Risk management for monetary policy near the zero lower bound.

Evans, Charles ; Fisher, Jonas ; Gourio, Francois 等

ABSTRACT With projections showing inflation heading back toward target and the labor market continuing to improve, the Federal Reserve has begun to contemplate an increase in the federal funds rate. There is, however, substantial uncertainty around these projections. How should this uncertainty affect monetary policy? In many standard models uncertainty has no effect. In this paper, we demonstrate that the zero lower bound (ZLB) on nominal interest rates implies that the central bank should adopt a looser policy when there is uncertainty. In the current context this result implies that a delayed liftoff is optimal. We demonstrate this result theoretically through two canonical macroeconomic models. Using numerical simulations of our models calibrated to the current environment, we find that optimal policy calls for a delay in liftoff of two to three quarters relative to a policy that does not take into account uncertainty about policy being constrained by the ZLB. We then use a narrative study of Federal Reserve communications and estimated policy reaction functions to show that risk management is a long-standing practice in the conduct of monetary policy.

**********

To what extent should uncertainty affect monetary policy? This classic question is relevant today as the Federal Reserve considers when to start increasing the federal funds rate. In the March 2015 "Summary of Economic Projections," most Federal Open Market Committee (FOMC) participants forecast that the unemployment rate would return to its long-run neutral level by late 2015 and that inflation would gradually rise, returning to its 2 percent target. This forecast could go wrong in two ways. First, the FOMC may be overestimating the underlying strength in the economy or the tendency of inflation to return to target. Guarding against these risks would call for cautious removal of accommodation. Second, the economy could be poised for stronger growth and inflation than currently projected. This second risk would call for more aggressive rate hikes. How should policy manage these divergent risks?

If the FOMC misjudges the impediments to growth and inflation and reduces monetary accommodation too soon, it could find itself in the uncomfortable position of having to reverse course and being constrained by the zero lower bound (ZLB) again. It is true that the FOMC has access to unconventional policy tools at the ZLB, but these appear to be imperfect substitutes for the traditional funds rate instrument. In contrast, if the Fed keeps rates too low and inflation rises too quickly, most likely inflation could be brought back into check with modest increases in interest rates. Since the unconventional tools available to counter the first scenario may be less effective than the traditional tools available to counter the second scenario, the costs of premature liftoff may exceed those of delay. It therefore seems prudent to refrain from raising rates until the FOMC is highly certain that growth is sustainable and inflation is returning to target. (1)

In this paper we establish theoretically that uncertainty about monetary policy being constrained by the ZLB in the future implies an optimally looser policy today, which in the current context means delaying liftoff--the risk management framework just described. We formally define risk management as the principle that policy should be formulated taking into account the dispersion of shocks around their means. Our main theoretical contribution is to provide a simple demonstration, using standard models of monetary policy, that the ZLB implies a new role for such risk management through two distinct economic channels.

The first channel, which we call the expectations channel, arises because the possibility of a binding ZLB tomorrow leads to lower expected inflation and output today, and hence dictates some counteracting policy easing today. The second channel, which we call the buffer stock channel, arises because, if inflation or output is intrinsically persistent, building up output or inflation today reduces the likelihood and severity of hitting the ZLB tomorrow. Optimal policy when either of these channels is operative should be looser whenever a return to the ZLB remains a distinct possibility. In simulations calibrated to the current environment, we find that optimal policy prescribes two to three quarters of delay in liftoff relative to a policy that does not take this uncertainty into account. However, under the optimal policy the central bank must also be prepared to raise rates quickly as the threat of being constrained by the ZLB recedes.

Would it be unusual for the Fed to take uncertainty into account in setting its policy rate? The second part of this paper argues that risk management has been a long-standing practice in U.S. monetary policy. Therefore, advocating it in the current policy environment would be consistent with a well-established approach of the Federal Reserve. Of course, because the ZLB was only recently perceived as an important constraint, the theoretical rationales for risk management were different in the past. It is true that in a wide class of models that abstract from the ZLB, optimal policy involves adjusting the interest rate in response to the mean of the distribution of shocks, and information on higher moments is irrelevant (the so-called "certainty equivalence" principle). However, there is an extensive literature covering departures from this result based on nonlinear economic environments or uncertain policy parameters that justify taking a risk management approach away from the ZLB.

We explore whether policymakers actually practiced risk management prior to the ZLB period in two ways. First, we analyze Federal Reserve communications over the period 1987-2008 and find numerous examples when uncertainty or the desire to insure against important risks to the economy were used to help explain the setting of policy. Confirmation of this view is found in the statements of Alan Greenspan, who during his tenure as Federal Reserve chair noted, "the conduct of monetary policy in the United States has come to involve, at its core, crucial elements of risk management." (2) Second, we estimate a conventional forecast-based monetary policy reaction function augmented with a variety of measures of risk based on financial market data, Federal Reserve Board staff forecasts, private-sector forecasts, and narrative analysis of the FOMC minutes. We find clear evidence that when measured in this way, risk has had a statistically and economically significant impact on the interest rate choices of the FOMC. For the FOMC, risk management appears to be old hat.

If the monetary policy toolkit contained alternative instruments that were perfect substitutes for changing the policy rate, then the ZLB would not present any special economic risk and our analysis would be moot. We do not think this is the case. Even though most central bankers believe unconventional policies such as large-scale asset purchases (LSAPs) or more explicit and longer-term forward guidance about policy rates can provide considerable accommodation at the ZLB, few argue that these tools are on an equal footing with traditional policy instruments. (3)

One reason for this is that the effects on the economy of unconventional policies are, naturally, much more uncertain than those of traditional tools. There are divergent empirical estimates of their effects, and there is uncertainty about the theoretical mechanism behind those effects. Various studies of LSAPs, for example, provide a wide range of estimates of their ability to put downward pressure on private borrowing rates and influence the real economy. Furthermore, the effects on interest rates of both LSAPs and forward guidance are complicated functions of private-sector expectations, which make their economic effects highly uncertain as well. (4)

Uncertainty about the transmission mechanism of LSAPs is reflected in Arvind Krishnamurthy and Annette Vissing-Jorgensen's (2013) discussion of the various hypotheses that have been proposed. Unconventional tools also carry potential costs. The four most commonly cited costs are these: (i) the large increases in reserves generated by LSAPs risk unleashing inflation; (ii) a large balance sheet may make it more difficult for the Fed to raise interest rates when the time comes; (iii) the extended period of very low interest rates and Federal Reserve intervention in the long-term Treasury and mortgage markets may induce inefficient allocation of credit and financial fragility; and (iv) the large balance sheet puts the Federal Reserve at risk of incurring financial losses if rates rise too quickly, and such losses could undermine its support and independence. (5) Costs reduce the incentive to use any policy tool. Moreover, because the costs of unconventional tools are very hard to quantify, the level of uncertainty associated with them is naturally elevated as well.

A consequence of this uncertainty over the benefits and costs of unconventional tools is that they are likely to be used more cautiously than traditional policy instruments, as suggested by William Brainard's (1967) classic analysis. For example, then Federal Reserve Chairman Ben Bernanke emphasized in 2012 that because of their uncertain costs and benefits, "the hurdle for using nontraditional policies should be higher than for traditional policies." (6) In addition, some of the benefits of unconventional policies may be decreasing, and their costs may be increasing in terms of balance sheet size or amount of time spent in a very low interest rate environment. (7) Accordingly, policies that had widespread support early on in a ZLB episode might be difficult to extend or expand with an already large balance sheet.

So, while they can be valuable, unconventional policies also appear to be less-than-perfect substitutes for changes in short-term policy rates. Accordingly, the ZLB presents a different set of risks to policymakers than those they face during more conventional times, and thus they are worthy of consideration in their own right. We abstract from unconventional policy tools for the remainder of our analysis.

I. Rationales for Risk Management Near the ZLB

The canonical framework of monetary policy analysis assumes that the central bank sets the nominal interest rate to minimize a quadratic loss function of the deviation of inflation from its target and the output gap, and that the economy is described by a set of linear equations. In most applications, uncertainty is incorporated as additive shocks to these linear equations, capturing factors outside the model that lead to variation in economic activity or inflation. (8) A limitation of this approach is that, by construction, it denies that a policymaker might choose to adjust policy in the face of changes in uncertainty about economic fundamentals. However, the evidence discussed below in sections II and III suggests that in practice, policymakers are sensitive to uncertainty and respond to it by following what appears to be a risk-management approach. Understanding why a central banker should behave in this way requires some departure from the canonical framework. The main contribution of this section is to consider a departure associated with the possibility of a binding ZLB in the future. We show that when a policymaker might be constrained by the ZLB in the future, optimal policy today should take account of uncertainty about fundamentals. We focus on two distinct channels through which this can occur. First, we use the workhorse forward-looking New Keynesian model to illustrate the expectations channel, in which the possibility of a binding ZLB tomorrow leads to lower expected inflation and an output gap occurring today, thus necessitating policy easing today. We then use a backward-looking "Old" Keynesian setup to illustrate the buffer stock channel, in which it can be optimal to build up output or inflation today in order to reduce the likelihood and severity of being constrained by the ZLB tomorrow. Both of these channels operate in modern DSGE (dynamic stochastic general equilibrium) models such as those described by Lawrence Christiano, Martin Eichenbaum, and Charles Evans (2005) and by Frank Smets and Rafael Wouters (2007), but they are more transparent if we consider them in separate, although related, simple models. After describing these two channels we construct some numerical simulations to assess their quantitative effects.

I.A. The Expectations Channel

The simple New Keynesian model has well established micro-foundations based on price stickiness. Given that excellent expositions of these foundations have been offered many times, for example by Michael Woodford (2003) and Jordi Gall (2008), we simply state our notation without much explanation. The model consists of two main equations, the Phillips curve and the IS curve.

The Phillips curve is specified as

(1) [[pi].sub.t] = [KX.sub.t] + [beta][E.sub.t][[pi].sub.t+1] + [u.sub.t],

where [[pi].sub.t] and [x.sub.t] are both endogenous variables and denote inflation and the output gap at date t; [E.sub.t] is the date t conditional expectations operator with rational expectations is assumed; [u.sub.t] is a mean zero exogenous cost-push shock; and 0 < [beta] < 1, [kappa] > 0. For simplicity we assume the central bank has a constant inflation target equal to zero, so [[pi].sub.t] is the deviation of inflation from that target. The cost-push shock represents exogenous changes to inflation such as an independent decline in inflation expectations, dollar appreciation, or changes in oil prices.

The IS curve is specified as

(2) [[pi].sub.t] = [E.sub.t][x.sub.t+1] - 1 / [sigma] ([i.sub.t] - [E.sub.t][[pi].sub.t+1] - [[rho].sup.n.sub.t]),

where [sigma] > 0, [i.sub.t] is the nominal interest rate controlled by the central bank, and [[rho].sup.n.sub.t] is the natural rate of interest given by

(3) [[rho].sup.n.sub.t] = [bar.[rho]] + [sigma][g.sub.t] + [sigma][E.sub.t] ([z.sub.t+1] - [z.sub.t]).

The variable [g.sub.t] is an exogenous mean zero demand shock, and [z.sub.t] is the exogenous log of potential output. Since [g.sub.t] and [z.sub.t] are exogenous, so is the natural rate. Equation 2 indicates that [[rho].sup.n.sub.t] corresponds to the setting of the nominal interest rate consistent with expected inflation at target and the output gap equal to zero. (9) If potential output is constant and the demand shock equals zero, then the natural rate equals the constant [bar.[rho]] > 0.

Our analysis is centered on uncertainty in the natural rate. (10) 11 From equation 3 we see that this uncertainty derives from uncertainty about [g.sub.t] and [E.sub.t] ([z.sub.t+1] - [z.sub.t]). We interpret the former as arising due to a variety of factors, including fiscal policy, foreign economies' growth, and financial considerations such as deleveraging. (11) The latter source of uncertainty is over the variety of factors that can influence the expected rate of growth in potential output, for example as emphasized in the recent debate over so-called secular stagnation.

We adopt the canonical framework in assuming that the central bank acts to minimize a quadratic loss function with the understanding that private-sector behavior is governed by equations 1 through 3. The loss function is

(4) L = 1/2 [E.sub.0] [[infinity].summation over (t=0)] [[beta].sup.t] ([[pi].sup.2.sub.t] + [lambda][x.sup.2.sub.t]),

where [lambda] [greater than or equal to] 0. We further assume the ZLB constraint, that is, [i.sub.t] [greater than or equal to] 0, abstracting from the possibility that the effective lower bound on [i.sub.t], is slightly negative. The short-term interest rate is the central bank's only policy instrument, and it is set by solving for optimal policy under discretion. In particular, in each period the central bank sets the nominal interest rate with the understanding that private agents anticipate that it will re-optimize in the following periods.

We focus on optimal policy under discretion for two reasons. First, the case of commitment with a binding ZLB already has been studied extensively. In particular, it is well known from the contributions of Paul Krugman (1998), Gauti Egertsson and Michael Woodford (2003), Woodford (2012), and Ivan Weming (2012) that commitment can reduce the severity of the ZLB problem by creating higher expectations of inflation and the output gap. One implication of these studies is that the central bank should commit to keeping the policy rate at zero longer than would be prescribed by discretionary policy. By studying optimal policy under discretion we find a different rationale for a policy of keeping rates "lower for longer" that does not rely on the central bank having the ability to commit to a time-inconsistent policy. (12) Nevertheless, below we discuss our intuition for why our main result should extend to the case of commitment. Second, discretion may better approximate the institutional environment in which the FOMC operates.

A ZLB SCENARIO We study optimal policy when the central bank is faced with the following simple ZLB scenario. The central bank observes the current value of the natural rate, [[rho].sup.n.sub.0], and the cost-push shock [u.sub.0]; moreover, there is no uncertainty in the natural rate after t = 2, [[rho].sup.n.sub.t] = [bar.[rho]] > 0 for all t [greater than or equal to] 2, nor in the cost-push shock after t = 1, [u.sub.t] = 0 for all t [greater than or equal to] 1. However, there is uncertainty at t = 1 regarding the natural rate [[rho].sup.n.sub.1]. (13) The variable [[rho].sup.n.sub.1] is assumed to be distributed according to the probability density function [f.sub.[rho]](*).

This very simple scenario keeps the optimal policy calculation tractable while preserving the main insights. We also think it captures some key elements of uncertainty faced by the FOMC today; notably, our formulation allows us to consider the optimal timing of liftoff. We do not have to take a stand on whether the ZLB is binding before t = 0, but one possibility is that the natural rate p" was sufficiently negative for r < 0 so that the optimal policy rate was set at zero, [i.sub.t] = 0 for t < 0, but because the economy has been improving the natural rate is close to zero by t = 0. The question is whether to raise the policy rate at t = 0, t - 1, or t = 2.

ANALYSIS To find the optimal policy, we solve the model backwards from t = 2 and focus on the policy choice at t = 0. First, for t [greater than or equal to] 2, it is possible to perfectly stabilize the economy by setting the nominal interest rate equal to the (now positive) natural rate, [i.sub.t], = [[rho].sup.n.sub.t] = [bar.[rho]]. This leads to [[pi].sub.t], = [x.sub.t] = 0 for t [greater than or equal to] 2. (14) The optimal policy at t = 1 will depend on the realized value of the natural rate [[rho].sup.n.sub.1]. If [[rho].sup.n.sub.1] [greater than or equal to] 0, then it is again possible (and optimal) to perfectly stabilize by setting [i.sub.1] = [[rho].sup.n.sub.1], leading to [x.sub.1] = [[pi].sub.1], = 0. However if [[rho].sup.n.sub.1] < 0, the ZLB binds and consequently [x.sub.1] = [[rho].sup.n.sub.1]/[sigma] < 0. The expected output gap at t = 1 is [E.sub.0][x.sub.1] = [[integral].sup.0.sub.-[infinity]] [rho] [f.sub.[rho]]([rho])[d.sub.[rho]]/[sigma] [less than or equal to] 0 and expected inflation is [E.sub.0][[pi].sub.1] = [kappa][E.sub.0][x.sub.1] < 0.

Because agents are forward-looking, this low expected output gap and inflation feed backward to t = 0. A low output gap tomorrow depresses output today by a wealth effect via the IS curve. Low inflation tomorrow depresses inflation today, since price-setting is forward-looking in the Phillips curve, and it also depresses output today by raising the real interest rate via the IS curve. The optimal policy at t = 0 must take into account these effects. This implies that optimal policy will be looser than if there were no chance that the ZLB would bind tomorrow.

Mathematically, substituting for [[pi].sub.0] and [i.sub.0] using equations 1 and 2, and taking into account the ZLB constraint, optimal policy at t = 0 solves the following problem:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Two cases arise, depending on whether the ZLB binds at t = 0 or not. Define the threshold value

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [[rho].sup.n.sub.0] > [[rho].sup.*.sub.0], then the optimal policy is to follow the standard monetary policy response to an inflation shock to the Phillips curve, [beta][E.sub.0][[pi].sub.1] + [u.sub.0], leading to

(7) [x.sub.0] = - [kappa]/[lambda] + [[kappa].sub.2] ([beta][E.sub.0] [[pi].sub.1] + [u.sub.0]); [[pi].sub.0] = - [lambda]/[lambda] + [[kappa].sub.2] ([beta][E.sub.0] [[pi].sub.1] + [u.sub.0]).

The corresponding interest rate is

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As long as [[integral].sup.0.sub.-[infinity]] [rho][f.sub.[rho]]([rho])dp < 0, equation 8 implies that the optimal interest rate is lower than if there were no chance of a binding ZLB tomorrow, that is, if [f.sub.[rho]] ([rho]) = 0 for p [less than or equal to] 0. The interest rate is lower today to offset the deflationary and recessionary effects of the possibility of a binding ZLB tomorrow. If [[rho].sup.n.sub.0] < [[rho].sup.*.sub.0] then the ZLB binds today and optimal policy is [i.sub.0] = 0. In this case,

(9) [x.sub.0] = [[rho].sup.n.sub.0]/[sigma] + (1 + [kappa]/[sigma]) [E.sub.0][x.sub.1]; [[pi].sub.0] = [kappa] [[rho].sup.n.sub.0]/[sigma] + [(1 + [beta])[kappa] + [[kappa].sub.2]/[sigma]] [E.sub.0] [x.sub.1].

Notice from equation 6 that higher uncertainty makes it more likely that the ZLB will bind at t = 0. Specifically, even if agents were certain that the ZLB would not bind at t - 1, [E.sub.0][x.sub.1] = [E.sub.0][[pi].sub.1] = 0 and [i.sub.0] = 0 if [[rho].sup.n.sub.0] [less than or equal to] - [sigma][kappa][u.sub.0]/([lambda] + [[kappa].sup.2]).

So the possibility of the ZLB binding tomorrow increases the chances of being constrained by the ZLB today.

Since [E.sub.0][x.sub.1] is a sufficient statistic for [[integral].sup.0.sub.-[infinity]] [rho][f.sub.[rho]]([rho])d [rho] in equation 8, the optimal policy has the flavor of a traditional forward-looking policy reaction function that only depends on the conditional expectations of output and inflation gaps. However [E.sub.0][x.sub.1] is not independent of a mean-preserving spread or any other change in the distribution of [[rho].sup.n.sub.1]. Accordingly, optimal policy here departs from the certainty equivalence principle, which says that the extent of uncertainty in the underlying fundamentals (in our case [[rho].sup.n.sub.1]) does not affect the optimal interest rate. (15) Furthermore, as a practical matter the central bank must infer private agents' [E.sub.0][x.sub.1] in order to determine optimal policy. Since [E.sub.0][x.sub.1] depends on the entire distribution of p", so must the central bank's estimates of it, which is a much more difficult inference problem than in the certainty equivalence case.

Turning specifically to the issue of uncertainty, we obtain the following unambiguous comparative static result:

Proposition 1: Higher uncertainty, that is, a mean-preserving spread in the distribution of the natural rate p" tomorrow, leads to a looser optimal policy today.

To see this, rewrite the key quantity [[integral].sup.0.sub.-[infinity]] [rho][f.sub.[rho]]([rho])d [rho] = E min([rho],0). Since the min function is concave, higher uncertainty through a mean-preserving spread about [[rho].sup.n.sub.1] leads to lower, that is, more negative, [E.sub.0][x.sub.1] and [E.sub.0][[pi].sub.1],. Hence, higher uncertainty leads to lower [i.sub.0]. (16)

The effect of higher uncertainty on [i.sub.0] is unambiguous, but the effect on the output gap and inflation is more subtle. If the ZLB does not bind at t = 0 initially, higher uncertainty leads to lower [E.sub.0][x.sub.1] and [E.sub.0][[pi].sub.1] and consequently to higher [x.sub.0] and lower [[pi].sub.0] according to equation 7. On the other hand, if the ZLB does bind at t = 0 initially, then higher uncertainty leads to lower [x.sub.0] and lower [[pi].sub.0] according to equation 9. (17) Overall, the effect of higher uncertainty on [[pi].sub.0] is unambiguously negative, but the effect on [x.sub.0] may be positive or negative.

Another interesting feature of the solution is that the distribution of the positive values of [[rho].sup.n.sub.1] is irrelevant for policy. That is, policy today is adjusted only with respect to the states of the world in which the ZLB might bind tomorrow. The logic is that if a very high value of [[rho].sup.n.sub.1] is realized, monetary policy can adjust to it and prevent a bout of inflation. This is a consequence of the standard principle that, outside the ZLB, natural rate shocks can and should be perfectly offset by monetary policy.

DISCUSSION Proposition 1 has several predecessors. Perhaps the closest are Klaus Adam and Roberto Billi (2007), Taisuke Nakata (2013a,b), and Anton Nakov (2008), who demonstrate numerically how, in a stochastic environment, the ZLB leads the central bank to adopt a looser policy. Our contribution is to provide a simple analytical example. (18) This result has been correctly interpreted to mean that if negative shocks to the natural rate lead the economy to be close to the ZLB, the optimal response is to lower the interest rate aggressively to reduce the likelihood that the ZLB becomes binding. The same logic applies to liftoff. Following an episode where the ZLB has been a binding constraint, the central bank should not raise rates as if it were sure the ZLB constraint would never bind again. (19) Even though the best forecast may be that the economy will recover and exit the ZLB--that is, in the context of the model, that [E.sub.0]( [[rho].sup.n.sub.1]) > 0--it can be optimal to have zero interest rates today. Note that policy is looser when the probability of being constrained by the ZLB in the future is high or the potential severity of the ZLB problem is large; that is, when [[integral].sup.0.sub.-[infinity]] [rho][f.sub.[rho]]([rho])d [rho] is a large negative number; the economy is less sensitive to interest rates (high [sigma]); and the Phillips curve is steep (high [kappa]).

With higher uncertainty, the increase in interest rates will be faster on average from t = 0 to t = 2. This follows since the / = 2 interest rate is unaffected by uncertainty whereas at t = 0 it is lower. More generally, when uncertainty about being constrained by the ZLB in the future dissipates, the interest rate can rise quickly because the effects holding it down disappear along with the uncertainty.

While we have deliberately focused on a very simple example, our results hold under more general conditions. For instance, the same results still hold if [{[[rho].sup.n.sub.1]}.sub.t [greater than or equal to]2] follows an arbitrary stochastic process, as long as it is positive. In the online appendix we consider the case of optimal policy with uncertainty about cost-push inflation. (20) We show that optimal policy also is looser if there is a chance of a binding ZLB in the future due to a low cost-push shock. Furthermore, the risk that inflation picks up due to a high cost-push shock does not affect policy today. If such a shock were to occur tomorrow, it would lead to some inflation; however, there is nothing that policy today can do about it. Finally, while the model chosen is highly stylized, the core insights would likely continue to hold in a medium-scale model with a variety of shocks and frictions.

Intuitively, we expect a version of Proposition 1 to still hold with commitment as well. Optimal policy with commitment involves promising at t = 0 that should the ZLB bind at t = 1, the central bank would keep interest rates lower for t [greater than or equal to] 2 than it would otherwise. As is well known, this policy reduces the size of the inflation and output gaps at t = 1, but it does not eliminate them entirely. These gaps then could generate negative expected inflation and output gaps at t - 0 that become more negative the larger the t = 1 uncertainty. Higher uncertainty should therefore lead to looser policy at t = 0, just as in the case of discretion.

One obvious limitation to these results is that we have assumed (and will continue do so when studying the backward-looking model below) that there is no cost to raising rates quickly if needed. For example, our welfare criterion does not value interest-rate smoothing. Smoothing has been rationalized by Marvin Goodfriend (1991) and others as facilitating financial market adjustments or as a signaling tool. It is true also that estimated reaction functions include lagged funds rate terms to fit historical data. Nonetheless, there have been instances when the FOMC has moved quickly. Some of these occurred as recessions unfolded, but not all: between February 1994 and February 1995 rates were tightened by 300 basis points and between November 1988 and February 1989 by nearly 165 basis points. Moreover, as Brian Sack (2000) and Glenn Rudebusch (2002) argue, interest rate smoothing might reflect learning about an uncertain economy rather than a desire to avoid large changes in interest rates per se. The policy prescriptions derived from our models are specifically aimed at addressing such uncertainty.

I.B. The Buffer Stock Channel

The buffer stock channel relies not on forward-looking behavior but on the view that the economy has some inherent momentum, for instance due to adaptive inflation expectations, inflation indexation, habit persistence, adjustment costs, or hysteresis. Suppose that output or inflation has a tendency to persist. If there is a risk that the ZLB binds tomorrow, building up output and inflation today creates some buffer against hitting the ZLB tomorrow.

This intuition does not guarantee that it is optimal to increase output or inflation today. In particular, the benefit of higher inflation or output today in the event that a ZLB event arises tomorrow must be weighed against the costs of excess output and inflation today, as well as tomorrow's cost to bring down the output gap or inflation if the ZLB turns out not to bind. So it is important to verify that our intuition holds up in a model.

To isolate the buffer stock channel from the expectations channel we focus on a purely backward-looking "Old" Keynesian model. Purely backwardlooking models do not have micro-foundations as the New Keynesian model does, but backward-looking elements appear to be important empirically. (21) Backward-looking models have been studied extensively in the literature, including by Thomas Laubach and John Williams (2003), Athanasios Orphanides and Williams (2002), David Reifschneider and Williams (2000), and Rudebusch and Lars Svensson (1999).

The model we study simply replaces the forward-looking terms in equations 1 and 2 with backward-looking terms:

(10) [[pi].sub.t] = [xi][[pi].sub.t-1] + [kappa][x.sub.1] + [u.sub.1];

(11) [x.sub.t] = [delta][x.sub.t-1] - 1/[sigma] ([i.sub.t] - [[rho].sup.n.sub.t] - [[pi].sub.t-1]),

where 0 < [xi] < 1 and 0 < [delta] < 1. This model is essentially the same as the simple example Reifschneider and Williams (2000) use to motivate their analysis of monetary policy constrained by the ZLB. Unlike in the New Keynesian model, it is difficult to map p'/ directly to underlying fundamental shocks as we do in equation 3. For simplicity, we continue to refer to this exogenous variable as the natural rate and use equation 3 as a guide to interpreting it, but it is perhaps better to think of it as simply a "demand" shock or "IS" shock.

ANALYSIS We consider the ZLB scenario described in section I.A (under "A ZLB Scenario") and again solve the model backwards from t = 2 to determine optimal policy at t = 0 and how this is affected by uncertainty in the natural rate at t = 1. After t = 1 the economy does not experience any more shocks, but it inherits initial lagged inflation and output terms [[pi].sub.1], and [x.sub.1], which may be positive or negative. The output gap term can be easily adjusted by changing the interest rate [i.sub.t] provided the central bank is not constrained by the ZLB at t = 2, that is, if [[rho].sup.n.sub.2], = [bar.[rho]] is large enough, an assumption we will maintain. (22) Given the quadratic loss, it is optimal to smooth this adjustment over time so that the economy will converge back to its steady-state slowly. The details of this adjustment after t = 2 are not very important for our analysis. What is important is that the overall loss of starting from t = 2 with lagged inflation [[pi].sub.1], and output gap [x.sub.1] is a quadratic function of [[pi].sub.1] only; we can write it as W[[pi].sup.2.sub.1]/2, where W is a constant that depends on [lambda], [kappa], and [beta] and is calculated in the online appendix.

Turn now to optimal policy at t = 1. Take the realization of p,' and last period's output gap [x.sub.0] and inflation [[pi].sub.0] as given. Substituting for [[pi].sub.1], and [i.sub.1] using equations 10 and 11, and taking into account the ZLB constraint, optimal policy at t = 1 solves the following problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the policymaker now anticipates the cost of having inflation [[pi].sub.1], tomorrow, and her choices are affected by yesterday's values [x.sub.0] and [[pi].sub.0].

Depending on the value of [[rho].sup.n.sub.1], two cases can arise. Define the threshold value:

(12) [[rho].sup.*.sub.1] ([x.sub.0], [[pi].sub.0]) = - [(1 + [beta]W) [kappa][xi]/(1 + [beta]W)[[kappa].sup.2] + [lambda] + 1] [[pi].sub.0] - [sigma][delta][x.sub.0].

For [[rho].sup.n.sub.1] [greater than or equal to] [p.sup.*.sub.1] ([x.sub.0], [[pi].sub.0]) the ZLB is not binding; otherwise it is. Hence the probability of hitting the ZLB is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [f.sub.[rho]] ([rho])d[rho]. In contrast to the forward-looking case, the probability of being constrained by the ZLB constraint is now endogenous at t = 1 and can be influenced by policy at t = 0. As indicated by equation 12, a higher output gap or inflation at t = 0 will reduce the likelihood of hitting the ZLB at t = 1.

If [[rho].sup.n.sub.1] [greater than or equal to] [[rho].sup.*.sub.1] ([x.sub.0], [[pi].sub.0]) optimal policy at t = 1 yields

[x.sub.1] = - (1 + [beta]W) [kappa][xi]/(1 + [beta]W) [[kappa].sup.2] + [lambda] [[rho].sub.0]; [[pi].sub.1] = [lambda][xi]/(1 + [beta]W) [[kappa].sup.2] + [lambda] [[pi].sub.0].

This is similar to the forward-looking model's solution, which reflects the trade-off between output and inflation, except that optimal policy now takes into account the cost of having inflation away from target tomorrow, through W. The loss for this case is V([x.sub.0], [[pi].sub.0], [[rho].sup.n.sub.1]) = W[[pi].sup.2.sub.0]/2, since in this case the problem is the same as the one faced at t = 2. If [[rho].sup.n.sub.1] < [p.sup.*]([x.sub.0], [[pi].sub.0]) the ZLB binds, in which case

[x.sub.1] = [delta][x.sub.0] + [[pi].sub.0] + [[rho].sup.n.sub.1]/[sigma]; [[pi].sub.1] = [kappa][delta][x.sub.0] + [[pi].sub.0] ([xi] + [kappa]/[sigma]) + [kappa] [[rho].sup.n.sub.1]/[sigma].

The expected loss from t = 1 on as a function of the output gap and inflation at t = 0 is then given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This expression reveals that the initial conditions [x.sub.0] and [[pi].sub.0] matter by shifting the payoff from continuation in the non-ZLB states, W[[pi].sup.2.sub.0]/2 the payoff in the case where the ZLB binds (the second integral); and the relative likelihood of ZLB and non-ZLB states through [[rho].sup.*.sub.1] ([x.sub.0], [[pi].sub.0]). Since the loss function is continuous in p, even at pf(jt0,7t0), this last effect is irrelevant for welfare at the margin.

The last step is to find the optimal policy at time 0, taking into account the effect on the expected loss tomorrow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We use this expression to prove the following, which is analogous to Proposition 1:

Proposition 2: For any initial condition, a mean-preserving spread in the distribution of the natural rate p, tomorrow leads to a looser optimal policy today.

From equations 10 and 11, higher uncertainty also leads to larger [x.sub.0] and [[pi].sub.0]. The proof of Proposition 2 is in the appendix. Note that it incorporates the case of uncertainty regarding cost-push shocks at t = 1 and shows that a mean-preserving spread in the cost-push shock tomorrow leads to looser policy today as well.

Our model also implies that an increase in uncertainty over the initial output gap will lead to looser policy. Specifically we have:

Proposition 3: Suppose the initial output gap [x.sub.-1], is unknown at t = 0 but becomes known at t= 1 and the central bank has a prior distribution over [x.sub.-1]. Then a mean-preserving spread in this prior distribution leads optimal policy to be looser at t = 0.

The proof of this proposition is similar to the one for Proposition 2. This result is particularly germane to the current policy environment where there is uncertainty over the amount of slack in the economy. Therefore, Proposition 3 provides an additional rationale for delaying liftoff.

DISCUSSION As far as we know, Proposition 2 is a new result, but its implications are similar to those of Proposition 1. As in the forward-looking case, liftoff from an optimal zero interest rate should be delayed today with an increase in uncertainty about the natural rate or cost-push shock that raises the odds of the ZLB binding tomorrow. Similarly, even if not constrained by the ZLB today, an increase in uncertainty about the likelihood of being constrained by the ZLB tomorrow leads to a reduction in the policy rate today. So the buffer stock channel and the expectations channel have very similar policy implications, though for very different reasons. The expectations channel involves the possibility of being constrained by the ZLB tomorrow feeding backward to looser policy today. The buffer stock channel has looser policy today feeding forward to reduce the likelihood and severity of being at the ZLB tomorrow. Note that as in the forward-looking model, optimal policy prescribes that interest rates rise as the likelihood of being constrained by the ZLB in the future falls, even if the output gap or inflation does not change.

It is useful to compare the policy implications of the buffer stock channel to the argument developed in Olivier Coibion, Yuriy Gorodnichenko, and Johannes Wieland (2012). Their paper studies the tradeoff between the level of the inflation target and the risk of hitting the ZLB using policy reaction functions instead of optimal policy. (23) Our analysis does not require a drastic change in monetary policy in order to improve outcomes. It is achieved through standard interest-rate policy rather than a credibility-damaging change to the inflation target.

I.C. Quantitative Assessment

We now assess the quantitative significance of the expectations and buffer stock channels using calibrated versions of the forward- and backward-looking models that we solve numerically. With parameters drawn from the literature and initial conditions calibrated to early 2015, we compare equilibrium outcomes under optimal discretion to alternative policies that do not take into account uncertainty. Our numerical methods are described in the online appendix. Importantly, and in contrast to most of the literature, they allow uncertainty to affect policy and to be reflected in welfare.

PARAMETER VALUES The parameter values are reported in table 1. We use the same values for parameters that are common to both models. The time period is one quarter, with t = 1 taken to be 2015Q1. The natural rate [[rho].sup.n.sub.1] is the sum of deterministic and random components. We assume the deterministic component rises linearly between t = 1 and t = T > 1, after which it remains constant at [bar.p] = 1.75 percent, which corresponds with the median long-run funds rate in the March 2015 FOMC Summary of Economic Projections, less the FOMC's inflation target [[pi].sup.*] = 2. The random component is AR(1) with auto-correlation coefficient pf and innovation standard deviation [[sigma].sub.[epsilon]]. We also assume there is an i.i.d. cost-push shock with standard deviation [[sigma].sub.u]. There is no uncertainty for t > T.

The degree of uncertainty we assume is central to our findings. The particular values of [[rho].sub.[epsilon]] and [[sigma].sub.[epsilon]] are not as important to our results as the unconditional volatility they imply. There is wide variation in estimates of volatility in the natural rate, corresponding to differences in theoretical concepts, models and empirical methods used. Our calibration implies that the unconditional standard deviation of the natural rate is 2.5 percent at an annual rate. This lies within the range of estimates in Robert Barsky, Alejandro Justiniano, and Leonardo Melosi (2014), Vasco Curdia and others (2015), and Laubach and Williams (2003). The auto-correlation coefficient is set midway between the values in Adam and Billi (2007) and Curdia and others (2015). We set the standard deviation of the cost-push shock [[sigma].sub.u] close to the value used in Adam and Billi (2007). Assuming serial correlation or a moderately different unconditional standard deviation of the cost-push shock is not very important for our results. Finally, by assuming that the economy is not subject to shocks for t > T and that the long run natural rate [bar.[rho]] is a known constant, we have been conservative in our specification of uncertainty.

The Phillips curve slope, elasticity of intertemporal substitution, and discount factor are all set to values common in the New Keynesian literature. For the backward-looking model we set the coefficient on lagged inflation in equation 10 to [xi] = 0.95, reflecting the fact that inflation has been very persistent in recent years. (24) The coefficient on lagged output in equation 11 is [delta] = 0.75, in order to generate significant persistence in the output gap. For the backward-looking model we assume an initial inflation rate of 1.3 percent, a recent reading for core PCE inflation, and an initial output gap [x.sub.0] = - 1.5 percent, based on a simple calculation using the 2014Q4 unemployment rate (5.7 percent), an estimate of the natural rate of unemployment (5.0 percent), and Okun's law. As indicated by Proposition 3, adding uncertainty about the initial output gap would only strengthen our results. (25)

We measure the quantitative effect of uncertainty on policy by comparing equilibrium outcomes under optimal discretion to a scenario in which we solve for optimal discretion when the central bank observes the current natural rate and cost-push shocks but acts as if there will be no more shocks. Private agents understand this policy but take into account the true nature of uncertainty. Actual outcomes will be inconsistent with the central bank's assumptions, so we call this the "naive" policy. We also compare equilibrium outcomes under optimal discretion to those obtained assuming the central bank follows a reaction function with weights on inflation and the output gap as in John Taylor (1993), and a constant term equal to 3.75 percent corresponding to [rho] + [[pi].sup.*].

RESULTS FOR THE FORWARD-LOOKING MODEL Figure 1 displays representative paths of the nominal interest rate, inflation, and the output gap under optimal discretion, the naive policy, and the Taylor rule, calculated by setting the ex post realized shocks to zero, the modal outcome. Under the modal outcome, the interest rate under the naive policy follows the natural rate exactly. The difference between the interest rate paths indicates the substantial impact uncertainty has on optimal policy; the naive policy is between 50 and 150 basis points above the optimal policy for 2 years. This difference in policy has little impact on the output gap, but under optimal policy the inflation gap is closed much faster. The inflation gap is more negative under the naive policy because the interest rate is higher both initially and in the future since it does not take into account uncertainty about the ZLB. (26) The Taylor rule prescribes rates above both the optimal and naive policies for most of the simulation period, and because agents are forward looking this feeds backward to cause much more negative gaps. (27)

Table 2 summarizes the distribution of outcomes under the three different policies based on simulating 50,000 paths drawn from the calibrated distributions of the shocks. Optimal discretion implies one-third the loss expected under the naive policy and one-eighth the loss expected under the Taylor rule. (28) One way to interpret these losses is to calculate the per-period reduction in the output gaps and inflation gaps that would make the central banker indifferent between the outcomes under the optimal policy and those under the alternatives. Both gaps would have to be 43 percent and 65 percent smaller under the naive policy and the Taylor rule, respectively, to achieve this indifference. Under optimal discretion, the median liftoff (defined as the nominal interest rate exceeding 25 basis points) is delayed by 2 quarters compared to the other policies; the mean liftoff is delayed by more than 3 quarters, reflecting skewness in the outcomes. At the time of liftoff, inflation and output are much closer to the target under optimal discretion compared to the two alternative policies.

[FIGURE 1 OMITTED]

When comparing policies it is also important to assess how well each balances the risks of bad outcomes. We do this by comparing the 75th percentile across simulations of the maximum inflation gap and the 25th percentile of the lowest output gap over the first 6 years. Under optimal policy, the bad output outcomes are much lower than under either alternative policy. The bad inflation outcomes do not seem particularly high under any of the policies.

The statistic in the bottom row is the median standard deviation of changes in the nominal interest rate. By comparing interest rate volatility under the Taylor rule in our model with that implied by the same Taylor rule in the data, we can determine whether the uncertainty underlying our results is reasonable. If the volatility were much higher in our simulations we would conclude that it is unreasonably large. In fact, the 0.97 standard deviation in our Taylor rule simulations is only a little larger than the 0.88 standard deviation we find in our data. (29) Interest rates are more volatile under both the optimal and naive policies because they respond to all fundamental shocks rather than to inflation and output alone. (30)

RESULTS FOR THE BACKWARD-LOOKING MODEL Figure 2 is the analog of figure 1 for the backward-looking model. The dynamics of return to target are quite different from those in the forward-looking model, but the key qualitative results are the same. As in the forward-looking model, optimal policy is substantially looser than both the naive policy and the Taylor rule. Here the optimal policy prescribes much more delay in lifting off from the ZLB. Delay now occurs under the naive policy because it is optimal to stimulate output strongly in order to return inflation to target, but this delay is shorter than under the optimal policy. The optimal policy also has a sharper liftoff than the naive policy. However, the increases under optimal policy are equivalent to just 25 basis points at each FOMC meeting, the same as the "measured pace" followed during the Fed tightening over 2004-06. Qualitatively, the differences in the output and inflation outcomes across the three policies are similar to those in the forward-looking model as well. Taking into account uncertainty about the ZLB leads the optimal policy to return inflation to target faster than the naive policy, and it achieves this by allowing the output gap to overshoot more in order to build a buffer against the possibility of bad shocks in the future.

[FIGURE 2 OMITTED]

Table 3 is constructed analogously to table 2. It shows that optimal policy provides only a marginal improvement over the naive policy in terms of expected losses, due to the offsetting effects of the inflation and output gaps. The median gaps are roughly closed at liftoff under both the optimal and naive policies, but they are quite large under the Taylor rule. The bad outcomes are similar across the three scenarios. Finally, note that the volatility of the interest rate under the Taylor rule is lower here compared to the data and the forward-looking model, so the underlying uncertainty is not excessive.

We conclude by illustrating one of the risks the optimal policy is able to address, namely the possibility that a shock will drive up inflation before the baseline liftoff. Figure 3 depicts a particular simulation where there is a large positive cost-push shock before the liftoff under the optimal policy shown in figure 2. The shock triggers earlier liftoff under the optimal policy so that the inflation response is mild. The implication is that staying at zero longer under the optimal policy does not impair the ability of the central bank to respond to future contingencies. However, it does have to be prepared to raise rates promptly. We obtain similar results with the forward-looking model.

II. Historical Precedents for Risk Management

The previous section demonstrates that the ZLB justifies a risk management approach to monetary policy. One may question whether following such an approach would be a departure from past FOMC behavior. Clearly, concerns about the ZLB are a relatively recent phenomenon. Nevertheless there are many reasons why a risk management approach can be justified when away from the ZLB, and we begin this section by reviewing them. We then demonstrate that the Federal Reserve has used risk management to justify its policy decisions over the period 1987-2008.

[FIGURE 3 OMITTED]

The FOMC minutes and other Federal Reserve communications reveal a number of episodes when uncertainty or insurance were used to justify the Fed's policy decisions. Sometimes the FOMC indicated that it had a wait-and-see approach to taking further actions or muted a funds rate move due to its uncertainty over the course of the economy or the extent to which early policy moves had yet shown through to economic activity and inflation. At other times the FOMC said its policy stance was taken in part as insurance against undesirable outcomes; during these times, the FOMC often noted that the potential costs of a policy overreaction likely were modest compared to the scenario it was insuring against.

Two episodes are particularly revealing. The first is the hesitancy of the FOMC to raise rates in 1997 and 1998 to counter inflationary threats because of uncertainty generated by the Asian financial crisis and the subsequent rate cuts following the Russian default. The second is the loosening of policy over 2000 and 2001, when uncertainty over the degree to which growth was slowing and the desire to insure against downside risks appeared to influence policy. Furthermore, in late 2001 the FOMC's aggressive actions also seemed to be influenced by attention to the risks associated with the ZLB on interest rates.

While the historical record is replete with references suggesting that the policy stance was influenced by uncertainty or insurance motives, this does not establish that risk management actually had a material impact on policy. Therefore, we conclude this section by quantifying these references into variables that we then use in section III to assess the importance of risk management for actual policy decisions.

II.A. Rationales for Risk Management Away from the ZLB

Policymakers have long emphasized the importance of uncertainty in their decision making. As Alan Greenspan (2004) put it: "The Federal Reserve's experiences over the past two decades make it clear that uncertainty is not just a pervasive feature of the monetary policy landscape; it is the defining characteristic of that landscape." (31) This sentiment seems at odds with linear-quadratic models in which optimal policy involves adjusting the interest rate in response to only the mean of the distribution of shocks away from the ZLB. What kinds of factors cause departures from such conditions and justify the risk management approach?

Relaxing the assumption of a quadratic loss function is perhaps the simplest way to generate a rationale for risk management. The quadratic loss function is justified by Woodford (2003) as being a local approximation to consumer welfare. However, it might not be a good approximation when large shocks drive the economy far from the underlying trend; alternatively, it might simply be an inadequate approximation of FOMC behavior. Examples of models with asymmetric loss functions include those described by Paolo Surico (2007), Lutz Kilian and Simone Manganelli (2008), and Juan J. Dolado, P. Ramon Marfa-Dolores, and Francisco Ruge-Murcia (2004). (32) The model studied by the last authors implies that the optimal policy rule can involve nonlinear output gap and inflation terms if policymakers are less averse to allowing output to run above potential than below it. The relevance of higher moments in the distribution of shocks for optimal policy is an obvious by-product of these nonlinearities.

Nonlinearities in economic dynamics are another natural motivation. For example, suppose recessions are episodes when self-reinforcing dynamics amplify the effects of downside shocks. This could be modeled as a dependence of current output on lagged output, as in our backward-looking model, but with such dependence being concave rather than linear. Intuitively, negative shocks have a more dramatic effect on reducing future output than positive shocks have on increasing it, so greater uncertainty leads to looser optimal policy to guard against the more detrimental outcomes. Alternatively, suppose the Phillips curve is convex, perhaps owing to downward nominal wage rigidities that become more germane with low inflation. Here, a positive shock to the output gap leads to a significant increase of inflation above target while a negative shock leads to a much smaller decline in inflation. The larger the spread of these shocks, the greater the odds of experiencing a bad inflation outcome. Optimal policy guards against this, leading to a tightening bias. (33) The risk management approach also appears in the large literature on how optimal monetary policy should adjust for uncertainty about the true model of the economy. Brainard (1967) derived the important result that uncertainty over the effects of policy should lead to caution and smaller policy responses to deviations from target. In contrast, the robust control analysis of Lars Hansen and Thomas Sargent (2008) has been interpreted to mean that uncertainty over model mis-specification should generate aggressive policy actions. As explained by Gadi Barlevy (2011), both the attenuation and aggressiveness results depend on the specifics of the underlying environment. Nonetheless, these analyses still often indicate that higher moments of the distribution of shocks can influence the setting of optimal policy.

II.B. 1997-98

The year 1997 was a good one for the U.S. economy: real GDP increased 3 3/4 percent (the March 1998 third estimate), the unemployment rate fell to 4.7 percent, and core CPI inflation was 2 1/4 percent. With solid growth and tight labor markets, the FOMC clearly was concerned about a buildup in inflationary pressures. As noted in the Federal Reserve's February 1998 Monetary Policy Report:

   The circumstances that prevailed through most of 1997 required that
   the Federal Reserve remain especially attentive to the risk of a
   pickup in inflation. Labor markets were already tight when the year
   began, and nominal wages had started to rise faster than
   previously. Persistent strength in demand over the year led to
   economic growth in excess of the expansion of the economy's
   potential, intensifying the pressures on labor supplies. (34)

Indeed, over much of the period between early 1997 and mid-1998, the FOMC directive maintained a bias indicating that it was more likely to raise rates to battle inflationary pressures than it was to lower them. Nonetheless, the FOMC left the funds rate unchanged at 5.5 percent from March 1997 until September 1998. Why did it do so?

Certainly the inaction in large part reflected the forecast for growth to moderate to a more sustainable pace as well as the fact that actual inflation had remained contained despite tight labor market conditions. Based on the funds rate remaining at 5.5 percent, the Board of Governors' staff forecast in the August 1998 Greenbook projected GDP growth to slow from 2.9 percent in 1998 to 1.7 percent in 1999. The unemployment rate was projected to rise to 5.1 percent by the end of 1999 and core CPI inflation was projected to edge down to 2.1 percent. Additionally, however, on several occasions heightened uncertainty over the outlook for growth and inflation apparently reinforced the decision to refrain from raising rates. The following quote from the July 1997 FOMC minutes is a revealing example:

   While the members assessed risks surrounding such a forecast as
   decidedly tilted to the upside, the slowing of the expansion should
   keep resource utilization from rising substantially further, and
   this outlook together with the absence of significant early signs
   of rising inflationary pressures suggested the desirability of a
   cautious "wait and see" policy stance at this point. In the current
   uncertain environment, this would afford the FOMC an opportunity to
   gauge the momentum of the expansion and the related degree of
   pressure on resources and prices. (35)

Furthermore, the FOMC did not regard "waiting and seeing" as having a high cost. They thought any increase in inflation would be slow and that, if needed, a limited tightening would be sufficient to rein in any emerging price pressures. This is seen in the following quote from the same meeting:

   The risks of waiting appeared to be limited, given that the
   evidence at hand did not point to a step-up in inflation despite
   low unemployment and that the current stance of monetary policy did
   not seem to be overly accommodative.

   ... In these circumstances, any tendency for price pressures to
   mount was likely to emerge only gradually and to be reversible
   through a relatively limited policy adjustment.

Thus, it appears that uncertainty and associated risk management considerations supported the FOMC's decision to leave policy on hold.

Of course, the potential fallout for the U.S. economy of the Asian financial crisis was a major factor underlying the uncertainty about the outlook. The baseline scenario was that the associated weakening in demand from abroad and a stronger dollar would be enough to keep inflationary pressures in check but would not be strong enough to cause inflation or employment to fall too low. As Chairman Greenspan noted in his February 1998 Humphrey-Hawkins testimony to Congress, there were substantial risks to this outlook, with the delicate balance dictating unchanged policy:

   However, we cannot rule out two other, more worrisome
   possibilities. On the one hand, should the momentum to domestic
   spending not be offset significantly by Asian or other
   developments, the U.S. economy would be on a track along which
   spending could press too strongly against available resources to be
   consistent with contained inflation. On the other, we also need to
   be alert to the possibility that the forces from Asia might damp
   activity and prices by more than is desirable by exerting a
   particularly forceful drag on the volume of net exports and the
   prices of imports. When confronted at the beginning of this month
   with these, for the moment, finely balanced, though powerful
   forces, the members of the Federal Open Market Committee decided
   that monetary policy should most appropriately be kept on hold.
   (36)

By late in the summer of 1998, this balance had changed, as the strains following the Russian default weakened the outlook for foreign growth and tightened financial conditions in the United States. The FOMC was concerned about the direct implications of these developments for U.S. financial markets, already evident in the data, as well as their implications for the real economy, which were still just a prediction. The staff forecast prepared for the September FOMC meeting reduced the projection for growth in 1999 by about 1/2 percentage point to 1 1/4 percent, predicated on a 75 basis-point reduction in the funds rate spread out over three quarters. Such a forecast was not a disaster--indeed, at 5.2 percent the unemployment rate projected for the end of 1999 was still below the staff's estimate of its natural rate. Nonetheless, the FOMC moved much faster than the staff assumed it would, lowering rates 25 basis points at its September and November meetings as well as making an inter-meeting rate cut in October. According to the FOMC minutes, the rate cuts were made in part as insurance against a worsening of financial conditions and weakening activity. As they noted in September of that year:

   Such an action was desirable to cushion the likely adverse
   consequences on future domestic economic activity of the global
   financial turmoil that had weakened foreign economies and of the
   tighter conditions in financial markets in the United States that
   had resulted in part from that turmoil. At a time of abnormally
   high volatility and very substantial uncertainty, it was impossible
   to predict how financial conditions in the United States would
   evolve ... In any event, an easing policy action at this point
   could provide added insurance against the risk of a further
   worsening in financial conditions and a related curtailment in the
   availability of credit to many borrowers. (37)

While the references to insurance are clear, a case also can be made that these policy moves were undertaken largely to realign the misses in the expected paths for growth and inflation from the FOMC's policy goals. At that time, the prescriptions to address the risks to their policy goals were in conflict: risks to achieving the inflation mandate called for higher interest rates while risks to achieving the maximum employment mandate called for lower rates. As the above quote from Chairman Greenspan's February 1998 testimony indicated, in early 1998 the FOMC thought that a 5 1/2 percent funds rate kept these risks in balance. Subsequently, as the odds of economic weakness increased, the FOMC cut rates to bring the risks to the two goals back into balance. As Chairman Greenspan said in his February 1999 Humphrey-Hawkins testimony:

   To cushion the domestic economy from the impact of the increasing
   weakness in foreign economies and the less accommodative conditions
   in U.S. financial markets, the FOMC, beginning in late September,
   undertook three policy easings.... These actions were taken to
   rebalance the risks to the outlook, and, in the event, the markets
   have recovered appreciably. (38)

Were the late 1998 rate moves a balancing of forecast probabilities, insurance against a downside skew in possible outcomes, or some combination of both? There is no easy answer. This motivates our econometric work in section III, which seeks to disentangle the normal response of policy to expected outcomes from uncertainty and other related factors that may have influenced the policy decision.

II.C. 2000-01

In the end, the economy weathered the fallout from the Russian default well. The strength of the U.S. economy and underlying inflationary pressures led the FOMC to execute a series of rate hikes that brought the funds rate up to 6.5 percent by May of 2000. At the time of the June 2000 FOMC meeting, the unemployment rate stood at 4.1 percent and core PCE inflation, which the FOMC was now using as its main measure of consumer price inflation, was running at about 1 3/4 percent, up from 1 1/2 percent in 1999. The staff forecast that growth would moderate to a rate near or a little below potential, the unemployment would remain near its current level, and inflation would rise to 2.3 percent in 2001--and this forecast was predicated on another 75 basis points tightening. Despite this outlook, the FOMC decided to leave rates unchanged. What drove this pause? It seems likely to us that risk management was an important consideration.

In particular, the FOMC appeared to want to see how uncertainty over the outlook would play out. First, the incoming data and anecdotal reports from committee members' business contacts pointed to a slowdown in growth, although how much it was slowing was unclear. Second, with rates having risen substantially over the previous year, and given the lags from policy changes to economic activity, it was unlikely that the full effects of the hikes had yet been felt. Given the relatively high level of the funds rate and the slowdown in growth that appeared in train, the FOMC seemed wary of over-tightening. Third, despite the staff forecast, the FOMC apparently considered the costs of waiting, in terms of inflation risks, to be small. Accordingly, the FOMC thought it better to put a rate increase on hold and see how the economy evolved. The June 2000 FOMC minutes contain a good deal of commentary supporting this interpretation: (39)

   The increasing though still tentative indications of some slowing
   in aggregate demand, together with the likelihood that the earlier
   policy tightening actions had not yet exerted their full retarding
   effects on spending, were key factors in this decision. The
   uncertainties surrounding the outlook for the economy, notably the
   extent and duration of the recent moderation in spending and the
   effects of the appreciable tightening over the past year ...
   reinforced the argument for leaving the stance of policy unchanged
   at this meeting and weighting incoming data carefully.... Members
   generally saw little risk in deferring any further policy
   tightening move, particularly since the possibility that underlying
   inflation would worsen appreciably seemed remote under prevailing
   circumstances. (40)

In the second half of 2000 it became increasingly evident that growth had slowed to a pace somewhat below trend and inflation was moving up at a slower pace than the staff had projected in June. The FOMC's response was to hold the funds rate at 6.5 percent through the end of 2000. But the data around the turn of the year proved to be weaker than anticipated. In a conference call on January 3, 2001, the FOMC cut the funds rate to 6 percent, and then at its end-of-month meeting it lowered the rate again, to 5 1/2 percent. (41)

In justifying the aggressive ease, the minutes stated:

   Such a policy move in conjunction with the 50 basis point reduction
   in early January would represent a relatively aggressive policy
   adjustment in a short period of time, but the members agreed on its
   desirability in light of the rapid weakening in the economic
   expansion in recent months and associated deterioration in business
   and consumer confidence. The extent and duration of the current
   economic correction remained uncertain, but the stimulus ... would
   help guard against cumulative weakness in economic activity and
   would support the positive factors that seemed likely to promote
   recovery later in the year ... In current circumstances, members
   saw little inflation risk in such a "front-loaded" easing policy,
   given the reduced pressures on resources stemming from the sluggish
   performance of the economy and relatively subdued expectations of
   inflation. (42)

According to this quote, not only was the actual weakening in activity an important consideration in the policy decision, but uncertainty over the extent of the downturn and the possibility that it might turn into an outright recession seemed to spur the FOMC to make a large move. The "help guard against cumulative weakness" and "front-loaded" language could be read as the FOMC taking out some additional insurance against the possibility that the weakening activity would snowball into a recession. This could have reflected a concern about the kinds of nonlinear output dynamics or perhaps non-quadratic losses associated with a large recession that we discussed in section II.A.

The FOMC steadily brought the funds rate down further over the course of 2001, against a backdrop of weakening activity, and the economy seemed to be skirting a recession. Then the tragic events of September 11 occurred. There was, of course, huge uncertainty over how international developments, logistics disruptions, and the sentiment of households, businesses, and financial markets would affect spending and production. By November the board staff was forecasting a modest recession: growth in the second half of 2001 was projected to decline 1 1/2 percent at an annual rate and rise at just a 1 1/4 percent rate in the first half of 2002. By the end of 2002 the unemployment rate was projected to rise to 6.1 percent and core PCE inflation was projected to be 1 1/2 percent. These forecasts were predicated on the funds rate remaining flat at 2 1/4 percent.

However, in the aftermath of the terrorist attacks the FOMC was worried about something more serious than the shallow recession forecast by the staff. Furthermore, a new risk came to light, namely the chance that dis-inflationary pressures might emerge that, once established, would be more difficult to fight with the funds rate already low. In response, the FOMC again acted aggressively, cutting the funds rate 50 basis points in a conference call on September 17 and again at their regular meetings in October and November. The November 2001 FOMC meeting minutes note:

   ... members stressed the absence of evidence that the economy was
   beginning to stabilize and some commented that indications of
   economic weakness had in fact intensified. Moreover, it was likely
   in the view of these members that core inflation, which was already
   modest, would decelerate further. In these circumstances
   insufficient monetary policy stimulus would risk a more extended
   contraction of the economy and possibly even downward pressures on
   prices that could be difficult to counter with the current federal
   funds rate already quite low. Should the economy display
   unanticipated strength in the near term, the emerging need for a
   tightening action would be a highly welcome development that could
   be readily accommodated in a timely manner to forestall any
   potential pickup in inflation. (43)

This passage suggests that the large rate cuts were not only aimed at preventing the economy from falling into a serious recession with deflationary consequences, but that the FOMC was also concerned that such an outcome "could be difficult to counter with the current funds rate already quite low." Accordingly, the aggressive policy moves could in part also have reflected insurance against the future possibility of being constrained by the ZLB, precisely the policy scenario and optimal policy prescription described in section I.

II.D. Quantifying References to Uncertainty and Insurance in FOMC Minutes

We have shown that Federal Reserve communications contain many references suggesting that uncertainty or insurance motives influenced the stance of policy. But the question remains: Has risk management had a material impact on policy? We now show how we quantified these references into variables that can be used to assess the importance of risk management for actual policy decisions.

In the spirit of the narrative approach pioneered by Christina Romer and David Romer (1989), we built judgmental indicators based on our reading of the FOMC minutes covering the period from the beginning of Greenspan's chairmanship in 1987 to 2008. We concentrated on the paragraphs that describe the FOMC's rationale for its policy decision, reading these passages for references to when uncertainty or insurance considerations appeared closely linked to the FOMC's decision. Other portions of the minutes were excluded from our analysis in order to better isolate arguments that directly influenced the policy decision from more general discussions of unusual data or forecast uncertainty.

We constructed two separate judgmental variables, one for uncertainty (hUnc) and one for insurance (hIns), where "h" stands for "human-coded." The uncertainty variable was coded to plus (minus) one if we judged that the FOMC appealed to uncertainty to position the funds rate higher (lower) than it otherwise would be based on the staff forecast alone. If uncertainty did not appear to be an important factor influencing the policy decision, we coded the indicator as zero. We coded the insurance variable similarly by identifying when the minutes cited insurance against some adverse outcome as an important consideration in the stance of policy. (44)

As an example of our coding, consider the June 2000 meeting discussed above when the FOMC decided to wait to assess future developments before taking further policy action. The commentary below highlights the role of uncertainty in this decision (our italics):

   The increasing though still tentative indications of some slowing
   in aggregate demand, together with the likelihood that the earlier
   policy tightening actions had not yet exerted their full retarding
   effects on spending, were key factors in this decision. The
   uncertainties surrounding the outlook for the economy, notably the
   extent and duration of the recent moderation in spending and the
   effects of the appreciable tightening over the past year, including
   the 1/2 percentage point increase in the intended federal funds
   rate at the May meeting, reinforced the argument for leaving the
   stance of policy unchanged at this meeting and weighting incoming
   data carefully. (45)

We coded this meeting as a minus one for hUnc--rates were lower because uncertainty over the economic outlook and the effects of past policy moves appear to have been important factors in the FOMC's decision not to raise rates. Similarly, the January and November 2001 quotes cited above led us to code hIns as a minus one for those meetings, since, as we noted in the narrative, the FOMC appeared to be making aggressive rate moves in part to insure against downside risks to the baseline scenario.

We did not code all mentions of uncertainty or insurance as a plus or minus one. For example, the March 1998 minutes referred to uncertainties over the economic outlook and said that the FOMC could wait for further developments before tightening to counter potential inflation developments. However, at that time the FOMC was not obviously in the midst of a tightening cycle; the baseline forecast seemed consistent with the funds rate setting at the time; and the commentary over the need to tighten was in reference to an indefinite point in the future. So, in our judgment, uncertainty did not appear to be a very important factor holding back a rate increase at that meeting, and we coded it as a zero. (46)

Of course, this coding of the minutes is inherently subjective, and there is no definitive way to judge the accuracy of the decisions we made. Consequently we also constructed objective measures of how often references to uncertainty or insurance appeared in the policy paragraphs of the minutes. In particular, we constructed variables which measure the percentage of sentences containing words related to uncertainty or insurance in conjunction with references to economic activity, inflation, or both. (47) The measures for uncertainty and insurance are denoted mUnc and mIns, where "m" indicates these variables are "machine-coded." Figures 4 and 5 show plots of our minutes-based uncertainty and insurance variables.

Non-zero values of the human-coded variables are indicated by dots and the bars indicate the machine-coded sentence counts. The uncertainty indicator hUnc "turns on" in 31 out of the 128 meetings between 1993 and 2008. Indications that insurance was an actor in shading policy are not as common, but still show up 14 times in hIns. Most of the time--24 for uncertainty and 11 for insurance---we judged that rates were set lower than they otherwise would have been to account for these factors.

The hUnc and hIns codings are not always reflected in the sentence counts. There are also meetings where the sentence counts are positive but we did not judge them to indicate that rates were set differently than they normally would have been. For example, in March 2007 hUnc is coded zero for uncertainty whereas mUnc finds uncertainty referenced in nearly one-third of the sentences in the policy section of the minutes. Inspection of the minutes indicates that the FOMC was uncertain over both the degree to which the economy was weakening and whether their expectation of a decline in inflation, which was running uncomfortably high at the time, actually would materialize. In the end, they did not adjust current policy in response to these conflicting uncertainties. Hence we coded hUnc to zero in this case.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Note that we did not attempt to measure a variable for risk management per se. The minutes often contain discussions of policies aimed at addressing risks to attaining the FOMC's goals. However, many times this commentary appears to surround policy adjustments aimed instead at balancing (possibly conflicting) risks to the outlook for output and inflation, not unlike the response to changes in economic conditions prescribed by the canonical framework for studying optimal policy under discretion. Such risk balancing was discussed in our narrative of the 1997-98 period. (48)

III. Econometric Evidence of Risk Management

So far we have uncovered clear evidence that risk management considerations have been a pervasive feature of Federal Reserve communications. But it is not clear at this stage whether risk management has had a material impact on the FOMC's policy decisions. If it has, then calling for a risk management approach in the current policy environment would be consistent with a well established approach to monetary policy. In this section we describe econometric evidence suggesting that risk management has had a material impact on the FOMC's funds rate choices in the pre-ZLB era.

We estimate monetary policy reaction functions of the kind studied by Clarida, Gall, and Gertler (2000) and many others. These have the funds rate set as a linear function of output gap and inflation forecasts; there is no role for risk management unless risk feeds directly into the point forecasts. To quantify the role of risk beyond such a direct influence we add variables that proxy for risk to the reaction function. (49)

III.A. Empirical Strategy

Let [R.sup.*.sub.t] denote the notional target for the funds rate in period t. We assume the FOMC sets this target according to

(13) [R.sup.*.sub.t] = [R.sup.*] + [beta]([E.sub.t][[[pi].sub.t,k]] - [[pi].sup.*]) + [gamma][E.sub.t][[x.sub.t,q]] + [mu][s.sub.t],

where [[pi].sub.t,k] denotes the average annualized inflation rate from t to t + k, [[pi].sup.*] is the FOMC's target for inflation, [x.sub.t,q] is the average output gap from t to t + q, [s.sub.t] is a risk management proxy, and [E.sub.t] denotes expectations conditional on information available to the FOMC at date t. The coefficients [beta], [gamma], and p are fixed over time. [R.sup.*] is the desired nominal rate when inflation is at target, the output gap is closed, and risk does not influence policy other than through the forecast, [mu] = 0. If the average output and inflation gaps are both zero and the FOMC acts as if the natural rate is constant and out of its control, then [R.sup.*] = [r.sup.*] + [[pi].sup.*], where [r.sup.*] is the real natural rate of interest. (50)

We make two more assumptions to arrive at our estimation equation. First, the FOMC has a preference for interest rate smoothing and so does not choose to hit its notional target instantaneously, and as a practical matter it is necessary to include lags of the funds rate to fit the data. Second, the FOMC does not have perfect control over interest rates, which gives rise to an error term, [v.sub.t]. These assumptions lead to the following specification for the actual funds rate, [R.sub.t]:

(14) [R.sub.t] = (1 - A(1)) [R.sup.*.sub.t] + A(L[).sub.Rt-1] + [v.sub.t]),

where A(L) - [[summation].sup.N-1.sub.j=0][a.sub.j+1][L.sup.j] is a polynomial in the lag operator L with N denoting the number of funds rate lags. The error term [v.sub.t] is assumed to be mean zero and serially independent. Combining equations 13 and 14 yields our estimation equation:

(15) [R.sub.t] = [b.sub.0] + [b.sub.1][E.sub.t][[[pi].sub.t,k]] + [b.sub.2][E.sub.t][[x.sub.t,q]] + A(L) [R.sub.t-1] + [b.sub.3][s.sub.t] + [v.sub.t],

where [b.sub.i], i = 0, 1, 2, 3 are simple functions of A(1), [beta], [gamma], [mu], [r.sup.*] and [[pi].sup.*]. (51)

We use the publicly available Federal Reserve Board staff forecasts of core CPI inflation (in percentage points) and the output gap (percentage point deviations of real GDP from its potential) to measure [E.sub.t] [[[pi].sub.t,k]] and [E.sub.t] [[x.sub.t,q]] with k = q = 3. (52) These forecasts are available for every FOMC meeting. We estimate equation 15 both meeting-by-meeting and quarterby-quarter. (53) When we estimate it at the quarterly frequency we use staff forecasts corresponding to FOMC meetings closest to the middle of each quarter. (54) We measure R, at the meeting frequency using the funds rate target announced (or estimated) at the end of the day of a meeting, and we measure it at the quarterly frequency using the average effective funds rate over the 30 trading days following the meeting closest to the middle of the quarter. Provided the error term [v.sub.t] is serially uncorrelated and is orthogonal to the forecasts and the risk proxies, we can obtain consistent estimates of [beta], [gamma], and [mu] by estimating equation 15 by ordinary least squares. We keep N sufficiently large to ensure that [v.sub.t] is serially uncorrelated.

To quantify the role of risk we study the magnitude and statistical significance of estimates of p in equation 13. An insignificant estimate of p cannot be interpreted as evidence against a role for risk management, because risk might operate by influencing point forecasts as in our forward-looking model. We also could find no effect because risk might tilt policy in opposite directions depending on the circumstances. With the exception of our human-coded FOMC-based variables, none of our risk proxies accounts for the fact that perceived risks to the forecast might have different effects on policy depending on the nature of the risk and the state of the economy. For example, an increase in uncertainty about the inflation outlook should lead to tight policy if this increase occurs during a period of heightened concerns about rising inflation, but to looser policy if concerns are over unwanted disinflation. As such, estimates of the effect of any given proxy will at best reflect the nature of the risk and the circumstances in which it has arisen that have predominated over the sample period.

Finally, we do not allow for the coefficients on the forecasts to depend on our risk proxies as is suggested by the work of Brainard (1967) and others. However, we show in the online appendix that if these forecast coefficients are linear functions of risk, then the null hypothesis that a given proxy's coefficient is zero in our now mis-specified model encompasses the null that the forecast coefficients are invariant to risk as measured by that proxy.

III.B. Proxies for Risk Management

In addition to our human- and machine-coded FOMC-based variables we consider several proxies for risk management that do not rely on interpreting the FOMC minutes. Two of these variables are constructed using the Federal Reserve Board staff's forecast, which is seen by the FOMC at its regular meetings, and we study them using our meeting frequency reaction functions. The remaining variables are measured at the quarterly frequency and can be divided into two groups based on whether they primarily reflect variance or skewness in the forecast.

The two additional FOMC-based proxies involve revisions to the Federal Reserve Board staff's forecasts for the output gap (frGap) and core CPI inflation (frInf). The revisions correspond to changes between meeting m and m - 1 in the forecasts over the same one-year period that starts in the quarter of meeting m - 1. A big change in the forecast is usually triggered by unusual events that may be difficult to interpret and hence generate uncertainty about the forecast. If the FOMC were only worried about these events in making its point forecast, then the post-shock forecasts of the output gap or inflation would be sufficient to describe the policy setting. However, if uncertainty has a separate effect on policy the forecast revisions might enter significantly.

Three of the quarterly proxies exploit financial market data: VXO, SPD, and JLN. VXO is the Chicago Board Options Exchange's measure of market participants' expectations of volatility in the S&P 500 stock index over the next 30 days. Since the S&P 500 reflects earnings expectations, VXO should, at least in part, measure market participants' uncertainty about the economic outlook. (55) SPD is the difference between the quarterly average of daily yields on BAA corporate bonds and 10-year Treasury bonds. Gilchrist and Zakrajsek (2012) demonstrate that this variable measures private-sector default risk plus other factors that may indicate downside risks to economic growth. (56) JLN is Kyle Jurado, Sydney Ludvigson, and Serena Ng's (2015) measure of the common variation in the one-year-ahead unforecastable components of a large number of activity, inflation, and financial indicators. Given its basis in measuring uncertainty about macroeconomic forecasts, JLN is a natural risk proxy to consider. But, unlike VXO and SPD, it does not measure real-time uncertainty, and similar to these two measures it confounds macroeconomic and financial uncertainty.

The remaining proxies are based on the Survey of Professional Forecasters (SPF) which surveys forecasters about their point forecasts of GDP growth and GDP deflator inflation and their probability distributions for these forecasts. We use both kinds of information to construct measures of variance and skewness in the economic outlook one year ahead. (57) Variance is measured using the median among forecasters of the standard deviations calculated from each individual's probability distribution (vGDP and vInf) and the interquartile range of point forecasts across individuals (DvGDP and DvInf.) (58) Skewness is measured using the median of the individual forecasters' mean minus mode (sGDP and sInf) and the difference between the mean and the mode of the cross-forecaster distribution of point forecasts (DsGDP and DsInf). Consequently, a positive (negative) value for one of these proxies represents upside (downside) risk to the modal forecast.

The principal advantage of these proxies is that they are real-time measures of perceived risks in the forecast. The main drawback of the measures based on survey respondents' forecast distributions is that the bins they are asked to put probability mass on are relatively wide, so statistics based on them may contain substantial measurement error. The proxies based on the cross-section of forecasts are properly thought of as measuring forecaster disagreement rather than variance or skewness in the outlook per se. However, there is a large literature that uses forecaster disagreement as a proxy for perceived risk. (59)

All estimates are based on samples that end in 2008 to avoid the ZLB period but begin at different dates to address idiosyncratic features of the data. The benchmark start date is determined by the onset of Alan Greenspan's tenure as chairman of the FOMC in 1987, but later dates are used in several cases. The sample for the FOMC-based indicators starts in 1993 because inter-meeting changes in the target funds rate were much more common prior to that year than afterwards; the FOMC often voted on a bias to future policy moves and the chairman subsequently acted at his discretion. We cannot use inter-meeting moves because we lack contemporaneous staff forecasts. Furthermore, the change in the frequency of intermeeting moves raises the spectre of instability in the reaction function. (60) The pre-1993 inter-meeting moves are less of a concern for our quarterly models, because in these specifications the funds rate is not as closely tied to any particular meeting. So we chose to include these data points to maximize the number of observations, except when considering the proxies based on individuals' forecast distributions from the SPF. In the latter cases, the first observation is 1992Q1, to coincide with a discrete change in SPF methodology. (61)

Tables 4 and 5 display summary statistics for Federal Reserve Board staff forecasts of inflation and the output gap and the various proxies for risk management at the meeting and quarterly frequencies. What is most worth noting in these tables is that no risk proxy displays a particularly large positive or negative correlation with either the output gap or inflation forecast. This suggests that our proxies contain information that is not already incorporated into these forecasts. Nevertheless, some variables have moderately large correlations in absolute value, so the forecasts do somewhat reflect underlying risks to the outlook. Interestingly, skewness in forecasters' GDP forecasts (GDP) is negatively correlated with the outlook for activity.

Tables 6 and 7 display cross-correlations of the FOMC-based and quarterly proxies, respectively. As suggested by figures 4 and 5, the human- and machine-coded FOMC variables for uncertainty and insurance are essentially uncorrelated. These variables also appear unrelated to the forecast revision variables. However, several correlations among the quarterly proxies are worth noting. Forecaster variance and disagreement about the GDP growth outlook (vGDP and DvGDP) are both positively correlated with VXO and SPD, suggesting that the financial variables do reflect some uncertainty about the growth outlook. Also, the relatively high correlation of SPD with sGDP suggests that the former to some extent captures skewness in the growth outlook. The correlation of vGDP with vlnf and DvGDP with DvInf are both fairly large, suggesting that uncertainty about inflation and uncertainty about GDP often move together. The correlations of the corresponding forecaster uncertainty and disagreement variables (vGDP with DvGDP and vInf with DvInf) are somewhat large too. Evidently, the amount of disagreement among forecasters is similar to the median amount of uncertainty they see. Finally Jurado, Ludvigson, and Ng's (2015) measure of macroeconomic uncertainty, JLN, is highly correlated with VXO and SPD and to some extent with DvGDP, but much less so with any of the other risk proxies.