2.1. Main structural relationships

As in CÚrdia et al. (2015), the workhorse New Keynesian model can be described by the following pair of log-linearized equations:

(1) \begin{align}\tilde{x}_{t} & = \mathbb{E}_{t}\tilde{x}_{t+1}-\left( 1-\beta \eta \right)\left( 1-\eta \right) \left( i_{t}-\mathbb{E}_{t}\pi _{t+1}-r_{t}^{n}\right), \end{align}

(2) \begin{align}\tilde{\pi}_{t} & = \beta \mathbb{E}_{t}\tilde{\pi}_{t+1}+\xi _{p}\left(\omega x_{t}+\left( \left( 1-\beta \eta \right) \left( 1-\eta \right)\right) ^{-1}\tilde{x}_{t}\right) +\mu _{t}, \end{align}

where

(3) \begin{align}\tilde{y}_{t} &\equiv y_{t}-\eta y_{t-1}-\beta \eta \mathbb{E}_{t}\left(y_{t+1}-\eta y_{t}\right) , \end{align}

(4) \begin{align}\tilde{y}_{t}^{n} &\equiv y_{t}^{n}-\eta y_{t-1}^{n}-\beta \eta \mathbb{E}_{t}\left( y_{t+1}^{n}-\eta y_{t}^{n}\right) , \end{align}

(5) \begin{align}\tilde{x}_{t} &\equiv \tilde{y}_{t}-\tilde{y}_{t}^{n}=x_{t}-\eta x_{t-1}-\beta \eta \mathbb{E}_{t}\left( x_{t+1}-\eta x_{t}\right) , \end{align}

(6) \begin{align}\tilde{\pi}_{t} &\equiv \pi _{t}-\iota _{p}\pi _{t-1}. \end{align}

Here, the one-period nominal interest rate ( $i_{t}$ ) is the policy rate, inflation ( $\pi _{t}$ ) is the first-difference on the consumption price level in logs, and the output gap ( $x_{t}$ ) is defined as $x_{t}\equiv y_{t}-y_{t}^{n}$ , that is, as the log-deviation of actual output ( $y_{t}$ ) from its potential level absent all nominal rigidities ( $y_{t}^{n}$ ).

Equation (1), often referred to as the dynamic Investment-Savings (IS) equation, describes the aggregate demand of the economy arising from the optimal decisions (the intertemporal Euler equation) of households. Equation (1) together with (3)--(5) implies that the current output gap ( $x_{t}$ ) depends on expected one-period and two-period ahead output gaps, the lagged output gap, the current nominal interest rate ( $i_{t} $ ), the expected one-period ahead inflation rate ( $\mathbb{E}_{t}\left( \pi _{t+1}\right) $ ), and the natural rate ( $r_{t}^{n}$ ) which is the real rate of interest that would prevail absent all nominal rigidities. Here, the intertemporal rate of substitution is set to one. There exists habit formation in consumption given by the parameter $0\leq \eta \leq 1$ , and households’ intertemporal discount rate is given by the parameter $0<\beta <1 $ .

Equation (2) denotes the New Keynesian Phillips Curve (NKPC) and follows from the optimizing decision of firms. These firms are owned by the households and are operated in a monopolistically competitive environment with Calvo (Reference Calvo1983) staggered price-setting behavior and Yun (Reference Yun1996) price indexation, similar to Christiano et al. (Reference Christiano, Eichenbaum and Evans2005). Consequently, equation (2) shows that inflation ( $\pi _{t}$ ) depends on lagged inflation, the expected one-period ahead inflation ( $\mathbb{E}_{t}\left(\pi _{t+1}\right) $ ), the current output gap ( $x_{t}$ ), the lagged output gap, the expected one-period ahead output gap, and a cost-push shock ( $\mu_{t}$ ).

A fraction of firms given by the parameter $0\leq \theta \leq 1$ are assumed to be unable to adjust their prices every period, while the remaining fraction $(1-\theta )$ of firms can. The non-reoptimizing firms index their prices to past inflation with the degree of indexation determined by the parameter $0\leq \iota _{p}\leq 1$ . Furthermore, the parameter $\omega >0$ is the inverse of the Frisch elasticity of labor supply, while the composite coefficient $\xi _{p}$ is defined as $\frac{\left( 1-\theta \beta \right)\left( 1-\theta \right) }{\theta }$ with $\beta $ being the household’s intertemporal discount factor and $\theta $ the constant fraction of non-reoptimizing firms per period.

We use (5) to re-express the system of equations given by (1)--(2) that describe the dynamics of the economy in terms of actual and potential output as follows:

(7) \begin{eqnarray}&&\left. \tilde{y}_{t}=\mathbb{E}_{t}\left( \tilde{y}_{t+1}\right) -\left(1-\beta \eta \right) \left( 1-\eta \right) \left( i_{t}-\mathbb{E}_{t}\pi_{t+1}-r_{t}^{n}\right) -\mathbb{E}_{t}\left( \Delta \tilde{y}_{t+1}^{n}\right) ,\right. \end{eqnarray}

(8) \begin{eqnarray}&&\left.\begin{array}{l}\tilde{\pi}_{t}=\xi _{p}\left( \omega y_{t}+\left( \left( 1-\beta \eta\right) \left( 1-\eta \right) \right) ^{-1}\tilde{y}_{t}\right) +\beta\mathbb{E}_{t}\tilde{\pi}_{t+1}+\mu _{t} \\\left. \left. \left. \left. \left. \left. {}\right. \right. \right. \right.\right. \right. -\xi _{p}\left( \omega y_{t}^{n}+\left( \left( 1-\beta \eta\right) \left( 1-\eta \right) \right) ^{-1}\tilde{y}_{t}^{n}\right) .\end{array}\right. \end{eqnarray}

Based on the output potential transformation in (4), we can further re-write the system of equations in (1)--(2) to obtain that:

(9) \begin{align}& \left.\begin{array}{l}\tilde{y}_{t}=\mathbb{E}_{t}\left( \tilde{y}_{t+1}\right) -\left( 1-\beta\eta \right) \left( 1-\eta \right) \left( i_{t}-\mathbb{E}_{t}\pi_{t+1}-r_{t}^{n}\right) \\\left. \left. \left. \left. \left. \left. {}\right. \right. \right. \right.\right. \right. -\left( \eta y_{t-1}^{n}-\left( 1+\eta +\beta \eta^{2}\right) y_{t}^{n}+\left( 1+\beta \eta +\beta \eta ^{2}\right) \mathbb{E}_{t}\left( y_{t+1}^{n}\right) -\beta \eta \mathbb{E}_{t}\left(y_{t+2}^{n}\right) \right) ,\end{array}\right. \end{align}

(10) \begin{align}& \left.\begin{array}{l}\tilde{\pi}_{t}=\xi _{p}\left( \omega y_{t}+((1-\beta \eta )\left( 1-\eta\right) )^{-1}\tilde{y}_{t}\right) +\beta \mathbb{E}_{t}\tilde{\pi}_{t+1}+\mu _{t}\\\left. \left. \left. \left. \left. \left. {}\right. \right. \right. \right.\right. \right. -\xi _{p}\left(\begin{array}{c}-\left( \left( 1-\beta \eta \right) \left( 1-\eta \right) \right) ^{-1}\eta y_{t-1}^{n}+\left( \omega +\left( \left( 1-\beta \eta \right) \left( 1-\eta\right) \right) ^{-1}\left( 1+\beta \eta ^{2}\right) \right) y_{t}^{n} \\[5pt]-\left( \left( 1-\beta \eta \right) \left( 1-\eta \right) \right) ^{-1}\beta\eta \mathbb{E}_{t}\left( y_{t+1}^{n}\right)\end{array}\right) ,\end{array}\right. \end{align}

with the same structural relationships as the system of equations given by (7)--(8). This showcases that the dynamic IS and NKPC equations can be expressed in terms of three observable macro variables: output ( $y_{t}$ ), inflation ( $\pi _{t}$ ), and the policy rate ( $i_{t}$ ), that is, in terms of the three-variable vector $Y_{t}=\left[ y_{t},\pi _{t},i_{t}\right] ^{^{\prime }}$ . Moreover, these equations also show that cost-push shocks ( $\mu _{t}$ ) as well as exogenously-driven shifts in the output potential ( $y_{t}^{n}$ ) and the natural rate of interest ( $r_{t}^{n}$ ) drive the dynamics of output ( $y_{t}$ ) and inflation ( $\pi _{t}$ ).

Frictionless allocation. The potential output allocation ( $y_{t}^{n}$ ) and the natural real rate ( $r_{t}^{n}$ ) are important constructs in our analysis and represent the levels of output and of the real interest rate that would prevail absent all nominal rigidities. In that counterfactual scenario, output potential ( $y_{t}^{n}$ ) evolves according to the following equation:

(11) \begin{align}\omega y_{t}^{n} & +\frac{1}{\left( 1-\beta \eta \right) \left( 1-\eta \right) }\left( y_{t}^{n}-\eta y_{t-1}^{n}\right) -\frac{\beta \eta }{\left( 1-\beta\eta \right) \left( 1-\eta \right) }\left( \mathbb{E}_{t}\left(y_{t+1}^{n}\right) -\eta y_{t}^{n}\right) \nonumber \\[5pt]& =\frac{\eta }{\left( 1-\beta \eta \right) \left( 1-\eta \right) }\left(\beta \mathbb{E}_{t}\left( \gamma _{t+1}\right) -\gamma _{t}\right) .\end{align}

The previous equation follows from CÚrdia et al. (2015). Equation (11) implies that output potential is a linear combination of current, lagged, and future expected values of output potential as well as current and future expected values of exogenous productivity growth, $\gamma _{t}\equiv \Delta \ln\left( A_{t}\right) $ where $A_{t}$ denotes total factor productivity (TFP). Given the efficient allocation of the output potential ( $y_{t}^{n}$ ) in (11), the household’s intertemporal Euler equation implies that the natural rate of interest ( $r_{t}^{n}$ ) can be expressed as:

(12) \begin{equation}r_{t}^{n}=\mathbb{E}_{t}\left( \gamma _{t+1}\right) -\omega \mathbb{E}_{t}\left( \Delta y_{t+1}^{n}\right) . \end{equation}

Equations (11) and (12) highlight the close connection between output potential and the natural rate of interest both of which respond to a common shock—the exogenous shock to productivity growth ( $\gamma _{t}$ ).

Here, we observe that the natural rate of interest depends: (a) positively on the forecastable components of next period’s exogenous productivity growth ( $\gamma _{t}$ ), and (b) negatively on the forecastable component of next period’s growth rate of output potential ( $\Delta y_{t+1}^{n}$ ) which itself depends on the exogenous productivity growth ( $\gamma _{t}$ ) through equation (11). Intuitively, point (b) captures the negative effect on the real interest rate of a higher expected growth rate of marginal utility which, under standard market clearing conditions, directly influences potential hours worked and in turn potential output as well.

Exogenous (Non-monetary) shock processes. The exogenous shock to productivity growth ( $\gamma _{t}$ ) and the cost-push shock ( $\mu _{t}$ ) are assumed to follow standard AR $\left( 1\right) $ processes:

(13) \begin{eqnarray}\gamma _{t} &=&\rho _{\gamma }\gamma _{t-1}+\varepsilon _{t}^{\gamma }, \end{eqnarray}

(14) \begin{eqnarray}\mu _{t} &=&\rho _{\mu }\mu _{t-1}+\varepsilon _{t}^{\mu },\end{eqnarray}

where $\varepsilon _{t}^{\gamma }\overset{iid}{\sim }N\left( 0,\sigma_{\gamma }^{2}\right) $ and $\varepsilon _{t}^{\mu }\overset{iid}{\sim }N\left( 0,\sigma _{\mu }^{2}\right) $ . ^{Footnote 9} The persistence of the productivity growth and cost-push shocks is given by the parameters $0<\rho _{\gamma }<1$ and $0<\rho _{\mu }<1$ , respectively. Similarly, the volatility of the productivity growth and cost-push shocks is given by $\sigma _{\gamma }^{2}>0$ and $\sigma _{\mu }^{2}>0$ , respectively. We do not consider spillovers between productivity growth and cost-push shocks and assume that their respective innovations are uncorrelated at all leads and lags.

2.2 Monetary policy

The monetary policymaker relies on the short-term nominal interest rate ( $i_{t}$ ) as its policy instrument. A Taylor (Reference Taylor1993)-type monetary policy rule is generally viewed as a simple and practical guide for the conduct of monetary policy in the US. ^{Footnote 10} Henceforth, we assume that the central bank follows a variant of the Taylor (Reference Taylor1993) rule whereby the nominal interest rate responds to inflation deviations from its zero-inflation target ( $\pi _{t}$ ) and possibly also to fluctuations in the output gap ( $x_{t}\equiv \left(y_{t}-y_{t}^{n}\right) $ ), that is,

(15) \begin{equation}i_{t}=\rho i_{t-1}+\left( 1-\rho \right) \left( \chi _{\pi }\pi _{t}+\chi_{x}\left( y_{t}-y_{t}^{n}\right) \right) +\varepsilon _{t}^{MP}.\end{equation}

This policy rule ensures the determinacy of the equilibrium whenever the policy parameters satisfy the Taylor principle, that is, whenever $\chi_{\pi }>1$ and $\chi _{x}\geq 0$ . The rule also includes lagged interest rates with a smoothing parameter given by $0\leq \rho <1$ and an unanticipated monetary policy shock ( $\varepsilon _{t}^{MP}$ ).

We recognize that the strategy by which the Federal Reserve has pursued forward guidance has evolved over time, so we introduce contingent forward guidance in the Taylor (Reference Taylor1993) rule flexibly in the form of anticipated monetary policy shocks (news) following the approach of LasÉen and Svensson (Reference Laséen and Svensson2011) and Del Negro et al. (Reference Del Negro, Giannoni and Patterson2012). This approach has more recently been used in related work by Cole (Reference Cole2020a) and Cole (Reference Cole2020b). Specifically, the monetary policy rule in (15) is augmented as follows:

(16) \begin{equation}i_{t}=\rho i_{t-1}+\left( 1-\rho \right) \left[ \chi _{\pi }\pi _{t}+\chi_{x}\left( y_{t}-y_{t}^{n}\right) \right] +\varepsilon_{t}^{MP}+\sum\nolimits_{l=1}^{L}\varepsilon _{l,t-l}^{FG}, \end{equation}

where the unanticipated (surprise) monetary policy shocks ( $\varepsilon_{t}^{MP}$ ) are combined with forward guidance (news) shocks ( $\varepsilon_{l,t-l}^{FG}$ for all $l=1,...,L$ ). ^{Footnote 11} The length of the forward guidance horizon provided by the news shocks is defined by the horizon $1\leq L<+\infty $ implying that there is a finite number of L forward guidance shocks in the summation term in equation (16).

Monetary policy surprises and forward guidance shocks are assumed to be purely transitory or i.i.d., that is,

(17) \begin{eqnarray}&&\left. \varepsilon _{t}^{MP}\overset{iid}{\sim }N\left( 0,\sigma_{MP}^{2}\right) ,\right. \end{eqnarray}

(18) \begin{eqnarray}&&\left. \varepsilon _{l,t-l}^{FG}\overset{iid}{\sim }N\left( 0,\sigma_{l}^{2,FG}\right) ,\text{ }\forall l=1,...,L,\text{ and }1\leq L<+\infty.\right.\end{eqnarray}

Each $\varepsilon _{l,t-l}^{FG}$ in equation (16) represents anticipated or news shocks that private agents know about in period $t-l$ but do not affect the interest rate until l periods later, that is, until period t. The volatility of the unanticipated and anticipated monetary policy shocks is given by $\sigma _{MP}^{2}>0$ and $\sigma _{l}^{2,FG}>0$ for all $l=1,...,L$ , respectively. The innovations of anticipated and unanticipated monetary policy shocks are uncorrelated with each other and with the cost-push shock and productivity growth shock innovations at all leads and lags.

Following LasÉen and Svensson (Reference Laséen and Svensson2011) and Del Negro et al. (Reference Del Negro, Giannoni and Patterson2012), the following recursive representation is added to the model’s system of equations to describe the news shocks: ^{Footnote 12}

(19) \begin{align}v_{1,t} &= v_{2,t-1}+\varepsilon _{1,t}^{FG}, \end{align}

(20) \begin{align}v_{2,t} &= v_{3,t-1}+\varepsilon _{2,t}^{FG}, \end{align}

(21) \begin{align}&\vdots \notag \\v_{L,t} &= \varepsilon _{L,t}^{FG}. \end{align}

Each component of the vector $v_{t}=\left[ v_{1,t},v_{2,t},\dots ,v_{L,t}\right] ^{\prime }$ represents all past and present central bank announcements to change the interest rate $1,2,\dots ,L$ periods later that private agents know in period t. In addition, we define $\psi _{t}=\left[\varepsilon _{1,t}^{FG},\varepsilon _{2,t}^{FG},\dots ,\varepsilon_{L,t}^{FG}\right] ^{^{\prime }}$ as the vector containing all current-period forward guidance shocks known today that affect the monetary policy rule $1,2,\dots ,L$ periods later. Equations (19)--(21) can be simplified to show that $v_{1,t-1}$ corresponds to the last term in equation (16), that is, the summation of all anticipated monetary policy shocks realized at time t, $v_{1,t-1}=\sum\nolimits_{l=1}^{L}\varepsilon _{l,t-l}^{FG}$ .

Accordingly, the policy rule in (16) can be re-expressed more compactly as:

(22) \begin{equation}i_{t}=\rho i_{t-1}+\left( 1-\rho \right) \left[ \chi _{\pi }\pi _{t}+\chi_{x}\left( y_{t}-y_{t}^{n}\right) \right] +\varepsilon _{t}^{MP}+v_{1,t-1}.\end{equation}

The method of using equations (22) together with (17)--(21) provides a tractable way to incorporate anticipated monetary policy (forward guidance) news shocks as well as conventional unanticipated (surprise) monetary policy shocks into the monetary policy rule. Forward guidance shocks, therefore, can be interpreted as the means by which the central bank communicates (announces) the path of future policy rates.

We choose to model forward guidance shocks in this way for a number of conceptual and practical reasons. First, we model forward guidance here following in the footsteps of the prior literature starting with LasÉen and Svensson (Reference Laséen and Svensson2011) and Del Negro et al. (Reference Del Negro, Giannoni and Patterson2012). That provides a degree of comparability. Second, the Federal Reserve’s forward guidance strategy has evolved. In the aftermath of the $2007-2009$ global financial crisis, the U.S. central bank forward guidance communication was “Odyssean” in nature but moved from calendar-based into state-contingent commitments over time. By 2012, the Fed also started to issue its own fed funds rate projections on top of the macro projections already being released since 2008 through the Fed’s Summary of Economic Projections (SEP) (“Delphic” forward guidance). ^{Footnote 13} All of these types of forward guidance share the same theme of communicating about the future course of policy. And equation (16) captures this feature in a reduced but flexible form. We use this reduced-form representation as a simple and flexible way to encompass the different approaches of forward guidance followed by the Federal Reserve.

Expectations augmented vector of observable variables. The state equations that describe the dynamics of the economy in (19)--(10), together with (3)--(4) and (6), pin down the solution to the vector of three observable macro variables given by $Y_{t}=\left[ y_{t},\pi _{t},i_{t}\right] ^{^{\prime }}$ which includes actual output ( $y_{t}$ ), inflation ( $\pi _{t}$ ), and the policy rate ( $i_{t}$ ). However, with monetary policy shocks split into unanticipated (surprise) and anticipated (news) shocks, the vector of observable variables $Y_{t}$ lacks fundamentalness in the sense of Hansen and Sargent (Reference Hansen and Sargent1980) and Martinez-Garcia (2018). In other words, these three observable macro variables do not contain enough information to pin down the vector of unobserved structural shocks $\varepsilon _{t}=( \gamma _{t},\mu_{t},\varepsilon _{t}^{MP},\{ \varepsilon _{l,t-l}^{FG}\}_{l=1}^{L}) ^{\prime }$ . Without additional observable variables, we can only recover residuals that are linear combinations of the underlying structural shocks.

Given the monetary policy rule in equation (16), we can show that the expected future path of the policy rate at time t can be written as follows: ^{Footnote 14}

(23) \begin{equation}\mathbb{E}_{t}\left( i_{t+s}\right) =\left\{\begin{array}{l}\rho i_{t-1}+\left( 1-\rho \right) \left[ \chi _{\pi }\pi _{t}+\chi_{x}\left( y_{t}-y_{t}^{n}\right) \right] +\varepsilon _{t}^{MP}+v_{1,t-1},\text{ for }s=0, \\[4pt]\rho i_{t-1+s}+\left( 1-\rho \right) \left[ \chi _{\pi }\mathbb{E}_{t}\left(\pi _{t+s}\right) +\chi _{x}\mathbb{E}_{t}\left( y_{t+s}-y_{t+s}^{n}\right) \right]\\[4pt]\quad +v_{s+1,t-1}+\varepsilon _{s,t}^{FG},\text{ }\forall s\in \left\{1,2,...,L-1\right\} ,\text{ for } 0<s<L, \\[4pt]\rho i_{t-1+s}+\left( 1-\rho \right) \left[ \chi _{\pi }\mathbb{E}_{t}\left(\pi _{t+s}\right) +\chi _{x}\mathbb{E}_{t}\left( y_{t+s}-y_{t+s}^{n}\right) \right] +\varepsilon _{s,t}^{FG},\text{ for }s=L, \\[4pt]\rho i_{t-1+s}+\left( 1-\rho \right) \left[ \chi _{\pi }\mathbb{E}_{t}\left(\pi _{t+s}\right) +\chi _{x}\mathbb{E}_{t}\left( y_{t+s}-y_{t+s}^{n}\right) \right] ,\text{ }\forall s>L.\end{array}\right. \end{equation}

Hence, the expression in (23) shows that—consistent with expectations about inflation and economic activity—expectations on the future path of the interest rate should shift in response to announcements of anticipated (forward guidance) monetary policy shocks helping us tease them apart from unanticipated (surprise) monetary policy shocks. Given this, we adopt the survey-based identification strategy explored by Doehr and Martinez-Garcia (Reference Doehr and Martínez-García2015) in a VAR setting and employed by Cole and Milani (Reference Cole and Milani2017) within a DSGE model which consists in augmenting the vector of observables $Y_{t}=[ y_{t},\pi _{t},i_{t}] ^{\prime}$ with survey-based forecasts with which to disentangle anticipated from unanticipated monetary policy shocks. ^{Footnote 15}

We expand the vector of observables $Y_{t}$ with expectations as follows:

(24) \begin{equation}\overline{Y}_{t}=\left[ y_{t},\pi _{t},i_{t},\mathbb{E}_{t}\left( \Delta y_{t+1}\right) ,\mathbb{E}_{t}\left( \Delta y_{t+2}\right) ,\mathbb{E}_{t}\left( \pi _{t+1}\right) ,\mathbb{E}_{t}\left( i_{t+1}\right) ,...,\mathbb{E}_{t}\left( i_{t+L}\right) \right] ^{^{\prime }}, \end{equation}

where $\Delta y_{t+j}=\Delta x_{t+j}+\Delta y_{t+j}^{n}$ denotes the growth rate of actual output at time $t+j$ (for $j=1,2$ ) and, by analogy, we define $\Delta y_{t+j}^{n}=( y_{t+j}^{n}-y_{t+j-1}^{n}) $ to be the corresponding growth rate of output potential at time $t+j$ . ^{Footnote 16} Given the structure of the economy described by equations (9)--(10), the non-monetary shock processes in (11)--(14), the Taylor (Reference Taylor1993) rule in (22), and the unanticipated and anticipated monetary policy shocks given by (17)--(21), the vector of expectations-augmented observables $\overline{Y}_{t}$ in (24) suffices to ensure that we can identify all structural shocks $\varepsilon_{t}=\left( \gamma _{t},\mu _{t},\varepsilon _{t}^{MP},\left\{ \varepsilon_{l,t-l}^{FG}\right\} _{l=1}^{L}\right) ^{\prime }$ .

2.3. Central bank credibility

Forward guidance opens up the possibility of central banks managing expectations but is also inherently prone to time-inconsistency problems (Kydland and Prescott (Reference Kydland and Prescott1977)). That is because, at the end of the day, forward guidance are promises about future monetary policy that the central bank may find beneficial to renege from unless future policymakers could be bound somehow to credibly honor those commitments when the time comes.

This is partly because, while the vector of observables $Y_{t}=[y_{t},\pi_{t},i_{t}]^{^{\prime }}$ can be monitored with observable data, neither announcements about the expected future path of the policy rate (news shocks) nor the central bank’s own public forecasts—if used to communicate the forward guidance policy—can be monitored and validated in real-time with the current observed data $Y_{t}$ or with its lags at the time the announcement is made. It is also partly because central banks may have incentives to deviate from those commitments.

Simply put, private agents realize that there is neither a full-proof verification mechanism nor a way to enforce those promises to guarantee that the monetary policymakers deliver on the future policy path that has been promised and, therefore, private agents must form expectations accordingly. In other words, private agents have to factor the credibility of the central bank’s forward guidance future policy commitment in forming their own expectations about the future.

We assume that private agents believing the central bank’s commitments form their expectations under the FIRE paradigm recognizing the aggregate implications of imperfect credibility on those policy commitments. Private agents believing the central bank’s commitments not to be credible form their expectations about the observables using a standard VAR model and commit themselves to forecast the future path of the economy’s macro aggregates in that way (ignoring all announcements until they materialize—if they do—at a later time). That is, private agents forecast the observable vector $Y_{t}=\left[ y_{t},\pi _{t},i_{t}\right] ^{\prime }$ with the following parsimonious structural VAR $\left( 1\right) $ process in mind:

(25) \begin{equation}Y_{t}=A+BY_{t-1}+u_{t}, \end{equation}

which captures well the historical dynamics of $Y_{t}$ in our sample. Here, A and B are reduced-form matrices of conforming dimensions, and $u_{t}$ is a vector of (non-structural) residuals. ^{Footnote 17}

Following on the footsteps of the axiomatic approach to aggregate heterogeneous beliefs introduced by Branch and McGough (Reference Branch and McGough2009), aggregate expectations ( $\mathbb{E}_{t}\left(Y_{t+1}\right) $ ) are a weighted sum of expectations from private agents who believe the central bank to be credible and those who do not. ^{Footnote 18} Specifically, we define aggregate expectations as follows:

(26) \begin{equation}\mathbb{E}_{t}\left( {{Y}_{t+1}}\right) =\tau \mathbb{E}_{t}^{C}\left( {Y_{t+1}}\right) +\left( 1-\tau \right) \mathbb{E}_{t}^{D}\left( {Y_{t+1}}\right) {,} \end{equation}

where $\mathbb{E}_{t}^{C}\left( Y_{t+1}\right) $ represents the FIRE forecasts of macro variables of private agents that view the central bank’s commitments as fully credible and $\mathbb{E}_{t}^{D}\left( Y_{t+1}\right) $ denotes the expectations of private agents who believe the monetary authority’s commitment to be not credible. As stated above, the latter form expectations based on equation (25). Equation (26) also follows Haberis et al. (Reference Haberis, Harrison and Waldron2019) who model aggregate expectations as a weighted average of FIRE and VAR forecasts.

The parameter $0<\tau <1$ determines the odds placed on $\mathbb{E}_{t}^{C}\left( Y_{t+1}\right) $ or, alternatively, the share of private agents that forms FIRE expectations. We refer to $\tau $ as the credibility parameter of the model. ^{Footnote 19} In the limiting case where $\tau = 1$ , all private agents in the economy believe the central bank to be perfectly credible and expectations to be homogeneous and FIRE across all agents (i.e. $\mathbb{E}_{t}{Y}_{t+1}=\mathbb{E}_{t}\left( Y_{t+1}^{C}\right) $ ). In the opposite polar case where $\tau \rightarrow 0$ , the monetary authority is considered not to be credible and aggregate beliefs imply that $\mathbb{E}_{t}\left( {{Y}_{t+1}}\right) \rightarrow \mathbb{E}_{t}^{D}\left( {Y_{t+1}}\right) $ .

Our specific choice of modeling credibility with equation (26) is partly motivated by prior research and comparability reasons. Bernanke et al. (Reference Bernanke, Kiley and Roberts2019) study imperfect credibility with FRB/US, the main simulation model of the Federal Reserve. Similar to our equation (26), aggregate expectations in the previous authors’ paper are assumed to be composed of model-consistent FIRE expectations and forecasts based on a VAR that ignores Federal Reserve announcements (as in our model with imperfectly credible guidance). The description of credibility in Yellen (Reference Yellen2006) also seems akin to our definition. Yellen (Reference Yellen2006) describes central bank credibility as agents correctly anticipating policy in response to economic shocks. Thus, lack of credibility could be understood as not incorporating Federal Reserve future policy announcements into forecasts, similar to what we do in equation (25). Blinder (Reference Blinder2018) also states that agents not incorporating monetary policy communication into their forecasts are effectively acting as if they do not believe the monetary authority.

In addition, we argue that incorporating heterogeneous expectations is crucial to reconcile the model with the survey evidence. Survey evidence seems to run contrary to a model with homogeneous expectations where all agents are fully informed and rational, and their forecasts align with those of the central bank. As a way of illustrating this point, we look at how since 2012 the Federal Reserve has communicated its “Delphic” guidance via dot plots for the fed funds rate as part of the Fed’s SEP. ^{Footnote 20} We compute the median SEP value of the fed funds rate from those dot plots for the current year and the following two years. We match each observation of the median SEP with the time-consistent individual forecast of the 3-month interest rate from the SPF for each forecasting horizon. We then calculate the pooled median and the 75th–25th percentile range of the individual SPF forecasts for each forecasting horizon and illustrate how those ranges relate with the SEP median in Figure 1.

Figure 1.

Range of pooled 3-month interest rate projections from Survey of Professional Forecasters (SPF) vs. Federal reserve’s median Summary of Economic Projections (SEP) on the fed funds rate.

Note: Bars represent the 75th and 25th percentile while the dot indicates the median (50th percentile) of the pooled individual SPF forecasts conditional on each given value of the SEP median forecast over the sample period. Data covers from 2012 : Q1 till 2018 : Q4. The dashed line plots the 45 degree line for reference. 2015 : Q1 is the most significant outlier between the policy path indicated in the SEP and the SPF forecasts. This occurred as the Fed’s language shifted to prepare markets for liftoff from the ZLB. SOURCES: Federal Reserve’s Summary of Economic Projections (SEP); Survey of Professional Forecasters (SPF); authors’ calculations.

The results shown in this figure display notable discrepancies between the median SEP and median SPF forecasts and a sizeable dispersion of the individual SPF forecasts themselves. We argue that allowing for heterogeneous expectations would not be enough to match the evidence, as relying on the central bank’s communication about the interest rate path is the costless path to follow for everybody irrespective of whether or not agents are fully informed and rational, unless agents are also concerned about the credibility of those announcements (or for some other unmodeled reason choose to act sub-optimally relying on other likely worse forecasts of the policy rate). Even if forward guidance announcements are deemed perfectly credible, the forecasts of agents who are neither fully rational nor fully informed would tend to coalesce around the policy path announced by monetary authorities as this does not require any costly effort to either acquire or process the information. However, as shown in Figure 1, the empirical forecasts for private agents do not appear to coalesce around their SEP median counterparts.^{Footnote 21}

We must recognize that, while heterogeneous beliefs can result from different causes, our structural model only allows heterogeneous expectations to arise as a result of imperfect credibility about the forward guidance announcements. How do we know this? Because the VAR representation accounted for in equation (25) is just a reduced-form representation of the solution of the structural model absent the forward guidance news shocks except at the point at which they materialize (see, e.g. Martinez-Garcia (2018)). In effect, the VAR solution representation in equation (25) is consistent with the structural solution that would characterize the equilibrium of such a model (when a unique solution exists).

Forward guidance credibility and the modeling structure. In general, aggregate expectations of private agents would be a convex combination weighed by a parameter $\tau $ that lies within the unit interval. This is the benchmark model we estimate in this paper and inevitably leads to an economy where forward guidance loses some of its power if $\tau $ is strictly less than one.

We choose to study forward guidance and central bank credibility with a relatively smaller scale New Keynesian model to avoid the danger of overparameterization. Using a larger scale model (e.g. Smets and Wouters (Reference Smets and Wouters2007)) would incorporate not only a richer set of equilibrium relationships but also would necessitate additional observables and shocks in the estimation to help identify a larger set of deep structural parameters. In that larger scale model, the number of structural parameters to estimate in equation (25) would expand. As discussed in Stock and Watson (Reference Stock and Watson2001), the potential issue with too many parameters is unreliable estimates. Thus, while it is indeed important to consider a larger-scale DSGE model that captures more of the equilibrium relationships between the variables, we limit the scope of our paper to examine the efficacy of forward guidance under imperfect credibility through the benchmark model presented earlier. We leave the exploration of a larger-scale DSGE model for future research. ^{Footnote 22}

3.1. Data sources

We utilize Bayesian estimation techniques with US macroeconomic time series variables at a quarterly frequency. Data for output, inflation, and interest rates correspond to US real GDP growth, the growth rate in the GDP deflator, and the fed funds rate. The relevant acronyms are GDPC1, GDPDEF, and FEDFUNDS with the data retrieved from the FRED database of the Federal Reserve Bank of St. Louis. We also employ observations for expectations of future macroeconomic variables. Specifically, we utilize expectations regarding one-quarter and two-quarters ahead output growth, one-quarter ahead inflation, and one-quarter to five-quarters ahead interest rates. These forecast series are retrieved from the SPF database of the Federal Reserve Bank of Philadelphia (FRB of Philadelphia (2019)). ^{Footnote 23} The relevant acronyms are RGDP, PGDP, and TBILL. In addition, our dataset spans the period from $1981\,:\,Q3$ through $2017\,:\,Q3$ with forecasts going up to $2018\,:\,Q4$ . ^{Footnote 24}

3.2. Estimation strategy

3.2.1 Observation equations

The observation equations mapping the model variables into the data are given by the following system of equations:

(27) \begin{equation}\left[\begin{array}{c}g_{t}^{obs} \\[3pt]\pi _{t}^{obs} \\[3pt]i_{t}^{obs} \\[3pt]\mathbb{E}_{t}^{obs}\left( g_{t+1}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( g_{t+2}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( \pi _{t+1}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+1}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+2}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+3}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+4}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+5}\right)\end{array}\right] =\left[\begin{array}{c}\Delta y_{t} \\[3pt]\pi _{t} \\[3pt]i_{t} \\[3pt]\mathbb{E}_{t}\left( \Delta y_{t+1}\right) \\[3pt]\mathbb{E}_{t}\left( \Delta y_{t+2}\right) \\[3pt]\mathbb{E}_{t}\left( \pi _{t+1}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+1}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+2}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+3}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+4}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+5}\right)\end{array}\right] +\left[\begin{array}{c}\bar{\gamma}^{g}+\gamma _{t} \\[3pt]\bar{\gamma}^{\pi } \\[3pt]\bar{\gamma}^{r} \\[3pt]\bar{\gamma}^{g^{1}}+\mathbb{E}_{t}\left( \gamma _{t+1}\right) \\[3pt]\bar{\gamma}^{g^{2}}+\mathbb{E}_{t}\left( \gamma _{t+2}\right) \\[3pt]\bar{\gamma}^{\pi } \\[3pt]\bar{\gamma}^{r^{1}} \\[3pt]\bar{\gamma}^{r^{2}} \\[3pt]\bar{\gamma}^{r^{3}} \\[3pt]\bar{\gamma}^{r^{4}} \\[3pt]\bar{\gamma}^{r^{5}}\end{array}\right] +\left[\begin{array}{c}\mathbf{0}_{3\times 8} \\[3pt]\mathbf{I}_{8\times 8}\end{array}\right] \left[\begin{array}{c}o_{t}^{g_{t+1}} \\[3pt]o_{t}^{g_{t+2}} \\[3pt]o_{t}^{\pi _{t+1}} \\[3pt]o_{t}^{i_{t+1}} \\[3pt]o_{t}^{i_{t+2}} \\[3pt]o_{t}^{i_{t+3}} \\[3pt]o_{t}^{i_{t+4}} \\[3pt]o_{t}^{i_{t+5}}\end{array}\right] , \end{equation}

where $g_{t}\equiv \Delta y_{t}$ represents the growth rate of output at time t. Observations for expectations include i.i.d. measurement error terms, that is, $o_{t}^{g_{t+1}}$ , $o_{t}^{g_{t+2}}$ , $o_{t}^{\pi _{t+1}}$ , $o_{t}^{i_{t+1}}$ , $o_{t}^{i_{t+2}}$ , $o_{t}^{i_{t+3}}$ , $o_{t}^{i_{t+4}}$ , and $o_{t}^{i_{t+5}}$ . This mapping is similar to that of Cole and Milani (Reference Cole and Milani2017) and consistent with the expectations-augmented approach to disentangle between news and surprises about monetary policy proposed by Doehr and Martinez-Garcia (Reference Doehr and Martínez-García2015).

It is important to clarify how the SPF expectations align with model implied expectations. From the SPF documentation (FRB of Philadelphia (2019)), the respondents of the SPF usually have to report their forecasts before the middle of the current quarter. For instance, in regard to forecasts for Q1, the deadline submission date is the second to third week of February. The nowcast is approximately a 2-month ahead forecast, while the one-quarter ahead is a 5-month ahead forecast and so on. For that reason, a number of papers in the literature most closely connected to ours map the nowcast of the forecasted variables of the SPF as the one-quarter ahead forecast in the model (e.g. that of Cole and Milani (Reference Cole and Milani2017)). In our baseline analysis, we also treat the SPF nowcast as corresponding to the one-quarter ahead forecasts in our benchmark model to ensure comparability. ^{Footnote 25} However, in Subsection 5.2, we analyze the baseline results under a different (but still plausible) timing convention using instead one-quarter ahead SPF forecasts as the one-quarter expectations in the model.

4.1 Estimates of central bank credibility

We now proceed with our main empirical exercise to investigate the effects that central bank credibility has on the efficacy of forward guidance. When we estimate $\tau$ , we refer to this case as the not perfectly credible central bank scenario and denote the estimate as $\widehat{\tau }$ . When we do not estimate $\tau$ and assume private agents perceive the monetary authority to be perfectly credible, we simply set the credibility parameter to be $\tau =1$ in the estimation. The results are shown via three channels: posterior point estimates, variance decomposition, and impulse response functions under both perfectly credible and imperfectly credible central bank scenarios.

Tables 1 and 2 display the posterior mean and 90% highest posterior density interval estimates. Table 3 shows the conditional variance decomposition upon impact of the forward guidance and all other structural shocks with parameter values at their posterior mean. The last line calculates the sum of all the variation in the macroeconomic variable due to the forward guidance shocks. Figures 2 and 3 display the impulse response functions to forward guidance shocks at different horizons while Figure 4 displays the impulse response functions for the productivity growth, cost-push, and unanticipated monetary policy shocks. Each panel shows the mean response of the model-implied output and the other macro observables (inflation and interest rate). The solid line represents the scenario where $\tau =1$ , while the dashed line denotes $\widehat{\tau }$ .

Figure 2.

Impulse response functions. Mean response of model-implied output and observables (inflation and interest rate) to one-period ahead forward guidance and four-period ahead forward guidance shocks. Solid Line: Perfectly Credible C.B. (i.e. $\tau=1$ ). Dashed line: Not Perfectly Credible C.B. (i.e. Benchmark $\hat{\tau}$ ). C.B.: Central Bank.

Figure 3.

Impulse response functions. Mean response of model-implied output and observables (inflation and interest rate) to eight-period ahead forward guidance and twelve-period ahead forward guidance shocks. Solid Line: Perfectly Credible C.B. (i.e. $\tau=1$ ). Dashed line: Not Perfectly Credible C.B. (i.e. Benchmark $\hat{\tau}$ ). C.B.: Central Bank.

Figure 4.

Impulse response functions. Mean response of model-implied output and observables (inflation and interest rate) to productivity growth, cost-push, and unanticipated monetary policy shocks. Solid Line: Perfectly Credible C.B. (i.e. $\tau=1$ ). Dashed line: Not Perfectly Credible C.B. (i.e. Benchmark $\hat{\tau}$ ). C.B.: Central Bank.

Table 1.

Prior & Posterior estimates of structural parameters

Note: C.B.: Central Bank, G: Gamma Distribution, N: Normal Distribution, B: Beta Distribution, U: Uniform Distribution, IG: Inverse-Gamma Distribution.

Table 2.

Prior & Posterior estimates of measurement errors

Note: C.B.: Central Bank, G: Gamma Distribution, N: Normal Distribution, B: Beta Distribution, U: Uniform Distribution, IG: Inverse-Gamma Distribution.

Table 3.

Variance decomposition upon impact

Note: This table computes the conditional variance decomposition upon impact of the structural and forward guidance shocks with parameter values at their posterior mean. Each column displays the percentage contribution of each shock to model-implied output and observables (inflation and interest rates). Total FG denotes the sum of all of the forward guidance shocks. The measurement errors are not shown as their contribution concerns expected values of observables.

We first examine the case in which the monetary authority is perceived to be perfectly credible. In Tables 1 and 2 the first three columns under “Posterior Distribution” show that the estimates of the main structural parameters largely align with prior literature. The estimated value of the inverse of the Frisch elasticity of labor supply $\omega=0.97 $ follows closely that of CÚrdia et al. (2015). The interest rate smoothing parameter is estimated to be high at $0.88$ which closely aligns with the value found in Milani (Reference Milani2007) using a model with FIRE expectations. The estimates for the degree of productivity growth inertia ( $\rho _{\gamma }$ ) and inflation indexation ( $\iota _{p}$ ) roughly follow the results found in CÚrdia et al. (2015) under their “W” rule. ^{Footnote 28}

The solid lines in Figures 2 and 3 show the mean impulse response whenever $\tau =1$ to a one standard deviation increase in a shock. Specifically, given that agents are forward looking, news that the interest rate will increase 1, 4, 8, or 12 periods ahead affects agents’ intertemporal decisions by (noticeably) lowering output and inflation on impact. When the shock is realized on the economy, output (roughly) reaches its trough. Since those who perceive the monetary authority to be perfectly credible form their expectations under FIRE, these agents completely understand the shock has already materialized, and thus, output, inflation, and interest rates proceed to return back to steady state. ^{Footnote 29}

What are the predicted effects if the central bank is not assumed to be perfectly credible? To answer this question, we first analyze the posterior estimates in the last three columns in Tables 1 and 2. Overall, the values of the main structural parameters do not drastically differ from the perfectly credible case, but do display a few slight differences. For instance, in the last three columns under “Not Perfectly Credible C.B.,” the estimated value of the autoregressive parameter on the cost-push shock is relatively lower than in the $\tau =1$ scenario. However, this lower persistence could instead be picked up by the higher estimates for the inflation indexation parameter relative to the perfectly credible central bank case. ^{Footnote 30}

More importantly, when allowing private agents the option of not fully believing forward guidance statements about the path of interest rates, the estimate of the credibility parameter $\tau $ is $0.77$ . This value indicates a certain level of trust in the Federal Reserve implying some effectiveness of forward guidance on the economy. ^{Footnote 31} However, the fact that this estimated value is not close to $\tau =1$ suggests that private agents do not believe the monetary authority to be perfectly credible (or at least act in forming their expectations as if monetary policy announcements were not fully credible).

The ramifications of this result are a dampening of the power of forward guidance on the economy. Figures 2 and 3 display this outcome. The impulse responses under $\widehat{\tau }$ (dashed line) follow similar paths as under $\tau =1$ (solid line). However, the dashed line is not as reactive to central bank forward guidance as the solid line. Specifically, the initial impact of output and inflation to forward guidance news is larger under the perfectly credible case than under the imperfectly credible scenario. When the forward guidance shock is realized on the economy l periods later, the responses of output and inflation are also overall larger under $\tau =1$ than $\widehat{\tau }$ . The reason for the discrepancies is that private agents believe central bank statements about future interest rates under the $\tau =1$ scenario, and thus, fully internalize the effects of forward guidance. In contrast, agents who do not fully believe forward guidance commitments do not incorporate the full effects of forward guidance, and thus, macroeconomic variables are not as responsive.

Variance decomposition results also display the diminished effects of forward guidance on the economy under an imperfectly credible monetary authority. In Table 3 we compute the conditional variance decomposition upon impact of the shocks with parameter values at their posterior mean. The combined contribution of the forward guidance shocks to output and inflation is less under $\widehat{\tau }$ than $\tau =1$ . Under a central bank that is perceived as imperfectly credible, the total contribution of $\varepsilon _{1,t}^{FG}$ , $\varepsilon _{2,t}^{FG}$ ,…, $\varepsilon _{12,t}^{FG}$ to output and inflation is $7.53\%$ and $0.00\%$ , respectively. ^{Footnote 32} Under a monetary authority perceived as perfectly credible, the combined contribution is $23.14\%$ and $1.36\%$ . Thus, if a central bank is perceived as more credible, there exist greater effects on the real economy from forward guidance but little effect on inflation. To put it another way, if the central bank is less credible, the effects on output are not as great relative to the perfectly credible scenario.

The results show that modeling forward guidance credibility contributes to dampen the large effects that led Del Negro et al. (Reference Del Negro, Giannoni and Patterson2012) to question the magnitudes of the anticipation impact in the New Keynesian model (which these authors termed the “forward guidance puzzle”). In their paper, Del Negro et al. (Reference Del Negro, Giannoni and Patterson2012) explain that a standard New Keynesian model similar to the one presented in Section 2 produces surprisingly large responses of the macroeconomic variables to forward guidance shocks. In contrast, our paper allows for private agents to not perceive the monetary authority as perfectly credible. As discussed in the previous paragraphs, the results show that the reaction of macroeconomic variables to forward guidance shocks is dampened and therefore is not as sizeable under $\widehat{\tau }$ relative to $\tau =1$ .

The prior literature has offered different views on the “forward guidance puzzle”. For instance, Del Negro et al. (Reference Del Negro, Giannoni and Patterson2012) show unusually large responses of the macroeconomic variables to forward guidance via a constant interest rate scenario. Bundick and Smith (Reference Bundick and Smith2020) and Kiley (Reference Kiley2016) utilize elasticities of output relative to a change in future interest rates when comparing model- and data-implied forward guidance effects. ^{Footnote 33} McKay et al. (Reference McKay, Nakamura and Steinsson2016) present the puzzle via a one-period anticipated interest rate shock occurring at time T in the future. However, they analyze the effects of forward guidance under complete and incomplete asset markets, whereas in our paper we focus on the role of central bank forward guidance credibility (or lack thereof).

In all of these cases, the point of reference is somewhat different and the quantification and interpretation are also a bit different as a result. In all papers cited, the results are model-dependent and, to our knowledge, none of the related papers have explored the potential role of credibility and heterogeneous expectations as we do. Hence, it is hard to make an apples-to-apples comparison across the different contributions to the literature to neatly show how our results compare. Nevertheless, we compute elasticities of output and inflation to a change in expected interests rates four-periods ahead which is a similar conceptual exercise to what Bundick and Smith (Reference Bundick and Smith2020) do. Specifically, we divide the trough response of output (inflation) to the four-period ahead forward guidance shock seen in Figure 2 by the response of (observed) expected interest rates four-periods ahead. We compute these statistics across both the $\tau=1$ and $\hat{\tau}$ cases.

Table 4 shows that the elasticities of output are $-5.28$ and $-0.65$ under the $\tau=1$ and $\hat{\tau}$ scenarios, respectively. In other words, under perfect credibility, a one percentage point increase in expected four-quarter ahead interest rates is associated with a $5.28$ percentage point drop in output at its trough point. Under imperfect credibility, the maximum decrease in output is $0.65$ , confirming our benchmark results that show the assumption of a perfectly credible central bank can overstate the predicted effects of forward guidance on the economy. The elasticities of inflation are $-0.10$ and $-0.03$ under the $\tau=1$ and $\hat{\tau}$ scenarios, respectively. We also should note that the elasticity numbers for the $\hat{\tau}$ scenario are less than the values found in Bundick and Smith (Reference Bundick and Smith2020). We attribute the difference, at least in part, to the fact that their analysis implicitly assumes fully credible forward guidance commitments. In our paper, we have argued and shown that the credibility assumption greatly affects the efficacy of forward guidance on the economy as shown in Table 4 as well as in Figures 2 and 3.

Table 4.

Posterior estimates of structural parameters without frictions

Note: B: Beta Distribution.

Table 5.

Estimated elasticity of output and inflation with respect to 1-year ahead expected interest rates

Overall, the results of our main exercise suggest a number of takeaways. Our estimate of the Federal Reserve’s credibility is high at $\widehat{\tau }=0.77$ . However, this estimated value is below the fully credible case which leads to a substantial attenuation of the power of forward guidance. If the central bank is perceived as less credible, there exist less immediate and overall effects on the economy from forward guidance. Hence, the integration of imperfect central bank credibility with heterogeneous expectations into a standard macroeconomic model can be another feature of the economy that contributes to mute the effects of forward guidance. Thus, accounting for imperfect credibility is important to model the formation of expectations in the economy and the transmission mechanism of forward guidance announcements.

4.2. Evidence that $\tau$ reflects central bank credibility

Our paper models aggregate expectations as a weighted average of those who take central bank forward guidance statements as fully credible and those that do not. The former group forecasts macroeconomic variables under the FIRE assumption, while the latter uses a data-driven VAR $\left( 1\right) $ model. In equation (26), the parameter $\tau$ measures the degree of central bank credibility in terms of forward guidance. This parameter was taken to be constant and exogenous for simplicity, similar to what the prior literature does (e.g. Haberis et al. (Reference Haberis, Harrison and Waldron2014)). However, a perennial question is to what extent $\tau$ reflects central bank credibility? To try to explore this question further, we perform three exercises that give us additional confidence that $\tau$ is capturing a measure of monetary authority credibility. Specifically, we examine the influence of frictions on the estimate of $\tau$ , the influence of the reduced-form forecasting model on $\tau$ , and the comovements between forecast errors and forecast disagreements in the SPF dataset relative to those implied by our benchmark model with imperfect credibility.

4.2.2 Alternative reduced-form forecasting model

We also examine the estimation of the central bank credibility parameter $\tau$ under alternative specifications of the forecasting models used by the private sector. The benchmark case in Section 4 assumed private sector agents who do not believe the monetary authority to be credible would form expectations from a VAR $\left( 1\right) $ based on (25). In turn, private agents who believe the central bank follow FIRE expectations. However, a natural question arises regarding whether the estimated value of $\tau $ is sensitive to the information set in the forecasting model of private sector agents?

Here, we examine the case when private agents that do not believe the central bank know more about the true structure of the economy. That is, we explore the case when not credible expectations ( $\mathbb{E}_{t}\left(Y_{t+1}^{D}\right) $ ) become better informed than under equation (25). Private agents are assumed to know now the productivity and cost-push shock realizations, that is, $w_{t}=\left[\gamma _{t},\mu _{t}\right] ^{^{\prime }}$ , when formulating their expectations of future macroeconomic variables. Thus, equation (25) is replaced with:

(28) \begin{equation}Y_{t}=A+BY_{t-1}+Cw_{t}+e_{t},\end{equation}

where A, B, and C are coefficient matrices of appropriate dimensions, and $e_{t}$ is a vector of white noise (nonstructural) residual terms.

Tables 6 and 7 produce two main takeaways. First, additional knowledge (or information) about the true structure of the economy seems to have minimal effect on the posterior estimates of the parameters. In Table 6, the estimate of our central bank credibility parameter is $0.78$ . This value is approximately the same as our baseline estimate of $0.77$ . ^{Footnote 34} Thus, even if private agents utilize a forecasting model with more information, the results do not substantially change. In particular, there still are noticeable effects of forward guidance on the economy as private agents are estimated to believe the Fed to be highly credible, although below the fully credible case of $\tau =1$ .

Table 6.

Prior & Posterior estimates of structural parameters with alternative non-credible expectations

Note: G: Gamma Distribution, N: Normal Distribution, B: Beta Distribution, U: Uniform Distribution, IG: Inverse-Gamma Distribution.

Table 7.

Prior & Posterior estimates of measurement errors with alternative non-credible expectations

Note: G: Gamma Distribution, N: Normal Distribution, B: Beta Distribution, U: Uniform Distribution, IG: Inverse-Gamma Distribution.

The second takeaway is that, since the estimate of $\tau $ does not significantly change when private agents are more informed, $\tau $ does not seem to be capturing private agents’ lack of knowledge about the true structure of the economy. In other words, it does not seem biased due to not knowing productivity growth and cost-push shocks. Thus, because the value of $\tau $ does not seem to depend on these other non-forward guidance elements, this latter result suggests once more that what ultimately underpins $\tau $ must be something else like central bank credibility on forward guidance.

4.2.3 Predictability of forecasting errors

Our benchmark model assumes that expectations in the economy are a weighted sum of private agents who believe the central bank to be perfectly credible and private agents who believe the monetary authority not to be credible. The former forecasts macroeconomic variables under the assumption of FIRE expectations, while the latter uses a data-driven VAR $\left( 1\right) $ model. However, a question naturally arises as to whether heterogeneous expectations (with a non-FIRE component) aggregated by $\tau $ or the perfectly credible, homogeneous and FIRE expectations case is the more plausible way to describe the observed SPF expectations?

Standard macroeconomic models often consider only the $\tau =1$ case, that is, homogeneous FIRE expectations. In this scenario, if we take the model to be the true data-generating process, forecast errors will be random and on average equal to zero (see, e.g. Sims (Reference Sims2002) and Cole and Milani (Reference Cole and Milani2017)). Moreover, in this case expectations are by construction homogeneous, there are no model-implied forecasting disagreements and thus forecasting errors would not be dependent or correlated with forecasting disagreements among private agents. However, do the forecast errors from the data agree with those implications of the $\tau =1$ scenario? Or does the imperfect credibility scenario with $\widehat{\tau }$ capturing the estimated central bank credibility fit the SPF data better? To explore these questions further, we compare the forecast errors from our model with imperfect central bank credibility to what is found in the observed SPF forecasting data. ^{Footnote 35}

We proceed in the following manner. We define forecasting errors of the interest rate at the one-quarter ahead horizon as $FE_{t}^{1}=\mathbb{E}_{t-1}\left( i_{t}\right) -i_{t}$ where the expectations $\mathbb{E}_{t-1}\left( i_{t}\right)$ are being matched to the mean SPF forecast. The remaining forecast errors are defined as $FE_{t}^{2}=\mathbb{E}_{t-2}\left( i_{t}\right) -i_{t}$ , $FE_{t}^{3}=\mathbb{E}_{t-3}\left( i_{t}\right) -i_{t}$ , $FE_{t}^{4}=\mathbb{E}_{t-4}\left( i_{t}\right) -i_{t}$ , and $FE_{t}^{5}=\mathbb{E}_{t-5}\left(i_{t}\right) -i_{t}$ at the two, three, four, and five-quarters ahead horizons, respectively. Forecasting disagreement ( $DEV_{t}^{1}$ ) at the one-quarter ahead horizon is specified as the difference between the 75th and 25th percentile (i.e., the interquartile range of the SPF data). The remaining forecasting disagreements are given by $DEV_{t}^{2}$ , $DEV_{t}^{3}$ , $DEV_{t}^{4}$ , and $DEV_{t}^{5}$ at the two, three, four, and five-quarters ahead horizons, respectively.

We collect data from the US economy. Forecasting errors are computed with respect to the mean forecast with SPF data. The relevant acronym is TBILL. Forecasting disagreements are measured with the interquartile range of the cross-sectional distribution of individual forecasts to make the empirical results less sensitive to outliers. ^{Footnote 36} The data span $1981\,:\,Q3-2018\,:\,Q4$ for the baseline case implying 150 observations. We run separate regressions of $FE_{t}^{h}$ on $DEV_{t}^{h}$ at the one, two, three, four, and five-quarters ahead horizons ( $h=1,2,3,4,5$ ):

(29) \begin{align}FE_{t}^{1} &= \delta _{0}^{1}+\delta _{1}^{1}DEV_{t}^{1}+e_{t}^{1}, \end{align}

(30) \begin{align}FE_{t}^{2} &= \delta _{0}^{2}+\delta _{1}^{2}DEV_{t}^{2}+e_{t}^{2}, \end{align}

(31) \begin{align}FE_{t}^{3} &= \delta _{0}^{3}+\delta _{1}^{3}DEV_{t}^{3}+e_{t}^{3}, \end{align}

(32) \begin{align}FE_{t}^{4} &= \delta _{0}^{4}+\delta _{1}^{4}DEV_{t}^{4}+e_{t}^{4}, \end{align}

(33) \begin{align}FE_{t}^{5} &= \delta _{0}^{5}+\delta _{1}^{5}DEV_{t}^{5}+e_{t}^{5}.\end{align}

The regression error terms are described by $e_{t}^{1}$ , $e_{t}^{2}$ , $e_{t}^{3}$ , $e_{t}^{4}$ , and $e_{t}^{5}$ at the one, two, three, four, and five-quarters ahead horizons, respectively.

We perform the same exercise using the simulated counterparts from our benchmark model with imperfect central bank credibility and heterogeneous expectations. Specifically, we simulate the model at the posterior mean for a time period of 2,000 observations discarding the first 100 simulated observations. We then perform the same regressions given by equations (29)-(33) on a random window of 150 observations. ^{Footnote 37} We compute our results on 10,000 different draws of the simulated data.

The full range of estimates are represented in Figure 5, which shows a box-and-whisker plot that displays their min, max, median, and interquartile range. The point estimates obtained from the SPF data are also shown in Figure 5. The solid line with circle markers denote estimates of $\delta _{1}^{1}$ , $\delta _{1}^{2}$ , $\delta _{1}^{3}$ , $\delta _{1}^{4}$ , and $\delta _{1}^{5}$ over the full sample (i.e. $1981\,:\,Q3-2018\,:\,Q4$ ). The dashed line with diamond markers represent estimates from the non-ZLB period of our dataset, that is, from the $1981\,:\,Q3-2008\,:\, Q4$ subsample.

Figure 5.

Estimates of forecasting errors response to forecasting disagreement, Survey of Professional Forecasters (SPF) vs. Simulated Data. Note: Forecasting errors are computed with respect to the mean forecast with SPF data from 1981 : Q3 until 2018 : Q4 (150 observations). Forecasting disagreements are measured with the interquartile range of the cross-sectional distribution of individual forecasts in order to make the empirical results less sensitive to outliers. We regress forecasting errors on an intercept and this measure of forecasting disagreement for all available time horizons. We perform the same exercise on a random window of 150 observations from a simulated sample data of 2000 observations under $\hat{\tau}$ . We represent the full range of estimates over 10000 draws with box-and-whisker plots that show their min, max, median, and interquartile range. SOURCES: Survey of Professional Forecasters (SPF), simulated data, authors’ calculations.

We first examine what the US data would postulate regarding the relationship between forecasting errors and forecasting disagreements. By using SPF data, estimates of $\delta _{1}^{1}$ , $\delta _{1}^{2}$ , $\delta _{1}^{3}$ , $ \delta _{1}^{4}$ , and $\delta _{1}^{5}$ show that forecasting errors are positively correlated with forecasting disagreement. The circle and diamond markers in Figure 5 are positive and notably display an upward trajectory as the forecast horizon increases. This challenges the assumption that forecasting errors ought to be unpredictable under the perfect credibility case with FIRE expectations whenever $\tau =1$ .

Another implication of the SPF results in Figure 5 is that the mean SPF forecast tends to overestimate the expected path of the US short-term interest rate more whenever disagreement among forecasters exists, and this tends to be more apparent at longer horizons. What occurs when private agents perceive the central bank as not perfectly credible in the model? The simple answer is that the model under $\widehat{\tau }$ seems to match the SPF data fairly well on that account too.

In Figure 5, the estimates $\delta _{1}^{1}$ , $\delta _{1}^{2}$ , $\delta _{1}^{3}$ , $\delta _{1}^{4}$ , and $\delta _{1}^{5}$ using SPF data lie toward the median of their respective box-and-whisker plots under $\widehat{\tau }$ . The median of the box-and-whisker plots is also in positive territory which matches their counterparts in the data. In addition, under both SPF and simulated data, estimates of $\delta _{1}^{1}$ , $\delta _{1}^{2}$ , $\delta _{1}^{3}$ , $\delta _{1}^{4}$ , and $\delta _{1}^{5}$ display an upward trend the longer the forecasting horizon. This finding suggests that a model of imperfect credibility and heterogeneous expectations is better suited to capture forecasting disagreements and the predictability of forecasting errors based on disagreement that we observe in the SPF data.

In short, as noted before, our model of central bank credibility and heterogeneous expectations offers a way to account for the imperfect comovement between the Fed’s forward guidance and the individual projections of private forecasters in the SPF. We hinted at that in Figure 1 for the Fed’s most recent phase of “Delphic” forward guidance. Our imperfectly credible central bank model extends that idea in a flexible manner to explore the full sample since the onset of the Great Moderation in the 1980s and tease out a common measure of credibility across different strategies (or phases) in the Fed’s use of forward guidance. What we find in Figure 5 is that this theoretical framework matches well key facts of the SPF data. In the SPF data, there exists indeed notable disagreements among forecasters which the fully credible central bank model (i.e. the $\tau =1$ scenario) cannot capture. Furthermore, the fully credible central bank model cannot account for the robust evidence found in the SPF that forecasting errors are predictable on the basis of forecasting disagreements either.

This is not to say that monetary policy credibility explains all forecasting disagreements, but it shows that imperfect credibility is a plausible reason for forecasting heterogeneity that can bring the workhorse New Keynesian model closer to the empirical evidence that we observe in the SPF data.

5.1 Subsamples

5.1.1 Non-ZLB

Monetary policy was instrumented with the fed funds rate until hitting the ZLB in the aftermath of the $2007-2009$ global financial crisis. It is important to compare the effect of central bank credibility on forward guidance during the non-ZLB and ZLB periods to assess whether our findings are sensitive across subsamples. Thus, in this subsection, we re-estimate the model over the subsample $1981\,:\,Q3-2008\,:\,Q4$ and compare the results to our benchmark outcomes.

The “Non-ZLB” column in Tables 8 and 9 displays the results. Even during an era where the interest rate does not bind at zero, our baseline result still holds. Specifically, the value of $\tau$ is estimated to be $0.78$ , which is about the same as our benchmark estimate of $0.77$ . The values of the other parameters do not considerably change either. Therefore, noticeable effects of forward guidance on the economy exist during the non-ZLB period when a high degree of Federal Reserve credibility is estimated to exist. However, the estimate of $\tau$ remains below the fully credible central bank scenario.

Table 8.

Prior & Posterior estimates of structural parameters under robustness checks

Note: G: Gamma Distribution, N: Normal Distribution, B: Beta Distribution, U: Uniform Distribution, IG: Inverse-Gamma Distribution.

Table 9.

Prior & Posterior estimates of measurement errors under robustness checks

Note: G: Gamma Distribution, N: Normal Distribution, B: Beta Distribution, U: Uniform Distribution, IG: Inverse-Gamma Distribution.

5.2 Alternative priors on the credibility parameter

Our baseline prior assumption for $\tau $ assumed a high degree of central bank credibility. This value seems reasonable and fits well with the Federal Reserve’s strong reputation and the related results in Park (Reference Park2018) who reports high values for Federal Reserve credibility as well. We have also alluded previously to the evidence in Figure 1. This evidence shows that while private forecasts do not seem to perfectly coalesce around the Fed’s own projections as we would expect them under the assumption of perfect credibility, there is nonetheless evidence of robust comovement that suggests that the Fed indeed has the ability to influence the expectations of private forecasters to a significant degree. Our prior for $\tau $ recognizes that.

However, it is also important to examine our results under a different light, that is, if we were to adopt a more agnostic view about the prior distribution of $\tau $ . In Tables 8 and 9, we examine the results when the prior distribution for $\tau $ changes to an uninformative uniform distribution on the unit interval. The results show that the estimate for $\tau $ in this case is $0.60$ , which is somewhat smaller than our benchmark of $0.77$ . ^{Footnote 38} Therefore, if one is agnostic about the true value of $\tau $ and adopts a U(0,1) prior distribution, similar to the benchmark results in Subsection 4.1, the estimated value of $\tau $ is still below the fully credible case (i.e. $\tau =1$ ). Moreover, there continues to exist effects of forward guidance on the US economy as the public still believes the US central bank to be credible, albeit both the credibility and the magnitude of those effects would be weakened. ^{Footnote 39}

Furthermore, the posterior distributions seem to be similar in shape regardless of the diffuseness of the prior density even when adopt a less informative (more disperse) beta prior instead of the uniform distribution. The right panel of Figure 6 displays the posterior (solid line) and prior (dashed line) distributions with a $B(0.80, 0.1)$ density assumed a priori. The left panel displays the baseline that we use in our estimation, that is, $B(0.80, 0.01)$ for $\tau$ . While the mean estimates differ somewhat, the posterior distributions between the more informative and more diffuse priors do not appear to overwhelm the posterior distribution on $\tau$ . A plot similar to the one for the diffuse prior beta arises also under the uninformative uniform prior.

Figure 6.

Posterior and prior distributions under alternative priors on $\tau$ . Left Panel: $B(0.80,0.01)$ Prior, Right Panel: $B(0.80, 0.1)$ Prior. Solid Line: Posterior Distribution, Dashed Line: Prior Distribution.

While there is a concern that the parameter $\tau $ might be weakly identified, we believe there are valid reasons to defend our benchmark prior—reasons to which we have appealed again at the beginning of this subsection—suggesting the prior should put more mass on higher values of $\tau $ . In any event, we interpret the results discussed here as suggesting that the power of forward guidance is overstated under perfect credibility and, if anything, that our baseline estimate of the effects of forward guidance should be viewed as rather conservative given that we favor a prior that places more mass on higher values of the credibility parameter. Our conservative benchmark estimate of $\tau $ already suggests that the power of forward guidance can be significantly dampened under imperfect credibility as seen in Table 5. From our perspective, then, the main takeaway from this is not so much whether the benchmark estimate of $\tau $ is overstated or not. The main takeaway is that even though we favor the more conservative estimate with somewhat higher estimated credibility which is what we report as our benchmark, we still find that the higher degree of credibility that we recover from the data (i.e. $0.77$ ) is not high enough to produce an impact on economic activity within range of what we would estimate under full credibility.

5.3 Alternative mapping of SPF forecasts

Section 3 described the data and observables that we included for the estimation of our model. In our benchmark analysis, we utilized the SPF nowcast for our model’s one-quarter ahead expectations and so on. As explained in Subsection 3.2.1, we believe this assumption made sense given the actual submission dates and timing of SPF forecasts. However, we recognize that questions can emerge regarding this mapping. Specifically, what if SPF one-quarter ahead forecasts were used to correspond with our model’s one-quarter ahead expectations instead of the nowcasts?

This section performs a robustness check to analyze the results when such possibility is taken into account. The model’s observation equations are modified as follows:

(34) \begin{equation}\left[\begin{array}{c}g_{t}^{obs} \\[3pt]\pi _{t}^{obs} \\[3pt]i_{t}^{obs} \\[3pt]\mathbb{E}_{t}^{obs}\left( g_{t+1}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( g_{t+2}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( \pi _{t+1}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+1}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+2}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+3}\right) \\[3pt]\mathbb{E}_{t}^{obs}\left( i_{t+4}\right)\end{array}\right] =\left[\begin{array}{c}\Delta y_{t} \\[3pt]\pi _{t} \\[3pt]i_{t} \\[3pt]\mathbb{E}_{t}\left( \Delta y_{t+1}\right) \\[3pt]\mathbb{E}_{t}\left( \Delta y_{t+2}\right) \\[3pt]\mathbb{E}_{t}\left( \pi _{t+1}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+1}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+2}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+3}\right) \\[3pt]\mathbb{E}_{t}\left( i_{t+4}\right)\end{array}\right] +\left[\begin{array}{c}\bar{\gamma}^{g}+\gamma _{t} \\[3pt]\bar{\gamma}^{\pi } \\[3pt]\bar{\gamma}^{r} \\[3pt]\bar{\gamma}^{g^{1}}+\mathbb{E}_{t}\left( \gamma _{t+1}\right) \\[3pt]\bar{\gamma}^{g^{2}}+\mathbb{E}_{t}\left( \gamma _{t+2}\right) \\[3pt]\bar{\gamma}^{\pi } \\[3pt]\bar{\gamma}^{r^{1}} \\[3pt]\bar{\gamma}^{r^{2}} \\[3pt]\bar{\gamma}^{r^{3}} \\[3pt]\bar{\gamma}^{r^{4}}\end{array}\right] +\left[\begin{array}{c}\mathbf{0}_{3\times 7} \\[3pt]\mathbf{I}_{7\times 7}\end{array}\right] \left[\begin{array}{c}o_{t}^{g_{t+1}} \\[3pt]o_{t}^{g_{t+2}} \\[3pt]o_{t}^{\pi _{t+1}} \\[3pt]o_{t}^{i_{t+1}} \\[3pt]o_{t}^{i_{t+2}} \\[3pt]o_{t}^{i_{t+3}} \\[3pt]o_{t}^{i_{t+4}}\end{array}\right] . \end{equation}

Two differences are apparent between the observation equations in this section (i.e. equation (34) and the baseline observation equations (i.e. equation (27). First, the $t+1$ timing in our model now corresponds to one-period ahead expectations in the SPF dataset. Second $\mathbb{E}_{t}^{obs}\left(i_{t+5}\right) $ is not in equation (34). Since this exercise utilizes one-quarter ahead SPF expectations (and not the nowcast) for $t+1$ expectations in our model, we only have data up to four-quarters ahead from the SPF under this alternative mapping of survey-based forecasts and model-based expectations. Tables 8 and 9 display the estimated values of the structural and measurement error parameters for this exercise.

The main takeaway is that the benchmark results are robust to the timing assumption matching SPF forecasts to our model’s expectations. The estimated value of our parameter of interest, $\tau$ , is $0.78$ , which does not notably change relative to the benchmark value reported in Section 4. In addition, the estimates of the other parameters do not considerably change either. However, the value of the marginal likelihood is lower at $631.36$ compared to $901.18$ from the baseline case. Since the marginal likelihood depends on the data, the discrepancy could be due to this implementation using one less observable than that of Section 4 as described in the previous paragraph.