Download Forecasting Fed Funds Target Changes with

Forecasting Fed Funds Target Changes with Large Datasets
Travis J. Berge
Research Division
Federal Reserve Bank of Kansas City
Michael T. Owyang
Research Division
Federal Reserve Bank of St. Louis
Keywords: Dynamic probit, factor probit, monetary policy, fed funds futures
February 12, 2014
Abstract
Most interest rate rules are continuous functions of deviations of output from its potential
and expected in‡ation from its target. In practice, central banks move the target rate in discrete
increments and base their decisions on a wide-range of data. We estimate a dynamic ordered
probit model of movements in the federal funds rate. In our model, monetary policy reacts to
a large dataset that is summarized by small set of dynamic factors. We then use the model
for out-of-sample forecasting and evaluate these forecasts using methods unique to problems of
classi…cation.
[JEL classi…cation: C32, E58]
1
Introduction
The Taylor (1993) rule is one of the most frequently used models of Federal Reserve behavior and
posits that the federal funds rate should move as a function of in‡ation relative to a target and
output relative to potential. While the ubiquity of the Taylor Rule is in no small part due to its
simplicity, the reality is that the Fed employs large sets of data forecast future in‡ation and output
and to decide when to move the fed funds target. Moreover, the funds target does not adjust
with small ‡uctuations in the objective variables; instead, the target moves in discrete increments
implying some nonlinearity in the response of policy to the targets.
Each of these two issues has been identi…ed and addressed in the literature, albeit not simultaneously. Bernanke & Boivin (2003) recognized that the Fed uses copious amounts of data to form
The authors bene…tted from conversations with Michael Dueker, Chris Otrok, and Jeremy Piger. Kate Vermann
provided research assistance. The views expressed herein do not re‡ect the o¢ cial positions of the Federal Reserve
Banks of Kansas City or St. Louis or the Federal Reserve System.
1
forecasts for policy. They …rst show that the large datasets (e.g., Stock & Watson (2002a)) summarized into a small number of dynamic factors can improve over Greenbook forecasts alone. They
then integrate the factors into a linear Taylor-type rule. Any forward-looking Taylor rule (perhaps
with the exception of van den Hauwe, van Dijk & Paap (2013)) must make assumptions about the
variables that are used for expectations formulation. Some models assume rational expectations,
use future data, and estimate the model with instrumental variables(e.g., Clarida, Galí & Gertler
(2000)). Other models use forecasts (say, from the Greenbook) or forecasting models to measure
expectations (e.g., Orphanides (2004), Monokroussos (2011)). Bernanke and Boivin argue that the
factors can be thought of as representing the forecasts of the future in‡ation and economic activity
in a forward-looking rule such as that found in Clarida et al. (2000).
Eichengreen, Watson & Grossman (1985) recognized central banks tend to move policy instruments in discrete increments. They developed a model— the dynamic ordered probit— in which
changes in the target variables a¤ect the probability of a change in the policy rate target (in
their case, the Bank of England’s Prime Rate). Hamilton & Jorda (2002) take a slightly di¤erent
approach by modeling the duration between changes in the Fed Funds rate as an autoregressive
process. The non-linearity of their setup uncovers an asymmetry in the e¤ects of monetary policy on
the economy— increases in the target rate a¤ect employment and prices di¤erently than decreases.1
In this paper, we construct a model that combines these two observations about monetary policy. We utilize a large panel of macroeconomic variables, a subset of the variables likely used by
the Fed to conduct policy. Information in the large panel is summarized by a small set of factors
that represent the expectations of future in‡ation and real economic activity. Factor models are
well-known to provide useful information for forecasting purposes in a wide-range of applications,
including output and in‡ation (see, e.g. Stock & Watson (2002a) Stock & Watson (2002b) Stock
& Watson (2006))), asset prices (e.g. Ludvigson & Ng (2009)), and as descriptions of the macroeconomy (e.g. Bernanke, Boivin & Eliasz (2005)).2 The factors that we estimate are used as an
1
More recently, and building on the work of Cargnoni, Muller & West (1997) and Dueker (1999), Monokroussos
(2011) …ts a discrete Taylor rule using as inputs the Federal Reserve Board’s own forecasts of in‡ation and real GDP
growth. Monokroussos’ primary interest is in looking for behavioral di¤erences in the Fed’s response function preand post- Volcker.
2
As shown by Stock and Watson, an important advantage of factor models is that they are robust to structural
breaks, meaning that our forecasting model is robust to changes in the particulars of the Federal Reserve reaction
function and the behavior of the macroeconomy.
2
input to a discrete interest rate rule modeled as a dynamic ordered probit.3
While other papers have, in general, been interested in estimating the parameters of the Taylor
rule, our objective is to determine how and what information can be used to forecast future changes
in the fed funds target. Speci…cally, we produce both in-sample forecasts of the probability of a
change in the Fed policy rate. We also perform a pseudo real-time forecasting exercise, forecasting out-of-sample all FOMC meetings between the period of September 1982 to December 2007.
We evaluate forecasts produced by the model by computing the area under the receiver operator
characteristic curve which explicitly focuses on the discrete nature of the problem.
Perhaps the paper that is most closely related to our own is that of van den Hauwe et al.
(2013), who combine an ordered probit model with a Bayesian model selection procedure to produce
forecasts of Fed Funds rate changes. The estimation focuses only on the direction-of-change of the
Target rate and not the magnitude of the change although the setup allows the authors to focus
on extracting which macroeconomic and …nancial variables the Fed responds to most closely.
The rest of the paper is organized as follows: Section 2 outlines the factor-augmented discrete
Taylor rule model. Section 3 describes the Gibbs sampler used to estimate the model, while section
3.3 describes the forecasting algorithm and the data. Section 4 presents the baseline empirical
results. Section 5 concludes.
2
The Empirical Model
A few papers have taken estimated monetary policy rules when the central bank only changes in the
policy instrument discretely. Our contribution is to consider allow the policymaker to be forward
looking in an environment where large datasets can used to form expectations about the future
paths of in‡ation and output growth. In this section, we propose a model that yields a desired
funds rate as a function of a potentially large dataset and a discrete target rate adjustment as a
function of the desired rate. Then, we consider the data available to form these expectations as
well as the data on the federal funds target.
3
In other contexts, factor-probit models of this type have been used to analyze business cycle phases (e.g., (Fossati
2011)). These models di¤er in how they are estimated.
3
2.1
The Discrete Taylor Rule
Interest rate rules often assume that changes in the objective (targeted) variables result in a proportional change in the policy instrument. In practice, however, the fed funds target changes in
discrete increments. Thus, changes in in‡ation and output (or unemployment) are generally not
associated with proportional changes in the funds target.4
We model discrete changes in the funds target by estimating the underlying changes in the
desired rate. Let rt represent the desired (latent) value of the fed funds rate and let Rt represent
the (observed) value of the fed funds target.5 Suppose that we believe the desired fed funds rate is
a convex combination between a Taylor-type rule and its own lags:
rt = (1
)
h
t
+r +e (
y
) + e (yt
t
i
y ) + e (L) rt
1;
(1)
where
t
is the in‡ation rate, r is the equilibrium real interest rate, yt is the output growth
rate,
is the in‡ation target, and y is potential output. The …rst term suggests that the policy
instrument is moved as the two target variables ‡uctuate around their objectives. Because the rule
is linear, small ‡uctuations in the target variables imply small changes in the policy instrument;
large ‡uctuations imply large changes. The second term re‡ects the policymaker’s desire smooth
interest rates. For exposition, in (1), we have assumed that the target and the deviation from
potential are time invariant; this assumption is straightforward to relax but will have little e¤ect
on forecasting.
While changes to the desired rate are continuous, changes to the observed fed funds rate,
Rt ,
occur only in discrete steps. Studying changes to the Bank of England’s Bank Rate, Eichengreen
et al. (1985) noted that the rate changed only in 25 basis point increments. They argued that
observed changes are based on the di¤erence between today’s desired rate (rt ) and the previous
period’s observed rate (Rt
1)
and occur if the di¤erence between the too are su¢ ciently large
described by a threshold. Notice that rt
Rt
in the desired rate at time t. The value rt
1
1
=
Rt
rt + rt
1,
1
Rt
1,
where
rt is the change
then, describes the change the target rate
4
Estimation of interest rate rules require lag terms for persistence and error terms to map the desired rate back
into the e¤ective fed funds rate.
5
Note that the desired rate is not equivalent to the e¤ective fed funds rate, which re‡ects small movements around
the target values.
4
that is desired but unful…lled because the move to the next target increment would not shrink the
di¤erence. EWG estimate the thresholds for which rt
Rt
1
1
triggers a Bank Rate change; later,
Dueker (1999) uses prede…ned thresholds when modeling the Fed’s behavior.
For our discrete characterization of the policy instrument, it will be convenient to rewrite (1)
in terms of the change in the desired rate:
rt =
where
+ (L) rt
1
+
t
+
collects the time-invariant components and (L),
y
y t + "t ;
,
y
(2)
are rewritten to suppress , and
"t is a normally distributed error term. In principle, we can include any number of target variables.
Let Zt represent a vector of period t target variables. Then, we can rewrite the rule to include Zt
as
rt =
where
+ (L) rt
1
+ Zt + "t ;
(3)
collects the coe¢ cients on the various elements of Zt .
Because the desired rate, rt , is latent, we only observe the amount the Fed moved the target
in time t,
Rt , and the previous level of the target, Rt
1.
We need a rule that infers (or bounds)
the desired level of the target from these two observables. Unfortunately, changes in the fed funds
target are not restricted to 25 basis point increments, at least for the early part of the sample.
Prior to November 1989, the fed funds target typically changed in increments of 6.25 basis points;
after this date, the funds target has only changed in multiples of 25 basis points. In this paper, we
de…ne “su¢ ciently large”similarly to Dueker (1999) and Monokroussos (2011). For shorter horizon
5
forecasts, we using the following rule:
Rt <
0:125 >
0
0<
0:25
Rt
Rt >
!
0:25
0:125
rt
Rt
1
<
!
0:215 > rt
Rt
!
0:05
Rt
rt
Rt = 0
!
Rt
!
0:05
rt
Rt
!
0:215 < rt
Rt
!
rt
0:25
0:125
Rt < 0:125
Rt > 0:25
0:05 < rt
Rt
0:375
0:215
1
Rt
1
0:375
1
1
1
1
< 0:05
:
(4)
0:215
0:275
> 0:275
For longer horizon forecasts, we will use a rule with larger windows.
The …rst column of the table shows the period t change in the fed funds target. The second
column shows bounds on the level of the desired rate as a function of the past target. A move in
the target of the amount on in column 1 implies that the deviation of the desired target from the
last period target must be within the bounds of the past target given in column 2.
2.2
The Factor-Augmented Dynamic Ordered Probit
The standard forward-looking linear Taylor rule makes the policy instrument a function of the
di¤erence between (1) the period-t expectation of future in‡ation and the in‡ation target and
(2) the period-t expectation of the future output gap. There are, in principle, three methods of
estimating these models: (i) assume rational expectations, use forward data, and estimate the TR
coe¢ cients using instrumental variables; (ii) use actual forecasts of future in‡ation and output
growth; or (iii) use a forecasting model for in‡ation and output growth. The discrete interest rate
rule literature has similar options. For example, Monokroussos (2011) uses Greenbook forecasts of
in‡ation and output growth.
Unfortunately, the Greenbook forecasts are not available in real time, making forecasting fed
funds target changes using them problematic. We know that the Fed employs a large number
of economists to analyze a vast panel of world, industry, and regional data. Thus, in addition to
accounting for the discreteness of the fed funds target changes, we are interested in using a large
number of predictors summarized by the (standardized) N
6
1 period t vector Xt . The predictive
content of a large vector of indicators can be condensed into a small set of factors, Ft , where
Xt = Ft + et ;
Ft is a period-t (K
and
1) vector of factors, K << N ,
(5)
is a (N
K) matrix of loadings, et
N (0; ),
is diagonal. The diagonality assumption implies that the observed correlation in the Xt is
produced primarily by the factors. We assume that the factors evolve as independent AR(p)’s:
Fkt =
where
k
k
(L) Fk;t
1
+ vkt ;
(6)
(L) is a pth order polynomial in the lag operator and vkt
N 0; $2k .
Introducing the factors produces two identi…cation issues. First, the sign of Ft is identi…ed but
not the sign of the individual elements. Second, the scale of Ft is identi…ed but not the individual
elements.6 We can solve the former by imposing the sign of the …rst (nonzero) loading on each factor.
We then calibrate the variance of each factor to account for the latter. In addition, we may want
to interpret the estimated factors as characterizing expected in‡ation and expected output growth.
We include both macro and …nancial variables which could be thought to a¤ect either in‡ation
expectations, output expectations, or both. In order to separately identify the factors, we need to
impose some restrictions. These restrictions typically are imposed on the factor loadings. In this
case, we impose a set of zero restrictions on each factor’s loadings for small non-overlapping subsets
of Xt . We assume that real macro variables such as (contemporaneous) output and industrial
production do not load on the in‡ation factor. Similarly, we assume that contemporary price series
–e.g., CPI in‡ation, commodity prices –do not load on the output growth factor.
We can then rewrite (2) replacing Zt with the factors:
rt =
+ (L) rt
1
+
X
k Fkt
+ "t ;
(7)
k
where
k
now represents the responsiveness to the factor, rather than a single macroeconomic
variable.
6
In addition, we may want to interpret the estimated factors as characterizing expected in‡ation and expected
output growth.
7
Equations (4), (5), (6), and (7) comprise the factor-augmented dynamic probit. As we will show
below, the model can be estimated using Bayesian methods by …ltering both the latent factors and
the latent desired funds rate. The latent desired rate behaves similar to a standard Taylor rule,
but augmented with factors. The factors can be thought of as capturing all of the expected future
or contemporaneous variation in the desired rate implied by the data in the panel. If we viewed
the in‡ation target, the equilibrium interest rate, and potential output as changing over time, the
factors could be thought of as capturing this time variation.
2.3
Data
The FOMC of the Federal Reserve currently meets at eight scheduled dates per year in order to
conduct monetary policy. However, it is not uncommon for the FOMC to meet at an unscheduled
date in order to conduct monetary policy, often following events that have the potential to be
major disruptions to the U.S. economy. As a consequence, these meetings are likely to result in
large movements in the target rate. For this reason, our dependent variable is the end-of-month
value of the target federal funds rate.7 Our data span the period September 1982 to December
2007, before the onset of the zero-lower-bound period. Specifying our model in calendar time and
not at the FOMC frequency means that the model forecasts the sum of the changes for each month,
irrespective of whether (or how frequently) the FOMC actually met in a given month.8
Table 1 gives summary statistics for the FOMC target rate over this period. The most common
outcome for the target rate month-to-month is no change to the target rate, which is unsurprising
since our dataset covers many months during which the FOMC did not meet. Unconditional on
the macroeconomic conditions that prevailed during this period, as well as the starting point for
our data, it appears that the FOMC is slightly more willing to decrease than increase the target
rate. One sees a similar asymmetry in the willingness of the FOMC to make large changes in the
target rate. Among movements of at least 50 basis points, there were twice as many were decreases
relative to increases in the target rate.
7
While we will show some full sample results as a baseline for comparison with other models, our interest is in
forecasting changes in the funds rate. Thus, the end-of-month assumption is not a problem if we assume that it is
the forecaster’s object of interest. Alternative timing assumptions (e.g., using only periods in which there was an
FOMC meeting or an unscheduled change in the funds target) could be used to estimate the discrete Taylor rule.
8
Monokroussos (2011) used an extended sample back to the 1960s. His sample used fed funds target rates from
1972 constructed by Cook & Hahn (1989). Prior to 1972, Monokroussos used the average rounded monthly funds
rate as a target.
8
End-of-month change
Outcome
Count Frequency
Rt < 0:50
6
2.0%
Rt = 0:50
21
6.9
Rt = 0:25
34
11.2
Rt = 0
193
63.7
Rt = 0:25
36
11.9
Rt = 0:50
12
4.0
Rt > 0:50
1
0.3
Total
303
100%
Fed Funds Rate
Nobs: 304
Mean: 5.5%
Std. Deviation: 2.4%
Minimum: 1.0%
Maximum: 11.5%
Table 1: Frequency of FOMC target rate changes: Sept 1982–Dec 2007. Observed variable is the
end-of-month target rate for Fed Funds rate for all months in the sample.
We estimate the macroeconomic factors from a balanced panel of 131 variables of macroeconomic interest. The dataset is the same that is used in Ludvigson & Ng (2009), and is an extended
version of the datasets used by Stock & Watson (2002a, 2006). The series include macroeconomic variables that fall into one of several broad categories of macroeconomic variables: real
output and income; employment and hours; real retail, manufacturing and sales data; international
trade; consumer spending; housing; inventories and inventory sales ratios; orders and un…lled orders; compensation and labor costs; capacity utilization measures; price indexes; interest rates and
interest rate spreads; stock market indicators; and foreign exchange measures. All variables have
been transformed such that each is stationary. For a complete list of the variables included in the
dataset and transformations applied thereto, see Ludvigson & Ng (2009).
3
Estimation and Forecasting
The model speci…ed above can be estimated using standard Bayesian techniques (Albert & Chib
(1993); Holmes & Held (n.d.)). We employ the Gibbs sampler [see Gelfand & Smith (1990); Casella
& George (1992); Carter & Kohn (1994)]. The sampler can be broken down into blocks: the slope
coe¢ cients,
= f ; ; g; the variance of the desired change,
factor loadings,
=f
0;
1 g;
2;
the factor AR coe¢ cients,
; the
a block consisting of the augmented data, frt gTt=1 ; the variances the
variables in the factor equation,
; and a block containing the factors, fFt gTt=1 .9 Let
9
denote the
Other papers (e.g., Fossati (2011)) that have estimated versions of the factor-augmented probit estimate the
model in two steps: …rst, they extract the factor using principal components and then they estimate the probit
conditioning on the factor.
9
full set of parameters, including the augmented data and the factors. Let
represent the full set of
parameters; the goal of the Gibbs sampler is to iteratively draw from the conditional distribution
of one block m, conditional on each other block,
m,
the factors, and the latent desired rates.
Each conditional distribution depends on the block’s prior. We assume a normal prior for both the
set of slope parameters,
, and the factor AR coe¢ cients. The diagonal elements of the innovation
variance-covariance matrix have an inverse gamma prior. Table 2 summarizes the prior and gives
the hyperparameters used for estimation.
Priors for Estimation
Prior Distribution
Hyperparameters
N ( 0; 0)
0 = 0K+q+1 1 ;
0 = IK+q+1
G (s0 ; S0 )
s0 = 1; S0 = 1
N ( 0; 0)
=
1
K+1 1 ; 0 = IK+1
0
N (d0 ; D0 )
d0 = 0p 1 ; D0 = Ip
G ( 0; 0)
0 = 1; 0 = 1
Parameter
2
k
1
ii
Table 2: Priors. Notes: K is the total number of factors; N is the number of variables in the vector
X; and P is the number of lags in the factor equation. The index ii represents the ith diagonal
element in the variance-covaraince matrix.
Most draws from the sampler are straightforward as the prior produces conjugate conditional
distributions for the model parameters. Details for the model draws can be found in the Appendix.
The two remaining blocks are the factors and the latent desired rate. The former can be drawn
from a standard Kalman …lter, exploiting the fact that, conditional on the desired rate, the state
space is linear. Details for this draw are also available in the Appendix.
3.1
Drawing rt Conditional on
rt ; FT ; RT ; XT
We can draw the latent variable, conditional on the factors, via an algorithm similar to that
proposed by Dueker (1999). Dueker argues that drawing the whole vector of latent variables is
intractable because it requires evaluation from the full likelihood. Instead, he advocates drawing
each period’s latent variable, conditional on the latest draws of both the past (t
(t + 1) values, f (rt jrt
De…ne
t
=
1 ; rt+1 ;
+ (L) rt
1
Rt ).
P
+ k
k Fkt .
1) and future
Then, rt can be drawn sequentially from t = 1; :::; T
from a truncated multivariate normal:
10
rt
TN
rbt ;
1
2
2
; l; u ;
where l and u are the lower and upper truncation points, respectively. The mean of the desire rate
is a function of the average between the past and future desired rates and the di¤erence between
the current economic conditions and the past:
rb =
1
(rt
2
1
+ rt+1 ) +
1
2
t 1
t
;
where the future value is taken from the last iteration of the sampler. That is, we condition on
the current draw of rt
1
and the past draw of rt+1 . The truncation points, l and u, depend on the
realized values of policy. Recall that the desired value of the funds target is de…ned to lie within a
band around the actual value, where the band is de…ned in (4).
3.2
The Forecast Environment
We use the model speci…ed above to produce both in- and out-of-sample Bayesian forecasts of the
FOMC target rate decisions. The sampler described in section 2 produces a series of draws that
form the joint posterior distribution p( ;
rj R; Z). From these draws, we obtain a predictive
probability for each category j at each iteration of the sampler i. Speci…cally, de…ne rbti to be the
P
i ) + ri
desired change in the target rate at time t, rbti = rti Rt 1 ( i + i (L)rti 1 + k ik Fkt
t 1
Rt
1.
Then,
Pr [ Rt 2 j] =
where
(cj+1
rbti ;
2 i
)
(cj
rbti ;
2 i
)
(8)
(:; :) denotes the normal cumulative distribution function. The posterior mean probability
of Pr [ Rt 2 j] is calculated by averaging over the draws from the posterior.
Smoothed in-sample forecasts evaluate the above expression using all available data, t = 1; :::; T .
Out-of-sample forecasts are produced for horizons h = 1; 6; 12 at each time t > t0 , where t0 denotes
the initial estimation sample. We set t0 = 120 months. Because the sampler is computationally
demanding, we reestimate the model every …ve years instead of at each t.
11
3.3
Evaluating the classi…cation ability of the model
Forecasts are evaluated using methods speci…c to the classi…cation literature and that build on the
application of the Receiver Operating Characteristic (ROC) curve.10 Suppose that fyi gN
i=1 ; yi 2 R
is a real-valued classi…er intended to determine the true underlying binomial state, Si 2 f0; 1g.11
Further suppose that yi on average takes a high value conditional on Si = 1 and a low value
otherwise. A ROC curve is the set of points that lay in the unit square de…ned by fF P (c); T P (c)g,
where F P (c) is the false positive rate of the classi…er (the signi…cance, or type I error), T P (c) is
the true positive rate of the classi…er (alternatively, the speci…city, or 1-type II error ), and the set
fF P (c); T P (c)g is de…ned as one varies the threshold c from negative to positive in…nity.12 ROC
curves themselves are visually intuitive, and the area underneath the ROC curve (or AUC ) is a
summary statistic commonly used to measure the e¤ectiveness of a classi…er. Because a completely
uninformative classi…er will have that T P (c) = F P (c) for any c, AU C 2 [0:5; 1], with a value of 0.5
indicating that the classi…er performs no better than random assignment to the two groups and a
value of 1 indicating perfect classi…cation ability.
The AUC is a Mann-Whitney-Wilcoxon U-statistic that can be interpreted as the probability
that the value of a random draw of yi from the diseased group (Si = 1) is higher than the value of
a random draw of yi from the healthy (Si = 0) and therefore can be estimated non-parametrically
(Bamber 1975). Inference is also straightforward. Hsieh & Turnbull (1996) have shown that under
mild conditions the AUC is distributed normally in large samples, and Hanley & McNeil (1982)
provide formulas for the variance.
ROC curves have been generalized to accommodate classi…cation problems of dimension k.
As in the two-dimensional case, a decision rule is based on k
c1 < c2 < ::: < ck
1.
1 thresholds c1 ; c2 ; :::; ck
Doing so produces k true positive rates and k(k
1
where
1) false class rates.
A summary statistic analogous to the area under the ROC curve can be found by calculating the
volume under the hyper-surface generated by the k true positive rates. Consider the case when k = 3
10
See Pepe (2003) for a detailed monograph on the application and estimation of ROC curves. For an application
to economics, see Berge & Jordà (2011).
11
For example, Si could be the true status of a potentially malignant tissue from individual i, while yi is a predictor
of this outcome such as a PSA blood test score. Examples more germane to economics could be Si as the state of the
economy (recession/expansion) in quarter i and yi real GDP growth in that quarter; or Si as individual i ’s default
status and yi as some predictor of default, say a credit score.
12
That is, T P (c) = P [yi > cjSi = 1] and F P (c) = P [yi > cjSi = 0]. Alternatively, one can de…ne the ROC curve
as the set of points fT N (c); T P (c)g, which Jordà & Taylor (2011) denote the correct classi…cation frontier.
12
(Mossman 1999). As before, the ROC surface is generated by varying the k
1 thresholds c1 < c2
across the domain of the classi…er yi and documenting the resulting true positive classi…cation rates
for each class k, T Pk (c); k = 1; 2; 3. The class-k true positive rates range from 0 to 1 as the thresholds
are varied, so that the corner coordinates of the ROC surface are f(1; 0; 0); (0; 1; 0); (0; 0; 1)g. An
uninformative classi…er in this three-class problem has a volume under the surface of 1/6. More
generally, a classi…cation problem of dimension k will have a null test statistic of
1
k! .
Mossman
(1999) and Dreiseitl, Ohno-Machado & Binder (2000) provide details on three-way ROC analysis;
see Nakas & Yiannoutsos (2004) for a discussion on higher-order ROC analysis. As before, the
volume under the surface is intimately connected to U-statistics, allowing the variance to be derived
from standard statistical theory, although again bootstrap methods can also be used for inference.
The AUC can be interpreted as a probability: Consider the sequence of 0/1 outcomes for the
“no change” bin. De…ne a vector z, such that zt = 1 if the FOMC does not change the funds
rate in period t, and zt = 0 if the FOMC did change the funds rate. We also can de…ne the
b t = i . Take one random draw of Pr z
b t = ijzt = 0 , and a random
probabilistic …tted value, Pr z
b t = ijzt = 1 . The AUC turns out to be:
draw of Pr z
b t = ijzt = 1 > Pr z
b t = ijzt = 0 :
AU C = Pr Pr z
That is, the AUC is the probability that a random draw of an observation when the Fed Funds
Rate did not change is greater than an observation from when the Fed did change the funds rate.
4
4.1
Empirical Results
In-sample Nowcasts
As a baseline for comparison, we constructed in-sample nowcasts of the fed funds target using the
full sample data from 1982 to 2007. Figure 2 shows the mean values of the two factors. The factors
are constructed using draws of the unsmoothed posterior densities obtained from the Kalman …lter;
the NBER recessions are shown as shaded areas. The blue line is the factor with the loadings on
price information restricted to be zero; the green line is the factor with output information restricted
to be zero.
13
Figure 3 shows the posterior mean of the latent variable along with the realized change in the
fed funds target. Recall that the realized values of the fed funds target truncates the support over
which the latent desired rate resides. Thus, the in-sample latent should …t closely to the realized
target values.
Figure 4 shows the in-sample probabilities of the seven bins from (4) with the top panel representing the probability of a downward move of more than 25 basis points.
Figure 5 shows the ROC curves for the in-sample forecasts. These curves indicate whether
the factors provide information over a random draw. Each of the curves lies above the 45 degree
line, suggesting that there is (in-sample) information in the factors. The full ROC curve would
be multidimensional object. For each draw of the MCMC sampler, we produce a probability that
the FOMC will move the fed funds target rate according to that bin (e.g., no change, +/- 25 bps,
etc). For each MCMC draw of the probabilities, we calculate the AUC of that bin’s estimated
probabilities against the actual FOMC moves (the 0/1 variable for each bin). Figure 5 shows
conditional posterior mean of these AUC values for three of the possible moves. The y-axis shows
the ratio of true positives for that move in the fed funds target; the x-axis shows the false positive
ratio.
4.2
Out-of-sample Forecasts
[Alternative horizons to be added]
5
Conclusions
[To Be Added]
14
A
The Gibbs Sampler
Here, we provide a detailed description of the sampler used for estimation.
A.1
Drawing
; FT ; RT
Conditional on
Conditional on fFt gTt=1 , (7) is an independent linear regression. Let
and
j
represent the vector of stacked
; RT
N( ;
t ’s.
t
0
1 ; :::; rt q ; F1t ; :::; FKt ]
= [1; rt
Then, given the prior, a draw of
can obtained from
), where
=
1
1
=
0
1
0
+
0
0
+
0
;
r ;
and r is the stacked vector of the di¤erence in the latent variables.
A.2
Drawing
2
Conditional on
2
; FT ; RT
Let u re‡ect the stacked vector of errors, ut , where ut =
the prior, we can draw
2
rt
(L) rt
1
Zt . Then, given
from
2
Gamma (s; S) ;
where s = s0 + T =2 and S = (S0 + u0 u) =2.
A.3
Drawing
Conditional on
; RT
Similar to the draw above, conditional on fFt gTt=1 , (6) is an independent linear regression for each k.
Let vkt = [Fk;t
0
1 ; :::; Fk;t p ]
and vk represent the vector of stacked vkt ’s from t = p+1 to t = T
Then, given the prior, a draw of
for factor k can obtained from
0
D = D0 1 + vk vk
15
1
;
[k] j
; RT
1.
N (dk ; Dk ), where
0
d = D D 0 1 d0 + v k Fk ;
and Fk = [Fk;T ; :::; Fk;p+1 ]0 .
A.4
Drawing
; FT ; RT ; XT
Conditional on
Conditional on fFt gTt=1 , (5) is also a set of independent linear regressions. Let Xyt = [F1t ; :::; FKt ]0
and let Xy represent the vector of stacked Xt ’s. Then, given the prior and the data, a draw of
can obtained from j
;Z
T N ( ; ), where
1
=
0
1
=
1
0
+ Xy Xy
0
0
;
0
+ Xy Z ;
T N (:; :) is the multivariate truncated normal, and the (positive) truncation is imposed on …rst
elements in each column of
to prevent the signs of the factors from changing across Gibbs
iterations. The truncation is necessary because
k FkT
does not identify the sign of the factors.
We also impose additional zero restrictions. For those variables with zero restrictions on the factor
loadings, the appropriate factor is omitted from the linear regression.
A.5
Drawing
; RT
Conditional on
Each diagonal element of the variance-covariance matrix
can be drawn independently under the
assumption that the Xt are uncorrelated conditional on Ft . The prior for each diagonal element of
eiT = [Xi1 ; :::; XiT ], FeT = [F1 ; :::; FT ], and Ft = [F1t ; :::; FKt ]0 . Then,
is inverse gamma. De…ne X
we can sample
1
ii
from a gamma posterior
1
ii
where
i
=(
0
j
G ( i;
i) ;
+ T ) =2 represents the degrees of freedom and the scale parameter is
16
i
A.6
=
1
2
0
eiT
+ X
Drawing F Conditional on
eiT
X
FeT
; R T ; XT
FeT
0
:
Conditional on the model parameters, the latent probit variable, rt , and the data, we can draw the
K factors recursively using probability distributions obtained from the Kalman …lter.13 It will be
convenient to rewrite the model in its state-space representation:
Yt = H& t +
&t = W &t
1
t;
(9)
+ $t ;
0
where Yt = [Xt0 ; rt
(L) rt 1
]0 , & t = F0t ; :::F0t p+1 , Ft = [F1t ; :::; FKt ]0 ,
i0
h
[v1t ; :::; vKt ] $ = vt0 ; 00(p 1) 1 . The state-space coe¢ cient matrices are
2
6
H=4
and
2
where each
i
is a (K
6
W =4
0N
K(p 1)
01
K(p 1)
IK(p
1)
0K(p
= [e0t ; ut ]0 , vt =
3
7
5
p
1
t
1) pK
3
7
5;
(10)
K) diagonal matrix collecting the ith lag coe¢ cients in (6).
Given a set of starting values of & 0j0 and P0j0 , the …lter iterates prediction and update steps
forward for t = 1; ::; T . We use a version of the RTS smoother to obtain the smoothed posterior
densities for the factors.14 As we alluded to above, the factor and the factor loading are not identi…ed
to sign – that is, a joint draw of
and
k
and Fk produces the same likelihood as a joint draw of
k
Fk . In order to resolve this identi…cation issue, we impose restrictions on the signs of the
13
We cannot simultaneously draw the factors and the latent variable, rt , in the state equation because the two
are contemporaneously correlated.
14
Note that the state space results in a singular Q; we follow Kim and Nelson (1999) and use only the nonsingular
subblocks of Q and W .
17
some elements of
k.
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