Download Short-rate expectations and unexpected returns in Treasury bonds

Short-rate expectations and
unexpected returns in Treasury bonds
Anna Cieslak∗
This paper studies how agents form expectations of the short-term interest rate, and implications
thereof for measuring bond risk premium. I uncover persistent and large differences between the
expected real rate perceived by agents in real-time and its full-sample counterpart estimated by
the econometrician. Entering recessions, agents systematically overestimate the real rate, and
underestimate future unemployment and the degree of monetary easing. These forecast errors
induce a predictable component in realized bond excess returns by lagged measures of real activity
that is uncorrelated with survey-based and statistical proxies of bond risk premium.
This version: May 2016
Cieslak is with Duke University, Fuqua School of Business, e-mail: [email protected]. I thank Snehal
Banerjee, Ravi Bansal, Gadi Barlevy, Luca Benzoni, Greg Duffee, Martin Eichenbaum, Yuriy Gorodnichenko,
Sam Hanson, Lorena Keller, Arvind Krishnamurthy, Scott Joslin, Viktor Todorov, Ken Singleton, Annette
Vissing-Jorgensen, Mark Watson, and participants at the AFA Meetings, Caltech, Chicago Fed, CEPR
ESSFM Gerzensee focus session, SF Fed conference “The Past and Future of Monetary Policy,” CFTC,
Duke Fuqua, ERID Macro Conference, EFA Meetings, Northwestern Kellogg, Minnesota Mini Asset Pricing
Conference, and the Red Rock Finance Conference for comments on an earlier draft of this paper. The
previous version of this paper was titled “Expecting the Fed.”
∗
This paper studies how agents form expectations of the short-term interest rate in real time,
and derives implications for the measurement of bond risk premia. Separating short-rate
expectations from risk premia in interest rates serves as a key input for understanding the
economic determinants of the yield curve. Such a decomposition provides insights about the
market’s perceptions of the future course of monetary policy, economic activity, inflation
and their associated risks, upon which agents base their economic choices. The importance
of being able to reliably disentangle these two components is summarized in the following
quote by the Federal Reserve Board Governor Donald Kohn:
“Investors’ expectations are reflected in asset prices, but so are risk premiums, and inferences about future economic conditions obtained from market prices are conditional
on estimates of those premiums. Neglecting or grossly misestimating risk premiums
will lead to misperceptions of the market’s outlook and thus potentially to market
moves that we did not anticipate. (...) To what extent are long-term interest rates
low because investors expect short-term rates to be low in the future due to some
underlying softness in aggregate demand, and to what extent do low long rates reflect
narrow term premiums, perhaps induced by well-anchored inflation expectations or
low macroeconomic volatility? Clearly, the policy implications of these two alternative
explanations are very different.” Governor Donald L. Kohn (July 21, 2005)
While recent research has extensively studied bond risk premia, relatively less is known about
how investors form expectations about the future path of the short rate. This focus is justified
in light of the common assumption of full-information rational expectations (FIRE) implying
that the information set of agents equals that of an econometrician. Based on this premise,
one interprets full-sample predictive regressions of bond returns on various conditioning
variables as a way to capture the time-varying bond risk premium, i.e. the compensation that
investors expected and required to be willing to hold Treasuries at different moments in time.
I show that the frictionless view of short-rate expectations, which underlies this approach, is
inconsistent with real-time survey measures of expectations and with the observed behavior
of interest rates.
Using regressions of bond excess returns on current yields as a point of departure (e.g.,
Cochrane and Piazzesi, 2005), I first revisit evidence on bond return predictability in the recent literature. Various authors augment the yields-only specification with auxiliary macroeconomic variables and document their predictive power in addition to yields.1 The pre1
E.g., Ludvigson and Ng (2009), Cooper and Priestley (2009), Joslin, Priebsch, and Singleton (2014),
Cieslak and Povala (2015). I review this literature in Section II. See also Duffee (2012) for a review.
1
dictability with auxiliary variables is surprising given that today’s cross-section of yields
reflects investors’ expectations of future short rates and of future excess returns, and thus
it should subsume all information relevant for forecasting (e.g., Duffee, 2011). I argue that
auxiliary predictors can be split into two categories—measures of realized real activity and
of inflation expectations, respectively—by means of substantively different estimates of bond
risk premia that they imply. While expected inflation helps forecast bond returns across the
full maturity spectrum, the significance of real variables is located mostly at short maturities.
Fitted values from regressions that condition on real variables would suggest that risk premia
at short maturities are countercyclical, i.e. investors require a higher compensation for
holding short-term bonds in recessions. For example, in the months following the collapse
of Lehman Brothers, a regression using yields and a lagged change in unemployment as
predictors would imply an almost 100 basis points (per annum) premium on a two-year
Treasury, while that using yields and expected inflation would suggest that risk premia
actually became negative.
Relying on survey expectations of short-term rates, I decompose realized excess returns into
an expected return (risk premium) and an ex-ante unexpected return (forecast error). I find
that real variables predict the unexpected return component and are essentially uncorrelated
with the survey-implied risk premium. The opposite result holds true for measures of risk
premium based either only on information in yields, or on yields and expected inflation,
both of which are highly correlated with the survey-based risk premium and neither of
which forecasts unexpected returns.
These results point to distinct mechanisms that underlie the predictive power of real variables
and expected inflation proxies for bond excess returns. I delineate two parallel mechanisms
differing in terms of assumptions about the information of investors versus econometrician,
and I discriminate between predictors along this dimension. The first explanation, which can
be used to rationalize the significance of expected inflation, is that yields may in fact contain
all relevant information about bond risk premium but are contaminated by measurement
error (Cochrane and Piazzesi, 2005; Duffee, 2011). Indeed, Cieslak and Povala (2015) argue
that yields-only regressions are sensitive to seemingly negligible amounts of noise. They show
2
that using a proxy for long-horizon expected inflation (trend inflation) is an effective strategy
for uncovering the variation in bond risk premium when measurement error is present. In
this case, the information set of an econometrician and of investors differ just by noise:
Neither the econometrician’s risk premium estimate nor the auxiliary variable (a measure of
expected inflation) should predict agents’ forecast errors of the short rate, which I confirm
empirically.
Measurement error cannot plausibly justify why real-activity variables do in fact predict
agents’ forecast errors of the short rate while, at the same time, being uncorrelated with the
survey-based risk premium. To explain this empirical result, I focus on a second, alternative,
mechanism. Specifically, I argue that predictive regressions which condition on real-activity
variables recover a predictable element of realized returns that is not in expectations of
investors, and thus cannot be understood as a risk premium. Using survey forecasts of the
federal funds rate (FFR) going back to the early 1980s, I find that short-rate forecast errors
are particularly large around turning points in monetary policy which coincide with the
economy entering into a recession. In those episodes, forecasters overpredict the FFR four
quarters out by as much as 400 basis points. I document that FFR forecast errors co-move
strongly and negatively with errors of unemployment. Thus, times when forecasters are
overoptimistic about the employment outlook coincide with times when they underestimate
the extent of monetary easing that is about to take place. Forecast errors of FFR and of
unemployment are both predictable by the same set of lagged variables. No corresponding
relationship exists between forecast errors of FFR and of inflation. This evidence implies
that deviations from the FIRE assumption may manifest themselves particularly strongly in
expectations about the real short rate.
I construct a measure of such deviations by comparing the real-rate expectations estimated
with access to a full sample of data versus expectations formed by agents in real time. I refer
to the difference between the two information sets as the expectations wedge. The wedge
has persistent and strongly cyclical dynamics over the business cycle, widening in periods of
economic downturns. Approaching recessions, agents expect the next year’s level of the real
3
rate to be as much as 200 basis points higher than what an econometrician would predict
with the benefit of hindsight.
The expectations wedge forecasts bond excess returns with a declining strength across
maturities, and independently from statistical risk premium measures such as the Cochrane
and Piazzesi (2005) factor or the cycle factor of Cieslak and Povala (2015). The declining
pattern across maturities is consistent with the intuition that the expected real rate, i.e.
the marginal product of riskless capital, is mean-reverting and thus affects more the shortend of the yield curve than the long end. Accordingly, errors in expectations about the
real rate reveal themselves predominantly in realized (and ex-ante unexpected) returns on
short-maturity bonds.
The above evidence favors the interpretation of the expectations wedge as capturing information advantage of an econometrician rather than a rational variation in the risk premium
required by bond investors. One concern, however, is that the evidence relies on survey
data which may not reflect the true expectations of market participants. To further support
the expectations frictions interpretation, I analyze the ex-post predictability of identified
monetary policy shocks. I specifically focus on shocks extracted from interest rate futures
at the frequency of the Federal Open Market Committee (FOMC) meetings. This narrow
focus ensures that those shocks can be treated as innovations relative to the information set
of investors. I show that the expectations wedge predicts up to 47% of variation in monetary
policy shocks realized over the subsequent year, in line with the notion that it contains more
information than investors had access to in real time.
The difficulties in forecasting economic activity and the systematic nature of forecast errors
are openly recognized by the current and former Fed officials. For example Greenspan (2004)
writes: “In hindsight, the paths of inflation, real output, stock prices, and exchange rates
may have seemed preordained, but no such insight existed as we experienced it at the time.”
I look into the transcripts of FOMC meetings to provide a narrative record of the challenges
of real-time forecasting faced by Fed staff economists and FOMC members when making
policy decisions. Furthermore, examining FFR forecasts produced by the Fed staff before
4
each FOMC meeting, I show that their forecast errors display similar properties to those of
the private sector, and are likewise predictable by the expectations wedge.
Several forms of information rigidities may interact to produce these results, which I collectively term as expectations frictions. I document that the predictability of FFR forecast
errors can be partially, but not entirely, rationalized with rigidities such as sticky or noisy
information as in Mankiw and Reis (2002) and Woodford (2003). I also investigate an
alternative scenario that involves agents’ learning about the parameters in a standard monetary VAR. Using simulations, I show that in such a setup an econometrician with access to
full sample of data would uncover predictability of forecast errors in the magnitude found
empirically.
My work builds on a large literature estimating the conditional risk premia in Treasury
bonds. A subject of an active discussion in this line of research is the observation that
bond excess returns are predictable by variables that are weakly correlated with current
yields. While the economic sources of this empirical regularity are still debated, the only
way it can be formalized within affine term structure models without giving up on the FIRE
assumption is via so-called hidden or unspanned factors in term premia (Duffee, 2011; Joslin,
Priebsch, and Singleton, 2014). Alternatively, I argue that unspanned factors represent a
wedge between short-rate expectations of a real-time forecaster and the econometrician.
A growing body of research in finance and macroeconomics emphasizes the role of expectations formation by relaxing the FIRE assumption (see Mankiw and Reis (2011) and
Woodford (2013) for an overview).
There exists a large theoretical literature studying
nominal information frictions and their implications for monetary policy (e.g. Orphanides
and Williams, 2005; Woodford, 2010; Wiederholt and Paciello, 2012). Using survey data,
Coibion and Gorodnichenko (2012, 2015) provide empirical support for such frictions. More
recently, Angeletos and La’O (2012) extend the theoretical model by introducing real-side
imperfections. Empirical evidence for the real-side frictions is relatively less developed. One
exception is the recent work by Greenwood and Hanson (2015), who show that expectational
errors of firms lead to capital overinvestment, causing investment boom and bust cycles and
excess volatility in prices that cannot be rationalized as a risk premium. In a related way,
5
my results point to significant frictions in expectations about the real rate dynamics, which
is the key input in savings and investment decisions.
Survey data have been used to study expectations formation in financial markets, e.g. foreign
exchange (Frankel and Froot, 1987), bonds and stocks (Froot, 1989; Bacchetta, Mertens,
and van Wincoop, 2009). This research shows that forecast errors are predictable with
past information. Drawing on multiple surveys of equity investors, Greenwood and Shleifer
(2013) document investors’ extrapolation of past returns and highlight a negative relationship
between the statistical and survey-based equity risk premia. Piazzesi and Schneider (2011)
argue that bond risk premia implied by interest rate surveys are more persistent than those
obtained with statistical approaches such as the Cochrane-Piazzesi regressions. The focus of
my study is different and lies in identifying the expectational errors in short-rate expectations.
Those errors give rise to a predictable element in realized bond returns at short maturities
that is orthogonal both to statistical or survey-based risk premia.
Deviations from the FIRE assumption have been recently invoked in studies of other major
asset markets.
Singleton (2014) emphasizes informational frictions in the commodities
market to explain the speculative pricing of oil. Vissing-Jorgensen (2004) and Malmendier
and Nagel (2015) show that lifetime experiences influence the way individuals form expectations. Building on this work, Nagel (2012) links the effect of lifetime experiences to
overextrapolation bias in expectations of housing market, stock market and inflation.
I. Data and notation
Survey data. To measure short-rate expectations in real time, I use forecasts of the
FFR from the Blue Chip Financial Forecasts (BCFF) survey. It is the longest consistently
compiled survey of FFR forecasts, and a primary source of private sector expectations of
other interest rates as well.2 The survey is conducted monthly.3 Forecast of the FFR are
2
Every month, the BCFF survey collects forecasts of about 45 leading financial institutions. The Blue
Chip forecasts are frequently discussed in the meetings of the Federal Open Markets Committee (FOMC).
Between 1994 and 2010, Blue Chip forecasts are mentioned 174 times in transcripts of 74 FOMC meetings.
3
The responses are collected over a two-day period, usually between the 23rd and 27th of each month, and
published on the first day of the following month. An exception is the survey for the January issue which
6
available since March 1983, forecasts of inflation—since June 1984, and forecasts of Treasury
yields at several maturities—since December 1987. Consistent time series of these forecasts
back to the 1980s can be constructed for horizons up to four quarters ahead.
Forecasters predict average effective FFR in a given quarter. To account for the difference in
frequencies of the survey (monthly) and the outcome variable (quarterly), I use the notation:
Ets (F F Rt,hQ ), where Ets (·) stands for the survey forecast formed in month t, and F F Rt,hQ
denotes the quarterly average effective FFR observed h calendar quarters ahead relative to
month t. This is to distinguish from F F Rt , which is the average effective FFR in month t.
For example, for a survey conducted in January 1990, Ets (F F Rt,4Q ) represents the expected
average effective FFR four quarters ahead, i.e. in the first quarter of 1991. Forecast errors
are defined as:
F Et (F F Rt,hQ ) = F F Rt,hQ − Ets (F F Rt,hQ ),
(1)
where the forecast error, conditional on a forecast made in month t, becomes known h
calendar quarters after month t. Similarly, I denote the forecast of the one-year nominal
(1)
Treasury yield as Ets (yt,hQ ).
The survey design implies a shrinking forecast horizon: e.g., both in January and February
1990, the four-quarter-ahead forecast pertains to the same average value of the FFR realized
in the first quarter of 1991. In practice, I show that this has little impact on forecasts beyond
one quarter ahead. When shrinking horizon may be a concern, I rely on quarterly sampling,
in which case I use the survey from the middle month of each quarter.
I also use survey forecasts of inflation and unemployment. From BCFF, I obtain monthly
forecasts of CPI inflation (seasonally adjusted annualized rate of change in total CPI).
Ets (∆CP It,hQ ) is the expected inflation rate between the quarter of month t and h quarters
out. For unemployment (UNE), I rely on the Survey of Professional Forecasters (SPF)
from Philadelphia Fed, as unemployment is not part of BCFF. SPF reports forecasts for
the quarterly average seasonally adjusted unemployment rate, and is released in the middle
month of each quarter. I use the notation Ets (UNEt,hQ ) in analogy to the BCFF survey.
generally takes place between the 17th and 20th of December. The BCFF does not publish precise dates
as to when the survey was conducted.
7
Both inflation and unemployment forecasts are available up to (at least) four quarters
ahead, consistent with the FFR forecasts. Forecast errors of macro variables are defined
as in equation (1), with quarterly CPI and unemployment being the average of monthly
observations within the quarter.
A simple combination of models/forecasters is known to increase forecast precision (e.g.
Timmermann, 2006). I use the median of individual forecasts, but the results are essentially
unchanged if the mean forecast is used instead.
Macro data. I download unemployment rate, Chicago Fed National Activity Index (CFNAI), the monthly average effective FFR, and core and total CPI from the FRED database at
the Federal Reserve Bank of St. Louis. CPI series are considered to be unrevised (Croushore
and Stark, 1999), and revisions to the unemployment rate are negligible (Stark, 2010).4
In contrast, CFNAI is subject to material revisions and its vintages are only available
starting from 2001 on the Chicago Fed website. I denote the year-on-year percent change
c
in unemployment rate as ∆UNEt−1,t , and annual core and total CPI inflation as ∆CP It−1,t
and ∆CP It−1,t , respectively.
Yield curve and bond return data. I use zero-coupon nominal Treasury yields from
the Gürkaynak, Sack, and Wright (2006) dataset, which is regularly updated on the Federal
Reserve Board website. The 3-month T-bill rate is from the FRED database. The data is
sampled at the end of month. The term spread, denoted St , is the difference between the
10-year and 3-month yield. The log excess return on a zero-coupon bond with n years to
(n)
(n)
(n−h)
(h)
maturity for a h-year holding period is computed as rxt+h = nyt −(n−h)yt+h −hyt . For
robustness, I also use non-overlapping monthly returns (h = 1/12) on actual bond portfolios
from CRSP Fama files in excess of the one-month risk free rate, also from CRSP.
Factor structure in realized bond excess returns. To summarize the factor structure
in realized bond excess returns, I construct a short-end return factor, denoted rxSt+1 , as
4
To verify this, I obtain vintage data from the Philadelphia Fed. Annual changes in real-time unemployment
rate and its current vintage have a correlation above 0.99; the root mean squared difference between the
real-time and the final vintage of unemployment rate is less than 10 basis points. I also verify that my
subsequent results are essentially identical when using the series from the FRED or the vintages from
the Philadelphia Fed.
8
a residual from regressing an one-year excess return on the two-year bond onto the one(2)
(20)
year excess return on the 20-year bond: rxSt+1 ≡ rxt+1 − â − b̂ · rxt+1 where â, b̂ are OLS
regression coefficient. Based on the intuition that long-term bonds are more affected by
the risk premium variation than short-term bonds (e.g. Cieslak and Povala, 2015), rxSt+1
aims to capture in a simple way the independent dynamics in realized excess returns at
(20)
short maturities. Indeed, over the period of my study, rxt+1 has a correlation of only 0.55
(2)
(5)
(10)
(20)
with rxt+1 , 0.78 with rxt+1 , and 0.94 with rxt+1 . Jointly, rxSt+1 and rxt+1 explain more
than 97% of variation in annual bond excess returns across maturities. The details of this
decomposition are provided in the online Appendix.
Statistical bond risk premium proxies. Since the risk premium is not directly observable, I rely on two proxies proposed in the literature. The first one is the linear combination
of forward rates from Cochrane and Piazzesi (2005), CPt ; the second one is the cycle factor
from Cieslak and Povala (2015), c
cf t . Importantly, these two measures are closely related
and aim to capture the same source of variation in the bond risk premium, but do so in
statistically different ways. The Cochrane-Piazzesi factor is constructed using only yieldcurve information. The cycle factor, in addition to yields, conditions on a measure of longhorizon inflation expectations. Specifically, Cieslak and Povala (2015) decompose yields
into a trend inflation component, an expected real rate component and a risk premium
factor (the cycle factor, c
cf t ). Trend inflation determines the overall level of yields across
maturities and is measured with a discounted moving average of past core CPI inflation,
which I denote with τtCP I .5 The authors show that the cycle factor subsumes the information
in the Cochrane-Piazzesi factor and predicts bond excess returns both in and out of sample.
Following Cochrane and Piazzesi (2008), I construct both risk-premium measures starting
in November 1971 when yields with maturities of ten years and above become available.
5
PN
i
i=0 v πt−i
P
,
N
i
i=0 v
CP Itc
ln( CP I c ).
t−1
I follow the construction in their paper, i.e. τtCP I =
months and year-over-year core inflation: πt =
9
with discount factor v = 0.987, N = 120
Sample period. The main empirical analysis is based on data from June 1984 through
August 2012, a period when the FFR was the main operating target of the Fed.6 The
sample covers 327 months (109 quarters): the first annual return (and four-quarter ahead
forecast error) is realized in June 1985 conditioning on data from June 1984, the last one is
realized in August 2012 conditioning on data from August 2011. The start of the sample is
determined by the availability of the CPI inflation forecasts in the BCFF survey. Whenever
I rely on survey forecasts of Treasury yields other than the FFR, the sample covers 285
months beginning in December 1987 when these forecasts are first released. The end of the
sample is when the zero-lower bound (ZLB) becomes binding for FFR expectations.7
II. Measuring the variation in bond risk premia with predictive regressions
This section reviews the evidence of predictability of bond excess returns with auxiliary
variables. I highlight a key difference between the predictive power of variables associated
with real macroeconomic activity versus measures of expected inflation. I show that the
significance of real variables stems from their ability to predict the short-rate forecast errors,
i.e. the component of returns that agents did not expect in real time.
II.A. Theoretical basis
Consider realized one-period excess return on a two-period zero-coupon bond:
(2)
(1)
(2)
(1)
rxt+1 = −yt+1 + 2yt − yt ,
(2)
where yt
(2)
(1)
denotes a continuously compounded two-period yield, and yt
is a one-period
(short) rate. Rearranging (2), the two-period yield can be expressed as:
6
The post-1985 sample is the focus of several recent studies of bond risk premia, e.g. Joslin, Priebsch, and
Singleton (2014). The benchmark term premium estimates published by the Federal Reserve Board using
the methodology of Kim and Wright (2005) are also based on the post-1985 sample.
7
In August 2011, the Fed provided a dated statement promising to keep interest rates near zero for the
next two years. Swanson and Williams (2014) argue that Treasury yields were as responsive to news
throughout 2008–2010 as in the earlier part of the sample, and until late 2011, market participants
expected the funds rate to lift off from zero within about four quarters, which is confirmed by the BCFF
survey expectations as well.
10
(2)
yt =
1
1 (1)
(1)
(2)
yt + yt+1 + rxt+1 .
2
2
(3)
Equation (3) follows from the definition of bond returns. Since it holds ex-post realizationby-realization it also holds ex-ante:
1 1 (1)
(1)
(2)
(2)
yt = Et yt + yt+1 + Et rxt+1 ,
2
2
(4)
where Et (·) is expectation conditional on all information available at time t. By recursive
argument, one obtains an analogous expression for the n-period yield:
(n)
yt
n−1
X
1
(1)
= Et
yt+i
n
i=0
!
n−2
X
1
(n−i)
rxt+i+1
+ Et
n
i=0
!
.
(5)
The current n-period yield is a sum of investors’ expectations about the average short rate
(the expectations hypothesis component) and expected excess returns to be earned over the
life of the bond (the risk premium component).
Suppose that all information that investors use to forecast future short rates and excess
returns is summarized in a vector xt . Since yields are conditional expectations, the current
yield curve can be fully described as a function of xt , yt = f (xt , J), where yt is a vector of
yields with J different maturities observed at time t. If the mapping f (·) is invertible, the
current yield curve reflects the state vector used by investors to form expectations. Therefore,
as long as investors’ expectations are FIRE, the yield curve contains all information useful
for forecasting future yields and returns. This argument implies that there is no immediate
reason to use variables other than current yields for forecasting future yields or bond returns,
a point emphasized by Cochrane and Piazzesi (2005) and, more recently, by Duffee (2011).
And yet, substantive empirical literature documents predictability of bond excess returns
with variables other than current yields.
11
II.B. Overview of empirical evidence on bond return predictability
The usual approach to measuring the risk premium variation in financial assets is through
full-sample predictive regressions. A general specification that embeds different regressions
estimated in the bond literature is:
(n)
rxt+1 = α + γ1′ yieldst + γ2′ macrot + γ3′ yieldst−h + εt+1 ,
(6)
(n)
where rxt+1 is an annual holding period return on an n-year zero-coupon bond. Cochrane
and Piazzesi (2005) assume γ2 = γ3 = 0 and project bond excess returns on a set of current
forward rates, which is equivalent to projections on yields or the principal components (PCs)
of yields. Various authors augment the yields-only specification with auxiliary variables
including a range of financial and realized macro variables (Ludvigson and Ng, 2009), output
gap (Cooper and Priestley, 2009), proxies for long-horizon inflation expectations or trend
inflation (Cieslak and Povala, 2015), CFNAI and four-quarter-ahead expected inflation from
the BCFF survey (Joslin, Priebsch, and Singleton, 2014), or lagged yield curve variables
(Cochrane and Piazzesi, 2005; Duffee, 2012). Those auxiliary variables are found to contain
predictive information about future returns in addition to current yields. I refer to this
feature of the data as “excess predictability,” i.e. predictability achieved with variables
other than just current yields.
To review this empirical evidence, I estimate predictive regressions of annual bond excess
returns on current yields and additional predictors. The auxiliary predictors are selected to
summarize the findings of previous studies in a simple and transparent way. As common
in the literature, I predict bond returns with an annual holding period using data sampled
at a monthly frequency, so each month I forecast excess return over the next 12 months.
I report conservative t-statistics based on reverse-regression (Hodrick, 1992) as well as
the usual Newey-West t-statistics with 18 lags.8
8
Table I presents regression estimates
Hodrick (1992) correction relies on predicting monthly returns with annual averages of predictors rather
than annual overlapping returns with month-t value of the predictor. I follow the delta method
implementation of reverse regressions by Wei and Wright (2013). Ang and Bekaert (2007) show that
Hodrick’s standard errors are well-behaved and less prone to overrejecting the null of no predictability
than the Newey-West or Hansen-Hodrick errors.
12
for maturities of 2, 5, 10 and 20 years (columns 2–5) and for the short-end return factor
rxSt+1 (column 1). Panel A contains the yields-only regressions. Following the literature
(Litterman and Scheinkman, 1991), I summarize information in the current yield curve
using the first three principal components (PCs) of yields with maturities between one and
20 years.9 The remaining panels extend the baseline specification from Panel A by including,
in addition to yield PCs, auxiliary regressors one at a time: CFNAI,10 year-on-year growth
in unemployment (∆UNE t−1,t ), one-year lagged term spread (St−1 ), BCFF survey forecast
of CPI inflation four quarters ahead (Ets (∆CP I t,4Q )), and the trend inflation (τtCP I ). The
first two regressors capture real economic activity: CFNAI is a more comprehensive measure
but it is also revised, whereas unemployment is not. The past term spread belongs to the
category of lagged term structure variables; Duffee (2012) considers multiple yields lagged
from one through 12 months. For transparency, I include just one lag of the term spread
that I find to summarize well the predictive power of regression with many yields and lags.
Finally, the last two variables aim to reflect real-time inflation expectations of investors.
Table I shows that while each of those variables contains information about future returns
beyond current yields, two distinct predictability patterns emerge. Real activity measures
and the lagged term spread are strongly significant and increase predictive R2 by more
than 10% at short maturities. However, their contribution dissipates at longer maturities.
Neither of these variables remain significant at maturities beyond five years. In contrast,
the significance of expected inflation proxies stays roughly constant across maturities, and
adds to the explained variation in excess returns both at the short and long end of the
term structure. The results obtained with the trend inflation τtCP I are particularly strong.
This is consistent with Cieslak and Povala (2015) who find that controlling for long-horizon
inflation expectations, which as they argue τtCP I reflects well, in predictive regressions offers
9
Three PCs account for 99.9% of cross-sectional variation in yields over my sample period. The residual
variation in yields which is unexplained by the three PCs is about 6 basis points on average across
maturities.
10
Using the approach of Stock and Watson (1999), Ludvigson and Ng (2009) perform a factor decomposition
of 132 economic and financial indicators. They find that the real activity factor is the most significant
predictor of bond excess returns among factors they extract. In my regressions, I use CFNAI because
it is easily available, regularly updated, and nearly perfectly correlated with their real activity factor
(correlation of 0.99 in the 1964–2003 sample used by Ludvigson and Ng (2009)). Indeed, CFNAI is
constructed by the Chicago Fed using the Stock-Watson methodology.
13
an efficient way of separating the expectations hypothesis term from the risk premium in the
yield curve.
Column 1 of Table I illustrates these differences among predictors by reporting their ability to
forecast the short-end return factor, rxSt+1 , defined in Section I. While measures of expected
inflation are insignificant, the significance of the remaining predictors for rxSt+1 actually
(2)
increases compared to the raw excess return on the two-year bond, rxt+1 , in column 2. This
suggests that CFNAI, ∆UNEt−1,t and St−1 capture a predictable element of bond excess
returns that is uncorrelated with sources of return predictability at the long end of the yield
curve. Importantly, those predictors all indicate that risk premia at the short end of the
yield curve are more countercyclical than those implied either by the yields-only or yieldsplus-expected-inflation regressions. According to Panels B and C, a one-standard deviation
decline in the real activity as measured by CFNAI (increase in ∆UNEt−1,t ) predicts a 50
[= 0.25 × 2] basis point (70 [= 0.35 × 2] basis point) rise in the risk premium on the two-year
Treasury. Notably, the negative coefficient on the lagged term spread in Panel D agrees
with those estimates, given the well-known forecast power of the term spread for the real
activity: A shrinking term spread predicts a future weakening of economic activity (Estrella
and Hardouvelis, 1991; Harvey, 1989).
The differences in the predictive behavior between the real factors and expected inflation
imply economically distinct interpretations of bond risk premium dynamics. For example,
using the fitted value from regression in Panel C with ∆UNEt−1,t as predictor, one would
conclude that in the six months after Lehman Brothers collapse, investors required more
than 90 basis points as compensation to hold a two-year Treasury bond (average fitted excess
return from October 2008 though March 2009). An analogous number based on regression
in Panel E and F with survey expected inflation and trend inflation would be −11 and −23
basis points, respectively.
II.C. Three hypotheses about the sources of excess return predictability
In an attempt to understand the sources of excess predictability with auxiliary variables,
I distinguish between three hypotheses, which I refer to as (i) the cancelation hypothesis,
14
(ii) the measurement error hypothesis and (iii) the expectations frictions hypothesis. Those
hypotheses should not be viewed as mutually exclusive, but rather as capturing the different
sources of predictability suggested by the empirical results above.
The cancelation hypothesis summarizes the argument made in the literature to formalize
excess predictability within no-arbitrage affine term structure models. Most term structure
models assume that Et (·) is formed under FIRE, i.e. the realized future short rate equals
(1)
(1)
yt+1 = Et (yt+1 ) + vt+1 , where the forecast error vt+1 is unpredictable by all information
available at time t. No distinction is made between the information set of an econometrician
and that of a real-time forecasters or investors. Thus, using equation (5), a variable can
forecast future returns without being revealed by time-t yields only when it impacts shortrate expectations and the risk premium in an exactly offsetting manner. Moreover, this
condition needs to hold maturity by maturity. An example of such a parametric restriction
is provided by Duffee (2011) in a context of a stylized term structure model. The cancelation
argument also implicitly underlies the approach of Joslin, Priebsch, and Singleton (2014)
who introduce a class of affine term structure models with unspanned real and nominal
macroeconomic risks.
The measurement error hypothesis maintains that a small amount of noise in observed yields,
due to bid-ask spreads or the splining of zero-coupon yields, can make it difficult to extract
the risk premium variation using yields-only regressions, even if true (measurement-error
free) yields perfectly span all relevant information. This explanation is pursued by Cieslak
and Povala (2015) to rationalize the significance of trend inflation in bond predictability
regressions. They show that augmenting yields-only predictive regressions with a proxy for
long-horizon expected inflation significantly improves the identification of the risk premium
in yields when measurement error is present. The magnitude of measurement error necessary
to produce this result is around 5 basis points, which is less than a plausible measurement
error that arises due to splining the zero-coupon yield curve (e.g., Bekaert, Hodrick, and
Marshall, 1997).
Finally, the expectations frictions hypothesis recognizes that the information sets of investors
and the econometrician may not coincide for reasons other than just small noise in yields. To
15
see how this could help explain excess predictability, it is worth going back to equation (4).
As highlighted by Fama and Bliss (1987), equation (4) is a tautology and is economically
void unless one makes an assumption about expectations Et (·). In particular, this equation
holds for any model of expectations formation and conditioning information set, as long as
the set contains yields. Identities (3) and (4) jointly imply:
h
i
(1)
(1)
(2)
(2)
− yt+1 − Et (yt+1 ) = rxt+1 − Et (rxt+1 ),
(7)
where the left-hand side is (the negative of) the forecast error about the short rate, and
the right-hand side is the unexpected return. From equation (7), any forecast error that
agents make when predicting the short rate must equal the unexpected returns that they
earn ex-post. This argument also holds for an n-period bond, for which we have:
−
n−2 h
n−2 h
i
i X
X
(n−j)
(n−j)
(1)
(1)
rxt+1+j − Et (rxt+1+j ) .
yt+1+j − Et (yt+1+j ) =
(8)
j=0
j=0
Clearly, yields reflect real-time expectations of investors. A variable that predicts forecast
errors in a full-sample predictive regression will have no effect on the current yield curve
when the econometrician uses a predictor that is not included in the time-t information set
of investors. A more nuanced possibility is that the agents have access to the particular
variable at time t but either deem it unimportant for the yield curve or the relationship
with the yield curve is difficult to estimate in real time. Next, I argue that deviations
between the information set of an econometrician and investors are particularly important
for understanding why real activity proxies predict bond excess returns, and why this
predictability is only present at short maturities.
II.D. Decomposing predictability of realized bond excess returns
Table I suggests that the sources of predictable variation in excess returns are different for
short- and long-maturity bonds. Realized bond returns are a sum of two components: a
return that investors expect to earn (risk premium) to be willing to hold the bond and an
ex-ante unexpected (surprise) return they earn in addition to what they have expected. For
16
the annual excess return on a two-year bond, we have the following identity:
h
i
h
i
(2)
(2)
(1)
(1)
(1)
rxt+1 = ft − Et (yt+1 ) − yt+1 − Et (yt+1 ) ,
|
{z
} |
{z
}
risk premium
unexpected return
(2)
(2)
Et (rxt+1 )
(2)
where ft
(9)
(2)
(2)
U rxt+1 ≡ rxt+1 − Et (rxt+1 )
(2)
is the one-year forward rate, ft
(2)
= 2yt
(1)
− yt . I focus on decomposing the
excess return on a two-year bond because it captures the segment of the yield curve where
(1)
all auxiliary regressors in Table I are significant. To proxy for Et (yt+1 ) in equation (9), I
(1)
use BCFF forecast of the one-year yield at the horizon of four quarters ahead, Ets (yt,4Q ).11
(2)
(2)
The survey-based expected excess return is defined as Ets (rxt+1 ) = ft
s,(2)
(1)
(1)
− Ets (yt,4Q ), and the
(1)
unexpected return is Urxt+1 = −[yt+1 − Ets (yt,4Q )].
In Table II, I regress the realized excess return on the two-year bond and its components
on auxiliary variables from Table I, as well as on the statistical measures of bond risk
c . The dependent
premium: the Cochrane-Piazzesi factor, CPt , and the cycle factor, cf
t
variables in columns 1 through 3 are the realized, expected and unexpected excess return:
(2)
(2)
s,(2)
rxt+1 , Ets (rxt+1 ) and Urxt+1 , respectively. The dependent variable in column 4 is the survey
forecast error of FFR, F Et (F F Rt,4Q ), which I include for future reference. FFR forecast error
is reported with a minus sign so that it co-moves positively with the unexpected and realized
s,(2)
return. The correlation between the unexpected return Urxt+1 and −F Et (F F Rt,4Q ) is 0.94.
Finally, in columns 5 and 6, the dependent variables are the four-quarter-ahead forecasts of
(1)
the one-year yield and of the FFR: Ets (yt,4Q ), Ets (F F Rt,4Q ), respectively. Regressions are
estimated at a monthly frequency on a sample starting in December 1987, when BCFF
forecasts for the one-year yield become available.
Explanatory variables in Table II form two distinct groups: those that predict forecast
errors and those that do not. I report them separately in Panels A and B. The first category
11
Forecasters in the BCFF survey predict the average level of the one-year yield four quarters ahead, which I
(1)
indicate with the notation Ets (yt,4Q ) as explained in Section I. I treat this forecast as an approximation for
investors’ expectation of the one-year yield in 12 months. This approximation does not have material effect
(1)
(1)
(1)
(1)
on the conclusions. The correlation between forecast error defined as yt+1 −Ets (yt,4Q ) and yt,4Q −Ets (yt,4Q )
(1)
(1)
is 0.996, where yt+1 is the end-of-month value of one-year yield realized in 12 months, and yt,4Q is the
average level of the one-year yield realized in four calendar quarters from month t.
17
(Panel A) contains real activity variables (CFNAI, ∆UNEt−1,t and St−1 ). Each of these
variables predicts unexpected returns and FFR forecast errors at the 10% confidence level
or better with an R2 between 13% and 18% (columns 3 and 4). None, however, appears to
(2)
be related to the expected return, Ets (rxt+1 ) (column 2): The R2 is zero for ∆UNEt−1,t and
St−1 , with regression coefficients not distinguishable from zero (t-statistics -0.28 and -0.41,
respectively). For CFNAI, the R2 is 4%, and the coefficient loading marginally significant
(t-statistic 1.92). However, the positive sign of the loading on CFNAI for the expected
return regression (column 2) contradicts the negative sign for the realized return regression
(column 1). Thus, the predictive power of CFNAI for the realized excess return is due to
the unexpected component of return (column 3).
The second category of regressors (Panel B) contains measures of expected inflation and
of risk premium. None of these variables predicts unexpected returns and FFR forecast
errors, but they correlate with the survey-based expected return. The significance of the
(2)
Cochrane-Piazzesi and the cycle factor for Ets (rxt+1 ) is consistent with their risk-premium
(2)
interpretation. The explanatory power of expected inflation proxies for Ets (rxt+1 ) needs to
be interpreted with caution as it stems entirely from a slight trend in the survey-based risk
premium which coincides with the trend in expected inflation over the post-1980 sample.
Neither realized returns nor statistical risk premia display such a trend.
Accordingly,
estimating regression in column (2) in monthly changes rather than in levels (not tabulated),
shows that neither of the expected inflation proxies is significantly related to Ets (rxt+1 ). At
the same time, the significance of the risk-premium measures actually increases compared
to the regressions estimated in levels.12
(2)
Figure 1 Panel A superimposes the realized excess return on the two-year bond, rxt+1 ,
against its decomposition into the expected and unexpected components. The graph makes
clear that the unexpected part accounts for the main portion of variation in the realized
excess return, and displays a business cycle pattern that is distinct from the dynamics of
12
For regressions run in changes, t-statistics are -0.6 and 0.05 for Ets (CP It,4Q ) and τtCP I , respectively, and
c explain 25% and 42% of changes in E s (rx(2) ), with t-statistics
both R2 are 0.00. Changes in CPt and cf
t
t
t+1
of 4.22 and 8.35, respectively. Changes in real activity variables from Panel A of Table II explain less
(2)
than 1% of variation in the changes of Ets (rxt+1 ).
18
the expected return. Panel B compares the survey-based risk premium with the cycle factor
showing that they trace each other closely (unconditional correlation is 0.54 in levels and
0.65 in monthly changes).
The results in Table II help assess the relative merit of the hypotheses in Section II.C
as possible explanations for excess predictability. First, the estimates do not support the
cancelation hypothesis. For the cancelation between risk premia and the expected short
rate to occur, one would need to find that an auxiliary variable 1) predicts realized returns;
2) predicts the expected excess return and the expected short rate each with a significant
coefficient and with an opposite sign; 3) does not predict forecast errors. None of the
predictors considered in Table II fulfills these conditions.
Second, the results for expected inflation and the statistical risk premium measures are in line
with these variables being in the information set of agents who price bonds. In particular,
Cieslak and Povala (2015) argue that expected inflation, and by extension the cycle factor,
are spanned by true (i.e. measurement-error free) yields. Consistently, neither variable
predicts forecast errors. The same result holds true for the Cochrane-Piazzesi factor. Both
c co-move positively with survey-based expected return, consistent with their
CPt and cf
t
interpretation as measuring the time-varying risk premium.
Finally, the expectations frictions hypothesis manifests itself in the predictability of forecast
errors and unexpected returns by the real-activity factors. The next section explores this
result in more depth.
III. Properties of short rate expectations
This section characterizes short-rate expectations formed by agents in real time. I discuss
the properties and the quality of survey-based forecasts of the FFR, and document a tight
negative link between forecast errors of FFR and of unemployment.
19
III.A. Summary statistics of FFR survey forecasts
Figure 2 presents BCFF survey forecasts of the FFR from three perspectives. Panel A shows
the time series of forecasts at horizons ranging from the current quarter out to four quarters
ahead. Panel B displays the term structures of FFR forecasts at different points in time,
conditioning on the current value of the FFR. For clarity, the graph displays forecasts made
in the middle month of each quarter. The distance between points along each term structure
and the solid line depicts the magnitude of the forecast error. Finally, Panel C plots the
time series of forecast errors.
Table III Panel A provides summary statistics. Over the 1984:6–2012:8 sample, forecast
errors are on average negative ranging from −12 basis points at one quarter ahead to −64
basis points at four quarters ahead.13 At four quarters, agents overpredict (underpredict) the
future level of FFR by as much as 440 (255) basis points. In line with the visual impression
from Figure 2, forecast errors are large and significantly negative following easing decisions by
the Fed, and positive (but not significant) following tightening decisions. This pattern lines
up with the view expressed by policy makers that the Fed eases in reaction to unexpected
events and tightens according to a well-communicated plan.14
Table III Panel B projects realized changes in the FFR on changes predicted by forecasters.
The results are presented both at the monthly frequency which uses all available observations
in the survey and at the quarterly frequency which avoids potential distortions due to
the shrinking forecast horizon. The slope coefficient of the regression is not statistically
significantly different from unity across sampling frequencies and horizons, implying that
13
The negative average error is consistent with Kuttner (2001) monetary policy shocks being negative on
average in the 1989–2008 period for which these shocks are available. Since Kuttner’s shocks are identified
in a narrow one-day window around the FOMC meeting, the magnitude of the surprise is naturally smaller
than the survey forecast error at horizon of a quarter or longer. I discuss the properties of monetary
policy shocks identified at high frequencies in Section IV.B.2.
14
To construct average error conditional on the direction of the monetary action, I use both scheduled and
unscheduled Fed policy decisions. The dating and description of these events is available in Bloomberg’s
U.S. Federal Funds Target Interest Rate History Table. If there has been more than one policy move
within a month, I use the direction of the last move. I compute the average of forecast errors conditional
on FFR expectations formed in easing and tightening months, respectively. The number of observations
in months with easing and tightening moves does not sum to the total number of months in the sample
because not all months had a policy decision and in some cases the Fed decided to leave rates unchanged.
20
forecasts are efficient in the sense that they are uncorrelated with both current predictions
and current realizations of the FFR (Mincer and Zarnowitz, 1969). The only significant
rejection of unit slope occurs at the monthly frequency at horizon of one quarter ahead,
where distortions due to the shrinking forecast horizon are most severe.
The alignment between forecasts across horizons, visible in Figure 2 Panel A, suggests that
forecasters anchor their predictions at current realizations of the FFR. However, they do
not just follow a naive random-walk forecasting rule. Table III Panel C compares forecast
precision of the survey against out-of-sample forecasts from three univariate models: a
random walk model and two autoregressive (AR) models with one and two quarterly lags,
respectively. Prior parameters for the AR models are selected using a burn-in period that
minimizes the out-of-sample root mean squared error (RMSE); the choice of the AR(2) model
is determined based on the Schwarz information criterion (SBIC) applied to post-1984 data.
Those choices endow statistical models with extra information that survey participants did
not have in real time in order to make the comparison conservative. Nevertheless, surveys
have the lowest RMSE across all forecast horizons. On average across horizons, the RMSEs
for random walk and AR(1) are about 12% higher, and for AR(2) about 6% higher than
those implied by survey forecasts.
The last result echoes the conclusion of Ang, Bekaert, and Wei (2007) who show that it
is hard to beat survey forecasts of inflation with statistical models. However, it is worth
distinguishing between survey forecasts of the short rate (i.e. the risk-free rate) versus survey
forecasts of longer term yields and excess bond returns. While surveys may perform better
than simple statistical models in terms of predicting the short rate, this needs not be true
for forecasting bond excess returns and longer-maturity yields. Indeed, Cieslak and Povala
(2015) argue that surveys provide a noisier measure of bond risk premium than the cycle
factor constructed in real time.
III.B. Do FFR forecast errors co-move with forecast errors of macroeconomic variables?
In trying to anticipate the future path of the short-term nominal interest rate, agents need
to form expectations about the evolution of the real economy, inflation, and how monetary
21
policy reacts to those variables. Using the intuition from the baseline Taylor (1993) rule,
according to which the Fed reacts to inflation during the past year and to the current
output gap, one would expect that FFR forecast errors are correlated with forecast errors
for macroeconomic variables. To study this question, I rely on survey forecasts of inflation
and unemployment.15
Figure 3 superimposes the time series of FFR forecast errors with forecast errors of unemployment (Panel A) and of inflation, both core and all-items (Panel B), at the fourquarter horizon.16 The graph shows a strong negative relationship between the forecast
errors of FFR and of unemployment (correlation of −0.67), and a much weaker relationship
between forecast errors of FFR and of inflation (correlation of 0.09 and 0.32 for core and total
inflation, respectively). This suggests that times when forecasters are too optimistic about
employment over the next year coincide with times when they underestimate the extent of
monetary easing that is about to take place.
Table IV studies whether forecast errors of FFR and of macro variables share a common
predictable component. Panel A simply verifies that macro forecast errors are orthogonal
to agents’ current forecasts, i.e. forecasts are not obviously inefficient. Panel B takes
the perspective of an econometrician and attempts to predict forecast errors with current
and lagged macro variables. I include ∆UNEt−1,t among regressors given its predictive
power for unexpected bond returns. Analogous regressions controlling for the lagged term
spread and CFNAI are deferred to the online appendix, as they deliver similar conclusions.
15
While public expectations of output gap are not directly observable via surveys, output gap and
unemployment are highly negatively correlated. The practice of approximating output gap with the
rate of unemployment, motivated by the Okun’s law, is common in policy makers’ deliberations (see
e.g., Yellen, 2015). Gali, Smets, and Wouters (2011) show that the correlation of unemployment with
measures of output gap is −0.95 over the 1965–2011 sample.
16
Although BCFF survey asks for forecasts of total inflation, I compute forecast errors for both the total and
c
core inflation: F Et (Xt,4Q ) = Xt,4Q − Ets (∆CP It,4Q ), where X = {∆CP It,4Q , ∆CP It,4Q
}. Core inflation
is what market participants and the Fed are more likely to focus on when evaluating implications of
inflation for future monetary policy. Moreover, since the mid-1980s total inflation has experienced
transitory volatility (due to the energy component) which may lead to spuriously large volatility of
inflation forecast errors. This transitory variation has not been reflected in the core inflation or in
interest rates (e.g. Ajello, Benzoni, and Chyruk, 2012; Stock and Watson, 2011). In the 1984:6–2012:8
period, the correlation between the total and core annual CPI inflation is 0.64. The online Appendix
shows that the transitory component of total inflation does not influence inflation expectations beyond
two quarters ahead. This is consistent with the evidence that from the mid-1980s the pass-through of
energy shocks onto the yield curve has been negligible.
22
The regressions show that the same variables that predict FFR forecast errors also predict
forecast errors of unemployment but not of inflation. In particular, ∆UNEt−1,t predicts
F Et (F F Rt,4Q ) with a standardized coefficient of −0.83 (t-statistic −3.40) and F Et (UNEt,4Q )
with a standardized coefficient of 0.63 (t-statistic 4.65).
Panel C of Table IV summarizes OLS and 2SLS projections of the form:
F Et (F F Rt,4Q ) = α + β1 F Et (UNEt,4Q ) + β2 F Et (∆CP It,4Q ) + ut,4Q .
(10)
The 2SLS estimates use variables from Panel B as instruments. A robust conclusion from this
exercise is that FFR and unemployment forecast errors are strongly negatively related. The
coefficient on F Et (UNEt,4Q ) is about −1, suggesting that one percentage point unexpected
increase in unemployment rate over the next year is associated with an unexpected decline
in the FFR by about 100 basis points.
In sum, over the last three decades the predictability of forecast errors of the nominal short
rate appears to be associated with the way agents form expectations about real variables
rather than inflation. This suggests that the main source of predictable variation in FFR
forecast errors stems from the real rate component of these errors.
IV. Expectations wedge
I quantify the discrepancy between the expectations of the real short rate formed by agents
in real time versus those of an econometrician who has access to a full sample of data. I show
that the expectations wedge is a powerful predictor or bond excess returns beyond measures
of the risk premium.
IV.A. Expected real rate: full sample versus real time expectations
I define the ex-post real rate as the FFR minus core inflation over the preceding 12 months:
c
rt+1 = F F Rt+1 − ∆CP It,t+1
.
23
(11)
This is the measure that market participants and Fed officials commonly look at when
thinking about the real rate dynamics (e.g., Hamilton, Harris, Hatzius, and West, 2015). In
an ex-ante form, the expected real rate is:
c
rte = Et (F F Rt+1 ) − Et (∆CP It,t+1
).
(12)
I compare two approaches to estimating (12). The first approach takes the viewpoint of an
econometrician using full-sample linear projections of rt+1 on instruments:
r̂te,F S = Et [ rt+1 | Instrumentst ] ,
(13)
where the instruments are unrevised, available at time t, and include FFR, core CPI inflation,
unemployment, as well as growth rate of unemployment and lagged term spread. The fitted
value from regression (13) is what I refer to as the full-sample ex-ante real FFR, and denote
it as r̂te,F S . Table V, Panel A, displays the results of estimating (13) by gradually expanding
the set of instruments. Adding just the past year’s growth rate of unemployment to FFR
and inflation captures 57% of the explained variation. The full set of instruments forecasts
63% of variation in rt+1 .
The second approach takes the perspective of a real-time forecaster, where I use survey
expectations of FFR and inflation to obtain:
rte,RT = Ets (F F Rt,4Q ) − Ets (∆CP It,4Q ).
(14)
I refer to rte,RT as the real-time ex-ante real FFR. A regression of rt+1 on rte,RT gives the
estimates: rt+1 = 0.01 + 0.79 rte,RT , with an R2 of 48%, compared to an R2 of 63% based on
(0.05)
(8.3)
the full-sample projection.
Figure 4, Panel A superimposes r̂te,F S and rte,RT . Absent statistical biases and/or expectations
frictions, the two measures of the ex-ante real rate should differ just by an unpredictable
noise component. However, the graph shows that the survey-based real rate expectations
systematically lag behind those based on the full-sample projections. To measure the gap
24
between the information set of the econometrician and the real-time forecaster, I define:
wedget = r̂te,F S − rte,RT ,
(15)
where r̂te,F S is based on the specification in column (5) of Table V using the full set of
instruments. This is also the specification that I rely on in the subsequent analysis. Figure
4 Panel B shows that the wedge is persistent, declines ahead of NBER-dated recessions
bottoming at about −200 basis points, and recovers after the recessions. During recessions
real-time forecasters expect a higher real rate compared to the full-sample estimates. As a
result, wedget predicts a significant part of the variation in ex-post FFR forecast errors. For
horizon of four quarters ahead, we have:
R̄2 = 0.33,
F Et (F F Rt,4Q ) = const. + 0.92 wedget + εt,4Q ,
[0.73, 1.17]
(16)
where to account for the generated regressor, in brackets I report the 5% bootstrap confidence
interval obtained with the Li and Maddala (1997) method.
Table V Panel B compares the predictability of forecast errors corresponding to different
estimates of wedget that vary by the set of instruments included in equation (13). The
explained variation increases from 4% with only the current FFR rate used as instrument
(column 1), in which case the wedge is insignificant, to 25% when past growth rate of
unemployment is included as an instrument (column 3), and to 33% when lagged term spread
is added (column 5). While the wedge is constructed using four-quarter-ahead forecasts,
Panel C of Table V shows that it consistently predicts FFR forecast errors at shorter horizons
as well.
It is worth noting that the real rate defined in equation (11) differs from the definition
of the ex-post real rate that is sometimes adopted in the literature: as the current oneperiod nominal interest rate less the rate of inflation between today and the next period. To
(1)
(1)
c
highlight the difference, I define: ret+1 = yt − ∆CP It+1
, where yt
is the one-year nominal
interest rate. The ex-ante real rate is then obtained from full-sample projections of ret+1
on a set of time-t instruments (e.g., Fama, 1975; Mishkin, 1981; Yogo, 2004), as a way to
25
extract the unobserved inflation expectations from the realized inflation. However, by using
the current nominal yield, ret+1 takes as given expectations about the real rate impounded
into the current yield curve. As such, it does not draw a distinction between the real-rate
expectations formed by agents in real time versus by the econometrician. For comparison
with the results presented above, I repeat the analysis with ret+1 in the online Appendix.
The expectations wedge constructed with ret+1 predicts at most 6% of variation in the FFR
forecast error with an insignificant coefficient. This is consistent with the conclusion that
the predictability of the FFR forecast errors is primarily driven by the difference between
real-time and full-sample estimates of the expected real rate rather than of the expected
inflation.
IV.B. Expectations wedge versus bond risk premium
The risk premium at time t should reflect compensation for risk that is currently expected
and demanded by investors in order to be willing to hold the bond. The expectations wedge
defined in (15) aims to summarize information that is not contained in the time-t information
set of real-time forecasters. If survey forecasts are a good approximation to expectations of
bond investors, the expectations wedge should not be spanned by the contemporaneous yield
curve, which summarizes those expectations. Indeed, projecting wedget on the first three
yield PCs explains 12% of its variance when regression is run in levels, and 2% when it is run
in changes, suggesting that shocks that drive yields are effectively uncorrelated with those
reflected in the wedge.
Next, I document that the expectations wedge predicts bond excess returns when controlling
for variation in the bond risk premium. To further argue that it contains information that is
not in the time-t information set of investors, I show that the wedge predicts monetary policy
surprises which are designed to capture shocks relative to time-t expectations of investors.
IV.B.1. Predictability of bond excess returns
I estimate predictive regressions of bond excess returns of the form:
26
(n)
rxt,t+h = α + β1 wedget + β2 RPt + εt,t+h ,
(17)
where RPt is a measure of the bond risk premium. I use both the Cochrane-Piazzesi and
the cycle factor as RPt . Table VI reports results for monthly excess returns (h = 1/12)
on CRSP bond portfolios. The focus on monthly returns is different from most of bond
predictability literature. The goal, however, is not to generate high R2 typical to predicting
long-horizon returns, but rather to provide a robust assessment of predictability in a way
that is least affected by statistical biases. In univariate regressions (Panel A), wedget is
a strong predictor of excess returns at short maturities. The negative coefficient is in line
with the positive coefficient for the FFR forecast error in regression (16), and with the fact
that the wedge picks up the countercyclical element of bond excess returns. The predictive
power is the strongest for returns on bonds with maturity below 12 months. It decays with
maturity and largely disappears for maturities above five years. The declining significance
agrees with the intuition that, in a world with a persistent but mean-reverting real short
rate, real-rate expectations should affect the short end of the yield curve more than the long
end.17 In this case, any frictions in real-rate expectations should manifest themselves most
visibly at short maturities.
Table VI Panel B contains bi-variate regressions (17). The predictability due to the expectations wedge is distinct from that achieved with the risk premium proxies, whose inclusion
does not materially affect the coefficient on the wedge. The Cochrane-Piazzesi factor and the
cycle factor predict excess returns at the longer maturities but they contain no information
about the independent variation in excess returns at the very short end of the yield curve.
Regressions for returns on zero-coupon bonds with an annual holding period (h = 1) are
provided in the online Appendix and confirm those conclusions.
IV.B.2. Ex-post predictability of identified monetary policy shocks
One concern with the interpretation of the previous results is the assumption that investors
form expectations in the same way as survey forecasters do. To assess the validity of this
17
See e.g. Fama (1990), Ireland (1996). This property is also confirmed by Cieslak and Povala (2015) using
their decomposition of the yield curve into trend inflation, real rate and term premium.
27
assumption, I focus on measures of monetary shocks extracted from fed fund futures at the
frequency of the FOMC meetings. By capturing updates to investors’ expectations about
the short rate within a narrow window surrounding an FOMC meeting, these shocks can be
treated as innovations relative to the time-t information set of investors. If the expectations
wedge contains information about the future short rate that is not in the time-t information
set of investors, it should also predict monetary policy shocks.
To verify this intuition, I consider monetary policy shocks from three studies: Kuttner (2001),
Gürkaynak, Sack, and Swanson (2005, GSS) and Campbell, Evans, Fisher, and Justiniano
(2012, CEFJ) as these studies differ in the range of futures’ maturities that they use, sample
period, and details of the identification strategy.18 Following GSS, shocks fall into two
categories—shocks to the current Fed’s interest rate target (target shocks) and shocks to
the future interest rate path (path shocks)—which provides a way to distinguish between
the effects of Fed actions versus the effects of Fed communication of their own expectations.
Kuttner (2001) captures the target shocks by focusing on the futures contract with the
shortest maturity, whereas GSS and CEFJ explicitly separate these two components by
including contracts with longer maturities. Summary statistics for these shocks are provided
in Table VII.
I analyze the predictability of monetary policy shocks by the expectations wedge. In Panel
A of Table VII, I project shocks observed in month t +
1
,
12
P
denoted εM
t+1/12 , on previous
month’s wedget . In Panel B, I report analogous regressions for cumulative shocks realized
P
MP
over the course of the following year, 12
i=1 εt+i/12 . The expectations wedge predicts 5.4% of
variation in next month Kuttner’s shocks, and 47% in the cumulative shocks. A one standard
deviation decline in the wedge predicts an easing shock of −29 basis points over the next year
(t-statistic 4.8). Similar estimates are obtained with GSS and CEFJ target shocks albeit the
economic magnitude is somewhat weaker (−18 and −25 basis points, respectively).
18
Thanks to Alejandro Justiniano and Eric Swanson for sharing their shock series. Kuttner’s shocks are
downloaded from Ken Kuttner’s website. Since FOMC meetings do not always take place at the end
of the month, I convert shocks to the monthly frequency following the approach of Romer and Romer
(2004). I first obtain daily cumulative shock series by adding shocks over time. I then average the daily
series within each month and difference the monthly series. The results are very similar if I just use the
shock in a given month as a monthly value and set the months that did not have an FOMC meeting to
zero.
28
Figure 5 superimposes the wedge in month t with the time series of cumulative Kuttner’s
1
shocks from month t + 12
though the end of year t + 1. The plot shows that the expectations
wedge widens ahead of some of the largest negative monetary policy shocks. Therefore, access
to the full sample of data gives the econometrician a significant information advantage over
real-time agents, especially around the turning points in monetary policy.
Two additional results suggest that the ex-post predictability of shocks is not a consequence
of the Fed’s information advantage compared to the private sector. First, contrary to
target shocks, the wedge does not predict path shocks (columns 4 and 6 of Table VII).
This is important because path shocks are news that the Fed provides about their expected
evolution of monetary policy. Second, the strong predictability of Kuttner’s target shocks
stems mostly from the unscheduled FOMC announcement days (compare columns 1 and 2 of
Table VII). Unscheduled monetary policy decisions (typically easings) happen in reaction to
economic events and thus reveal new information about the economy rather than represent
exogenous monetary policy shocks (Bernanke and Kuttner, 2005). For example, before 1994
unscheduled easings frequently coincided with a weaker than expected employment report,
and post 1994 they came after a significant turbulence in financial markets. Both of these
results suggest that the expectations wedge does not arise due to the Fed being able to
persistently surprise the market with exogenous monetary policy decisions. In fact, policy
makers themselves may exhibit persistent forecast errors about future economic conditions.
Narrative evidence provided below is consistent with this interpretation.
V. Short rate forecasts by the Fed
V.A. Narrative evidence on the nature of policy makers’ forecast errors
The challenges related to forecasting the business cycle and the associated path of interest
rates in real time are well-appreciated among monetary policy makers and Fed economists.
The former Chairman of the Federal Reserve Board, Alan Greenspan, explicitly recognizes
that the “success of monetary policy depends importantly on the quality of forecasting” and
later he concedes:
29
“As the transcripts of FOMC meetings attest, making monetary policy is an especially
humbling activity. In hindsight, the paths of inflation, real output, stock prices,
and exchange rates may have seemed preordained, but no such insight existed as we
experienced it at the time. In fact, uncertainty characterized virtually every meeting,
and as the transcripts show, our ability to anticipate was limited. From time to time
the FOMC made decisions, some to move and some not to move, that we came to
regret.” (Greenspan (2004), page 40)
Similar preoccupation with forecast accuracy is shown by the former Fed Chairman, Ben
Bernanke, who states:
“The accuracy of both central bank and private-sector forecasters has been extensively
studied and the results are not impressive. Unfortunately, beyond a quarter or two, the
course of the economy is extremely hard to forecast. That said, careful projections are
essential for coherent monetary policymaking, just as business plans and war strategies
are important in their spheres.” (Bernanke (2015), Kindle locations 904-907)
Transcripts of the FOMC meetings provide a real-time record of the difficulties that forecasters face. The FOMC members and staff economists at the Fed seem acutely aware of
the systematic nature of the forecast errors pertaining to various indicators of the economic
environment, and occurring at different stages of the business cycle. For example, in the
second half of the 1990s, the FOMC members struggled to reconcile the strong consumer
demand with the lack of inflationary pressures. While with hindsight it became evident that
the US economy had experienced a sequence of positive supply side shocks, in real time it
was much harder to realize that such development was taking place, let alone to anticipate
it (e.g., Greenspan, 2004; Meyer, 2009). The experience of those years is summarized by the
President of the Federal Reserve Bank of San Francisco Robert Parry in the transcript of
the October 5, 1999 meeting:
“(...) there is simply no way to determine the size or persistence of the current
supply shock. A review of forecast errors in recent years indicates that this shock
has consistently surprised us on the positive side.” (Transcript of October 5, 1999
FOMC meeting, page 22)
An analogous observation yet on the other side of the business cycle is made by the Fed’s
Governor Donald Kohn during the February 2, 2005 meeting:
“Finally, in making my forecast of real growth, I took account of my serial forecast
errors. I’ve been overpredicting growth since I got on the Committee [i.e. on Aug 5,
2002], so I used a sophisticated algorithm to compensate for this propensity: I decided
what I really wanted to forecast and I took a little off!” (Transcript of February 1-2,
2005 FOMC meeting, page 106)
30
Incidentally, Kohn’s 2005 remarks on the unwelcome consequences of confusing risk premia
with short-rate expectations, cited in the Introduction, comes just a few months after his
above realization.
The systematic pattern of forecast errors is a source of worry for Fed staff economists
producing so-called Greenbook forecasts, which serve as a point of departure for discussion
during each FOMC meeting. While forecast errors are frequently discussed at the meetings,
and despite realizing their systematic character, the ways to improve forecast performance
are not obvious in real time. Such doubts are reflected upon by the Fed economist, Michael
Prell, when summarizing the Greenbook outlook at the September 30, 1997 FOMC meeting:
“Going forward, I do not have the sense that our forecast implies an asymmetry of
risks in terms of aggregate demand being stronger or weaker. But it makes me nervous
that we keep making revisions in the same direction (...).” (Transcript of September
30, 1997 FOMC meeting, page 19)
More recently, direct evidence of the policy makers’ forecast errors comes from the FOMC’s
Summary of Economic Projections (SEP). Since 2012, the FOMC has been reporting the
projections of its members for the FFR target. Figure 6 displays the term structures of
weighted-average forecasts that FOMC members provided between 2012 and 2015, by each
vintage.19 The increasing pattern of the forecast term structures (with all forecasts situated
above the current upper bound for the FFR target range) suggests that FOMC members
have consistently anticipated the tightening to happen sooner than it actually did. Indeed,
only on December 16, 2015, for the first time since 2008, did the FOMC decide to increase the
FFR target to the 1/4–1/2 percent range. However, in January 2012, the FOMC members
on average predicted that the target would exceed 1.25% at the end of 2014, and 6 out of
17 members expected a policy tightening to take place by the end of 2013 or sooner. These
forecast were issued against the backdrop of the FOMC’s promise in August 2011 to keep the
interest rate unchanged at 0 to 1/4 percent at least through mid-2013. While there is some
19
The forecasts are available on the FRB website. The projections are usually made in March, June,
September and December. While individual projections are not disclosed, the FOMC reports the number
of its members that expect the target to be at a given level at the end of the current year, as well as
during the following two or three calendar years. Long-range forecasts are also available but are omitted
from Figure 6.
31
evidence of forecasts being adjusted downward in the course of the year, this adjustment has
been small and corrected only a fraction of the previous forecast errors.
V.B. Does the Fed have information advantage about the path of the short rate?
While FOMC’s projections of the FFR are available for a short sample when the ZLB was
binding, one may wonder whether Fed insiders were able to better predict the path of the
short rate compared to private sector during the period of my study. To answer this question,
I look into the expectations of the FFR target formed by the Fed staff economists. Before
each FOMC meeting, the staff prepares forecasts of the FFR target ranging from the current
quarter up to five quarters ahead, as part of the Greenbook. The Greenbooks are released
internally to FOMC meeting attendees a few days before the scheduled FOMC meeting
but become available to the public with a five-year lag.20 Greenbook forecasts have several
useful characteristics. The Fed staff has access to economic and confidential regulatory data
coming in from the regional Reserve Banks. The five-year publication lag, ability to observe
the current expectations of the market participants, and the Fed’s better understanding of
their own policy rule can lead to information asymmetries between the Fed and the private
sector (Romer and Romer, 2000).
Table VIII compares forecasts of the FFR by the Fed staff to the those from the BCFF
survey.21 Panel A reports regressions of the quarterly change in the FFR jointly on the
forecasts of the staff and the public. Staff forecasts drive out the private sector forecasts
at short horizons. However, their advantage weakens with the horizon. Staff forecasts are
generally more precise as measured by the RMSEs, but the relative precision of the public
20
The data are obtained from the Philadelphia Fed website and are available for the period 1981:01–2008:09.
As explained by Reifschneider, Stockton, and Wilcox (1997), Greenbook forecasts are judgemental. A
usual point of departure for the FFR forecast is the random walk assumption, unless this assumption is
“so at odds with the stated objective of most policymakers that such a projection would not serve as a
useful baseline for discussions” (page 8).
21
I merge the data so that a given Greenbook forecast is matched with the latest monthly BCFF survey
available to the staff at the time of their forecast. The staff predicts the FFR target while the BCFF
participants—the effective FFR. At the quarterly frequency this inconsistency is very minor. A regression
of quarterly changes in the effective FFR on the quarterly changes in the target has a slope coefficient of
0.99, intercept of less than half basis point and R2 = 0.95 (sample 1984:Q3–2011:Q3). For consistency
with previous results in Table VIII, I use the effective FFR as the variable being predicted.
32
increases with the horizon. At four quarters ahead, public forecasts contribute economically
and marginally statistically significant information about the future FFR changes.
Panel B of Table VIII displays projections of the staff FFR forecast errors on the expectations
wedge. The results are strikingly similar to those for the BCFF forecasts reported earlier.
This suggests that expectations frictions pertain to different groups of agents with likely
different access to information. The increasing alignment between expectations of the public
and the staff at longer horizons corroborates the earlier quote by Ben Bernanke that “beyond
a quarter or two the course of the economy is extremely hard to forecast.”
VI. Sources of short-rate forecast error predictability
Forecast errors can be predictable for different reasons. Agents may face frictions such
as noisy information as in Woodford (2003) or sticky information as in Mankiw and Reis
(2002). They may not know the parameters of the model driving the economy, and of the Fed
reaction function (Friedman, 1979). Alternatively, faced with complex underlying dynamics,
they may base their forecasts on simpler intuitive models that deviate from the truth in
a significant way but still imply a small utility loss (Cochrane, 1989; Fuster, Laibson, and
Mendel, 2010). Below, I show that under these scenarios an econometrician with an access
to full-sample information would find ex-post predictability of FFR forecast errors.
VI.A. Information rigidities
Models with rigidities such as sticky and noisy information assume that agents know the
structure and the parameters of the economy but the information they receive about the state
of the economy is imperfect. To test whether the predictive power of the expectations wedge
could be explained within such models, I use the approach of Coibion and Gorodnichenko
(2015). Coibion and Gorodnichenko (2015) show that sticky and noisy information models
can be tested by regressing the average (across agents) ex-post forecast error on the average
forecast revision. The baseline test has the form:
s
F Et (F F Rt,hQ ) = β0 + β1 Ets (F F Rt,hQ ) − Et−1/4
(F F Rt,hQ ) + εt,hQ ,
33
(18)
where the right-hand side variable is the update of FFR expectations between the last quarter
and the current quarter. The presence of information frictions implies that β1 > 0. The
estimates of equation (18) are reported in Table IX (columns 1, 3 and 5). The horizon
h ranges from one through three quarters ahead, the maximum horizon for which the
update can be constructed with the available survey data.22 I estimate the regressions
at the quarterly frequency using the middle month of a quarter. The coefficient on forecast
update is always positive and statistically significant. For example, in the context of a noisy
information model, the coefficient of 0.65 at h = 3 quarters implies that agents put a 60%
weight on new information and a 40% weight on their past forecasts.23 The update alone
explains between 7% and 12% of variation in the FFR forecast errors.
Models of information frictions imply that forecast updates should account for all predictable
variation in ex-post forecast errors. Therefore, I augment regression (18) with the expectations wedge to see whether its predictive power can be explained away by the forecast update.
The results of the extended test indicate that the expectations wedge has explanatory power
beyond forecast updates (columns 2, 4 and 6 of Table IX). For instance, at the threequarter horizon, wedget raises the R̄2 from 12% in the baseline test (18) to 35%, and is
highly statistically significant (t-statistic 4.83). The fact that the statistical and economic
significance of the forecast update changes little when the expectations wedge is included
suggests that these two variables capture different sources of forecast error predictability.
Therefore, information rigidities such as the ones posited by sticky and noisy information
models are an important, but not sole reason for the predictable variation in forecast errors
about the short rate.
VI.B. Parameter learning
Substantial empirical evidence points to a variation in the parameters of both the Fed’s
reaction function and of macroeconomic dynamics, with especially large changes documented
22
The estimates of (18) use the median forecast for consistency with the previous results. I verify that the
results are essentially identical when using the average forecast. The mean and median forecast errors
and updates are more than 0.99 correlated with each other at corresponding horizons.
23
In the noisy information model, the weight on new information is determined as G = 1/(1 + β1 ), where G
is the Kalman gain.
34
around the Volcker disinflation period between 1979 and 1983 (e.g., Primiceri, 2005; Boivin,
2006; Ang, Boivin, Dong, and Loo-Kung, 2011). This evidence suggests that in the early
1980s, while investors may have realized that a structural change had occurred, they had
few data points to form expectations of future economic conditions. In an environment
when uncertainty about the true data generating process is high and structural breaks are
present, Evans, Honkapohja, and Williams (2010) show that constant gain (CG) learning
is the maximally robust estimator. Thus, I quantify the effects of agents’ learning for the
ex-post predictability of forecast errors in the context of a standard monetary VAR where
agents use the CG learning rule to form their expectations. The VAR includes year-on-year
CPI inflation, unemployment rate and the fed funds rate as in Stock and Watson (2001). I
assume that agents do not know the parameters of the model. To form expectations of the
future short rate, they recursively estimate:
Y t = αt +
p
X
At Yt−i/4 + εt ,
(19)
i=1
where Yt = (∆CP It−1,t , UNEt , F F Rt )′ , p = 2.24 The data is sampled quarterly. Parameters
αt and At are estimated with the CG recursion (e.g., Branch and Evans, 2006), where agents
discount past observations at a rate (1 − γ), with γ being the gain parameter. The gain
parameter is calibrated to minimize RMSE for FFR forecast at a particular horizon; at four
quarters ahead γ = 0.012. I then ask whether the econometrician could predict forecast
errors from the VAR with lagged variables when estimating the following regression on the
full sample:
F EtCG (F F Rt,4Q ) = β0 + β1′ Yt + β2′ ∆Yt−1,t + ut,hQ ,
(20)
where ∆Yt−1,t is year-on-year change in Yt .
Table X Panel A presents RMSEs for forecasts from the model along with the estimates of
regression (20) at horizon of four quarters ahead. The RMSEs from the model are 9.5%
24
The second lag order p = 2 is selected using SBIC for the full sample 1954–2012 and individually for the
pre- and post-1984 subsamples.
35
higher than those of survey forecasts. Importantly, the econometrician could predict 33% of
the forecast error variation, and would find significant coefficients on lagged macro variables.
The true data generating process for the FFR is likely to be more complex than the above
VAR. To assess the effect of learning separately from model misspecification and from the role
of time-varying parameters, I conduct a Monte-Carlo experiment. I simulate 1000 artificial
data sets from a constant parameter VAR(2) model calibrated to historical data. In each
simulation, the agent learns parameters using the CG algorithm. Focusing on four-quarterahead forecasts, I set the gain to 0.012, as calibrated above. To avoid biases in starting values,
prior parameters are formed using 12 quarters of initial data in each simulation. Further,
as a simple way to assess the consequences of agents using an incorrect model, I compare
forecast errors under a correct model (VAR(2)) versus a misspecified model (VAR(1)).
Table X Panel B presents the distribution of statistics from regression (20) estimated on the
simulated data, where FFR forecasts are evaluated at the horizon of four quarters. In samples
of the size consistent with my empirical analysis (i.e. 109 quarters), an econometrician
with access to the full sample of data can predict on average about 40% of variation in
agents’ forecast errors, even if agents know the structure of the true model and correctly
estimate a VAR(2) specification. He or she would conclude 89% of times that at least one
of the regressors in (20) is statistically significant, and about 65% of times they would reject
the null hypothesis that all regressors are jointly zero at the 10% level. Not surprisingly,
forecast errors become more predictable if the agent estimates a misspecified model, VAR(1).
The econometrician’s advantage over the real-time forecaster is particularly visible in small
samples, and diminishes as the sample size increases. Even with 1000 quarterly observations,
forecast errors are ex-post predictable, although predictive R̄2 are half as large as in the
small sample.25 Due to an increased power of the test statistics, the fraction of cases with
statistically significant regression coefficients increases with the sample size.
25
In contrast to recursive least squares, with CG learning, the agent will not learn the true parameters
because they always put some positive weight on the surprise as the new data become available. However,
the estimates converge to a stationary distribution. CG learning is a close approximation to an optimal
learning rule when parameters change over time (Evans and Honkapohja, 2009).
36
These results suggest that agents’ learning about parameters could account for a nontrivial
part of the bond return predictability presented earlier. Importantly, by focusing only on the
FFR dynamics, the VAR specification abstracts from any variation in the term premium.
In a related way, the consequences of learning about the true data generating process of
fundamentals have been emphasized by Timmermann (1993) in the context of predictability
of equity returns.
VII. Conclusions
This paper studies how agents form expectations about the short-term interest rate in real
time, comparing them to expectations of an econometrician who has the benefit of hindsight
and access to historical data spanning several business cycles. I document a significant wedge
between the real-time and full-sample estimates of the expected real short rate. Investors
tend to systematically overestimate the level of the real rate when the economy enters into a
recession. This empirical regularity leads to a predictable pattern in realized excess returns
on short-maturity bonds that is independent of the variation in the bond risk premium.
Evidence suggests that it is challenging to forecast economic conditions beyond a quarter or
two. To the extent that the difficulties faced by real-time forecasters have direct implications
for the expectations of the real interest rates, they also affect economic decisions of firms
and households. As such, my results complement the conclusions of Greenwood and Hanson
(2015) who show that expectations errors can have real effects by generating excess volatility
in capital investment and prices. A further study of real-side information frictions and their
consequences for economic outcomes is a promising area of research.
37
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41
Table I: Predictive regressions of bond returns with auxiliary variables
The table presents regressions of annual bond excess returns on three yield PCs (P Ctyld ) and an auxiliary regressor Zt . The
regression is specified as
(n)
rxt+1 /n = γ0 + γ1′ P Ctyld + γ2 Zt + εt+1 ,
where the dependent variable is excess return for an annual holding period on a n-year bond. For easy comparison of coefficients
(n)
across maturities, excess returns are duration standardized, rxt+1 /n as return volatility scales proportionally with bond
duration. rxS
in
column
(1)
is
the
short-maturity
factor
defined
in Section I (also duration standardized). Panel A uses
t+1
only PCs as regressors. The row “St.dev. of rc
x(%)” reports the unconditional standard deviation of the fitted value from the
regression. Panels B–F show results for different variables Zt , and only γ2 coefficient is reported. All explanatory variables
are standardized to have a zero mean and unit standard deviation. Regressions are estimated at a monthly frequency, so each
month I forecast excess returns earned over the following 12 months. The data covers the period 1984:6–2012:8 for a total of
327 months (first annual return is realized in June 1985 and last in August 2012). T-statistics are reported for two types of
standard errors: Newey-West adjustment with 18 lags (row “t-stat[NW]”) and Hodrick reserve regression (row “t-stat[H]”).
(1)
(2)
(3)
(4)
(5)
rxS
rx(2)
rx(5)
rx(10)
rx(20)
A. Only yield PCs: rxt+1 = γ0 + γ1′ P Ctyld + εt+1
St.dev. of rc
x (%)
R̄2
0.18
0.08
0.28
0.14
0.38
0.14
0.45
0.23
0.45
0.26
-0.28
(-4.84)
[-3.90]
0.27
-0.25
(-3.37)
[-3.06]
0.25
-0.21
(-2.06)
[-1.74]
0.18
-0.07
(-0.80)
[-0.62]
0.23
0.05
( 0.80)
[ 0.45]
0.26
0.34
( 2.30)
[ 1.78]
0.23
0.12
( 0.93)
[ 0.78]
0.24
-0.07
(-0.73)
[-0.31]
0.27
-0.38
(-3.33)
[-2.05]
0.26
-0.21
(-1.98)
[-1.30]
0.27
-0.10
(-1.11)
[-0.81]
0.27
B. CFNAI, Zt = CF N AIt
CF N AIt
t-stat[NW]
t-stat[H]
R̄2
C. Year-on-year growth in unemployment, Zt = ∆U N Et−1,t
∆U N Et−1,t
t-stat[NW]
t-stat[H]
R̄2
0.38
( 4.04)
[ 3.38]
0.38
0.35
( 3.14)
[ 2.73]
0.32
D. One-year lagged term spread, Zt = St−1
St−1
t-stat[NW]
t-stat[H]
R̄2
-0.29
(-4.40)
[-3.68]
0.27
-0.34
(-4.57)
[-2.70]
0.32
E. Four-quarter-ahead inflation forecast from BCFF, Zt = Ets (∆CP It,4Q )
Ets (∆CP It,4Q )
t-stat[NW]
t-stat[H]
R̄2
-0.35
(-1.24)
[-1.11]
0.12
-0.79
(-2.86)
[-2.02]
0.27
-1.07
(-3.16)
[-2.05]
0.27
-0.98
(-3.06)
[-1.75]
0.35
-0.94
(-2.86)
[-1.57]
0.39
F. Trend inflation, Zt = τtCP I
τtCP I
t-stat[NW]
t-stat[H]
R̄2
-0.37
(-1.57)
[-1.12]
0.11
-1.05
(-3.95)
[-2.16]
0.34
-1.61
(-4.98)
[-2.49]
0.40
-1.64
(-6.31)
[-2.84]
0.52
-1.48
(-7.51)
[-3.24]
0.53
N (months)
327
327
327
327
327
42
Table II: Decomposing predictable variation of realized excess return on a
two-year Treasury bond
(2)
The table decomposes predictability of the realized excess return on the two-year bond, rxt+1 , into component stemming from
predictability of unexpected returns (forecast errors) and from the risk premium variation. The survey-based risk premium is:
(1)
(1)
(1)
s,(2)
(1)
(2)
(2)
Ets (rxt+1 ) = ft − Ets (yt,4Q ), and the corresponding unexpected return is U rxt+1 = −[yt+1 − Ets (yt,4Q )], where Ets (yt,4Q )
is the four-quarter-ahead forecast of one-year yield from the BCFF survey; F Et (F F Rt,4Q ) = F F Rt,4Q − Ets (F F Rt,4Q ) is
four-quarter ahead forecast error for the FFR. Panels A and B separately present results for variables that predict and that do
not predict forecast errors. Sample period is 1987:12–2012:8 for a total of 285 months. The beginning of the sample is dictated
by the availability of survey forecasts for the one-year yield. The t-statistics in parentheses use Newey-West adjustment with
18 lags. Right-hand side variables are standardized to have a zero-mean and unit standard deviation.
(1)
(2)
(3)
(4)
(5)
(6)
(2)
rxt+1
(2)
Ets (rxt+1 )
s,(2)
U rxt+1
-F Et (F F Rt,4Q )
(1)
Ets (yt,4Q )
Ets (F F Rt,4Q )
A. Variables that predict forecast errors
CFNAIt
R̄2
∆U N E t−1,t
R̄2
St−1
R̄2
-0.37
(-2.13)
0.08
0.11
( 1.92)
0.04
-0.48
(-2.92)
0.13
-0.52
(-2.42)
0.16
0.53
( 2.38)
0.07
0.56
( 2.16)
0.07
0.50
( 2.03)
0.15
-0.02
(-0.28)
0.00
0.51
( 2.30)
0.15
0.48
( 1.88)
0.14
-0.82
(-3.78)
0.19
-0.94
(-3.99)
0.20
-0.58
(-3.44)
0.20
-0.04
(-0.41)
0.00
-0.54
(-2.60)
0.17
-0.55
(-2.91)
0.18
-0.53
(-1.51)
0.08
-0.66
(-1.80)
0.10
B. Variables that do not predict forecast errors
Expected inflation proxies
Ets (∆CP I t,4Q )
R̄2
τtCP I
R̄2
0.11
( 0.37)
0.00
0.37
( 4.69)
0.31
-0.26
(-0.98)
0.02
-0.22
(-0.67)
0.01
1.87
( 7.36)
0.63
1.99
( 6.66)
0.59
0.10
( 0.27)
0.00
0.50
( 6.66)
0.46
-0.41
(-1.20)
0.05
-0.27
(-0.69)
0.02
2.04
( 5.88)
0.59
2.16
( 5.26)
0.55
0.61
( 3.02)
0.15
0.34
( 5.69)
0.29
0.27
( 1.23)
0.03
0.18
( 0.83)
0.01
0.22
( 0.83)
0.01
0.07
( 0.25)
0.00
0.23
( 1.04)
0.03
0.21
( 3.33)
0.15
0.02
( 0.11)
0.00
-0.05
(-0.20)
0.00
-0.29
(-1.02)
0.02
-0.47
(-1.48)
0.05
285
285
285
285
285
285
Risk premium proxies
c
cf
R̄2
CP t
R̄2
N (months)
43
Table III: Properties of FFR forecasts
The table summarizes statistical properties of FFR forecasts in the BCFF survey. Panel A reports summary statistics for
forecast errors, as well as average forecast errors across months in which the Fed has tightened or eased monetary policy. In
cases where there has been more than one monetary policy change within a month, I use the last move in that month. Not all
months have had a monetary policy change. Panel B reports projections of the realized FFR change on the expected change.
The results are presented both at the monthly frequency (i.e. frequency at which the BCFF survey is conducted) and at the
quarterly frequency (given that forecasters predict average FFR value in a quarter). The quarterly sampling uses forecasts for
the middle month in a quarter. Row “p-val (β1 = 1)” reports the p-value for the test that β1 = 1. T-statistics use Newey-West
adjustment with 18 lags for monthly data and 6 lags for quarterly data. Panel C compares the out-of-sample RMSEs from
univariate models (random walk, and AR(1) and AR(2)) with those from the survey forecasts. The models are estimated at
the quarterly frequency. The burn-in period for AR specifications is selected to minimize the sum of RMSEs across horizons
(in both cases 120 quarters), and the order of the AR(2) model is determined by SBIC on the post-1984 sample. Out-of-sample
forecasts start in 1984:Q3 with the last forecast made in 2011:Q3.
h = 1Q
h = 2Q
h = 3Q
h = 4Q
A. Summary statistics for FFR forecast errors F Et (F F Rt,hQ ) = F F Rt,hQ − Et (F F Rt,hQ )
Mean
t-stat
Std. Dev.
Min
Max
N (months)
Easing
Mean
N (months)
Tightening
Mean
N (months)
-0.12
(-2.26)
0.46
-2.64
1.05
327
-0.29
(-2.49)
0.80
-3.82
1.38
327
-0.46
(-2.64)
1.10
-4.48
2.05
327
-0.64
(-2.75)
1.37
-4.40
2.55
327
-0.28
(-3.14)
58
-0.68
(-3.83)
58
-1.01
(-3.94)
58
-1.29
(-3.91)
58
-0.0074
(-0.07)
46
0.10
(0.54)
46
0.14
(0.59)
46
0.16
(0.56)
46
B. Predicting FFR changes ∆F F Rt,hQ = β0 + β1 Et (∆F F Rt,h ) + ut,hQ
Monthly sampling
β1
p-val (β1 = 1)
β0
R̄2
N (months)
Quarterly sampling
β1
p-val (β1 = 1)
β0
R̄2
N (quarters)
0.70
(4.70)
[0.043]
-0.12
(-2.08)
0.25
327
0.93
(3.45)
[0.79]
-0.28
(-2.48)
0.22
327
1.06
(3.61)
[0.83]
-0.47
(-2.64)
0.22
327
1.08
(3.76)
[0.77]
-0.66
(-2.62)
0.20
327
1.03
(5.62)
[0.87]
-0.10
(-2.06)
0.27
109
1.18
(4.71)
[0.46]
-0.26
(-2.78)
0.27
109
1.26
(4.28)
[0.38]
-0.46
(-3.28)
0.25
109
1.23
(4.48)
[0.40]
-0.65
(-3.35)
0.22
109
1.15
1.27
1.26
1.23
109
1.48
1.57
1.55
1.54
109
C. RMSEs
BCFF survey
RW
AR(1)
AR(2)
N (quarters)
0.45
0.53
0.53
0.48
109
0.81
0.92
0.93
0.87
109
44
Table IV: Forecast errors of FRR and of macro variables
Panel A presents regressions of forecast errors of FFR and macro variables (unemployment and inflation) on the time t forecasts.
c
Panel B projects forecast errors on realized macro variables. ∆CP It−1,t is the annual all-items inflation, ∆CP It,t−1
is the
annual core inflation; they have a correlation of 0.64. Explanatory variables are standardized. Panel C estimates regressions of
FFR forecast errors on inflation and unemployment forecast errors. Columns labeled “2SLS” are estimated with two-stage least
squares using explanatory variables from Panel B as instruments. The regressions are estimated at the quarterly frequency due
to the availability of unemployment forecasts. T-statistics are Newey-West adjusted with 6 lags. All forecast errors are for the
horizon of four quarters.
A. Regressions of forecast errors on survey forecasts
F Et (Xt,4Q ), X :
FFR
Ets (F F Rt,4Q )
UNE
∆CP I
∆CP I c
-0.40
(-1.91)
1.02
(1.60)
0.10
109
0.0077
(0.07)
-0.29
(-0.96)
-0.0091
109
-0.080
(-0.91)
Ets (U N Et,4Q )
-0.042
(-0.62)
Ets (CP It,4Q )
Const.
-0.20
(-0.48)
0.012
109
R̄2
N (quarters)
0.30
(0.62)
-0.0052
109
B. Regressions of forecast errors on realized macro variables
F Et (Xt,4Q ), X :
F F Rt
U N Et
∆CP It−1,t
c
∆CP It−1,t
∆U N Et−1,t
Const.
R̄2
N (quarters)
FFR
UNE
∆CP I
∆CP I c
-0.90
(-1.88)
-0.17
(-0.75)
-0.58
(-3.28)
0.71
(1.58)
-0.83
(-3.40)
-0.59
(-2.86)
0.23
109
0.056
(0.33)
-0.15
(-1.45)
0.57
(3.26)
-0.28
(-1.51)
0.63
(4.65)
0.050
(0.52)
0.48
109
-0.22
(-0.54)
-0.25
(-1.05)
-0.31
(-0.99)
-0.092
(-0.29)
-0.26
(-1.30)
-0.22
(-1.26)
0.16
109
0.013
(0.07)
-0.10
(-0.92)
0.098
(0.94)
-0.027
(-0.19)
-0.033
(-0.44)
-0.27
(-3.95)
0.099
109
C. Regressions of FFR forecast errors on macro forecast errors
Dependent variable: F Et (F F Rt,4Q )
OLS
F Et (U N Et,4Q )
F Et (∆CP It,4Q )
2SLS
-1.00
(-3.29)
0.15
(0.83)
-1.07
(-3.28)
c
F Et (∆CP It,4Q
)
Const.
R̄2
N (quarters)
0.33
(0.93)
-0.45
(-2.65)
0.45
109
-0.50
(-3.28)
0.46
109
45
-0.70
(-1.65)
0.59
(1.72)
-0.42
(-2.16)
0.32
109
-0.94
(-2.93)
0.077
(0.11)
-0.52
(-1.78)
0.44
109
Table V: The expectations wedge
Panel A reports the projections of ex-post real rate on a set of time-t instruments that are listed in the rows of the table.
The ex-post real rate is defined in equation (11). Right-hand side variables are standardized to have unit variance and zero
mean. Panel B shows the predictability of four-quarter-ahead forecast error with wedget , where r̂te,F S corresponds to columns
in Panel A. Row “CI 5%” reports the 5% confidence interval based on a bootstrapped distribution of the coefficient to account
for wedget being pre-estimated. Newey-West t-statistics with 18 lags are reported in parentheses.
(1)
(2)
(3)
(4)
(5)
c
A. r̂te,F S = Proj(rt+1 |Instrumentst ), where rt+1 = F F Rt+1 − ∆CP It,t+1
F F Rt
1.28
( 6.20)
1.84
( 5.63)
-0.69
(-1.70)
1.21
( 2.64)
-0.28
(-0.68)
-0.65
(-2.02)
1.06
( 2.59)
-0.17
(-0.38)
-0.64
(-1.93)
-0.15
(-0.53)
0.45
327
0.49
327
0.57
327
0.57
327
c
∆CP It−1,t
∆U N Et−1,t
U N Et
St−1
R̄2
N (months)
1.39
( 3.56)
-0.31
(-0.82)
-0.20
(-0.60)
-0.58
(-2.26)
0.71
( 2.62)
0.63
327
B. F Et (F F Rt,4Q ) = α + β wedget + εt+1 , where wedget = r̂te,F S − rte,RT
wedget
CI 5%
R̄2
0.43
( 1.49)
[-0.09; 1.16]
0.04
0.72
( 2.38)
[0.21; 1.36]
0.09
0.90
( 3.26)
[0.59; 1.14]
0.25
0.97
( 3.77)
[0.65; 1.26]
0.27
0.92
( 5.39)
[0.73; 1.17]
0.33
C. F Et (F F Rt,hQ ) = α + β wedget + εt,4Q , where wedget is based on column (5) in Panel A
wedget
CI 5%
R̄2
N (months)
h = 1Q
h = 2Q
h = 3Q
h = 4Q
0.20
( 2.87)
[0.09; 0.36]
0.14
327
0.43
( 3.82)
[0.28; 0.64]
0.21
327
0.67
( 4.79)
[0.50; 0.93]
0.28
327
0.92
( 5.39)
[0.73; 1.17]
0.33
327
46
Table VI: Forecasting monthly excess returns on CRSP bond portfolios with
the expectations wedge
The table presents predictive regressions of realized excess returns on CRSP bond portfolios. rx(<Xm) denotes one-month
excess return on a portfolio of bonds with less than X months to maturity. Returns are in excess of the one-month Tbill rate
(also from CRSP). Panel A reports univariate regressions of excess returns on the expectations wedge. To account for the
generated regressor, “CI 5%” provides the 5% confidence interval obtained from the bootstrap distribution of the coefficient on
wedget . Panel B shows predictability of excess returns on selected portfolios by wedget and measures of the bond risk premium,
c , and Cochrane-Piazzesi factor, CPt . For ease of comparison, both left- and right-hand variables are
RPt : the cycle factor, cf
t
standardized. The data is monthly and covers the period 1984:6–2011:9 (i.e. last observation of the predictor is 2011:8; last
observation of the excess return in 2011:9). T-statistics in parentheses use Newey-West standard errors with 18 lags.
(<Xm)
A. With information wedge: rxt+1/12 = α + β1 wedget + εt+1/12
wedget
CI5%
R̄2
N (months)
rx(<12m)
rx(<24m)
rx(<36m)
rx(<48m)
rx(<60m)
rx(<120m)
rx(>120m)
-0.26
(-3.73)
[-0.44;-0.12]
0.07
327
-0.22
(-3.58)
[-0.41;-0.13]
0.05
327
-0.18
(-3.26)
[-0.36;-0.11]
0.03
327
-0.15
(-2.86)
[-0.31;-0.08]
0.02
327
-0.13
(-2.56)
[-0.29;-0.06]
0.01
327
-0.10
(-1.97)
[-0.28;-0.03]
0.01
327
-0.04
(-0.92)
[-0.19; 0.02]
0.00
327
(<Xm)
B. With controls for risk premium: rxt+1/12 = α + β1 wedget + β2 RPt + εt+1/12
wedget
RP t
R̄2
N (months)
Cycle factor
c
RPt = cf
t
Cochrane-Piazzesi factor
RPt = CPt
rx(<12m)
rx(<24m)
rx(<60m)
rx(>120m)
rx(<12m)
rx(<24m)
rx(<60m)
rx(>120m)
-0.23
(-3.46)
0.12
( 1.58)
0.08
327
-0.17
(-2.82)
0.20
( 2.95)
0.08
327
-0.07
(-1.15)
0.22
( 4.20)
0.06
327
0.03
( 0.51)
0.24
( 4.61)
0.05
327
-0.25
(-3.93)
0.13
( 1.55)
0.08
327
-0.20
(-3.53)
0.16
( 2.21)
0.07
327
-0.11
(-2.07)
0.16
( 2.58)
0.04
327
-0.02
(-0.46)
0.16
( 2.77)
0.02
327
47
Table VII: Predicting monetary policy shocks with the expectations wedge
The table reports the predictability of monetary policy shocks by the expectations wedge, wedget . Monetary policy shocks in
column (1) and (2) are from Ken Kuttner’s website (Kuttner (2001)), in column (3) and (4) from Gurkaynak, Sack and Swanson
(2005, GSS), and in column (5) and (6) from Campbell, Evans, Fisher and Justiniano (2012, CEFJ). Shocks are identified from
fed funds futures at the FOMC meeting frequency and converted into monthly frequency. Panel A reports predictability of
monthly shocks realized in month t+1/12, Panel B reports the predictability of shocks accumulated over the following year from
t + 1/12 to t + 1. T-statistics in parentheses are Newey-West adjusted with 18 lags. The wedget variable is standardized to have
a mean of zero and a standard deviation of one. Panel C reports the summary statistics for each monthly (i.e. non-cumulative)
shock in basis points.
(1)
Kuttner (all)
(2)
Kuttner (sched)
(3)
GSS (target)
(4)
GSS (path)
(5)
CEFJ (target)
(6)
CEFJ (path)
0.133
(0.21)
-0.0347
(-0.05)
179
-0.005
1.418
(2.98)
-0.0356
(-0.07)
227
0.042
-0.0336
(-0.09)
-0.0241
(-0.06)
227
-0.004
24.47
(4.84)
2.067
(0.44)
206
0.448
2.087
(0.58)
-1.566
(-0.40)
206
0.004
P
A. Monthly shocks: εM
= α + βwedget + ut+1/12
t+1/12
wedget
Const.
N
R̄2
1.904
(3.23)
-2.422
(-3.92)
228
0.054
B. Cumulative 12-month shocks:
wedget
Const.
N
Adjusted R2
28.68
(4.77)
-26.99
(-4.87)
216
0.472
0.617
(2.91)
-0.501
(-2.11)
228
0.017
P12
MP
i=1 εt+i/12
5.985
(2.83)
-6.021
(-2.51)
216
0.179
1.010
(2.62)
0.0814
(0.17)
179
0.018
= α + βwedget + ūt+1/12,t+1
17.76
(4.41)
1.730
(0.41)
167
0.411
5.125
(1.25)
-0.00723
(-0.00)
167
0.020
C. Summary statistics for monetary policy shocks (non-cumulative, in basis points)
wedget : Mean = −35.8; St.dev. = 86.7; N = 327; Sample: 1984:6–2011:8
Sample
Mean
Std.dev.
1989:6–2008:6
-2.54
7.89
-0.54
4.28
1990:2-2004:12
0.05
7.02
48
-0.04
10.04
1990:2-2004:12 & 2007:8-2011:12
-0.05
6.88
-0.02
7.10
Table VIII: Fed’s staff expectations of FFR
This table tests whether Fed’s staff is better able to forecast the FFR than the public in the BCFF survey. The staff forecasts
are from the Greenbook, and are available at the frequency of scheduled FOMC meetings (8 observations per year). Superscripts
“F” and “P” refer to the Fed staff and public forecasts, respectively. To align the BCFF forecasts (monthly frequency) with
the Greenbook frequency, I use the last available BCFF forecast prior to each FOMC meeting. Panel A regresses the change in
the FFR from the current quarter (in which the meeting is held) to h quarters ahead, ∆F F Rt,hQ , on the change expected by
the Fed staff and the public. Row “corr(F E F , F E P )” is the unconditional correlation between the Greenbook and the BCFF
forecast error at the corresponding horizon. Row “RMSE ratio F/P ” shows the ratio of the RMSEs for the forecasts of the
Fed staff relative to the public. A number less than one indicates a smaller RMSE of the Fed forecast. Panel B displays the
regressions of Fed’s forecast errors on the expectations wedge. The sample period is 1984:7–2008:9; the end of the sample is
when the last Greenbook forecast is available. T-statistics are Newey-West adjusted with 12 lags.
h = 1Q
h = 2Q
h = 3Q
F
P
A. ∆F F Rt,4Q = α + γF Et ∆F F Rt,4Q + γP Et ∆F F Rt,4Q + εt,hQ
EtF ∆F F Rt,4Q
0.95
0.86
0.85
(5.54)
(4.60)
(4.60)
EtP ∆F F Rt,4Q
-0.14
0.25
0.53
(-0.57)
(0.67)
(1.43)
Const.
-0.14
-0.34
-0.57
(-1.99)
(-2.40)
(-2.67)
R̄2
0.21
0.20
0.21
N (Greenbooks)
195
195
195
RMSE ratio F/P
corr(F E F , F E P )
0.85
0.86
h = 4Q
0.94
(4.49)
0.65
(2.01)
-0.79
(-2.77)
0.21
195
0.93
0.87
0.96
0.88
0.97
0.89
0.49
(4.42)
-0.13
(-1.22)
0.23
195
0.78
(5.08)
-0.17
(-1.06)
0.31
195
1.06
(5.79)
-0.19
(-0.90)
0.37
195
B. F EtF (F F Rt,hQ ) = α + β wedget + εt,hQ
wedget
0.24
(3.68)
-0.051
(-1.09)
0.17
195
Const.
R̄2
N (Greenbooks)
Table IX: Tests of sticky and noisy information models
The table tests information frictions following Coibion and Gorodnichenko (2015) according to equation:
h
i
s
F Et (F F Rt,hQ ) = β0 + β1 Ets (F F Rt,hQ ) − Et−1/4
(F F Rt,hQ ) + β2 wedget + εt,hQ .
(21)
The row “updatet ” refers to the first regressor. The data is sampled at the quarterly frequency using the middle month of each
quarter. T-statistics are Newey-West adjusted with 6 lags.
(1)
(2)
(3)
h = 1Q
updatet
0.23
(2.92)
wedget
Const.
R̄2
N (quarters)
-0.064
(-1.29)
0.07
109
(4)
(5)
h = 2Q
0.16
(2.25)
0.18
(3.88)
-0.011
(-0.26)
0.19
109
0.40
(2.92)
-0.17
(-1.61)
0.09
109
49
(6)
h = 3Q
0.31
(2.61)
0.37
(4.25)
-0.056
(-0.66)
0.25
109
0.65
(3.19)
-0.29
(-1.84)
0.12
109
0.56
(3.42)
0.60
(4.83)
-0.10
(-0.85)
0.35
109
Table X: Forecast errors in CG-VAR
Panel A reports estimates of regression (20) for the four-quarter-ahead FFR forecast error. FFR forecast error is obtained
from a monetary CG-VAR(2) model with inflation, unemployment and FFR, Yt = (∆CP It−1,t , U N Et , F F Rt ). The frequency
of data is quarterly. The burnin period for the VAR is 40 quarters to obtain prior parameters starting from 1954:Q3 when
effective FFR data become available. The gain parameter (γ = 0.012) in the CG recursion is selected to minimize the RMSE
of FFR forecast errors over the sample 1985:Q3–2012:Q3 for best comparison with survey forecasts. Forecast errors from CGVAR are then projected on Yt and on year-on-year change in Yt , ∆Yt−1,t , over this period. Panel B presents a Monte Carlo
simulation to study the properties of regression (20). The simulated model is a constant-parameter VAR(2) calibrated to the
historical estimates over the 1954:Q3–2012:Q3 sample. Agents use CG recursive least squares to estimate the model parameters
in real time with the same γ as above; they obtain prior parameters using 12 quarters of initial data in each simulation. Rows
“VAR(2)” are for the case when the agent estimates the correct statistical model; row “VAR(1)” are for the case when the agent
estimates VAR(1) while the true dynamics is VAR(2). The table reports the distribution of predictive R̄2 from regression (20)
for four-quarter-ahead forecast error. Column “Joint at 10%” shows the frequency of a rejection at the 10% level of the null
hypothesis that all coefficients in the regression are jointly insignificant (excluding constant). Column “Indiv. |t| ≥ 2” gives
the frequency at which at least one regressor in (20) has a t-statistics above 2 in absolute value. Simulations are based on 1000
repetitions. T-statistics in both panels are Newey-West adjusted with 6 lags.
A. Predicting four-quarter FFR forecast errors from the CG-VAR model
Yt
∆CP It−1,t
-0.60
(-2.57)
-0.10
(-0.64)
0.21
( 2.14)
1.21
( 1.05)
U N Et
F F Rt
Const.
R̄2 = 0.33;
∆Yt
∆2 CP I
t−1,t
∆U N Et−1,t
∆F F Rt−1,t
-0.09
(-0.50)
-0.54
(-2.13)
0.13
( 1.19)
RMSE = 1.62; N = 109 quarters
B. Simulation
Distribution of R̄2
Mean
Significance
Std
p5
p50
p95
Joint
at 10%
Indiv.
|t| ≥ 2
0.22
0.21
0.08
0.13
0.41
0.48
0.76
0.81
0.65
0.64
0.89
0.91
0.14
0.11
0.02
0.11
0.19
0.25
0.44
0.47
0.81
0.78
0.91
1.00
Small sample, 109 quarters
VAR(2)
VAR(1)
0.41
0.47
Large sample, 1000 quarters
VAR(2)
VAR(1)
0.20
0.27
50
(2)
A. Survey-based decomposition of rxt+1
4
%p.a.
2
0
-2
(2)
Realized return, rxt+1
(2)
Survey-based expected return, Ets (rxt+1 )
s,(2)
Survey-based unexpected return, U rxt+1
-4
1985
1990
1995
2000
2005
2010
2015
B. Survey-based vs. estimated risk premium
3
%p.a.
2
1
0
-1
-2
1985
(2)
Survey-based expected return, Ets (rxt+1 )
ct
Cycle factor, cf
1990
1995
2000
2005
2010
2015
Figure 1: Decomposing realized excess return on the two-year Treasury bond
(2)
(2)
Panel A shows the decomposition of the realized excess return on a two-year bond, rxt+1 = Ets (rxt+1 ) +
s,(2)
(2)
(2)
(1)
s,(2)
(1)
(1)
(1)
U rxt+1 , where Ets (rxt+1 ) = ft − Ets (yt,4Q ) and U rxt+1 = −[yt+1 − Ets (yt,4Q )], and Ets (yt,4Q ) is the
four-quarter-ahead forecast of the one-year yield from the BCFF survey. Panel B plots the survey-based
excess return against the cycle factor.
51
A. Time series of FFR forecasts
0Q
1Q
2Q
3Q
4Q
%p.a.
10
5
0
1985
1990
1995
2000
2005
2010
B. Conditional term structures of FFR forecasts
%p.a.
10
5
0
1985
1990
1995
2000
2005
2010
C. Time series of FFR forecast errors
%p.a.
2
0
1Q
2Q
3Q
4Q
-2
-4
1985
1990
1995
2000
2005
2010
Figure 2: Properties of FFR forecasts in the BCFF survey
Panel A plots the time series of FFR forecasts from the BCFF survey. The forecasts are for the current
quarter up to four quarters ahead. Panel B plots the term structures of FFR forecasts. For clarity, while the
forecasts are given monthly, the plot shows those made in the middle of each quarter, i.e. February, May,
August and November. Panel C displays the time series of forecast errors for horizons from one through four
quarters ahead. The shaded areas are NBER-dated recessions. The timing is such that the forecast error
realized at time t + hQ, F Et (F F Rt,hQ ), is plotted at time t in the figure.
52
4
4
A. Forecast errors of FFR and of unemployment
FE(FFR)
−1
−4
0
−2
FEt(FFRt,4Q)
0
1
2
FEt(UNEt,4Q)
2
3
FE(UNE)
1995
2000
2005
date
B. Forecast errors of FFR and of inflation
2010
4
1990
4
1985
FE(FFR)
FE(CPI) total
−4
−4
−2
−2
0
FEt(CPIt,4Q)
FEt(FFRt,4Q)
0
2
2
FE(CPI) core
1985
1990
1995
2000
2005
2010
Figure 3: Forecast errors of FFR, unemployment and inflation
The figure superimposes forecast errors of FFR against forecast errors of unemployment (panel A), and
against forecast errors of inflation (panel B). Forecast errors are measured at horizon of four quarters ahead
and are expressed in percent per annum.
53
A. Expected real FFR
r̂te,F S
rte,RT
% p.a.
4
2
0
-2
1985
1990
1995
2000
2005
2010
B. Expectations wedge: r̂te,F S − rte,RT
% p.a.
1
0
-1
-2
1985
1990
1995
2000
2005
2010
Figure 4: Real-rate expectations wedge
Panel A plots two estimates of the expected real FFR: r̂te,F S is obtained using full-sample projections in
equation (13) and the full set of instruments (corresponding to column (5) in Table V); rte,RT is constructed
using survey forecasts in equation (14). Panel B shows the real-rate expectations wedge defined as the
difference between the two estimates: wedget = r̂te,F S − rte,RT .
54
200
−200
basis points
−100
0
100
Cumulative Kuttner shock, t+1/12 to t+1
wedge, t
1985
1990
1995
2000
2005
2010
Figure 5: Expectations wedge vs. cumulative monetary policy shocks
The figure superimposes rolling sum of Kuttner’s surprises over 12 months (from t + 1/12 to t + 1) against
the expectations wedge in month t.
(Expected) FFR target (%p.a.)
4
SEP
SEP
SEP
SEP
3
2012
2013
2014
2015
2
1
0
2012
2013
2014
2015
2016
2017
2018
2019
Figure 6: FOMC members’ forecasts of the FFR target
The figure shows the weighted average of the FFR target forecasts by the FOMC members. The forecasts
are released by the Federal Reserve Board as part of the Summary of Economic Projections. Forecasts are
given for the current calendar year and two or three following years. In the figure, they are reported as of
December 31 of a particular calendar year.
55