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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. 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(2004): “Estimating the Elasticity of Intertemporal Substitution when Instruments are Weak,” Review of Economics and Statistics, 86, 797–810. 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