrix of shocks to the stochastic volatility processes. Shocks to the stochastic volatilities in the gap shocks can be correlated, which allows the model to pick up on commonalities in time-variation of business-cycle volatility across variables (Jurado et al., 2015; Clark et al., 2016).
All told, our model consists of equations (1), (2), (3), (4), (5), (6), (7), (8) and (9); out of these, equations (2), (3), (5), (6), (7), (8) are represented as a conditionally linear state-space whose estimation is described further below.
We conceptualize the shadow rate as the nominal interest rate that would prevail in the absence of the ELB. On a period-by-period basis, the interest rate is either equal to the shadow rate or equal to the ELB. The key distinction between shadow rates and interest rates is thus that shadow rates have unbounded support.
In the model above, the shadow-rate gap, as well as its lags, as part of a joint dynamic system, which allows the shadow rate to have the same persistence properties when the ELB is binding and when it is not. By contrast, Iwata and Wu (2006) and Nakajima (2011) model the variables in their models as functions of lagged observed interest rates. This means that, in those papers, at the ELB the value of the shadow-rate in the previous period has no direct effect on its value today. This assumption is in stark contrast to the properties of shadow-rate models in the dynamic term-structure literature as well as economic models.
Similar to the dynamic term-structure literature, we embed the shadow rate into a state vector with auto-regressive dynamics, such that the persistence of the shadow rate does not depend on whether the ELB binds. When the ELB is binding on observed interest rates, the shadow rate is intended to capture the hypothetical level of the nominal rate that would prevail in the absence of the ELB constraint; accordingly, we deem it beneficial that the estimated persistence of the shadow rate in our specification reflects to a large degree the persistence of observed interest rates when those are away from the ELB.
Because interest rates are truncated shadow rates, the expected interest rate is necessarily weakly larger than the expected shadow rate. In turn, it is also the case that,
In our model, s_{t} is the median forecast of lim_{h!1} i_{t+h}, offering a direct connection between far- ahead shadow rates and interest rates. Further, assuming that the Fisher hypothesis holds, this connection gives r_{t} the interpretation of the median forecast of the real interest rate in the long run. Notably, as a result of cointegration between interest rates of different maturities, the same relationship holds, up to a constant offset, for the other yields in our model as well. For the remainder of the paper, we refer to r_{t} as the trend real interest rate. Importantly, the assumed cointegration lets longer-term yields provide a signal about the trend of the short-term real rate, above and beyond variations in short-term interest rates, which is particularly valuable when the latter is at the ELB.
To estimate the parameters and unobserved states of the model, we use Bayesian methods. Our model has the form of an unobserved components model with stochastic volatility (UC-SV). The novel modeling contribution of this paper lies in the sampling of the unobserved trend and gap components of the data when the interest rate data are at the ELB, so we focus our presentation on this step of the estimation procedure, relegating a detailed description of our Markov Chain Monte Carlo (MCMC) sampler to the supplementary appendix. Conditional on parameter values and a sequence of volatilities, our model can be represented as follows
where £_{t} contains the stochastic trend and gap components of our model, as well as the appropriate number of lags to represent their dynamics in companion form, and T,^{1}/^{2} is a diagonal matrix of stochastic volatilities (as well as the constant volatility of real-rate trend shocks in the baseline model).^{18} The matrices A, B, and C are constructed from the parameters in our model,^{19} and the max operator encodes the ELB in the observation equation for the interest rate. We set the value of the ELB to 25 basis points, and assume that the three-month Treasury rate was at the ELB for every quarter in which the average annualized three-month yield was less than 25 basis points.
We collect the unobserved state variables of our model in the vector £, which contains each vector £_{t} stacked by time. Our approach for drawing from the posterior of £ is to first treat the interest rate data at the ELB as missing and take draws from the posterior distribution of £, which is straightforward using standard filtering and smoothing techniques. Knowing that the interest rate is at the ELB (instead of missing) in period t amounts to knowing that the values of £_{t} that are consistent with the data imply values of s_{t} that are less than the effective lower bound during that period. Thus, we can draw from the posterior of £ by first treating interest rate data at the ELB as missing, and then rejecting draws until we find a £ that is consistent with the ELB. Our rejection sampling amounts to sampling from the truncated posterior distribution for £ implied by the ELB.
Our estimation procedure is a generalization of Park et al. (2007) that applies the methodology of Hopke et al. (2001). Appendix A explains in further detail how to construct a draw from the posterior distribution of £, conditional on the data, in a conditionally linear Gaussian state-space model like ours. With a draw of £ in hand, the posterior distribution of the parameters can be sampled using standard methods in the literature on conditionally linear time series models with time-varying parameters and stochastic volatility, such as those used in Primiceri (2005) or Cog- ley and Sargent (2005). We jointly estimate the parameters and unobserved states of the model using Bayesian MCMC techniques; our priors and details of the MCMC steps are described in the supplementary appendix.
3. Shadow Rate and Trend Estimates
In this section we describe the posterior distribution of our model with regard to estimated shadow rates and trends. Our model is estimated using quarterly data from 1960:Q1 to 2017:Q2, which includes the recent period at the ELB. All data are publicly available from the FRED database maintained by the Federal Reserve Bank of St. Louis. Inflation is measured by the quarterly rate of change in the PCE headline deflator (expressed as an annualized percentage rate). Readings for the nominal yields are constructed as quarterly averages of the Treasury’s constant maturity rates. The output gap is computed as the log difference between real GDP and the CBO’s measure of potential real GDP for a given quarter. All computations are based on the vintage of FRED data that was available at the end of October 2017.
Our model delivers estimates of the shadow rate that are, by construction, less than the ELB during the period in which the bound is binding. Figure 1 shows results from our model when the cyclical factor is measured by the output gap; similar results based on the unemployment rate gap are reported in the supplementary appendix. Panel (a) of Figure 1 shows the posterior mean of our shadow rate, along with uncertainty bands. Panel (a) also shows estimates from Wu and Xia (2016), and Krippner (2013). Two features are worth noting. First, our estimated shadow rate, which also conditions on the business cycle measure and inflation, is lowest during 2009, near the trough of the Great Recession, according to the NBER. By contrast, the other estimates reach low points much later, and those low points are more negative than our estimate. Second, all three estimates are remarkably similar at the end of 2015, just before the Federal Reserve’s departure from the ELB.
[Figure 1 about here.]
Panel (b) of Figure 1 shows quasi real-time estimates of the shadow rate, which are conditioned solely on data through period t. The quasi real-time estimates of the shadow rate are notably similar to the full-sample estimates. This similarity indicates that the data are informative about the shadow rate, even in real time. Additionally, this similarity suggests that the length of the ELB period (7 years) does not heavily influence the particular values of the full-sample estimates of the shadow rate.
Figure 2 displays the posterior mean of our estimates of the trend real rate, r_{t}, along with uncertainty bands from our model, when the cyclical factor is measured as the output gap. Panel (a) of Figure 2 shows the smoothed estimates, in that the entire data sample is used to estimate the parameters and r_{t}. Panel (a) also shows smoothed estimates reported by Lubik and Matthes (2015) and Laubach and Williams (2003).^{23} Notably, these other measures decline markedly around the time of the onset of the global financial crisis. In contrast, our measure displays only a modest downward drift that extends back well into to 1990s. Panel (b) of Figure 2 shows quasi real-time estimates of r_{t}, which are conditioned solely on data through period t. Panel (b) also shows estimates reported by Laubach and Williams (2003) that are based on a one-sided filter. Again, our estimate does not fall by as nearly as much as the estimate of Laubach and Williams (2003) around the time of the global financial crisis. Consistent with results reported by Hamilton et al. (2015), Kiley (2015), and Lubik and Matthes (2015), the uncertainty bands surrounding our estimates of r_{t} are wide. Overall, the modest downward drift in our real-rate trend estimates is not significant.
[Figure 2 about here.]
The median estimate of r_{t} using the entire data sample declined by about half a percentage point from the mid 1990s to the end of the sample. Naturally, the quasi real-time estimate of r_{t} is more variable than the estimate from the entire data sample, but displays also a less discernable downward drift. Notably, the quasi real-time estimate of r_{t} did not decline with the onset of the global financial crisis but moved a little lower since 2014.
[Figure 3 about here.]
Panel (a) of Figure 3 reports our estimate of r_{t} over our entire data sample, along with the model-implied estimate of the actual real three-month Treasury rate over that period. The actual real three-month Treasury rate is computed as the nominal three-month Treasury rate minus expected inflation over the next three months, where expected inflation is computed from our model. Clearly real interest rates have been historically low since 2009. However, unlike Laubach and Williams (2003) and Lubik and Matthes (2015), our model estimates a wide and persistent deviations of the real short-term interest rate from its trend level over this period, rather than a precipitous decline in the trend.^{24} In good part, this is due to our model’s assessment of the time-varying volatility of shocks to the cyclical component of the real (shadow) rate, shown in Panel (b) of the figure. Similar to the heights of the Volcker disinflation, estimates of volatility in cyclical shocks to the real rate during the Great Recession rose quite substantially, thus lowering the signal taken by our model from incoming data about variations in the trend component of the real rate.
Stochastic Volatility in Real Rate Trend Shocks
In our benchmark model, the shocks to r_{t} have a constant volatility. To assess if this assumption is appropriate, we estimated a version of our model where r_{t} evolves so that
As with the inflation trend, we assume that e_{r},_{t} ~ N(0,1), p_{r},_{t} ~ N(0,1), and \p_{r}\ < 1. We compare the fit of the model with stochastic volatility in r_{t} to our benchmark specification using the marginal data density (MDD), p(Z), where Z is the observable data.
We estimate the MDD using the harmonic mean estimator of Geweke (1999), as presented by Herbst and Schorfheide (2014), given by
where N is the number of draws from the posterior distribution, ✓ is a vector that collects all of the estimated parameters and ✓ is a particular draw from the posterior distribution, f is a function of ✓ that integrates to one, p(0^{n}) is the prior density of 6^{n}, and p(Z\6^{n}) is the likelihood.^{26} For the different versions of our model, computation of the likelihood requires a particle filter because of the multiple layers of latent variables — trends and gaps as well as the stochastic volatilities); see also Fuentes-Albero and Melosi (2013) and Chan and Grant (2015). Our particle filter is an auxiliary particle filter that employs Rao-Blackwellization to handle the linear parts of the state space. As described in the supplementary appendix, to handle the ELB, the Rao-Blackwellization is dropped when the ELB binds to account for s_{t} < ELB during those periods.
We find substantially higher MDD for the model without stochastic volatility in the real rate
trend: When the output gap is used as the business cycle measure, the log-MDD is estimated at 307.7 as opposed to 239.8 for the model with stochastic volatility in real-rate trend shocks, providing very strong evidence in favor of our baseline against the alternative. Similarly, when the unemployment rate gap is used as the business cycle measure, we find that the log-MDD is substantially larger for the model without stochastic volatility (570.9 as opposed to 492.6 for the model with stochastic volatility). These results confirm our prior disposition, discussed in section 2.2, that r_{t} is best modeled without stochastic volatility.
4. Forecasting Interest Rates
Our shadow-rate approach has significant implications for forecasting interest rates. In this section we offer a number of ways to evaluate our shadow-rate approach by analyzing the model’s out-ofsample predictions.
To create out-of-sample forecasts from our model, at each date we use the median of the predictive densities from the posterior distribution (using data only up to that date) to forecast future interest rates. Our forecasting procedure thus captures uncertainty about both the parameter values and the unobserved states in our model. The posterior predictive distribution can be highly asymmetric because of the ELB. As a result, we follow Bauer and Rudebusch (2015) and use the posterior median as the point forecast for forecast evaluation, rather than the posterior mean.
[Figure 4 about here.]
So as to illustrate its skewness, Figure 4 display statistics from the posterior predictive density of the short-term nominal interest rate from our baseline model (with the output gap as the cyclical
factor) at different dates. The forecast horizon extends for five years, and, in addition to mean and median predictions, shaded areas indicate 50 and 90 percent uncertainty bands. The dashed lines that extend below the ELB indicate posterior quantiles of the shadow rate distribution (as opposed to the interest rate distribution). The predictive density of the interest rate is a truncated version of the predictive density of the shadow rate distribution, so the quantiles of the shadow rate distribution become exactly the quantiles of the interest rate distribution if the value is larger than the ELB. The truncation of the shadow-rate distribution causes substantial asymmetry in the interest rate distribution leading to marked differences in the predictive means and medians of our baseline model.
In 2008:Q4, the first period before the ELB (Panel (a) of Figure 4), the predictive density from our model takes the ELB already into account and produces interest rate forecasts based on the truncated distribution of future shadow rates. In doing so, the mean interest-rate forecast rises appreciably above the median for several periods.
In 2009:Q1, the period the ELB begins to bind (Panel (b) of Figure 4), accounting for the ELB produces interest rate forecasts that place substantial probability on remaining exactly at the ELB for several quarters. As in Panel (a), the truncation of the shadow rate distribution in order to produce interest rate forecasts creates a divergence of mean and median estimates of interest rates for several years.
In 2010:Q4, after the ELB had been binding for some time (Panel (c) of Figure 4), our baseline model still predicts substantial probability of interest rates at the ELB because of the estimated negative shadow rate. Moreover, the median interest rate forecast remains at the ELB for a number of quarters. Toward the end of the ELB period (2015:Q4, shown in Panel (d) of Figure 4), the forecasts for the short-term interest rate are similar to the forecasts for the shadow rate, in large part because our estimated shadow rate is only slightly less than the ELB at that point.
To assess the forecasting performance of our model, we compare the accuracy of forecasts from our model to the accuracy of forecasts from the SPF and the shadow-rate term-structure model of Wu and Xia (2016) (WX-SRTSM). We also compare our model to the no-change forecast from a random-walk model, which is a common benchmark in the literature. We report results for both the version of our model that uses the CBO’s output gap as the business cycle measure and the version of our model that uses the CBO’s unemployment rate gap as the business cycle measure. For the SPF, forecasters submit survey responses in the middle of the quarter. So as to not give our model an informational advantage, we date the forecasts from the SPF as being made in the previous quarter. For interest rate forecasts, forecasters in the SPF submit forecasts for the average value of the three-month and 10-year Treasury rates in the current and subsequent four quarters. These rates are part of our benchmark data set, and as such we compare forecasts from our model directly to the average forecasts across forecasters from the SPF. We use the same data set when we compare forecasts from our model to the random-walk model.
The WX-SRTSM is estimated at a monthly frequency using month-end data on fitted zero- coupon Treasury yields since 1990 from Gurkaynak et al. (2007). So as to be consistent, when comparing our model to the WX-SRTSM, we estimate our model on data since 1990 and use quarter-end interest rate data from the fitted zero-coupon Treasury yields from Gurkaynak et al. (2007) for the two-, five-, and 10-year Treasury rates in our model. We compare our model’s quarterly forecast for quarter-end interest rates to the median forecast from the WX-SRTSM in the final month of each quarter, and we use the fitted yields from Gurkaynak et al. (2007) as the realized data.
To assess performance, we focus on two statistics: the root-mean-squared error (RMSE) and the mean absolution deviation (MAD). Table 1 compares the forecasts from the SPF with our model for the post-2008 period.
[Table 1 about here.]
Except at very short horizons where the SPF has an informational advantage, our model performs better than the SPF for both the three-month and 10-year rates. To roughly assess the statistical significance of these differences, we use the statistic proposed by Diebold and Mariano (1995). The differences in forecast performance between our model and the SPF for the three- month yield are not statistically significant, however our model statistically outperforms the SPF for forecasting the 10-year yield at horizons greater than 3 quarters.
[Table 2 about here.]
Table 2 compares the forecasts from the WX-SRTSM with our model for the post-2008 period. Our model performs at least as well as the WX-SRTSM, on balance, for both the three-month and 10-year Treasury yields. Thus, even though our model does not impose the cross-equation restrictions associated with the no-arbitrage assumptions of the WX-SRTSM, our model does no worse over the ELB period in forecasting interest rates.
[Table 3 about here.]
Table 3 compares the forecasts from our model with the no-change forecasts from the random- walk model for the post-2008 period. By construction, point forecasts generated from a random walk will not violate the ELB; instead, when the ELB binds, the random-walk forecasts will predict this condition to persist forever. As it happens, the only ELB episode in our data turned out to be longer than generally predicted, and near-term forecasts from the random walk clearly outperform our model as well as the SPF and the WX-SRTSM. Considering the 10-year rate, our model performs about as well as the random walk model.
Naturally, the ELB has important effects on forecast uncertainty for nominal interest rates when the predictive density for shadow rates has non-negligible coverage below the ELB. To illustrate the relevance of these effects, Figure 5 compares forecast uncertainty about the shadow rate with forecast uncertainty about the short-term interest rate. For the purpose of this figure, we measure forecast uncertainty by the conditional interquartile range of the predictive densities described above for one and eight-quarters ahead. As the level of nominal rates has been trending down since the 1980s, the probability of reaching the ELB has become more and more non-negligible; this is particularly true for longer-horizons forecasts made since 2000, causing forecast uncertainty about the shadow rate to differ at times from forecast uncertainty about the short-term interest rate.
[Figure 5 about here.]
Not surprisingly, the onset of the last NBER recession in 2007 is reflected in higher estimated levels of stochastic volatility to all shocks in our model, leading to increased shadow-rate uncertainty. By accounting for the ELB, our model recognizes that during the last recession, increased shadow-rate uncertainty is accompanied by a marked downward shift of the shadow-rate distribution to values below the ELB such that the truncated distribution of actual short-term nominal interest rates almost collapses at values at or slightly above the ELB. Consequently, near-term uncertainty about short-term nominal rates declines during the last recession when properly accounting for the ELB, as shown in Panel (a) of Figure 5. In contrast, as shown in Panel (b) of the figure, medium-term uncertainty about nominal interest rates increases with the increasing shadow-rate uncertainty, though not by quite as much, as nominal rates are projected to return to their estimated non-negative trend level.
During the period in which the ELB was binding the United States, our model’s predicted path for future short-term interest rates implies, at each point in time, a forecast for the average number of quarters that the ELB will bind. Similarly, survey responses in the SPF about the short-term nominal interest rate imply a forecast for how many quarters survey participants expected the short-term rate to be less than 25 basis points. In this subsection, we compare our model’s implied expected duration of the ELB with that from the SPF.
The SPF asks participants to submit forecasts for the three-month Treasury rate in the subsequent four quarters. For each respondent, we compute the the number of consecutive quarters, beginning in the quarter after the quarter in which the survey is conducted, that the three-month Treasury rate is expected to be less than 25 basis points. We then average over respondents.
From our model, a comparable metric is generated as follows: We use the predictive density at time t (computed using only data up through t) to sample draws of the policy-rate path up to four quarters out. For each draw we measure how long the ELB has continuously been binding. The expected duration is then computed by numerical integration over these draws from the predictive density. Figure 6 compares forecasts of the duration of the ELB episode generated by our model against those from the SPF.
[Figure 6 about here.]
The forecasts from our model are similar to those from the SPF in that the forecasts of the longest duration of the ELB are at some point in late 2012 or 2013, indicating that it took some time for both the model and professional forecasters to grasp the length of time that interest rates would be at their ELB. In 2009 and 2010, our model predicts a longer stint at the ELB than the SPF, meaning that our model initially saw the recession and policy response as deeper and more dramatic than participants in the SPF. After 2013, the forecasts from our model and the SPF are similar in that the forecasts of the duration of the ELB fall precipitously. In 2015, the expected duration at the ELB falls somewhat more precipitously for the SPF than the model, suggesting that survey participants were more confident that rates would soon rise.
5.Impulse-Response Analysis
A common question of interest in empirical research is to measure the response of macroeconomic variables to identified economic shocks; for example, shocks due to monetary policy. Absent a binding lower-bound on nominal interest rates, the conventional tool for such impulse-response function (IRF) analysis are linear VARs. In this section, we show how to perform a similar analysis within our shadow-rate approach to time series modeling.
Our application is similar in spirit to the VAR analysis of Wu and Xia (2016) and others who use estimated shadow rates as observables in a (factor-augmented) VAR system and identify monetary policy shocks via short-run restrictions on VAR-implied forecast errors for the shadow rate.^{ }However, we also extend this approach in several dimensions: First, instead of a two-step approach that ignores uncertainty about the true shadow-rate values, we estimate impulse responses jointly with our shadow-rate inference. Second, our model implies a time-varying vector moving average (VMA) representation of the data with respect to observable forecast errors, which generates time- varying impulse responses with distinctively different patterns at and away from the ELB. Similar to Wu and Xia (2016), we focus on monetary shocks identified by short-run restrictions on surprise changes in the shadow rate. Specifically, our short-run restrictions are identical to those used by Christiano et al. (1999, “CEE”) for the short-term nominal interest rate. As in CEE, who build on Sims (1980), we define identified shocks as linear combinations of forecast errors in observable variables (as well as the shadow rate).
Section 5.1 lays out the identification of time-varying impulse responses to monetary policy shock from a UC-SV model, such as the model we described in Section 2.5. Impulse-response estimates are reported in Section 5.2.
Our identification applies principles known from standard VAR analysis to our UC-SV model while accounting for the effective lower bound on interest rates. Even absent an ELB constraint, the UC-SV structure generates time-varying impulse responses to shocks that are defined as linear combinations of forecast errors in observable variables. The time-variation in impulse responses reflects changes in the covariance structure of the reduced form forecast errors that are induced by the stochastic volatilities. After a brief recap of standard VAR analysis we derive these impulse responses from a UC-SV model that treats the shadow rate as observable and then turn to their application in an ELB environment.
In a standard VAR framework, the reduced-form dynamics of a vector of variables (X_{t}) are described by A(L)X_{t} = e_{t} where e_{t} = X_{t} — E(X_{t}|X^{t-1}), Var (e_{t}) = O, and A(L) = P°=_{0} A_{i}L^{i }with A_{0} = 1.^{32} Economic assumptions then relate the reduced-form forecast errors, e_{t}, to structural shocks z_{t} = Q^{-1}e_{t} where QQ' = O. For example, in their by-now canonical approach, CEE identify monetary policy shocks by placing a short-run restriction on Q: Monetary policy shocks are assumed to be given by the residual obtained from regressing forecast error in the policy rate on forecast errors in other macroeconomic (though not financial) variables. This scheme can be implemented by ordering the policy rate in Xt after other macroeconomic variables and before financial variables — similar to our definition of X_{t} above (ignoring, for now, the distinction between shadow rate s_{t} and actual policy rate i_{t}). Q can then be set equal to the lower-triangular Choleski factorization of the variance-covariance matrix O of forecast errors, and the monetary policy shock is given by the element of zt corresponding to the location of the policy rate in Xt. Impulse responses are then generated by the VMA representation X_{t} = A(L)^{-1}Qz_{t}.
However, when considering the ELB on nominal interest rates, such a VAR approach is hardly applicable: At the ELB, the dynamics of the actual rate are not linear and one-step ahead forecast
errors for the actual rate are unlikely to be a good starting point for the identification of monetary policy shocks. These limitations affect both the identification as well as propagation of shocks. As proposed by our model as well as other shadow-rate approaches, it is, however, conceivable that actual rates dynamics reflect an underlying shadow-rate process that is at least conditionally linear.
As shown below, in such a shadow-rate system the propagation of shocks is not only straightforward to simulate; we can also address suggestions from economic theory that the economy should react differently to policy shocks when the ELB binds. If so, impulse-responses generated from a time-invariant VAR will hardly provide an appropriate characterization. As shown next, our UC-SV model generates a time-varying VMA with respect to surprises in observable variables that provides a suitable alternative.
Moreover, innovations in the shadow rate can be used for the identification of monetary policy shocks. When the ELB is not binding, the policy rate and shadow rate are identical and well- established approaches, such as CEE, exist for the identification of policy shocks. When the ELB binds, the shadow rate is an unobserved state variable that matters for forecasting future outcomes in the policy rate and other variables. As in Wu and Xia (2016), unexpected variations in the shadow rate can be interpreted as reflecting changes in monetary policy implemented through unconventional tools (such as asset purchases or forward guidance); under this hypothesis, an identification scheme such as CEE could be applied to shadow rate surprises to tease out estimated monetary policy shocks. Compared to other shadow-rate shock approaches reviewed above, a novelty of our implementation is to identify monetary policy shocks jointly with the estimation of the shadow rate itself and to do so within a model that generates time-varying impulse responses.
For now, let us ignore the ELB and treat the shadow rate (and thus X_{t} rather than just Z_{t}) as observable and denote innovations of X_{t} relative to its own history by
where X_{t}|_{t-1} denotes the expectation of X_{t} relative to its own history X^{t-1}; similarly, let £_{t}|_{t-1} = E (£_{t}|X^{t-1}) .^{37} For now, when taking expectations, let the model’s parameters and trajectories for the stochastic volatilities be given, such that A, B, C, and E_{t} are known; the estimation of E_{t} will be integrated into the computation of the impulse responses in the next section.
We obtain the VMA of X_{t} with respect to e_{t} from the innovations representation of the state space given by (12) and (13) as follows
Time variation in the VMA D_{t} stems from K_{t}, which is a time-varying Kalman gain matrix induced by the stochastic volatilities embedded in E_{t}.^{38} Thus, time-varying IRF reflect variations in the signal-to-noise ratios involved in filtering out trends and gaps, which lead to time-varying persistence in the effects of e_{t} on X_{t}.
The stochastic volatilities also give rise to a time-varying innovations variance-covariance matrix of the forecast errors Var_{t} (e_{t}) = Q_{t} = Q_{t}Q'_{t} with lower-triangular Choleski factor Q_{t}. Q_{t} provides a linear mapping between the forecast errors and a vector of mutually uncorrelated
shocks z_{t}, e_{t} = Qz_{t}. The third element of z_{t}, denoted z_{t},_{m}, corresponds to the residual of projecting the shadow-rate innovation onto innovations in inflation and the output gap (our business cycle measure) and — as in the CEE approach — will be assigned a structural interpretation as monetary policy shock. Denoting the third column of Q_{t} by q’p, responses of X_{t+k} to a monetary policy shock at time t are summarized by the following lag polynomial
So far, the expectations used in this section conditioned on given values for model parameters as well as stochastic volatilities. This section briefly describes how to integrate inference on the stochastic volatilities in the IRF computation with a particle filter. Considerations of the ELB in IRF estimation as well as the construction of actual-rate response are deferred until the next section. Throughout, we maintain the assumption that parameters values — encoded in the state- space matrices A, B, and C — are known; in our empirical application they are set equal to their posterior median estimates from a full-sample MCMC estimation of the model.
Since time-variation in the impulse responses reflects the time-varying volatilities in our model, it is important to embed inference on the stochastic volatilities into the computation of the IRF. A particle filter is highly suitable for this task, since it combines estimation of X_{t}|_{t-1} and £_{t}|_{t} with filtered estimates of E_{t}|_{t} — in contrast, MCMC sampling only provides draws from the smoothed posteriors for volatilities and latent states, (£_{t}, E_{t}) |X^{T}, which are not useful for constructing forecast updates (a.k.a. IRF) for dates other than T.
We use a particle filter that approximates the filtered posterior density of the stochastic volatilities with a swarm of proposals. The proposed values for the stochastic volatilities are also known as “particles.” In light of the conditionally linear structure of our UC-SV model, the filter employs Rao-Blackwellization to construct — conditional on each particle — estimates of the latent states, £_{t}, by applying Kalman-filtering computations like those used in the construction of the IRF above. Each particle is then weighted by the resulting likelihood. Indexing particles by i = 1,2 , ...,N and denoting particle weights by w_{t}|_{t}__{1} we compute the response of X_{t+k} to a monetary policy shock at time t as
Accounting for the ELB requires three distinct adjustments the framework above. First, in order to calculate impulse responses for the actual rate, we need to characterize the full predictive density of the shadow rate before and after a monetary policy shock. Via the censoring function in equation (1), predictive densities for the shadow rate can then be transformed into their truncated counterparts for the actual rate. Actual-rate impulse responses are then given by the change in mean forecasts induced by the policy shock. As described in the supplementary appendix, the particle filter represents predictive densities for the shadow rate as a mixture of normal distributions from which we can simulate the corresponding actual-rate forecasts and impulse responses.
Second, the VMA and innovations representation derived above relied on a linear, Gaussian Kalman filtering framework. But, observing interest rates at the ELB at time t implies the nonlinear constraint s_{t} < ELB, and we embed this information via a mean-variance approximation. Specifically, for a given particle, let I_{t} represent all information available through time t except for s_{t} = h_{s}£_{t} < ELB and assume that £_{t}|I_{t} had a normal distribution: the posterior for £_{t}| (I_{t}, h_{s}£_{t} < ELB) is then a multivariate truncated normal distribution. As described in the supplementary appendix, we approximate this truncated normal by a normal distribution whose mean and variance-covariance matrix are set equal to mean and variance-covariance matrix of the truncated normal.
(i)
Third, as described in the supplementary appendix, computation of the particle weights w_{t }needs to be adjusted to properly reflect the likelihood, implied by each particle, that s_{t} < ELB at times when the ELB has been binding in the data.
With these three adjustments, we continue to identify monetary policy shocks by imposing the CEE short-run restrictions on innovations in X_{t} but where the innovations are defined relative to the history of observed data, Z^{t}, instead of X^{t}. So, for every particle, the impulse responses reflect forecast updates relative to Z^{t-1}
and t.i.m.p. refers to terms independent from the monetary policy shocks.
Using the identification scheme of CEE, this section discusses the impulse responses implied by our model to monetary policy shocks identified as surprise innovations in the shadow rate that are uncorrelated with contemporaneous forecast errors in the output gap and inflation. The shocks are jointly estimated with linear states of our model (£) as well as the stochastic volatilities by applying our particle filter (further described in the supplementary appendix). Time-variation in the volatilities affecting trends and gaps cause variations in the filter’s assessment of the permanent and transitory effects of shadow-rate shocks. We consider impulse-responses at four different dates that represent different points in time before, during and after the ELB had been binding in recent U.S. history: the fourth quarters of 2007, 2009, 2011, and 2016. For the ease of comparability, monetary policy shocks have been scaled to generate a 1 percentage point drop in the shadow rate on impact.
By applying a conventional identification scheme to our UC-SV model, we generate time- varying impulse responses without imposing restrictions that would be specific to one particular economic model. When the actual rate is at the ELB, declines in the shadow rate can be interpreted as standing in for the effects of other, unconventional policy tools — such as forward guidance or asset purchases — that are not directly modeled here. The effects of such tools are, however, captured indirectly, to the extent that they have affected outcomes for longer-term yields, inflation, and the output gap, which in turn inform our model’s estimate of the shadow rate.
[Figure 7 about here.]
Figure 7 reports impulse responses of the shadow rate and the short-term nominal rate to identified monetary policy shocks. Reflecting the unobserved trend-cycle structure of the model, monetary policy shocks are estimated to have transitory and permanent effects on nominal rates (and inflation). As shown in Panel (a) of the figure, it takes roughly a year for the shadow rate to settle on a new permanent level after a shock. Moreover, monetary policy shocks had much less permanent effects on the shadow rate during the ELB years compared to non-ELB times.
Actual rates depend on the shadow rate via the censoring function in equation (1). Consequently, the ELB has a particularly noticeable effect on the responses of actual rates to monetary
policy shocks as can be seen from Panel (b) of Figure 7. Because of the censoring, responses of the actual rate to shocks depend on the distance of the actual rate to the ELB as well as the size and the sign of the impulse: during the ELB years, the actual-rate responses to a negative shadow-rate shock are close to zero. In 2007:Q4 — when the three-month rate was above 3 percent and thus well away from the ELB — the actual rate response is very similar to the shadow rate; both decline strongly in response to the simulated cut in the shadow rate. But, even one year after the departure of U.S. monetary policy from the ELB, the actual-rate response to a one-percentage point drop in the shadow rate is still quite muted in 2016:Q4, as the three-month rate still just hovered closely above the ELB at that time.
[Figure 8 about here.]
It might seem surprising to see any decline at all in the estimated actual-rate responses to a drop in the shadow-rate during periods when the ELB was firmly binding. However, the impulse response paths reflect changes in forecasted trajectories, not changes relative to the initial level of the actual rate. Negative values for the actual-rate responses after a drop in the shadow rate indicate a slower return to trend levels — that have remained well above the ELB — than before, not a cut below the level of the actual rate that prevailed on impact. The resulting asymmetries are also illustrated in Figure 8, which contrasts the hypothetical effects of a positive and a negative shadow rate shock in 2015:Q4 (the quarter of the first rate hike since the economy had reached the ELB in 2009).
[Figure 9 about here.]
Figure 9 displays impulse responses of longer-term yields and macro variables to a monetary policy shock. Like the shadow rate, the permanent component of longer-term rates adjusts more dramatically to a monetary policy shock away from the ELB, as shown in Panels (a) and (b) of the figure. However, the transition paths of short- and long-term rates to their new trend levels are markedly different: while the shadow rate adjusts fairly swiftly (within a year and a half) and monotonically to its new steady state, it takes the 10-year rate about three times that long. Moreover, the 10-year response is hump-shaped, suggesting a temporary (and slightly delayed) decline in term premia after a negative monetary policy shock.
At least part of the permanent effect of a monetary policy shock on the level of interest rates is attributed to a shift in the inflation trend. Not surprisingly, given the long-run Fisher equation (5) embedded in the model, the long-run response of inflation to a monetary policy shock is “Fisherian” in that (permanent) declines in policy rates are expected to come along with (permanent) declines in inflation. As shown in Panel (c) of Figure 9, the response of inflation to a monetary policy shock is characterized by a largely monotonic adjustment toward the new trend level. In particular, as shown in Panel (d) of the figure, inflation adjusts only partially toward the new trend level after impact, temporarily leaving a positive inflation gap in response to a negative monetary policy shock, as suggested by standard Keynesian logic. If anything, at the ELB, the short-run response of inflation is more “Fisherian,” as it jumps almost immediately towards its new trend level (with mild transitory fluctuations thereafter). Similar to the nominal-rate responses, shadow-rate shocks have much less permanent effects on inflation when the ELB binds.
The responses of the output gap, shown in Panel (e) of Figure 9, are very persistent: after a monetary policy shock, it takes at least four years for the resulting movements in the output gap to dissipate. Moreover, the responses of the output gap to a monetary policy shock differ markedly between times when the ELB binds rather than when not. Near the trough of the most-recent recession, when the ELB was binding, monetary policy shocks would have generated a much more sizable pick-up in real activity than what is estimated for non-ELB dates. One rationale for these results can be drawn from the impulse responses of the spread between the 10-year rate and the two-year rate, that is the difference between the responses shown in Panels (b) and (a) as plotted in Panel (f) of the figure. At the ELB, monetary policy shocks lowered the spread by more than when away from the ELB, thus leading to easier financial conditions that correspond to the stronger stimulus in economic activity shown in Panel (e) for the years when the ELB was binding.
Our results suggest that monetary accommodation would have been more effective at stimulating real activity during the depths of the most-recent recession than at other times. The time- varying effect of monetary policy shocks on the yield spread indicates that, at the ELB, policy shocks identified from shadow rate innovations may indeed capture the effects of unconventional policies on longer-dated yields. Overall, these results are consistent with economic models that suggest monetary stimulus via unconventional policies, including forward guidance, is most effective when the economy is in a deep recession.
6.Conclusion
In this paper, we develop a methodology to account for the ELB in time series models of nominal interest rates. Our method makes linear Gaussian models amenable to the ELB, but can also be applied to time-varying-parameter models that are only conditionally linear. For instance, our empirical application is based on an unobserved components model with stochastic volatility. We demonstrate how to estimate the parameters and latent states of such a model with an otherwise standard Bayesian MCMC sampler.
Taking proper account of the ELB has, of course, drastic effects on interest rate forecasts. Even among predictors that adhere to the ELB, our shadow-rate approach does well compared to several competitors. We also estimate a common trend in real rates, defined as a long-term forecast of the real interest rate, and find no significant decline in the trend real rate since the 1990s.
Finally, our model generates time-varying impulse responses to monetary policy shocks. We find that monetary policy shocks identified from shadow rate innovations affect yield spreads more markedly when the ELB has been binding, consistent with the notion that shadow rates capture the effects of unconventional policies. Importantly, our results suggest that monetary accommodation during the depths of the most-recent recession would have provided more stimulus than at other times.
APPENDIX
A Sampling States with Censored Data
Our Gibbs sampling procedure is a generalization of Park et al. (2007) that applies the methodology of Hopke et al. (2001). This appendix briefly describes sampling the shadow rate (and related state variables) for given parameter values and trajectories for the stochastic volatilities in our model. A complete description of the MCMC sampler used in the estimation of our model can be found in the supplementary appendix to this paper. This MCMC sampler combines the shadow-rate sampling described below with other, commonly used, Gibbs sampling steps for the estimation of parameters and stochastic volatilities.
Assume that a vector of economic variables X_{t} is linearly related to a state vector £_{t}
where e_{t} is a vector of standard normal random variables, and the sequence of matrices {A_{t}}^{,}^_{=1 }and {B_{t}}^{T}=_{1}, {C_{t}}_{=1} are given — for example, by other steps of a MCMC sampler in which the shadow-rate sampling described here is included as well. In our specific application, described in the main body of the paper, A_{t} = A and C_{t} = C are constants while B_{t} is given by B_{t} = B s/^{/}^{2 }where St is a diagonal matrix of stochastic volatilities; further details about mapping model parameters and latent states into the state space are provided in the supplementary appendix.
We assume that X_{t} is partitioned into a vector shadow rates, (S_{t}), and a vector of “macro”
variables that are unconstrained by the ELB, (M_{t}).^{47} That is, we have
where the max operator is applied element by element. The ELB acts as a censoring function in the model through the max operator, though more general censoring functions could be used.
Define X = [X[, X'_{2},XT]^{0}, and Z = [Z[, Z'_{2},..., ZT]^{0}. We split Z into two parts, one part containing all non-interest rate data and all observations for interest rates that are not constrained by the ELB, Z^{NC}, and another part with interest rate data at the ELB, Z^{C}. The corresponding elements of X are X^{NC} and X^{C}, and X^{C} consists solely of shadow rates with values less than the ELB.
Given a normal distribution for £_{0}, it follows that the vectors X^{NC} and £ = £, £_{2},..., £_{T}]^{0 }have a multivariate normal (prior) distribution, and we can derive the posterior distribution of £
NC conditional on X
Kalman smoother provides a convenient way to recursively compute the posterior moments in equation (26) and recovers the distribution of £ conditional on observations for Z^{NC}.^{49}
So far, we have not yet conditioned on information contained in Z^{C}, which collects interest- rate observations at the ELB. The corresponding elements of X^{C} are linear combinations of £, X^{C} = C^{C}£, that correspond to shadow-rate realizations below the ELB. The posterior distribution
where the inequality is elementwise. To sample from the posterior distribution of £ conditional on all observations in Z, we first draw £ from Pr (£|X^{NC} = Z^{NC}). We then reject draws until we find a draw that satisfies the requirement that X^{C} < ELB. Rejection sampling is thus done on an entire draw of £, which corresponds to an entire trajectory of {£_{t} j^.
In our baseline framework, lagged values of £_{t} appear as explanatory variables and are not censored. A straightforward extension is to incorporate a given number of p lags of Z_{t}, which includes interest rate data that can be constrained the ELB. In this case, we change (24) to be
and {F_{t} j^ are known, conformable matrices. The posterior of £ can be constructed exactly as in our baseline model, treating {£_{t-1}jJ_{=1} as exogenous in every period because the rejection step will ensure that the sampled values of £ are consistent with £_{t-1} for all t. For comparison, the models of Iwata and Wu (2006) and Nakajima (2011) can be cast, conditional on parameter values, as special cases of this setup in which the matrix A_{t} = 0. A notable difference in the posterior simulation of the model is that the truncated distributions in Iwata and Wu (2006) and Nakajima (2011) can be cast as period-by-period truncated normals. By contrast, our posterior estimates require rejection sampling on an entire time series draw of £.
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