Notes on the Treasury Yield Curve Forecasts October 2017 Kara Naccarelli Moody s Analytics has updated its forecast equations for the Treasury yield curve. The revised equations are the Treasury yields for the one-month, three-month, six-month, one-year, two-year, three-year, five-year, seven-year, 20- year and 30-year tenors (FRTB1MQ.IUSA, FRTB3M.IUSA, FRTB6M.IUSA, FRTB1Y.IUSA, FRTB2Y.IUSA, FRTB3Y.IUSA, FRTB5Y.IUSA, FRTB7Y.IUSA, FRTB20Y.IUSA, FRTB30Y.IUSA). Those yield curve variables are key drivers of other variables in the financial block of Moody s Analytics macro model such as yields on corporate bonds, Libor and swap rates. To model the yield curve, we use two key tenors a short-term rate and a long-term rate to pin down overall shape of the curve, and we express the remaining tenors as a linear combination of adjacent tenors. The large literature on interest rate forecasting has demonstrated that changes in the yields of a short-term and long-term tenor (which combined describe the level and slope of the yield curve) account for the vast majority of the variability in the shape of the yield curve over time. Expressing yields for other tenors as a weighted average of adjacent tenors helps to ensure that the shape of the yield curve is well-behaved. In particular, yields of one tenor should be close to yields of adjacent tenors, and yields should generally increase with tenor when the yield curve overall is upward sloping. The adjacent tenor specification also helps to make the shape of the yield curve stay well-behaved in alternative macro scenarios when intermediate tenors are made exogenous. With the latest revision, the forecast equations have changed in the following key ways. 1. The three-month replaces the one-month as the key short-term tenor. In the new specification, the three-month yield is specified as a linear combination of the federal funds rate and the CBOE Volatility Index for the S&P 500 plus a constant. The one-month yield is specified as a linear function of the three-month yield. One reason that the threemonth bill rate is preferable as the key tenor is that it tends to be more focal in federal policy deliberations than the one-month rate. For example, the macro scenarios specified by the Federal Reserve for stress-testing financial institutions specify exogenous paths for the three-month rate and that is simpler to implement when the three-month is the key tenor. 2. We simplify the equations for the six-month, two-year and 20-year yields by removing lag terms. Those lags can cause implausible forecasts of yields (particularly relative to adjacent tenors) in the Moody s Analytics baseline and alternative scenarios. 3. With the exception of the end points of the curve (the one-month and the 30-year), we impose a restriction that the weights on adjacent tenors sum to one in each equation for the non-key tenors. For example, the yield on the sixmonth bill is expressed as a weighted average of the yield on the three-month tenor with a weight of 68% and the yield on the one-year tenor with a weight of 32% plus a constant 6 basis points. These restrictions help to make the yield curve retain a plausible shape in alternative scenarios. Together, these changes simplify the equations significantly and make them more consistent, which results in better behaved forecasts in the baseline and alternative scenarios at the small price of a modest increase of in-sample and out-of-sample forecast error for some of the short-term tenors. New equation specifications Dependent variable: FRTB1MQ_US Date: 09/28/17 Time: 16:27 Sample (adjusted): 2001Q3 2017Q2 Included observations: 64 after adjustments Variable Coefficient Std. error t-statistic Prob. C -0.034811 0.010580-3.290218 0.0017 FRTB3M_US 0.986622 0.005295 186.3389 0.0000 R-squared 0.998218 Mean dependent var 1.207280
Adjusted R-squared 0.998189 S.D. dependent var 1.544517 S.E. of regression 0.065731 Akaike info criterion -2.575733 Sum squared resid 0.267877 Schwarz criterion -2.508268 Log likelihood 84.42347 Hannan-Quinn criter. -2.549155 F-statistic 34722.19 Durbin-Watson stat 0.564581 Dependent variable: FRTB3M_US Date: 09/28/17 Time: 16:29 Sample (adjusted): 1954Q3 2017Q2 Included observations: 252 after adjustments C 0.548033 0.060498 9.058633 0.0000 FRFED_US 0.853294 0.006543 130.4069 0.0000 FSPVOL_US -0.310862 0.054122-5.743687 0.0000 R-squared 0.985600 Mean dependent var 4.457136 Adjusted R-squared 0.985484 S.D. dependent var 3.092440 S.E. of regression 0.372583 Akaike info criterion 0.875120 Sum squared resid 34.56574 Schwarz criterion 0.917137 Log likelihood -107.2652 Hannan-Quinn criter. 0.892027 F-statistic 8521.192 Durbin-Watson stat 0.776408 Dependent variable: FRTB6M_US-FRTB3M_US Date: 09/28/17 Time: 16:31 Sample (adjusted): 1958Q4 2017Q2 Included observations: 235 after adjustments C -0.022871 0.013211-1.731293 0.0847 FRGT1Y_US-FRTB3M_US 0.318412 0.020121 15.82504 0.0000 R-squared 0.518029 Mean dependent var 0.144219 Adjusted R-squared 0.515961 S.D. dependent var 0.174936 S.E. of regression 0.121708 Akaike info criterion -1.365914 Sum squared resid 3.451377 Schwarz criterion -1.336471 Log likelihood 162.4949 Hannan-Quinn criter. -1.354044 F-statistic 250.4319 Durbin-Watson stat 0.511813 Dependent variable: FRGT1Y_US-FRTB6M_US Date: 09/28/17 Time: 16:33 Sample (adjusted): 1976Q2 2017Q2 Included observations: 165 after adjustments
C 0.064408 0.027471 2.344551 0.0203 FRGT2Y_US-FRTB6M_US 0.480055 0.030610 15.68300 0.0000 R-squared 0.601425 Mean dependent var 0.416294 Adjusted R-squared 0.598979 S.D. dependent var 0.321508 S.E. of regression 0.203599 Akaike info criterion -0.333284 Sum squared resid 6.756753 Schwarz criterion -0.295636 Log likelihood 29.49590 Hannan-Quinn criter. -0.318001 F-statistic 245.9564 Durbin-Watson stat 0.149499 Dependent variable: FRGT2Y_US-FRGT1Y_US Date: 09/28/17 Time: 16:34 Sample (adjusted): 1976Q2 2017Q2 Included observations: 165 after adjustments C 0.007916 0.008302 0.953510 0.3417 FRGT3Y_US-FRGT1Y_US 0.615875 0.011353 54.25007 0.0000 R-squared 0.947522 Mean dependent var 0.316719 Adjusted R-squared 0.947200 S.D. dependent var 0.337829 S.E. of regression 0.077627 Akaike info criterion -2.261755 Sum squared resid 0.982230 Schwarz criterion -2.224108 Log likelihood 188.5948 Hannan-Quinn criter. -2.246473 F-statistic 2943.070 Durbin-Watson stat 0.301890 Dependent variable: FRGT3Y_US-FRGT2Y_US Date: 09/28/17 Time: 16:35 Sample (adjusted): 1976Q2 2017Q2 Included observations: 165 after adjustments C -0.022750 0.006334-3.591816 0.0004 FRGT5Y_US-FRGT2Y_US 0.399484 0.008555 46.69391 0.0000 R-squared 0.930441 Mean dependent var 0.184686 Adjusted R-squared 0.930014 S.D. dependent var 0.219218 S.E. of regression 0.057994 Akaike info criterion -2.844913 Sum squared resid 0.548215 Schwarz criterion -2.807266 Log likelihood 236.7054 Hannan-Quinn criter. -2.829631 F-statistic 2180.321 Durbin-Watson stat 0.434698 Dependent variable: FRGT5Y_US-FRGT3Y_US Date: 09/28/17 Time: 16:37 Sample (adjusted): 1969Q3 2017Q2 Included observations: 192 after adjustments
C -0.011371 0.005490-2.071455 0.0397 FRGT7Y_US-FRGT3Y_US 0.591900 0.007316 80.90972 0.0000 R-squared 0.971795 Mean dependent var 0.305374 Adjusted R-squared 0.971647 S.D. dependent var 0.316682 S.E. of regression 0.053324 Akaike info criterion -3.014481 Sum squared resid 0.540265 Schwarz criterion -2.980549 Log likelihood 291.3902 Hannan-Quinn criter. -3.000738 F-statistic 6546.383 Durbin-Watson stat 0.499359 Dependent variable: FRGT7Y_US-FRGT5Y_US Date: 09/28/17 Time: 16:39 Sample (adjusted): 1969Q3 2017Q2 Included observations: 192 after adjustments C 0.033847 0.005350 6.326958 0.0000 FRGT10Y_US-FRGT5Y_US 0.502125 0.009250 54.28088 0.0000 R-squared 0.939421 Mean dependent var 0.229759 Adjusted R-squared 0.939102 S.D. dependent var 0.221716 S.E. of regression 0.054714 Akaike info criterion -2.963039 Sum squared resid 0.568784 Schwarz criterion -2.929107 Log likelihood 286.4518 Hannan-Quinn criter. -2.949297 F-statistic 2946.414 Durbin-Watson stat 0.323362 Dependent variable: FRGT20Y_US-FRGT10Y_US Date: 09/28/17 Time: 16:39 Sample (adjusted): 1993Q4 2017Q2 Included observations: 95 after adjustments C 0.239279 0.023766 10.06810 0.0000 FRGT30Y_US-FRGT10Y_US 0.531407 0.034174 15.55020 0.0000 R-squared 0.722229 Mean dependent var 0.549832 Adjusted R-squared 0.719243 S.D. dependent var 0.236993 S.E. of regression 0.125574 Akaike info criterion -1.291012 Sum squared resid 1.466506 Schwarz criterion -1.237246 Log likelihood 63.32305 Hannan-Quinn criter. -1.269286 F-statistic 241.8086 Durbin-Watson stat 0.186396 Dependent variable: FRGT30Y_US Date: 09/28/17 Time: 16:40
Sample (adjusted): 1993Q4 2017Q2 Included observations: 95 after adjustments C 0.579716 0.043792 13.23808 0.0000 FRGT20Y_US 0.886678 0.008673 102.2331 0.0000 R-squared 0.991180 Mean dependent var 4.845203 Adjusted R-squared 0.991085 S.D. dependent var 1.372942 S.E. of regression 0.129629 Akaike info criterion -1.227457 Sum squared resid 1.562735 Schwarz criterion -1.173691 Log likelihood 60.30420 Hannan-Quinn criter. -1.205731 F-statistic 10451.60 Durbin-Watson stat 0.260664 Mnemonics referenced in the above equation, for example FET, can be defined using the Mnemonic 411 feature on DataBuffet. Please contact Help@economy.com for assistance.
Previous equation specifications Dependent variable: FRTB1MQ_US Date: 09/24/15 Time: 14:23 Sample: 2001Q3 2015Q2 Included observations: 56 FRFED_US 0.908439 0.011634 78.08195 0.0000 FSPVOL_US -0.068743 0.022093-3.111539 0.0030 R-squared 0.988291 Mean dependent var 1.336429 Adjusted R-squared 0.988075 S.D. dependent var 1.608976 S.E. of regression 0.175706 Akaike info criterion -0.604949 Sum squared resid 1.667118 Schwarz criterion -0.532615 Log likelihood 18.93859 Hannan-Quinn criter. -0.576906 Durbin-Watson stat 0.367479 Dependent variable: FRTB3M_US Date: 09/24/15 Time: 14:21 Sample: 2001Q3 2015Q2 Included observations: 56 C -0.015713 0.004691-3.349262 0.0015 FRTB1MQ_US 0.597445 0.020376 29.32151 0.0000 FRTB6M_US 0.407513 0.019901 20.47748 0.0000 R-squared 0.999805 Mean dependent var 1.386509 Adjusted R-squared 0.999798 S.D. dependent var 1.630963 S.E. of regression 0.023192 Akaike info criterion -4.637954 Sum squared resid 0.028507 Schwarz criterion -4.529453 Log likelihood 132.8627 Hannan-Quinn criter. -4.595888 F-statistic 135977.7 Durbin-Watson stat 1.287529 Dependent variable: FRTB6M_US Date: 09/24/15 Time: 14:22 Sample: 1962Q2 2015Q2 Included observations: 213 C 0.123661 0.021505 5.750477 0.0000 FRTB3M_US 0.914609 0.014357 63.70530 0.0000 FRGT1Y_US(-1) 0.078694 0.013515 5.822683 0.0000 R-squared 0.997287 Mean dependent var 5.033359 Adjusted R-squared 0.997261 S.D. dependent var 3.123872 S.E. of regression 0.163490 Akaike info criterion -0.770145
Sum squared resid 5.613086 Schwarz criterion -0.722803 Log likelihood 85.02049 Hannan-Quinn criter. -0.751013 F-statistic 38594.91 Durbin-Watson stat 0.968277 Dependent variable: FRGT1Y_US Date: 03/21/16 Time: 12:24 Sample (adjusted): 1976Q2 2015Q4 Included observations: 159 after adjustments FRTB6M_US 0.744476 0.016679 44.63645 0.0000 FRGT2Y_US 0.302033 0.014887 20.28811 0.0000 R-squared 0.999182 Mean dependent var 5.329651 Adjusted R-squared 0.999177 S.D. dependent var 3.797709 S.E. of regression 0.108976 Akaike info criterion -1.582878 Sum squared resid 1.864498 Schwarz criterion -1.544275 Log likelihood 127.8388 Hannan-Quinn criter. -1.567202 Durbin-Watson stat 0.391195 Dependent variable: FRGT2Y_US Sample: 1976Q3 2013Q1 Included observations: 147 C 0.186392 0.045951 4.056342 0.0001 FRGT1Y_US 0.748685 0.021630 34.61389 0.0000 FRGT3Y_US(-1) 0.250784 0.022914 10.94475 0.0000 R-squared 0.995017 Mean dependent var 6.023968 Adjusted R-squared 0.994948 S.D. dependent var 3.603226 S.E. of regression 0.256105 Akaike info criterion 0.133737 Sum squared resid 9.444908 Schwarz criterion 0.194766 Log likelihood -6.829636 Hannan-Quinn criter. 0.158533 F-statistic 14378.09 Durbin-Watson stat 0.979241 Dependent variable: FRGT3Y_US Sample: 1976Q3 2013Q1 Included observations: 147 C -0.084421 0.013751-6.139158 0.0000 FRGT2Y_US 0.574148 0.009529 60.25519 0.0000 FRGT5Y_US 0.433684 0.010363 41.84918 0.0000
R-squared 0.999758 Mean dependent var 6.196463 Adjusted R-squared 0.999754 S.D. dependent var 3.498605 S.E. of regression 0.054820 Akaike info criterion -2.949335 Sum squared resid 0.432750 Schwarz criterion -2.888306 Log likelihood 219.7761 Hannan-Quinn criter. -2.924538 F-statistic 297257.7 Durbin-Watson stat 0.458025 Dependent variable: FRGT5Y_US Sample: 1969Q3 2013Q1 Included observations: 175 C -0.074911 0.014309-5.235404 0.0000 FRGT3Y_US 0.382959 0.008873 43.16093 0.0000 FRGT7Y_US 0.624704 0.009825 63.58586 0.0000 R-squared 0.999714 Mean dependent var 6.591714 Adjusted R-squared 0.999710 S.D. dependent var 3.058282 S.E. of regression 0.052058 Akaike info criterion -3.055915 Sum squared resid 0.466130 Schwarz criterion -3.001662 Log likelihood 270.3926 Hannan-Quinn criter. -3.033908 F-statistic 300173.1 Durbin-Watson stat 0.545033 Dependent variable: FRGT7Y_US Sample: 1969Q3 2013Q1 Included observations: 175 C 0.003286 0.016909 0.194324 0.8462 FRGT5Y_US 0.483075 0.012533 38.54403 0.0000 FRGT10Y_US 0.520507 0.013768 37.80425 0.0000 R-squared 0.999636 Mean dependent var 6.804247 Adjusted R-squared 0.999632 S.D. dependent var 2.922415 S.E. of regression 0.056079 Akaike info criterion -2.907100 Sum squared resid 0.540924 Schwarz criterion -2.852846 Log likelihood 257.3712 Hannan-Quinn criter. -2.885093 F-statistic 236176.8 Durbin-Watson stat 0.322352 Dependent variable: FRGT20Y_US Method: ARMA conditional least squares (Marquardt - EViews legacy) Sample: 2006Q2 2013Q1 Included observations: 28 Convergence achieved after 7 iterations
FRGT10Y_US 0.443201 0.068553 6.465111 0.0000 FRGT30Y_US 0.615513 0.056632 10.86862 0.0000 AR(1) 0.624353 0.119204 5.237685 0.0000 R-squared 0.992605 Mean dependent var 4.008810 Adjusted R-squared 0.992013 S.D. dependent var 0.875295 S.E. of regression 0.078224 Akaike info criterion -2.157516 Sum squared resid 0.152976 Schwarz criterion -2.014780 Log likelihood 33.20523 Hannan-Quinn criter. -2.113881 Durbin-Watson stat 1.360990 Inverted AR Roots.62 Dependent variable: FRGT30Y_US Sample: 1977Q2 2013Q1 Included observations: 144 C 1.176821 0.054384 21.63906 0.0000 FRGT10Y_US 0.880988 0.007200 122.3665 0.0000 R-squared 0.990606 Mean dependent var 7.267814 Adjusted R-squared 0.990540 S.D. dependent var 2.702769 S.E. of regression 0.262884 Akaike info criterion 0.179587 Sum squared resid 9.813371 Schwarz criterion 0.220835 Log likelihood -10.93029 Hannan-Quinn criter. 0.196348 F-statistic 14973.55 Durbin-Watson stat 0.341101 Mnemonics referenced in the above equation, for example FET, can be defined using the Mnemonic 411 feature on DataBuffet. Please contact Help@economy.com for assistance.