Online Appendix for Forecasting the Equity Risk Premium: The Role of Technical Indicators

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1 Onlne Appendx for Forecastng the Equty Rsk Premum: The Role of Techncal Indcators Chrstopher J. Neely Federal Reserve Bank of St. Lous Davd E. Rapach Sant Lous Unversty Guofu Zhou Washngton Unversty n St. Lous and CEMA zhou@wustl.edu August 20, 2013 Jun Tu Sngapore Management Unversty tujun@smu.edu.sg Ths Onlne Appendx descrbes the wld bootstrap procedure used to compute emprcal p-values n Secton 2 of the paper. It also reports the complete structural stablty test results (Table A1) dscussed n footnote 9 of the paper, as well as the predctve regresson results for SENT (Table A2) and condtonal asset prcng model estmaton results (Table A3) dscussed n Secton 2.4 of the paper. We begn wth the wld bootstrap procedure. Let ˆε t+1 = r t+1 ( ˆα + N =1 ˆβ,x x,t + N =1 ˆβ,S S,t ), (A1) where ˆα, ˆβ,x ( = 1,..., N), and ˆβ,S ( = 1,..., N) are OLS parameter estmates for the general multple predctve regresson model that ncludes a constant and all of the N macroeconomc varables and N techncal ndcators as regressors. Followng conventon, we assume that each macroeconomc varable follows an AR(1) process: x,t+1 = ρ,0 + ρ,1 x,t + v,t+1 for = 1,..., N. (A2) Defne ˆv c,t+1 = x,t+1 ( ˆρ c,0 + ˆρc,1 x,t) for = 1,..., N, (A3) where ˆρ,0 c and ˆρc,1 are reduced-bas estmates of the AR(1) parameters n (A2). The reduced-bas estmates of the AR parameters are computed by teratng on the analytcal second-order bas expresson for the OLS estmates. Based on these AR parameter estmates and ftted resduals, we buld up a pseudo sample of observatons for the equty rsk premum and N macroeconomc varables under the null hypothess of no

2 return predctablty: r t+1 = r + ˆε t+1 w t+1 for t = 0,...,T 1, (A4) x,t+1 = ˆρ c,0 + ˆρc,1 x,t + ˆvc,t+1 w t+1 for = 1,..., N and t = 0,...,T 1, (A5) where r s the sample mean of the equty rsk premum, w t+1 s a draw from the standard normal dstrbuton, and x,0 = x,0 ( = 1,..., N). Observe that we multply ˆε t+1 n (A4) and each ˆv,t+1 c n (A5) by the same scalar, w t+1, when generatng the month-(t + 1) pseudo resduals, thereby makng t a wld bootstrap. In addton to preservng the contemporaneous correlatons n the data, the wld bootstrap accounts for general forms of condtonal heteroskedastcty. Employng reduced-bas parameter estmates n (A5) helps to ensure that we adequately capture the persstence n the macroeconomc varables. To generate a pseudo sample of observatons for the N techncal ndcators, we assume that each ndcator follows a frst-order, two-state, Markov-swtchng process wth the followng transton matrx: where P = p 0,0 p 1,0 p 0,1 p 1,1 for = 1,..., N, (A6) p j,k = Pr(S,t = k S,t 1 = j) for j, k = 0, 1, (A7) and p 0,0 + p 0,1 = p 1,0 + p 1,1 = 1. Based on estmates of the transton probabltes n (A6) and S,0 ( = 1,..., N), we can buld up a pseudo sample of observatons for the N techncal ndcators va smulatons. Usng the pseudo sample of observatons for the equty rsk premum [{rt+1 }T 1 t=0 ], N macroeconomc varables [{x,t }T 1 t=0 ( = 1,..., N)], and N techncal ndcators [{S,t }T 1 t=0 ( = 1,..., N)], we estmate the slope coeffcents and correspondng heteroskedastcty-robust t-statstcs for the bvarate predctve regressons gven by (1) and (10) n the paper for each, as well as the prncpal component predctve regressons gven by (11), (12), and (13) n the paper. Note that we compute the prncpal components n (11), (12), and (13) usng {x,t }T 1 t=0 and {S,t }T 1 t=0 ( = 1,..., N), so that the pseudo sample accounts for the estmated regressors n the prncpal component predctve regressons. We store the t-statstcs for all of the predctve regressons. Repeatng ths process 2,000 tmes yelds emprcal dstrbutons for each of the t-statstcs. For a gven t-statstc, the emprcal p-value s the proporton of the bootstrapped t-statstcs greater than the t-statstc for the orgnal sample. 2

3 Table A1 Predctve Regresson Structural Stablty Test Results, 1951: :12 Predctor qll Predctor qll Model qll DP 9.85 MA(1,9) PC-ECON DY MA(1,12) 7.62 PC-TECH EP MA(2,9) 7.02 PC-ALL 6.88 DE 7.08 MA(2,12) 6.86 RVOL MA(3,9) 7.53 BM 9.88 MA(3,12) 8.05 NTIS MOM(9) 8.01 TBL MOM(12) 7.50 LTY 9.80 VOL(1,9) 6.26 LTR 8.77 VOL(1,12) 7.30 TMS VOL(2,9) 8.57 DFY VOL(2,12) 9.29 DFR 8.76 VOL(3,9) 8.63 INFL VOL(3,12) 8.75 Notes. The second and fourth columns report the Ellott and Müller (2006) qll statstc for the bvarate predctve regresson model, r t+1 = α + β q,t + ε,t+1, where r t+1 s the log equty rsk premum (n percent) and q,t s one of the 14 macroeconomc varables (14 techncal ndcators) gven n the frst (thrd) column. The last column reports the qll statstc for a predctve regresson model based on prncpal components, r t+1 = α + K β k j + ε t+1, where j s the kth prncpal component extracted from the 14 macroeconomc varables ( j = ECON, PC-ECON), 14 techncal ndcators ( j = TECH, PC-TECH), or the 14 macroeconomc varables and 14 techncal ndcators taken together ( j = ALL, PC-ALL); k = 3 for PC-ECON, k = 1 for PC-TECH, and k = 4 for PC-ALL. The qll statstc s for testng the null hypothess that the ntercept and slope coeffcents are constant. The 10%, 5%, and 1% crtcal values for the qll statstcs n the second and fourth columns and for the PC-TECH model are 12.80, 14.32, and 17.57, respectvely; the 10%, 5%, and 1% crtcal values for the qll statstc for the PC-ECON (PC-ALL) model are 23.37, 25.28, and ( 28.55, 30.60, and 35.09), respectvely (where we reject for small values);,, and ndcate sgnfcance at the 10%, 5%, and 1% levels, respectvely.

4 Table A2 Predctve Regresson Estmaton Results for the Sentment-Changes Index, 1965:08 to 2010:12 Macroeconomc varables Slope Techncal ndcators Slope Predctor coeffcent R 2 R 2 EXP R 2 REC Predctor coeffcent R 2 R 2 EXP R 2 REC Panel A: Bvarate Predctve Regressons DP 0.12 [0.92] 0.26% 0.38% 0.25% MA(1,9) 0.20 [2.26] 0.94% 0.04% 4.94% DY 0.15 [1.10] 0.37% 0.45% 0.08% MA(1,12) 0.24 [2.48] 1.19% 0.24% 5.07% EP 0.09 [0.83] 0.17% 0.51% 1.25% MA(2,9) 0.16 [1.75] 0.56% 0.11% 3.27% DE 0.02 [0.15] 0.00% 0.05% 0.23% MA(2,12) 0.26 [2.83] 1.47% 0.62% 4.95% RVOL 0.41 [ 0.55] 0.05% 0.16% 0.40% MA(3,9) 0.11 [1.25] 0.28% 0.22% 2.31% BM 0.19 [1.06] 0.27% 0.55% 0.86% MA(3,12) 0.14 [1.50] 0.42% 0.06% 1.87% NTIS 0.27 [ 0.14] 0.00% 0.05% 0.18% MOM(9) 0.19 [2.01] 0.75% 0.35% 2.35% TBL 0.02 [1.13] 0.20% 0.13% 1.56% MOM(12) 0.20 [2.15] 0.84% 0.40% 2.64% LTY 0.02 [1.20] 0.24% 0.25% 0.22% VOL(1,9) 0.17 [1.88] 0.63% 0.02% 3.27% LTR 0.01 [ 0.63] 0.07% 0.04% 0.17% VOL(1,12) 0.18 [1.93] 0.70% 0.14% 2.97% TMS 0.01 [0.23] 0.01% 0.13% 0.57% VOL(2,9) 0.19 [2.10] 0.86% 0.31% 3.07% DFY 0.07 [0.77] 0.12% 0.24% 1.59% VOL(2,12) 0.20 [2.18] 0.92% 0.57% 2.35% DFR 0.04 [ 1.15] 0.30% 0.76% 1.56% VOL(3,9) 0.12 [1.23] 0.31% 0.10% 1.16% INFL 0.05 [0.45] 0.04% 0.08% 0.51% VOL(3,12) 0.15 [1.54] 0.47% 0.19% 1.60% Panel B: Prncpal Component Predctve Regressons 1 ECON 0.01 [0.42] 0.07% 0.00% 0.38% 1 TECH 0.03 [2.31] 0.96% 0.28% 3.72% 2 ECON 0.01 [0.32] 3 ECON 0.01 [0.25] Panel C: Prncpal Component Predctve Regresson, All Predctors Taken Together 1 ALL 0.03 [2.15] 1.11% 0.10% 5.21% 2 ALL 0.02 [0.93] 3 ALL 0.02 [0.61] 4 ALL 0.01 [0.26] Notes. Panel A reports estmaton results for the bvarate predctve regresson model, SENT t+1 = α + β q,t + ε,t+1, where SENT t+1 s the sentment-changes ndex and q,t s one of the 14 macroeconomc varables (14 techncal ndcators) gven n the frst (sxth) column. Panels B and C report estmaton results for a predctve regresson model based on prncpal components, SENT t+1 = α + K β k j + ε t+1, where j s the kth prncpal component extracted from the 14 macroeconomc varables ( j = ECON), 14 techncal ndcators ( j = TECH), or the 14 macroeconomc varables and 14 techncal ndcators taken together ( j = ALL). The brackets to the mmedate rght of the estmated slope coeffcents report heteroskedastcty-consstent t-statstcs;,, and ndcate sgnfcance at the 10%, 5%, and 1% levels, respectvely, based on one-sded (upper-tal) wld bootstrapped p-values; 0.00 ndcates less than n absolute value. The R 2 statstcs n the thrd and eghth columns are computed for the full sample. The REXP 2 (R2 REC ) statstcs n the fourth and nnth (ffth and tenth) columns are computed for NBER-dated busness-cycle expansons (recessons), as gven by (9) n the paper.

5 Table A3 Tests of Excluson Restrctons for the Condtonal Fama- French Three-Factor Model of Momentum Portfolo Excess Returns, 1951: :12 Portfolo () α,k = 0 for k = 2, 3, 4;,2 = 0 j; α,1 = 0,,3 = 0 j; α,k = 0 k,,1 = 0 j,4 = 0 j,k = 0 j, k # restrctons Low Hgh UMD Notes. The table reports heteroskedastcty-consstent χ 2 -statstcs for tests of parameter restrctons for the condtonal Fama-French three-factor model: R,t+1 R f,t+1 = α,t + β MKT,t MKT t+1 + β,,t SMB SMB t+1 + β,t HML HML t+1 + ε,t+1, where R,t+1 s the (smple) return on momentum portfolo, R f,t+1 s the rskfree return, MKT s the excess market return, SMB (HML) s the sze (value) premum, and α,t = α,0 + 4 α,k ALL,,t =,0 + 4 ALL,k for j = MKT, SMB, HML, where ALL s the kth prncpal component extracted from 14 macroeconomc varables and 14 techncal ndcators taken together. The momentum portfolos are from Kenneth French s Data Lbrary and formed from NYSE pror return decles; UMD s the up-mnus-down portfolo. The χ 2 -statstcs correspond to the excluson restrctons gven n the column headng;,, and ndcate sgnfcance at the 10%, 5%, and 1% levels, respectvely, based on wld-bootstrapped p-values.