Nonlinearities, Smoothing and Countercyclical Monetary Policy

Empirical analysis of the Fed’s monetary policy behavior suggests that the Fed smooths interest rates— that is, the Fed moves the federal funds rate target in several small steps instead of one large step with the same magnitude. We evaluate the effect of countercyclical policy by estimating a Vector Autoregression (VAR) with regime switching. Because the size of the policy shock is important in our model, we can evaluate the effect of smoothing the interest rate on the path of macro variables. Our model also allows for variation in transition probabilities across regimes, depending on the level of output growth. Thus, changes in the stance of monetary policy affect the macroeconomic variables in a nonlinear way, both directly and indirectly through the state of the economy. We also incorporate a factor summarizing overall sentiment into the VAR to determine if sentiment changes substantially around turning points and whether they are indeed important to understanding the effects of policy. [JEL codes: C24, E32] The authors benefitted from conversations with Michael McCracken, Valerie Ramey, Martin Sola, participants at the St. Louis Fed Applied Econometrics Workshop, the San Francisco Fed Research Department Seminar Series, and at the Bentley University Research Seminar Series. Diana A. Cooke, Hannah G. Shell, and Kate Vermann provided research assistance. Views expressed here are the authors’ alone and do not reflect the opinions of the Federal Reserve Bank of St. Louis or the Federal Reserve System. Corresponding author. Department of Economics. Bentley University. 175 Forest Street, Waltham, MA 02452. ljackson@bentley.edu. Research Division. Federal Reserve Bank of St. Louis. Department of Economics and Finance. University of North Carolina, Wilmington.


Introduction
Empirical analysis of the Fed's monetary policy behavior suggests that the Fed smooths interest rates-that is, the Fed moves the federal funds rate target in several small steps instead of one large step with the same magnitude. Smoothing has been characterized as an optimal monetary policy response in models that incorporate the private sector's expectations of future policy [e.g., Woodford (1999)]. In these types of models, the monetary authority's method of credibly altering expectations is important in determining the e¢cacy of the stabilization policy.
Because monetary policy is countercyclical, its e¤ectiveness is often measured by its ability to induce large responses in output growth and in ‡ation. However, the models used to measure these responses (e.g., VARs) are often linear and may exaggerate the e¤ectiveness of policy if the dynamics of the economy change across states. 1 Even when the VARs do incorporate some form of regime switching, the responses are often computed within-regime-i.e., the responses are computed assuming that the regime never changes. This assumption is problematic for evaluating countercyclical policy. In these models, during recessions, the Fed drops the funds target to raise output growth but has no e¤ect on the duration of the recession. 2 To evaluate the e¤ectiveness of countercyclical policy, we estimate a VAR with regime switching.
In our model, the probability of transitioning across regimes depends on the level of output growth. 3 Thus, changes in the stance of monetary policy a¤ect the macroeconomic variables nonlinearly, both directly and indirectly through the state of the economy. To this end, the estimated responses of macro variables to monetary shocks can depend on (1) the current state of the economy 4 , (2) 1 The existing literature provides mixed empirical evidence of asymmetry. Cover (1992) …nds that negative money supply shocks have larger e¤ects on output than positive shocks. Ravn and Sola (1996) …nd symmetric responses once they account for a break in late 1970's. Morgan (1993) …nds asymmetric responses of output to interest rate changes but the evidence is weaker when excluding the early 1980's when the Fed abandoned traditional rate targeting polices. 2 Garcia and Schaller (2002) extend Hamilton's (1989) regime-switching model to allow monetary policy to a¤ect the growth rate of output and the probability of switching between states. They …nd that changes in the fed funds rate have larger e¤ects during recessions than during booms and that policy has substantial e¤ects on the probability of switching between expansionary and recessionary regimes. 3 In related work, Weise (1999) uses a smooth-transition VAR to examine the asymmetric e¤ects of policy based on the three dimensions of interest: size, sign, and position in the business cycle. The author …nds evidence of size, but not sign, asymmetries and di¤erent e¤ects during periods of high or low growth. Monetary shocks have stronger output e¤ects and weaker price e¤ects when growth is initially low but have stronger price e¤ects and weaker output e¤ects when in a high growth state.
the history of the economy, (3) future shocks, and (4) the size of the (current) monetary shock. 5 We incorporate consumer and producer sentiment into the VAR to determine if con…dence and expectations change substantially around turning points and whether they are indeed important to understanding the e¤ects of policy. In addition, because the size of the shock is important in our model, we can evaluate the e¤ect of smoothing the interest rate on the path of macro variables.
We …nd empirically relevant di¤erences between the macroeconomic responses to contractionary and expansionary policy shocks, depending on the underlying state of the economy at the time of the shock. Small expansionary policy shocks induce responses with substantial variation in high and low output growth environments, but show less variation in periods of high and low in ‡ation. The responses to large expansionary shocks do not exhibit the same variation. We also …nd signi…cant di¤erences between gradual policy changes and one-time, large policy shocks, thus making a case for more aggressive policy intervention to combat recessions.
The balance of the paper is outlined as follows: Section 2 outlines the models. We start by …xing notation with the familiar single regime VAR. We then add the e¤ects of sentiment, modeled by a latent factor, and Markov-switching. Finally, we augment the Markov-switching with time-varying transition probabilities. Section 3 describes the data and the methods used to estimate the model. Details for the full sampler are left to the Appendix. Section 3.3 compares the di¤erent methods to compute the impulse responses to evaluate the e¤ectiveness of the shocks. In this section, we reiterate the importance of history, future, sign, and scale of the shock. Section 4 presents the baseline results. Section 5, in particular, focuses on the experiment comparing the e¤ect of a net 25-basis-point change in the federal funds rate implemented in a single step or in multiple steps. Section 6 o¤ers …nal thoughts.

Empirical Approach
One of the most commonly used models in the empirical analysis of monetary policy is the VAR.
A simple example of a monetary VAR is a three-variable model with measures of output growth and prices and a monetary policy instrument. The e¤ects of the policy shocks are determined by tracing out the impulse responses to identi…ed shocks. In this section, we construct a VAR that allows for asymmetric responses to shocks and di¤erences in shock volatilities. In addition, we model economic sentiment through a latent factor that is allowed to a¤ect or be a¤ected by macroeconomic aggregates in di¤erent ways depending on the state of the economy.

The VAR
Let y t represent the N 1 vector of period t variables of interest; then, the reduced-form VAR(P ) where we have suppressed the constant and any trends, e " t N 0; e is the reduced-form innovation, and e is left unrestricted. Inference on the e¤ect of shocks is derived from the structural form of the VAR:

Modeling Sentiment
Our desire is to augment the VAR with a broad measure of sentiment regarding the current strength of and outlook for the economy. Sentiment, however, is not easily quanti…able. We opt to include a factor (or vector of factors) F t representing overall sentiment in the VAR. Then, the (N + 1) 1 vector of variables of interest can be de…ned as Y t = [F t ; y 0 t ] 0 and the VAR rewritten as where the reduced form shocks " t N (0; t ) are now an (N + 1) 1 vector and we have imposed autoregressive dynamics on the factor. The factor F t summarizes the information in M series collected in a vector X t that contains observable information about consumer and producer sentiment.
The factor is related to X t = [X 1t ; :::; where & mt iidN 0; 2 m , which assumes that the innovations to the elements of X t are uncorrelated. This assumption imposes that the correlation across series are a result of the factor alone and is relatively common in the factor literature.

The Markov-Switching VAR
Recently, studies have investigated whether monetary policy has time-dependent e¤ects-for example, depending on the state of the economy. 6 For example, one could ask whether monetary policy has di¤ering e¤ects in recessions and expansions, when the Fed tightens or eases, or when the change in the fed funds target rate is large or small, etc. 7 One popular model used to determine the state-dependent e¤ects of monetary policy is the Markov-switching VAR, which has a reduced-form: where S t = f0; 1g follows an irreducible …rst-order Markov process with (constant) transition , and regimedependent heteroskedastic covariance matrix 6 See Hamilton (2015) for a detailed overview of regime-switching modeling techniques and applications within macroeconomics. 7 In recent work, Angrist, Jorda, and Kuersteiner (2013) …nd that contractionary policy can achieve reductions in output, employment, and in ‡ation but expansionary policy produces very little stimulus. Barnichon and Matthes (2014) …nd that contractionary policy shocks have strong adverse e¤ects on output while expansionary shocks do not have signi…cant e¤ects unless the shocks are large and occur speci…cally during recessions.
In this case, the economy takes on two alternative dynamics, dictated by the realization of the underlying state S t . When S t = 1, the economy has B 1 (L) dynamics and when S t = 0, the economy has B 0 (L) dynamics. Thus, the model is linear, conditional on S t being known. 8 The shock processes-by assumption-follow the same regime-switching process as the reduced-form VAR coe¢cients, making the contemporaneous e¤ects of the shocks regime-dependent. 9 The shocks are identi…ed using similar methods as above or, additionally, exploiting the regime-dependence [e.g., Rigobon and Sack (2004)].

Time-Varying Transition Probabilities
One drawback of the constant probability Markov-switching VAR is that the underlying regime is invariant to the model variables. Countercyclical policy then cannot a¤ect-either directly or indirectly-the state of the economy, making both the regimes and the impulse responses to changes in policy di¢cult to interpret. 10 One way to ameliorate this problem is to allow the state of the economy to depend, in part, on variables in the VAR. For example, if we want to interpret the state variable as business cycle regimes, we can make the transition probabilities functions of output growth.
We can accomplish this by assuming that the state process S t has time-varying, rather than constant, transition probabilities. Moreover, we assume that changes in the underlying state of the economy (and, thus, underlying changes in the dynamic responses to monetary shocks) are driven by (lags of) a variable z t . If, as in our case, z t is a variable in the VAR, shocks to the policy instrument a¤ect z t which, in turn, feed back into the regime. 11 Thus, more accommodative monetary policy in a recession can stimulate output and increase the probability of switching back 8 Based on multiple Lagrange Multiplier tests for linearity, Weise (1999) …nds that when using lagged output growth as the switching variable, the data prefer the non-linear model with time-variation in the coe¢cients of all equations in the VAR to a standard linear VAR with constant parameters. Furthermore, the parameter governing the speed of the transition between regimes is very large. This suggests a sharp transition between regimes and justi…es the use of a discrete regime-switching model. 9 It is straightforward to extend the model to allow the covariance switching process to vary from the coe¢cient switching process. The main drawback is that independent processes increases the number of regimes geometrically. 1 0 The constant transition probability model is also of limited use for forecasting. Conditional on the past regime being known, no additional data improves the forecast of the regime. 1 1 Potter (1995) called the system in which the transition variable is also in the VAR self-exciting.
to expansionary dynamics. We assume that the transition probabilities follow a logistic formulation: for each of the regimes with P k p ki (z t d ) = 1 for all i, t. We de…ne S t = 0 as the reference state; thus, all parameters governing the transition into expansion ( 0i and 0i for i = 0; 1) are normalized to 0.
We consider lagged output growth as the transition variable and set the delay parameter, d, to 1. Thus, output growth in the previous period will a¤ect the probability of switching between expansion and recession in the current period. In order to identify the two separate regimes, we impose that the coe¢cient on lagged output growth in ‡uencing the transition from expansion to recession, 10 , is negative. Therefore, if S t 1 = 0 (expansion) and output growth is above average, the probability that S t = 1 (recession) falls.

Empirical Analysis
In this section, we describe how the model is estimated, the data used in the estimation, and the methods for which we compute the impulse responses.

The Sampler
The model parameters, factors, regimes, and transition probabilities are estimated using the Gibbs sampler. Let the full set of parameters, including the regimes and the factors, be represented by: S T = fS t g T t=1 , and F T = fF t g T t=1 . The Gibbs sampler draws elements of , S T , and F T , conditional on the previous draw of each other elements. We sample from …ve blocks: (1) the VAR coe¢cients and covariance matrices; (2) the regimes; (3) the transition function parameters; (4) the factor; and (5) the factor loadings and residual variances. The joint posterior distribution of all the model parameters and the factors are obtained from these draws from the conditional distribution after discarding some draws to allow for convergence.
The Gibbs sampler is a Bayesian method and requires a prior. We assume a multivariate normal-inverse Wishart prior for the VAR parameters, a multivariate normal prior for the transition function parameters, independent normal-inverse Gamma priors for each of the factor loadings and their associated residual variance. Table 1 shows the hyperparameters of the prior distributions.
Given the prior and the data and conditional on the sequence of regimes and the factor, the posterior for each regime's VAR parameters is conjugate normal-inverse-Wishart. The regimes are drawn from the Hamilton …lter, modi…ed to account for time-variation in the transition probabilities. The parameters of the transition function are drawn employing the di¤erence in random utility model described in Kaufmann (2015). The factor is drawn from an application of the Kalman …lter; conditional on the factor, the loadings and variances of the factor equations have normalinverse-Gamma posterior densities. The blocks of the sampler and the derivation of the posterior distributions are described in detail in the Appendix.

Data
Our sample period runs from 1960:1 to 2008:12, when the federal funds rate approaches the zero lower bound. We exclude the zero lower bound period because of the di¢culty in assessing the stance of monetary policy in a single policy instrument. 12 The data for the baseline VAR are monthly and consist of a measure of output, prices, and policy. We use the change in the log of the Conference Board Coincident Indicators Index (ZCOIN), the change in the log of the personal consumption expenditures price level index (PCEPI), and the e¤ective federal funds rate.
The model also requires data that proxy for overall sentiment in the form of a factor. We utilize a small unbalanced panel of monthly data that includes multiple surveys and indices. We include the Conference Board Consumer Con…dence Index (CBCCI), the University of Michigan Consumer Sentiment Index (UMCSI), the Organization for Economic Cooperation and Development Consumer Con…dence Index (OECDCCI), and the Institute for Supply Management Purchasing Managers Index (PMI). 13 We order the sentiment factor …rst in the VAR, allowing the macro variables and the policy 1 2 One alternative that has been proposed is the shadow short rate of Krippner (2013) and Wu and Xia (2016). The shadow short rate exploits the Gaussian a¢ne term structure model and changes in the long rate to estimate the level of a hypothetical short rate that is allowed to fall below the zero lower bound. 1 3 All sentiment data are normalized to have mean zero and unit standard deviation.
rate to respond contemporaneously to shocks to overall (consumer and producer) sentiment. This restriction implies that the factor itself responds to policy shocks with a lag. The results are qualitatively similar and the overall conclusions unchanged if the factor is instead ordered last, after the policy rate, allowing it to respond to contemporaneous policy shocks.

Computing Impulse Responses
The e¤ects of monetary policy shocks from VARs are typically summarized using impulse responses. While these (conditionally) linear responses are often used to distinguish between the dynamics across the regimes, they do not take into account the future possibility that the economy exits the initial regime. In this sense, they can overestimate the di¤erences between shocks that are incident in the di¤erent regimes. Alternatively, Krolzig (2006) shows that simple constant probability transitions across regimes can be accounted for by computing the response as a weighted average of the two regime-dependent responses. The weight at any horizon is a function of the transition probabilities and the responses are computed conditional on the period t regime. For more com-plicated models with time-varying transition probabilities, we need to account for the response of the transition probability to the shock.
In our case, the transition probabilities depend on the variables in the VAR. Thus, simply propagating the transition probabilities out over time is insu¢cient to obtain any inferences about the e¤ect of shocks. One alternative is the generalized impulse response functions (GIRFs) suggested by Koop, Pesaran, and Potter (1996). They argue that an impulse response at horizon h can be viewed as the di¤erence between two conditional expectations, one conditional on the (structural) shock u t = occurring at time t and one conditional on no shock at time t: In the linear model, the di¤erence in the conditional expectation is invariant to the history up until time t and the future sequence of shocks up through t + h. In addition, the magnitude of acts only as a scaling factor and the response is symmetric with respect to the sign of . To compute the expectations, we average the expected paths of Y over all histories Y t 1 that correspond to a Gibbs draw of S t = i. Thus, we are computing the averages of separate responses for average shocks that occur in di¤erent regimes.
i represent the number of incidences of S t = i for the gth Gibbs iteration. In addition to the histories, the responses depend on the future sequence of shocks. We can account for variation in future shocks by computing the average response over Q draws of future shock paths. Finally, we average over a subsample of the Gibbs draws. The generalized response at horizon h is for each history starting with S t = i, i = 0; 1, and the superscript g indicates the gth Gibbs iteration. The error bands for the impulse responses can be constructed by computing the appropriate coverage over the G Gibbs draws.
In addition to the policy shocks, a change in regime can cause a response in the macroeconomic variables both through a change in the regime-dependent mean growth rate and a change in the dynamics. We can compute the response to a change in the regime at time t. In this case, we do not shock the system as in the GIRFs; the only di¤erence in the two conditional expectations is the change in regime.
We compute the response Y t+h of a change from S t 1 = j to S t = i by simulating the errors out to horizon h: for all histories in each Gibbs iteration for which S [g] t 1 = j.

Results
To   Table 2.

Baseline Results
The bottom panel of Table 2 provides the posterior means and 68-percent coverage intervals for the TVTP coe¢cient estimates. 14 The transition probabilities consist of a time-invariant component, ji , and a time-varying component, ji , which represents the e¤ects of lagged output on the regime. The posterior mean estimates of 10 and 11 are 4:08 and 1:22, suggesting that the expansionary regime is more persistent than the recessionary regime. We impose that lagged output's e¤ect on the transition from expansion to recession is negative (estimated to be equal to 1:02), which reduces the probability of switching from expansion to recession if output growth is above 1 4 The full set of parameter estimates are available from the authors upon request.
average. Additionally, the posterior mean estimate of the coe¢cient a¤ecting the persistence of recessions, 11 , is also negative ( 0:67). Therefore, an increase in output growth will reduce the probability of remaining in recession from one period to the next. The time-varying e¤ects are signi…cant, suggesting that lagged output growth is an important indicator for determining the transition between the two regimes.

Comparison to Linear FAVAR
To establish a basis for comparison and to gauge the overall value-added of allowing the model parameters to vary across regimes, we also estimated a linear FAVAR without Markov-switching in any of the components. 15 Figure 2  These comparisons indicate that much of the non-linearity and regime-switching nature of the data are driven by variation in the volatility and covariance of the data series in recessionary and expansionary phases. While we do uncover di¤erences in the systematic VAR parameters when S t = 1 versus S t = 0, most of the separate regime identi…cation comes through the regimedependent heteroskedastic covariance matrices 0 and 1 . 16 In both the pre-and post-Great Moderation subperiods, the magnitudes of all reduced-form variance and covariance terms are larger in the recessionary regime. 1 5 As we did with the MS-TVTP-FAVAR, we allow for a one-time structural break in the covariance matrix in 1984 to accommodate the Great Moderation. 1 6 This result is consistent with Sims and Zha (2006) who estimate a variety of structural VAR model speci…cations to describe the potentially regime-switching behavior of monetary policy and its e¤ects on the economy. They …nd that the model which best …ts the data is one in which only the variances of structural innovations change across regimes.  Figure 3 suggests that the behavior of the sentiment factor is similar in both regimes. The peak response to all shocks is reached after 9 months. In both regimes, the peak e¤ect on output (ZCOIN) growth is reached 12 months after the shock. With regards to in ‡ation in both regimes and to all shock sizes, the peak e¤ect is reached after 4 to 5 months. We see more volatility in the projected future path of in ‡ation but the response is not signi…cantly di¤erent from zero. The peak response of the policy rate is reached more quickly in recessions (2 months) than in the expansions (4 months).

Generalized Impulse Responses to Shocks of Varying Size and Sign
We focus speci…cally on the responses of the sentiment factor and output growth to the shocks of 25 basis points, the shock size most commonly analyzed in the literature. Figure 4 illustrates the posterior mean GIRFs in recession (left column) and expansion (right column), the 68-percent posterior coverage interval, and the posterior mean response from the linear model without regimeswitching. While the GIRFs of the sentiment factor in recession and expansion are similar, they both highlight that the regime-switching model suggests a larger response to policy shocks than that identi…ed by the linear model. The GIRFs of output growth suggest less variation between the linear response and that produced in either regime. This result is due to the fact that the recessionary regime is rather short-lived and the GIRF simulations quickly switch from the recessionary regime into the expansionary regime. The RDIR for the expansionary regime closely resembles that of the linear VAR.

Generalized Impulse Responses to a Change in Regime
If we think of S t = 0 and S t = 1 as two steady states, we can compute the transition path between them. The GIRFs to a change in regime represent the behavior of macroeconomic variables in the model, conditional on a di¤erence in regimes at time t. Figure 5 plots the mean response and the 68-percent posterior coverage intervals for these GIRFs. The left column of Figure 5 illustrates the e¤ects of switching from expansion to recession. The subsequent months see a slight decline in output growth and little noticeable change in in ‡ation. Additionally, the federal funds rate exhibits a shift upward in the mean but then declines following the regime shift. Sentiment falls at the time of the change and then takes more than two years to recover back to a level path.
The right column of Figure 5 depicts the macroeconomic behavior given a switch from the recession to the expansion regime. Due to the limited number of periods in recession, conditioning on this history when computing these GIRFs results in less-precise estimates. Based on the posterior mean path, output growth increases slightly at the time of the switch. In the subsequent months, the mean path of the federal funds rate rises and sentiment adjusts slightly downward. This could be indicative of precautionary behavior during the early stages of moderate recoveries.

Conditioning on Economic Conditions
In addition to computing GIRFs based on the economy either being in state S t = 0 or S t = 1 at the time of the shock, we can perform policy experiments that condition on the speci…c economic climate at the time of the policy action. For example, a recession during which in ‡ation is far above target may witness di¤erent policy e¤ects than a recession during which in ‡ation is controlled. We examine the state-dependent e¤ects of expansionary policies taken during recessions characterized by a variety of in ‡ation and output growth values. Figure 6 plots the responses of the sentiment factor and output growth to 25-and 6.25-basis-point reductions in the federal funds rate (left and right columns, respectively) during recessions in which output growth was either: (1) less than 1-standard-deviation below average, (2) between 0-and 1-standard-deviation below average, (3) between 0-and 1-standard-deviation above average, and (4) greater than 1-standard-deviation above average. We compute these responses at the posterior mean estimate of all model parameters. As seen in Figure 6, a small expansionary shock of 6.25 basis points results in responses with considerable variation, depending on the state at the time of the shock. This variation decreases with the size of the shock and is less apparent with the 25-basis-point rate cut.

Panel (A) of
Panel (B) of Figure 6 plots the responses of the sentiment factor and output growth to 25-and 6.25-basis-point reductions in the federal funds rate (left and right columns, respectively) during recessions in which in ‡ation was either above or below 3% at the time of the shock. The responses do not seem to show much variation depending on the level of in ‡ation when policymakers took action. Interestingly, the most noticeable di¤erences are seen for the sentiment factor when in ‡ation is low. When in ‡ation is less than 3%, expansionary policy shocks are more persistent over the medium-and long-term horizons and thus produce a larger boost to sentiment.

Sequential Shocks
Empirical evidence on Taylor rules and reaction functions suggests that the Fed smooths interest rates. For example, the Fed may anticipate that it will reduce the federal funds rate in the face of a recession. It can do so in one large move or make a series of smaller moves. We have argued before that, in the linear model, the response is invariant to the size of the shock, up to a scalar multiplethat is, a 25-basis-point shock produces the same response as a 1-basis-point shock multiplied by 25. Thus, a 25-basis-point shock produces a response equivalent to four consecutive 6.25-basispoint shocks, except for the slight variation in timing. On the other hand, altering the magnitude of the shock in the nonlinear model does not produce a scalar multiple response. Thus, there is no guarantee that the 25-basis-point shock will produce anything similar to a sequence of four 6.25-basis-point shocks.

Sequential Shock Responses
In order to evaluate the e¤ect of smoothing the shocks, we compare the responses of the economic variables to two sets of shocks: (i) a 25-basis-point change in the federal funds rate and (ii) four consecutive 6.25-basis-point changes in the federal funds rate. We then measure the expected paths of the macroeconomic variables, including the latent state, integrating over the histories, future shocks, and Gibbs iterations: Notice that we are conditioning on the same (structural) shocks for period t + 1 to t + h even though we have additional shocks for the second term in periods t + 1 to t + 3. Thus, the innovation to the federal funds rate can be thought of as a 6.25-basis-point shock above and beyond the set of Monte Carlo structural shocks. This conditioning ensures the shocks to both terms are the same except for the innovations that we are interested in. basis points at time t or in four consecutive 6.25-basis-points moves at times t, t + 1, t + 2, and t + 3 during an expansion and the sentiment factor is listed …rst in the VAR. These two contractionary policy sequences induce signi…cantly di¤erent behavior in the factor. Even after the four months it takes to fully implement both policy prescriptions, the factor is still signi…cantly lower after the one-time, large contractionary shock. The large shock results in a slightly deeper contraction in output growth, but this is not persistent. There is little discernible di¤erence in the response of in ‡ation. The di¤erences in the paths of the federal funds rate suggest that a series of smaller rate hikes leads to a higher path for the policy rate over the medium-to longer-term horizons, as represented by the GIRF taking on negative values after the four months it takes to implement the smoothed policy approach.
The right column of Figure 7 shows the responses if the federal funds rate is decreased in a single 25-basis-point move or four sequential 6.25-basis-point moves during a recession. Under these conditions, when the factor is listed …rst in the VAR, the di¤erence in responses of the factor stays positive after the large shock for longer than the four months witnessing small, incremental shocks.
This result suggests a longer-lasting, more favorable response of the sentiment factor after a large policy accommodation. Congruently, the large shock induces a slightly bigger boost to output growth but, again, little variation in the response of in ‡ation. After four months, by the time both policies have been fully implemented, any di¤erence between the paths for output growth and in ‡ation disappears. Therefore, the larger stimulus initially provides an immediate, stronger boost to output growth that is not surpassed by the smooth policy approach. By construction, for the …rst four months in which the sequential policy is enacted, the decline in the policy rate is more substantial after the initial large shock. Once both policies have had time to induce the same systematic changes in the federal funds rate, the di¤erence in paths becomes positive. This result suggests that, following the large rate cut, the federal funds rate takes on larger values in the medium term than if the Fed enacts a series of smaller rate cutes to achieve its target.  the GIRFs during recessions in which output growth was either: (1) less than 1-standard deviation below average, (2) between 0-and 1-standard deviation below average, (3) between 0 and 1 standard deviation above average, and (4) greater than one standard deviation above average. Any substantial di¤erences in the responses are seen after the four months it takes to cut the federal funds rate the full 25 basis points using incremental steps. When output growth is negative, the response of the sentiment factor exhibits greater persistence throughout the one-and-a-half years following the …rst four months of policy changes. Additionally, while we …nd less variation in the responses of output growth, we also see slightly greater persistence in the responses when output growth is negative. The right column shows the GIRFs based on whether in ‡ation was above or below 3% at the time of the initial policy shock. We …nd almost no variation in the GIRFs conditioning on in ‡ation levels and the responses appear similar to those using the full history of recessions identi…ed in the sample, as in the right column of Figure 6.

Conclusions
We estimate a self-exciting, TVTP-VAR in which lagged output growth a¤ects the underlying state of the economy. As a result, countercyclical policy a¤ecting the variables within the VAR also a¤ects the latent state explaining the transition between expansionary and recessionary regimes.
Additionally, we extract a factor representing overall sentiment regarding the health and outlook of the economy. We …nd that this factor declines in the months preceding each of the NBER-dated Finally, smoothing of policy rates in order to enact gradual adjustments induces di¤erent e¤ects than large, one-time policy shocks which ultimately result in changes in the policy rate of the same magnitude. The greater stimulus to overall sentiment and output from large policy shocks may suggest that more aggressive policy intervention, without as much emphasis on smoothing, may be appropriate to combat recessions.

A Sampler Details
The following subsections describe the draws of the estimation method.
Let " T re ‡ect the stacked vector of errors; then, given the prior, we can draw from 1 W ( ; $) ; A.2 Drawing S t j ; Y Let t = fy : tg collect all the data up to time t. From Chib (1993), the conditional density for S is The density p (S t j ; Y) is computed by Hamilton's modi…cation of the Kalman …lter, the last iteration yielding p (S T j ; Y). From Bayes Law, we have p (S t jS t+1 ; ; Y) = p S t+1 ;St p S t j t ; ; 2 ; P 3 j=1 p S t+1 ;j p (S t = jj t ; ; 2 ; ) ; where p j;i is the (time varying) transition probability. Combined, this allows us to generate S t recursively.
A.3 Drawing j ; Y; S The transition parameters are drawn using the di¤erence random utility model described in Frühwirth-Schnatter and Frühwirth (2010) and Kaufmann (2015). Under this speci…cation, the regime variable has an underlying continuous utility representation, U m;t . The period t latent state utility for regime k is and v k;t follows a Type 1 extreme value distribution. We assume the regime with the maximum utility at time t is the observed regime: Di¤erences in utility are given by ! k;t = 8 > < > : where, analogous to the case above, the observed regime is the one with the highest utility We can rewrite the state utilities as where k;t = exp (Z 0 t k ); Similarly, the di¤erence in state utilities can be rewritten as Logistic: For normalization purposes, we impose k = 0 to be the reference regime. This implies the restriction 0 = [0; 0; 0; 0] 0 . Thus, it is only necessary to draw the transition parameters for the regime k = 1. Practically, there are three substeps to the sampling technique for 1 . The …rst substep is to sample the latent state utility di¤erences outlined above for all time periods: W t U (0; 1) : Next, we estimate the logistic distribution of the true errors, , by a mixture of normal distributions with six components. The components, R t , are sampled from the distribution where the component weights, w r , and component standard deviation, s r , are given in Table 1 of Frühwirth-Schnatter and Frühwirth (2010).
Finally, given the prior 1 N (g 0 ; G 0 ), we generate the draw of 1 from the normal posterior and X mT = [X m1 ; :::; X mT ] 0 .
De…ne X T = [X 1T ; :::; X M T ] 0 . Then, we can sample and W = 2 where each B ip is a (N + 1 N + 1) matrix collecting the pth lag coe¢cients for the ith regime.
Given a set of starting values of & 0j0 and P 0j0 , the …lter iterates prediction and update steps forward for t = 1; ::; T . The prediction step computes a projection of the period t state variable based on information available at time t 1. The prediction density is typically written as From the prediction density, we can update the state vector using the next period realization of the data. De…ne the prediction error as where the variance can then be written as: The covariance with & tjt 1 as with variance An alternative is to use a multi-move sampler (Carter and Kohn (1994)), which draws the entire state vector at once from p & t j t ; & t+1 , where t represents the data known at time t and the superscript indicates the truncation of the state vector due to the singular covariance matrix. We and e N is a vector with a 1 as the (N + 1)th element and 0's everywhere else. The main di¤erence between (13) and (14) is that the former generates all the posterior distributions simultaneously while the latter forms them recursively, conditional on the t + 1 period draw.