University:
Boston University
Course:
CAS EC 203 | Empirical Economics 1
Academic year:

2020
Views:

90

Pages:

23
Author:

Jameson Hunt

Appendix A. Details of the arbitrageurs’ portfolio optimization problem. Let Pnt denote the price of a pure-discount n-period bond (with P0t = 1), Wt the total wealth of the arbitrageurs, and znt the portion of their wealth allocated to each bond maturity. Then the arbitrageurs’ wealth evolves according to Wt+1 = N X n=1 znt Pn−1,t+1 Wt Pnt with associated rate of return rt,t+1 N Wt+1 − Wt X Pn−1,t+1 znt = = −1 . Wt Pnt n=1 If the change in prices between t and t + 1 is small,22 the portfolio’s mean return and variance can be approximated Et rt,t+1 ≈ −z1t (a1 + 0 b1 f t ) + N X h i 0 0 0 znt an−1 + bn−1 (c + ρft ) + (1/2)bn−1 ΣΣ0 bn−1 − an − bn ft n=2 (36) Vart (rt,t+1 ) ≈ d0t ΣΣ0 dt (37) where the (J × 1) vector dt summarizes exposures to each of the J factor risks associated with holding the (N × 1) vector of bonds zt . The arbitrageurs thus choose zt so as to maximize (1) subject to (36), (37), (9), and PN n=1 znt = 1, for which the first-order condition is given by (7). 22 Suppose that qn,t+1 ≡ √ (Pn−1,t+1 − Pnt ) = exp µn h + hεn,t+1 − 1 Pnt where (ε1,t+1 , ..., εN,t+1 )0 ∼ N (0, Ω). Our approximation is derived from the limiting behavior as h becomes small, analogous to those obtained when considering a continuous-time representation of a discrete-time process. Thus as in Merton (1969) , ! N N X X znt [µn h + Ωnn h/2 + o(h)] Et znt qn,t+1 = n=1 n=1 Vart N X ! znt qn,t+1 = zt0 Ωzt h + o(h) n=1 for Ωnn the row n, column n element of Ω and zt = (z1t , ..., zN t )0 . Equations (36) and (37) are obtained by setting h = 1 and o(h) = 0. Specifically, Pn−1,t+1 0 0 = exp an−1 + bn−1 ft+1 − an − bn ft Pnt 0 0 µn = an−1 + bn−1 (c + ρft ) − an − bn ft 0 Ωnn = bn−1 ΣΣ0 bn−1 . Appendix B. Arbitrage-free equilibrium. 0 Note that ynt = −n−1 pnt = −n−1 (an + bn ft ) and suppose that ζnt = ζn + ϑ0n ft . If we multiply (10) by bn−1 and sum over n = 2, ..., N , we find using (9) that equilibrium requires dt = N X h i 0 bn−1 ζn + ϑ0n ft + (αn /n)(an + bn ft ) . n=2 Equation (11) is obtained from (8) with λ = γΣ0 N X bn−1 [ζn + (αn /n)an )] n=2 Λ = γΣ N X 0 h i 0 bn−1 ϑ0n + (αn /n)bn . n=2 Appendix C. ATSM estimation for a just-identified model. We first estimate the parameters of (6) and (17) by OLS:  ĉ ρ̂ = T X ! ft 0 ft−1 1 t=2 T X      t=2 0 Σ̂Σ̂ = (T − 1) −1 T X −1   1 ft−1      0 1 ft−1  (ft − ĉ − ρ̂ft−1 )(ft − ĉ − ρ̂ft−1 )0 t=2  Â B̂ = T X t=1 ! Y2t 1 ft0  −1  T X  1        t=1 ft  1 ft0   Σ̂e Σ̂0e =T T X −1 (Y2t − Â − B̂ft )(Y2t − Â − B̂ft )0 . t=1 The predicted value for row i of B̂ is given by B̂i0 = 0 n−1 i b1 i h Q ni −1 Q Q 2 + ··· + ρ IJ + ρ + ρ for i = 1, ..., M. For the just-identified case with M = J + 1, we solve this [(J + 1) × J] system of equations for the J(J + 1) unknowns ρQ and b1 using numerical search. Taking these values for ρQ and b1 as given, we can then use (12) to solve for bn for any desired n along with an = na1 + n X 0 b`−1 cQ + (1/2) `=1 n X 0 b`−1 ΣΣ0 b`−1 . `=1 The J + 1 values for a1 and cQ are then found by numerical solution of the J + 1 equations Âi = −n−1 i ani for i = 1, ..., M. Appendix D. ATSM estimation for an overidentified model. We estimated (32) by unconstrained OLS,  Â†2 B̂2† = T X t=1 ! Y2t 1 Y1t0  −1  T X  1        t=1 Y1t  1 Y1t0   for which the inverse of the usual variance matrix for the estimated coefficients is given by  R̂ = Ω̂−1 e ⊗T −1  T X  1    1 Y0   1t t=1 Y1t with Ω̂e given by diagonal elements of T −1 T X (Y2t − Â†2 − B̂2† Y1t )(Y2t − Â†2 − B̂2† Y1t )0 . t=1 The minimum-chi-square estimation procedure proposed by Hamilton and Wu (2010) estimates the structural parameters of interest θ = (π Q , a∗1 )0 or (π Q , a∗1 , a1 )0 by minimizing T [π̂ − g(θ)]0 R̂[π̂ − g(θ)] 0 where π̂ = vec (38) Â†2 B̂2† and g(θ) denotes the corresponding predicted value from (33). Under the null hypothesis that the model is correctly specified, the minimal value achieved for (38) should have an asymptotic χ2 (k1 − k0 ) distribution, where k1 = 14 is the number of parameters in Â†2 and B̂2† and k0 = 2 or 3 is the number of elements in θ. Appendix E. Details of data construction. Following Greenwood and Vayanos (2010), we started with CRSP data for outstanding Treasury debt by individual CUSIP number to estimate outstanding nominal Treasury debt at the end of each month. We calculated n for each issue by calculating the number of days between maturity and the last Friday of the month, and converted to weeks by rounding up. The raw source for these data appears to be the Monthly Statement of the Public Debt of the United States. We checked these data by summing all the maturities and comparing this sum with the sum of nominal bills, bonds, and notes recorded in the Haver database,23 which also comes from the same Monthly Statement. We found numerous discrepancies, which came from such factors as the CRSP files on occasion missing individual CUSIP series and at other times having incorporated assorted data entry errors. We were able to correct CRSP data errors so as to reduce almost all discrepancies to less than $200 M by hand comparison of the CRSP numbers with individual copies of the Monthly Statement itself. Although the Federal Reserve currently reports outright Treasury holdings for the System Open Market Account by individual CUSIP, we were unable to secure access to historical archives of these, and settled for rough estimates constructed as follows. The Federal Reserve’s weekly H41 release24 reports SOMA each Wednesday by rough maturity breakdowns (less than 15 days, 16-90 days, 91 days to 1 year, over 1 year to 5 years, over 5 years to 10 years, and over 10 years), and we matched up the last Wednesday of each month for SOMA holdings with the last calendar day of the month for Treasury marketable debt. Unfortunately, the reported SOMA maturity categories include both nominal Treasuries as well as TIPS, which we exclude from our analysis. Our solution was to assume that Fed holdings of TIPS as a fraction of the Fed’s total holdings of notes and bonds was the same across all maturity categories. Total Fed holdings of notes and bonds are reported on the H41, as are total TIPS holdings (though 23 We thank Christiane Baumeister for sharing these Haver data. Available in Table 2 of http://www.federalreserve.gov/datadownload/Choose.aspx?rel=H41. Prior to June 2003, we used the end-of-calendar month data compiled by Kuttner (2006) available at http://econ.williams.edu/people/knk1/research. 24 prior to December 2002, we had to read the latter by hand from the notes section of individual reports). We then multiplied each maturity category greater than 1 year by this ratio to get an estimate of total TIPS holdings in those categories. For maturity categories less than 1 year, we multiplied by the product of this ratio with the ratio of the Fed’s notes and bonds of maturity less than 1 year to the Fed’s total Treasury securities less than one year. We then subtracted the resulting estimates of TIPS holdings within each maturity category from the reported total holdings within each category to get our estimate of nominal Fed holdings for each maturity category. We then allocated this ratio evenly across total outstanding Treasury securities of each weekly maturity falling within that category to arrive at our estimate of how much of those securities were held by the Federal Reserve’s SOMA. Tables Table 1: Parameter estimates for the weekly affine term structure model Estimated parameters −0.0118 −0.0036 cQ 0.0116 (0.0002) (0.0005) (0.0007) ρQ 0.9990 0.0094 −0.0140 Implied parameters −0.1378 0.1604 −0.0564 Λ −0.0867 −0.0480 −0.0948 (0.0717) (0.0727) (0.0687) (0.0001) (0.0002) (0.0005) (0.0468) (0.0594) (0.1203) 0.0027 0.9870 0.0330 0.0847 −0.0266 0.1773 (0.0003) (0.0004) (0.0010) (0.0455) (0.0825) (0.1200) −0.0018 −0.0028 0.9867 −0.0567 0.0531 −0.1862 (0.0436) (0.0596) (0.1594) (0.0002) (0.0002) (0.0008) c −0.0034 −0.0003 0.0006 (0.0089) (0.0074) (0.0066) ρ 0.9895 0.0042 −0.0244 (0.0072) (0.0081) (0.0157) 0.0083 0.9826 0.0478 (0.0047) (0.0081) (0.0123) −0.0013 0.0055 0.9755 (0.0041) (0.0058) (0.0132) a1 × 5200 4.1158 b1 × 5200 1.0345 −0.6830 0.6311 (0.0058) (0.0081) (0.0189) 0.1094 0 0 0.0360 0.1027 0 (0.0100) (0.0045) −0.0670 0.0025 0.0968 (0.0188) (0.0130) (0.0149) 0.0978 0 0 0 0.0674 0 0 0.0531 0 Σ λ (0.0074) (0.0236) Σe × 5200 (0.0023) 0 (0.0016) 0 0 (0.0013) 0 0 0 0.1171 (0.0028) Estimated Jan 5, 1990 to July 27, 2007. Small-sample standard errors in parentheses. Sample size: T = 917. Bold indicates statistically significantly different from zero at the 5% significance level. Variable definitions: c, ρ, and Σ are the intercept, AR coefficients, and square root of variance for the VAR for factors (equation (6)); λ and Λ are the intercept and factor loadings for the price of risk (equation (11)); cQ and ρQ are the risk-adjusted intercept and AR coefficients (equations (15) and (13)); a1 and b1 are the intercept and factor loading for the short-term interest rate (equation (4)); Σe is the square root of the variance matrix for the measurement error (equation (17)). Table 2: Holding-return forecasting regressions Regressors 6m over 3m c, ft∗ 0.357 (0.000) A∗ c, ft , zt 0.410 (0.020) L∗ c, ft , zt 0.428 (0.003) pc∗ c, ft , zt 0.368 (0.001) ∗ c, ft , vt 0.385 (0.016) ∗ c, ft , qt 0.444 (0.002) pc ∗ c, ft , zt , qt 0.452 (0.002) ∗ c, ft , vt , qt 0.458 (0.001) A L ∗ c, ft , zt , zt , qt 0.476 (0.000) 1yr over 6m 2y over 1y 5y over 1y 0.356 0.331 0.295 (0.000) (0.000) (0.000) 0.373 0.300 0.420 (0.119) (0.311) (0.728) 0.501 0.524 0.398 (0.008) (0.006) (0.035) 0.361 0.333 0.297 (0.007) (0.062) (0.098) 0.409 0.388 0.339 (0.001) (0.006) (0.008) 0.568 0.714 0.617 (0.000) (0.000) (0.000) 0.571 0.717 0.618 (0.000) (0.000) (0.000) 0.595 0.737 0.640 (0.000) (0.000) (0.000) 0.597 0.741 0.670 (0.001) (0.000) (0.002) 10y over 1y 0.331 (0.000) 0.336 (0.665) 0.357 (0.196) 0.334 (0.051) 0.338 (0.227) 0.549 (0.001) 0.550 (0.002) 0.552 (0.002) 0.634 (0.054) R2 and hypothesis tests for holding-return forecasting regressions. Reported numbers are the R2 for the regressions, with p-values in parentheses, for tests of the null hypothesis that coefficients on starred variables are zero. All regressions also include a constant term and all hypothesis tests use Newey-West variance matrix with 20 lags. Bold indicates coefficients on starred variables are statistically significantly different from zero at the 5% significance level. Variable definitions: c is the constant term in the regression; ft is a (3 × 1) vector consisting of the level, slope, and curvature at time t as calculated in footnote 8; ztA is the average maturity of outstanding debt (equation (21)); ztL is the fraction of outstanding debt of 10-year maturity or longer (equation (22)); ztP C is a vector consisting of the first 3 principal components of {z1t , ..., zN t }; vt is the Cochrane-Piazzesi factor described in foonote 14; qt is the (3 × 1) vector of Treasury risk factors defined in equation (19). Table 3: Factor vector autoregression F test φ0i ∆ level 3.256 0.005 (0.023) (0.112) slope 4.415 −0.250 (0.005) (0.116) curvature 2.672 −0.073 (0.049) (0.116) Granger-causality tests and scenario impact estimates for factor vector autoregression. First column reports F test (p-value in parentheses) of null hypothesis that φi = 0 in regression fit = ci + ρ0i ft−1 + φ0i qt−1 + εit for fit the ith factor described in footnote 8. Second column reports estimate of φ0i ∆ for that regression (with standard error) for ∆ the average change in q under the alternative scenario (equation (24)). Table 4: R2 for post-crisis sample 3m 1y 5y 30y Contemporaneous Forecast restricted unrestricted restricted unrestricted 0.625 0.668 0.522 0.602 0.891 0.924 0.652 0.767 0.961 0.975 0.753 0.753 0.965 0.972 0.735 0.787 R2 for post-crisis sample (March 3, 2009 to Aug 10, 2010) for unrestricted regression of indicated yield on current or lagged yields and R2 for regression in which coefficients are restricted to be pre-crisis weights adjusted as in equation (26). Contemporaneous: prediction of ynt given current 6-month, 2-year and 10-year yields. Forecast: predictions of ynt given lagged 6-month, 2-year and 10-year yields. Table 5: Comparison of different estimates Study Measure Gagnon, et. al. 10 yr yield Greenwood-Vayanos 5yr-1yr spread 20yr-1yr spread D’Amico-King 10yr yield Deutsche Bank 10yr yield Original estimates Hamilton-Wu estimates Pre-crisis ZLB Pre-crisis ZLB -20 -14 -13 -39 -17 -9 -74 -25 -18 -67 -14 -13 -20 -14 -13 Comparison of different estimates of the effect of replacing $400 billion in long-term debt with short-term debt. Table 6: Risky securities Yield Factor loadings Normal level slope curvature effect 10-year Treasury 1.000 0.500 0.167 -14 Aaa Corporate 0.883 0.453 0.379 -14 Baa Corporate 0.888 0.441 0.535 -14 30-year Mortgage 0.933 0.363 0.325 -11 ZLB effect -13 -15 -17 -13 Empirical loadings of selected yields on Treasury level, slope and curvature factors, and predicted effect on yield (in basis points) of selling $400 billion in short-term Treasury debt and buying $400 billion in long-term Treasury debt. Figures Figure 1: Alternative measures of 5-year expected inflation 4 3 2 1 0 -1 -2 2005 2006 2007 2008 2009 2010 2008 2009 2010 3 2.5 2 1.5 1 0.5 0 2005 2006 2007 Top panel: 5-year break-even inflation rate, calculated as nominal yield of 5-year Treasury bond minus nominal yield on 5-year Treasury Inflation Protected Security, 2005:M1-2010:M6. Data source: FRED database of the Federal Reserve Bank of St. Louis. Bottom panel: 5-year expected CPI inflation, from the average response of the Survey of Professional Forecasters, 2005:Q3 to 2010:Q2. Data source: Federal Reserve Bank of Philadelphia. Figure 2: Federal Reserve assets 3000 Maiden 1 MMIFL 2500 TALF AIG ABCP 2000 PDCF discount 1500 swaps CPLF TAC 1000 RP MBS agency 500 misc other FR treasuries 0 Federal Reserve assets, in billions of dollars, Jan 3, 2007 to Aug 4, 2010, Wednesday values, seasonally unadjusted, from Federal Reserve H41 release. Maiden 1: net portfolio holdings of Maiden Lane LLC; MMIFL: net portfolio holdings of LLCs funded through the Money Market Investor Funding Facility; TALF: loans extended through Term Asset-Backed Securities Loan Facility; AIG: sum of credit extended to American International Group, Inc. plus net portfolio holdings of Maiden Lane II and III; ABCP: loans extended to Asset-Backed Commercial Paper Money Market Mutual Fund Liquidity Facility; PDCF: loans extended to primary dealer and other broker-dealer credit; discount: sum of primary credit, secondary credit, and seasonal credit; swaps: central bank liquidity swaps; CPLF: net portfolio holdings of LLCs funded through the Commercial Paper Funding Facility; TAC: term auction credit; RP: repurchase agreements; MBS: mortgage-backed securities held outright; agency: federal agency debt securities held outright; misc: sum of float, gold stock, special drawing rights certificate account, and Treasury currency outstanding; other FR: Other Federal Reserve assets; treasuries: U.S. Treasury securities held outright. Figure 3: Maturity structure of U.S. federal debt 100,000 80,000 60,000 40,000 20,000 0 1 105 209 313 417 521 625 729 833 937 1041 1145 1249 1353 1457 1561 Maturity structure of U.S. federal debt as of December 31, 2006. Horizontal axis: maturity in weeks. Black bars: face value of marketable nominal Treasury securities of that maturity, in millions of dollars. Light bars: imputed holdings of the System Open Market Account of the U.S. Federal Reserve. Figure 4: Average maturity 320 300 280 260 240 220 200 180 160 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 Average maturity in weeks of debt held by the public, plotted monthly from Jan 31, 1990 to Jan 31, 2011. Figure 5: Yields and factors Yields 10.0 Y6M Y2Y 7.5 Y10Y 5.0 2.5 0.0 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 Level 9 7 5 3 1 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 1999 2000 2001 2002 2003 2004 2005 2006 2007 2001 2002 2003 2004 2005 2006 2007 Slope 4 3 2 1 0 -1 1990 1991 1992 1993 1994 1995 1996 1997 1998 Curvature 2.5 1.5 0.5 -0.5 -1.5 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Yields and factors used in baseline estimation, weekly from Jan 5, 1990 to July 27, 2007. Figure 6: Impact on yields of lengthening maturity of Fed holdings 0.15 baseline with vt 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 0 200 400 600 800 1000 1200 1400 1600 Solid curve: predicted change in yn,t+1 (quoted in annual percentage points) as a function of weeks to maturity n in response to shift in qt of size ∆ using the baseline specification. Calculated from 5200b0n φ̂∆ as in footnote 16. Dashed curve: predicted change from specification in which Cochrane-Piazzesi yield factors are also included (calculated as 5200b0n φ̃∆ with φ̃ the estimate from equation (25)). Figure 7: Actual and model fitted interest rates 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 06-Mar-2009 26-Aug-2009 15-Feb-2010 07-Aug-2010 Actual (solid) and predicted (dashed) behavior of selected interest rates, weekly from March 7, 2009 to August 10, 2010. Rates shown (in order from top to bottom) are the 30 year, 5 year, 1 year, and 3 month. Figure 8: Factor loadings 1.5 1 0.5 0 0 200 400 600 800 1000 1200 1400 1600 0 200 400 600 800 1000 1200 1400 1600 0 200 400 600 800 1000 1200 1400 1600 1 0 −1 1 0.5 0 −0.5 Solid curves: normal loadings (plots of 5200bn as function of maturity n in weeks). Dashed curves: zero-lower-bound loadings (5200b∗n ). Top panel: level loadings; middle panel: slope loadings; bottom panel: curvature loadings. Figure 9: Impact of lengthening maturity of Fed holdings in normal times and at the ZLB 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 0 200 400 600 800 1000 1200 1400 1600 Predicted change in yn,t+1 (quoted in annual percentage points) as a function of weeks to maturity n in response to shift in qt of size ∆. Solid: effect in normal times (plot of 5200b0n φ∆ as a function of n); dashed: effect at the zero lower bound (plot of 5200b∗0 n φ∆). Figure 10: Assorted long-term yields 11 BAA 30−Year Fixed Rate Mortgage AAA 10 year Treasury 10 9 8 7 6 5 4 3 Jan/1990 Nov/1995 Assorted long-term yields, 1990:M1 to 2007:M7. Sep/2001 Jul/2007 Figure 11: Effects of two different maturity swaps 0.02 SWAP2-ZLB SWAP1-ZLB 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 0 200 400 600 800 1000 1200 1400 1600 Effects of two different maturity swaps when implemented at the zero lower bound. Dashed curve: Fed sells all its holdings of less than 1-year maturity and retires debt at the longest end of the maturity structure (plot of 5200b∗0 n φ∆ as a function of n). Solid curve: Fed sells all its holdings of less than 1-year maturity and retire debt evenly across 2-1/2 to 10 year maturities (plot of 5200b∗0 n φ∆2 ). Figure 12: Effects of QE2 on maturity structure Average maturity of publicly-held Treasury debt 228 226 224 222 220 218 216 214 212 Jan Feb Mar Apr May Jun Jul 2010 Aug Sep Oct Nov Dec Jan Long-term publicly-held Treasury debt as a percent of total publicly-held Treasury debt 8.0 7.9 7.8 7.7 7.6 7.5 7.4 Jan Feb Mar Apr May Jun Jul 2010 Aug Sep Oct Nov Dec Jan Top panel: average maturity of Treasury debt other than that held by the Federal Reserve (ztA ), 2010:M1-2011:M1. Bottom panel: fraction of outstanding Treasury debt not held by the Federal Reserve that is of 10 years or longer maturity (ztL ), 2010:M1-2011:M1.