Forecasting in the Presence of in and Out of Sample Breaks Appendix

Appendix In this appendix, we describe the detailed steps to compute the estimate using the methodology described in Section 4 for the random level shift model with and without shifts in the volatility process. Mixture Kalman filtering algorithm for the basic model (1)-(2): At t = 0, for (i) i = 1, ..., M, and given initial parameters Θ0 = (r0 , r1 , σ e , σ η , ρ), draw K0 ∼ Ber(p0 ), (i) (i) (i) β 0 , P0 ∼ f (β 0 ; Θ0 ) and set w0 = 1/M. For t = 1, ..., T and i = 1, ..., M: 1) Con(i) (i) (i) (i) (i) (i) struct the Kalman predictions: β t|t−1 = β t−1|t−1 + Kt−1 μη,t|t−1 and Pt|t−1 = Pt−1|t−1 + (i) (i) _ t−1,(i) (i) Kt−1 σ 2η , where μη,t|t−1 = ρ(β t−1|t−1 − β ); 2) Compute the forecast of yt : E[f (β t|t−1 )] = PM (i) (i) (i) (i) (i) (i) ∼ N(vt , Ft ), where vt = i=1 wt−1 f (β t|t−1 ); 3) Compute the importance weights wt (i) (i) (i) yt − Xt β t|t−1 , Ft = Xt Pt−1|t−1 Xt0 + σ 2u and set the normalized importance weights as (i) (i) P (j) (i) (i) (i) M ŵt = wt / M j=1 wt ; 4) Resample M samplers {β t−1|t−1 , Pt−1|t−1 , Kt−1 }i=1 with probabili(i) (i) (i) ties {ŵt }M i=1 and for i = 1, ..., M set wt = 1/M; 5) Draw Kt ∼ Ber(pt ) and construct the (i) (i) (i) (i) (i) (i) (i) following steps of the Kalman filter: β t|t−1 = β t−1|t−1 + Kt μη,t|t−1 , Pt|t−1 = Pt−1|t−1 +Kt σ 2η , (i) (i) (i) (i) (i) (i) (i) −1,(i) (i) (i) = Xt Pt−1|t−1 Xt0 + σ 2u , β t|t = β t|t−1 + Pt|t−1 Xt0 Ft vt , Pt|t = P (i) (i) −1,(i) (i) (i) (i) M Pt|t−1 ; 6) Compute the filtered estimate: E[f (β t )] = i=1 ŵt−1 f (β t|t ). Pt|t−1 − Pt|t−1 Xt0 Ft = yt − Xt β t|t−1 , Ft vt Mixture Kalman filtering algorithm for model (1)-(3) with stochastic volatility and Kσt = 1: At t = 0, for i = 1, ..., M, and given initial parameters Θ0 = (r0 , r1 , φ, σ v , σ η , ρ), (i) (i) (i) (i) (i) let lnσ 2²,t = zt , draw K0 ∼ Ber(p0 ), z0 ∼ f (z0 ; Θ0 ), β 0 , P0 ∼ f (β 0 ; Θ0 ) and set w0 = 1/M. The procedure is then the same as described above, except that in step 3, we have (i) (i) (i) (i) (i) Ft = Xt Pt−1|t−1 Xt0 + exp(zt−1 ) and in step 5, one also draws zt ∼ N(φzt−1 , σ 2v ) and uses (i) (i) (i) Ft = Xt Pt−1|t−1 Xt0 + exp(zt ). Particle smoothing algorithm for the basic model (1)-(2): Consider the weighted (i) (i) β(i) samplers obtained from the filtering algorithm {wt , β t , Kt }M i=1 for i = 1, . . . , M, and (j) (j) (j) (i) M t = 1, . . . , T . Let {sβ,t , sK1 ,t }j=1 be a set of particle smoothers. First set sβ,T = β T and (j) β(i) sK1 ,T = KT with probability (1/M). Then, for t = T − 1, T − 2, . . . , 1, compute (j) (i) wt|t+1 ∝ (i) (j) (i) wt p(sβ,t+1 |β t ) (i) (j) (sβ,t+1 − β t − μη )2 s(j) 1−sK ,t+1 K1 ,t+1 1 ∝ {pt+1 exp(− )} {1 − p } t+1 2σ 2η (j) (i) (j) β(i) for i = 1, . . . , M, and let sβ,t = β t and sK1 ,t+1 = Kt (i) with probability wt|t+1 . Repeat (j) (j) the steps above decreasing from t − 1 until 1 to obtain {sβ,t , sK β ,t+1 } as approximations to t p(β t , Ktβ |y (T ) ), for j = 1, . . . , M. The stochastic volatility model (1)-(3) with Ktσ unrestricted: 1) For i = 1, . . . , M, β,(i) (i) and given initial parameters Θ0 = (p1 , p2 , φ, σ η .σ v , ρ), generate K0 ∼ Ber(p1 ), then β 0 ∼ A-1 β(i) σ,(i) (i) σ,(i) K0 N(0, σ 2η ). Also generate K0 ∼ Ber(p2 ), then with ln σ 2²,t ≡ zt , z0 = K0 N(0, σ 2v ). (i) β,(i) ∼ Ber(p1 ) Set the initial weights to w0 = (1/M); 2) For t = 1, . . . , T , generate Kt (i),(t−1) (i),(t−1) (i) (i) β,(i) (i) 2 ), where β̄ is the average and β t = β t−1 + Kt N(μη,t , σ η ), μη,t = ρ(β t|t−1 − β̄ σ,(i) ∼ Ber(p2 ) and of all the particle filters from t = 1 to time t − 1. Also generate Kt σ,(i) (i) (i) (i) 2 −1/2 exp{−(yt − zt = φzt−1 + Kt N(0, σ v ); Compute wt ∝ p(yt |xt )wt−1 ∝ (2π exp(zt )) (i) 2 (i) Xt β t ) /(2 exp(zt ))}, for i = 1, . . . , M, and set the normalized importance weights as ŵt = (i) P (i) (i) β,(i) (i) σ,(i) (i) (i) , zt , Kt }M wt / M i=1 with probability ŵt , and set wt = i=1 wt ; 4) Resample {β t , Kt (1/M); 5) Repeat steps 1-4 increasing from t + 1 until T . MCEM algorithm: For the E-step, the complete likelihood of {β 1 , . . . , β T , K1β , . . . , KTβ , y1 ,. . . . , yT } is f (β, K1 , Y ) = QT β t=1 f (β t |β t−1 , Kt ) QT β t=1 f (Kt ) QT t=1 f (yt |β t , Ktβ ) T T (β t − β t−1 − μηt )2 K β Q 1 1 (yt − Xt β t )2 Ktβ 1−Ktβ Q t p p ={ exp(− )} pt (1−pt ) exp(− ) 2σ 2η 2σ 2e 2πσ 2η 2πσ 2e t=1 t=1 t=1 T Q The log-likelihood function is: −2logf (β, K β , Y ) = T P t=1 −2 Ktβ [log(σ 2η ) + T P (β t − β t−1 − μηt )2 ] σ 2η [Ktβ log(pt ) + (1 − Ktβ )log(1 − pt )] + t=1 T P [log(σ 2e ) + t=1 (yt − Xt β t )2 ] σ 2e The expectation of the complete log-likelihood function with respect to the unknown state variables β, K β given Y and the current parameter estimates Θ(k−1) is the objective function to be maximized. For the Monte Carlo EM algorithm, we approximate the expectation by Monte Carlo sample average with random samples drawn from p(β t , Ktβ |y T ) obtained using the particle smoothing algorithm. Then, Q(Θ, Θ(k−1) ) = E[−2logf (β, K β , Y )|Y, Θ(k−1) ] (i) (i) M P T (β t − β t−1 − μηt )2 1 P β(i) { Kt [log(σ 2η ) + ] = M i=1 t=1 σ 2η −2 T P β(i) [Kt t=1 β(i) log(pt ) + (1 − Kt )log(1 − pt )] + T P (i) [log(σ 2e ) + t=1 (yt − Xt β t )2 ]} σ 2e For the M-step, the objective function becomes the usual log-likelihood function of Θ. Hence, standard maximum likelihood estimates are obtained by solving the first order conditions. For the full model with stochastic volatility, the estimation methodology is the same. The difference is that instead of having two state variables, we now have three, namely A-2 {β t , Ktβ , lnσ 2²,t }. Similarly, if different parameters are allowed to vary independently, we β ). simply add the additional latent variables (β jt , Kjt Selection of the initial values and construction of the standard errors: In order to speed up the convergence of the estimation algorithm, we can use information from the data to provide better initial parameter values. Consider the simple model yt = β t + et , β t = β t−1 + Ktβ η t where η t ∼ N(0, σ 2η ), et ∼ N(0, σ 2e ) and Ktβ ∼ Ber(p). The initial parameter values are set (0) (0) (0) to p(0) , σ 2η = |var(y − y−2 ) − var(y − y−1 )| and σ 2e = (var(y − y−1 ) − p(0) σ 2η )/2. We set p(0) according to prior judgment about the frequency of the jumps. To construct the standard errors of the estimates, Duan and Fulop (2011) proposed a stable estimator of the information matrix applicable to the EM algorithm. They estimate the variance using the smoothed individual scores. Define at (Θ) = E[∂logf (xt |χt−1 , Θ)/∂Θ|Y, Θ], P then the estimate of the information matrix is Iˆ = Ω0 + lj=1 w(l)(Ωj + Ω0j ), where Ωj = PT −j 0 t−1 at (Θ̂)at+j (Θ̂) and w(j) = 1 − j/(l + 1). This method is easy to compute and does not require evaluations of the second-order derivatives of the complete data log-likelihood. A-3 Table 1: Forecasting Comparisons from Simulated Models Panel 1: RLS Basic (infrequent large breaks) h=1 h=4 h=12 h=24 RLS_m 0.98 1.00 1.00 0.99 RLS_SV 1.01 1.07 1.13 1.12 Average 7.20 19.13 26.04 24.60 Rolling 1.09 1.28 1.31 1.22 ARMA 1.01 1.02 1.07 1.15 ARIMA 1.01 1.03 1.09 1.19 MS 4.57 11.41 15.33 14.54 TVP 2.25 4.72 6.19 6.03 Panel 3: RLS mean reverting RLS_SV 1.03 1.03 1.01 0.98 Average 1.52 2.04 1.91 1.56 Rolling 1.45 2.05 2.14 1.75 ARMA 1.32 1.78 1.97 1.86 ARIMA 1.07 1.11 1.07 1.02 MS 1.02 1.06 1.17 1.32 TVP 1.43 2.04 2.44 2.68 Panel 5: RLS_SV RLS_m 1.00 0.99 0.98 0.96 Average 1.07 1.19 1.48 1.50 Rolling 1.01 1.04 1.14 1.15 ARMA 1.09 1.20 1.46 1.46 ARIMA 1.04 1.02 1.09 1.14 MS 1.07 1.18 1.45 1.46 TVP 1.75 3.95 8.67 10.09 Panel 6: ARMA (AR=0.95,MA=-0.5) RLS_m RLS_SV Average Rolling ARIMA MS TVP 1.06 1.09 4.23 4.88 1.03 3.50 1.01 1.12 1.15 4.30 5.01 1.10 3.64 1.05 1.07 1.09 2.47 2.76 1.28 2.21 1.22 1.04 1.06 2.15 2.02 1.62 1.83 1.59 Panel 2: RLS Basic (frequent small breaks) h=1 h=4 h=12 h=24 RLS_m 0.99 1.03 1.01 1.00 RLS_SV 1.01 1.03 0.96 1.00 Average 1.43 2.77 3.61 3.13 Rolling 1.07 1.32 1.57 1.46 ARMA 1.01 1.04 1.13 1.20 ARIMA 1.02 1.10 1.26 1.39 MS 2.87 8.61 14.13 14.08 TVP 1.90 4.81 6.60 6.29 Panel 4: RLS no mean reverting RLS_m 0.98 0.93 0.88 0.95 RLS_SV 1.52 2.33 2.82 2.49 Average 2.39 4.38 5.23 4.00 Rolling 1.61 2.54 3.11 2.72 ARMA 0.98 0.94 0.91 0.99 ARIMA 0.98 0.94 0.88 0.96 MS 2.40 4.40 5.23 3.97 TVP 3.18 6.35 7.77 6.01 Panel 7: ARIMA (AR=0.1,MA=-0.5,d=1) RLS_m 1.01 1.02 1.04 RLS_SV 1.05 0.99 0.95 Average 48.84 47.96 41.30 Rolling 3.69 3.31 3.02 ARMA 1.01 1.04 1.11 MS 8.78 8.56 7.88 TVP 1.14 1.19 1.23 1.07 0.92 31.08 2.45 1.23 6.27 1.18 Panel 8: Time Varying Parameter Model RLS_m 1.00 1.01 1.01 1.01 RLS_SV 1.62 1.65 1.35 1.24 Average 47.60 47.49 25.85 15.47 Rolling 7.09 7.18 4.37 3.02 ARMA 1.03 1.10 1.27 1.49 ARIMA 1.02 1.00 1.01 1.01 MS 37.44 37.25 20.18 11.99 Panel 9: Markov Switching Model RLS_m RLS_SV Average Rolling ARMA ARIMA TVP 1.02 1.06 1.00 1.02 1.02 1.01 1.43 1.00 1.17 1.00 1.08 1.00 1.02 3.41 1.01 1.26 1.01 1.23 1.01 1.05 7.93 1.06 1.66 1.02 1.40 1.02 1.08 17.83 Note: This table reports the relative cumulative MSFEs with respect to the true model. In each case, we generate 100 true data paths and 1000 observations for each path. We use the first 800 observations for in-sample estimation and the rest to evaluate out-of-sample forecasting accuracy. The forecasting horizon is 24. In Panels 1 and 2, the true model is the basic random level shift model with parameters (p, 𝜎𝑒 , 𝜎𝜂 ) as stated in the text. The differences lie in the parameters setting for the frequency and magnitudes of the breaks. In Panel 3, the true model is the RLS model with mean reversion with parameters (𝑟0 , 𝑟1 , 𝜎𝑒 , 𝜎𝜂 , 𝜌), as stated in the text. In Panel 4, the true model is the same as the one in Panel 3 except that the mean reverting parameter ρ is set to 0. In Panel 5, the true model is the RLS model with stochastic volatility. The benchmark model for each panel is the one with the correct model specification. Numbers smaller than 1 indicate a better forecasting performance than the corresponding benchmark model. The bold numbers in each panel stand for the smallest relative cumulative MSFEs. Table 2.1: Equity Premium Forecasting Comparisons for the Period 1998-2012 (Monthly Data; Shiller Dataset) Historical average Rolling 10 years Dividend_no break Dividend_rolling TVP Regime Switching Level Shift Dividend_K1t Dividend_K2t 0.14 (0.00) 0.12 (0.00) 0.09 (0.00) 0.01a (0.69) 0.26 (0.00) Cumulative MSFE h=6 h=12 h=18 356 1364 2955 (0.00) (0.00) (0.00) Relative Cumulative MSFE 0.16 0.18 0.19 (0.00) (0.00) (0.00) 0.13 0.14 0.15 (0.00) (0.00) (0.00) 0.10 0.11 0.11 (0.00) (0.00) (0.00) 0.03 0.04 0.06 (0.01) (0.00) (0.00) 0.30 0.33 0.35 (0.00) (0.00) (0.00) 0.01a (0.38) 0.01a,* (1.00) 0.01a (0.38) 0.03 (0.01) 0.01 (0.07) 0.01a,* (1.00) h=1 10.27 (0.00) 0.04 (0.00) 0.02 (0.01) 0.02a,* (1.00) 0.06 (0.00) 0.03 (0.00) 0.02a,* (1.00) h=24 5010 (0.00) h=30 7390 (0.00) h=36 9981 (0.00) 0.20 (0.00) 0.16 (0.00) 0.12 (0.00) 0.08 (0.00) 0.38 (0.00) 0.23 (0.00) 0.18 (0.00) 0.12 (0.00) 0.10 (0.00) 0.42 (0.00) 0.25 (0.00) 0.21 (0.00) 0.13 (0.00) 0.12 (0.00) 0.47 (0.00) 0.08 (0.00) 0.03 (0.00) 0.03a,* (1.00) 0.10 (0.00) 0.04 (0.00) 0.03a,* (1.00) 0.12 (0.00) 0.05 (0.00) 0.04a,* (1.00) Note: This table reports the relative MSFEs with respect to the benchmark model, which is the historical average. Numbers in the parentheses are p-values for the model confidence set of Hansen et al. (2011). Numbers with superscript “a” indicate models which belong to the 10% model confidence set using all comparisons. Numbers with an asterisk refer to the model with the smallest MSFE amongst all models. ‘Rolling 10 years’ refers to forecasting using historical averaged data with a window size fixed at 10 years. ‘Dividend_no break’ refers to an OLS-based regression with fixed parameter of the equity premium on a constant and the lagged dividend-price ratio using the full in-sample data to construct the forecasts; ‘Dividend_rolling’ uses the same OLS regression but with a 10 years rolling in-sample data; ‘TVP’ stands for the time varying parameter model in which the unconditional mean of the equity premium is modelled as a random walk; ‘Regime switching’ is the two-state Markov regime switching model; ‘Level shift’ is the unconditional mean model with level shifts and mean reversion; ‘Dividend_K1t’ is the conditional mean model with a constant term and the lagged dividend-price ratio as regressor with the constant term modelled as a level shift process with mean reversion; ‘Dividend_K2t’ is the conditional mean model with a constant term and the lagged dividend-price ratio as regressor with the coefficient of the lagged dividend-price ratio modelled as a level shift process with mean reversion. Table 2.2: Equity Premium Forecasting Comparisons for the Period 1988-1996 (Monthly Data; Shiller Dataset) Historical average Rolling 10 years Dividend_no break Dividend_rolling TVP Regime Switching Level Shift Dividend_K1t Dividend_K2t See Notes to Table 2.1 Cumulative MSFE h=1 h=12 h=24 22.86 3228 12445 (0.00) (0.00) (0.00) Relative Cumulative MSFE 0.076 0.081 0.090 (0.00) (0.00) (0.00) 0.303 0.272 0.235 (0.00) (0.00) (0.00) 0.019 0.021 0.028 (0.00) (0.07) (0.01) a, a 0.002 * 0.013 0.018a (1.00) (0.72) (0.37) 0.626 0.622 0.615 (0.00) (0.00) (0.00) 0.002 (0.00) 0.005 (0.00) 0.004 (0.00) 0.013a,* (1.00) 0.021 (0.07) 0.015a (0.22) 0.016a,* (1.00) 0.027 (0.01) 0.018a (0.48) h=36 27030 (0.00) h=48 47108 (0.00) h=60 72598 (0.00) 0.102 (0.00) 0.205 (0.00) 0.036 (0.00) 0.021a (0.17) 0.608 (0.00) 0.110 (0.00) 0.171 (0.00) 0.049 (0.00) 0.027a (0.19) 0.605 (0.00) 0.115 (0.00) 0.143 (0.00) 0.060 (0.00) 0.031a (0.25) 0.602 (0.00) 0.018a,* (1.00) 0.037 (0.00) 0.022a (0.28) 0.022a,* (1.00) 0.049 (0.00) 0.027a (0.37) 0.025a,* (1.00) 0.060 (0.00) 0.029a (0.44) Table 3.1: Equity Premium Forecasting Comparisons for the Period 1975-2009 (Post Oil Shock; Annual Data; Welch & Goyal Dataset) Historical average Rolling 10 years BM_no break BM_rolling TVP Regime Switching Level Shift BM_K1t BM_K2t Cumulative MSFE h=1 h=2 h=3 29.37 109.43 224.08 (0.01) (0.05) (0.01) Relative Cumulative MSFE 0.67 0.76 0.87 (0.02) (0.05) (0.06) 0.92 0.94 0.99 (0.01) (0.05) (0.02) a 0.38 0.39 0.43a (0.08) (0.35) (0.27) a, a 0.19 * 0.27 0.36a (1.00) (0.85) (0.81) 0.78 0.77 0.77a (0.01) (0.05) (0.12) a a, 0.19 0.27 * 0.35a (0.52) (1.00) (0.81) a a 0.27 0.29 0.32a,* (0.14) (0.85) (1.00) a 0.35 0.35 0.37a (0.08) (0.35) (0.48) h=4 363.85 (0.01) h=5 522.17a (0.19) 0.99 (0.01) 1.07 (0.01) 0.50a (0.47) 0.46a (0.47) 0.77a (0.28) 0.45a (0.47) 0.37a,* (1.00) 0.41a (0.47) 1.11 (0.00) 1.16 (0.00) 0.57a (0.33) 0.58a (0.29) 0.77a (0.28) 0.56a (0.37) 0.42a,* (1.00) 0.46a (0.37) Note: The models and notations are the same as in Table 2.1, except that ‘BM’ replaces the “dividend price ratio’. We use annual data from 1947-2014. Data before 1975 are used for in-sample estimation and the forecasting horizon is 5 years ahead. The Book-to-Market Ratio (BM) is the ratio of book to market value for the Dow Jones Industrial Average. The covariate used to forecast the level shift probability is the absolute changes in the earning-price ratio. Table 3.2: Equity Premium Forecasting Comparisons for the Period 1975-2009 (Post Oil Shock; Monthly Data; Welch & Goyal Dataset) Historical average Rolling 10 years BM_nobreak BM_rolling TVP Regime Switching Level Shift BM_K1t BM_K2t Cumulative MSFE h=1 h=12 h=24 h=36 29.19 4112 15895 34205 (0.00) (0.00) (0.00) (0.00) Relative Cumulative MSFE 0.526 0.597 0.674 0.751 (0.00) (0.00) (0.00) (0.00) 0.857 0.904 0.945 0.985 (0.00) (0.00) (0.00) (0.00) 0.201 0.226 0.236 0.243 (0.00) (0.00) (0.00) (0.00) a, a 0.014 * 0.065 0.132 0.196 (1.00) (0.16) (0.00) (0.00) 0.699 0.691 0.680 0.669 (0.00) (0.00) (0.00) (0.00) a a 0.014 0.065 0.132 0.196 (0.94) (0.16) (0.00) (0.00) a a, a, 0.014 0.054 * 0.106 * 0.155a,* (0.94) (1.00) (1.00) (1.00) a a a 0.015 0.067 0.127 0.192a (0.92) (0.16) (0.20) (0.10) h=48 56932 (0.00) h=60 82170 (0.00) 0.837 (0.00) 1.030 (0.00) 0.252 (0.09) 0.270 (0.00) 0.659 (0.00) 0.270 (0.00) 0.207a,* (1.00) 0.262 (0.09) 0.934 (0.00) 1.084 (0.00) 0.264a (0.78) 0.353 (0.00) 0.651 (0.00) 0.353 (0.00) 0.257a,* (1.00) 0.324 (0.07) Note: The models and notations are the same as in Table 2.1, except that ‘BM’ replaces the “dividend price ratio’. We use monthly data from 1921/03/31-2014/12/31. Data before 1975 are used for in-sample estimation and the forecasting horizon is 60 months ahead. Table 4: Equity Premium Forecasting Comparisons for the Period 2009-2015 (Monthly Data; Welch & Goyal Dataset) Historical average Rolling 10 years Dividend_no break Dividend_rolling TVP Regime Switching Level Shift DP_LS DP_VIX_SPX_LS 0.67 (0.00) 0.91 (0.00) 0.25 (0.00) 0.01a (0.38) 0.64 (0.00) Cumulative MSFE h=4 h=8 h=12 800 2859 5794 (0.00) (0.00) (0.00) Relative Cumulative MSFE 0.67 0.68 0.69 (0.00) (0.00) (0.00) 0.93 0.96 0.97 (0.00) (0.00) (0.00) 0.28 0.32 0.35 (0.00) (0.00) (0.00) (0.02a 0.04a 0.07a (0.51) (0.65) (0.96) 0.63 0.61 0.60 (0.00) (0.00) (0.00) 0.02 (0.02) 0.01a,* (1.00) 0.03 (0.02) 0.03 (0.00) 0.02a,* (1.00) 0.05 (0.07) h=1 54.45 (0.00) 0.06 (0.00) 0.04a,* (1.00) 0.06a (0.44) 0.10 (0.00) 0.06a,* (1.00) 0.07a (0.96) h=16 9244 (0.00) h=20 12843 (0.00) h=24 16176 (0.00) 0.70 (0.00) 0.98 (0.00) 0.37 (0.00) 0.10a (0.66) 0.58 (0.00) 0.72 (0.00) 0.99 (0.00) 0.40 (0.00) 0.16a (0.17) 0.57 (0.00) 0.74 (0.00) 0.99 (0.00) 0.42 (0.00) 0.24 (0.00) 0.55 (0.00) 0.16 (0.00) 0.10a (0.66) 0.08a,* (1.00) 0.24 (0.00) 0.16a (0.17) 0.10a,* (1.00) 0.36 (0.00) 0.24 (0.01) 0.12a,* (1.00) Note: The notation is the same as in table 2.1. We use monthly data from 1990/01/31-2015/07/31. Data before 2009 are used for in-sample estimation and the forecasting horizon is 24 months ahead. ‘Level Shift’ is the unconditional mean model with level shifts and mean reversion; ‘DP_LS’ is the conditional mean model with a constant term and the lagged dividend-price ratio as regressors with the constant term following a level shift process with mean reversion; ‘DP_VIX_SPX_LS’ is the conditional mean model with a constant term, the lagged dividend-price ratio, the VIX index and the returns on the monthly S&P 500 index option as regressors with the constant term following a level shift process with mean reversion. Table 5: Treasury Bill Rate Forecasting Comparisons Recursive OLS h=12 2.07 (0.06) h=24 2.67a Rolling 5 years Rolling 10 years TVP AR_K2t AR_K1t,K2t 1.12 1.04 1.72 1.02 0.97a,* (0.06) (0.06) (0.00) (0.06) (1.00) 1.38 1.03 3.44 1.01a 0.97a,* Recursive OLS 2.05 (0.01) 2.99 Rolling 5 years Rolling 10 years TVP AR_K2t AR_K1t,K2t 0.93a,* 0.97 0.98 1.03 0.97a (1.00) (0.35) (0.01) (0.00) (0.35) 0.78a,* 0.84 1.21 1.01 0.98 Recursive OLS 2.76a (0.55) 3.32a Rolling 5 years Rolling 10 years TVP AR_K2t AR_K1t,K2t 1.20a 1.02a 1.75 1.01a 0.98a,* (0.22) (0.53) (0.00) (0.55) (1.00) 1.73a 1.06a 3.68 1.00a 0.98a,* Recursive OLS 1.23 (0.04) 1.59 Rolling 5 years Rolling 10 years TVP AR_K2t AR_K1t,K2t 1.17 1.25 2.69 1.02 0.94a,* (0.00) (0.00) (0.00) (0.01) (1.00) 1.23 1.36 5.78 1.01 0.89a,* MSFE 1968-2002 h=36 (0.13) 3.10a Relative MSFE (0.00) 2.22 (0.00) 1.06 (0.00) 5.06 (0.13) 1.00a (1.00) 0.95a,* MSFE 1970s (0.00) 3.11 Relative MSFE (1.00) 0.76 a,* (0.00) 0.80 (0.00) 1.58 (0.00) 1.00 (0.00) 0.96 MSFE 1980s (0.66) 4.17a Relative MSFE (0.43) 2.76 (0.66) 1.07 (0.00) 4.80 (0.66) 1.00a (1.00) 0.95a,* MSFE 1990s (0.00) 1.74 Relative MSFE (0.00) 1.17 (0.00) 1.44 (0.00) 9.33 (0.00) 1.01 (1.00) 0.92a,* (0.11) h=48 3.32a (0.12) h=60 3.44 (0.08) (0.00) (0.00) (0.00) (0.11) (1.00) 4.06 1.10 6.79 1.00a 0.94a,* (0.00) (0.00) (0.00) (0.12) (1.00) 8.35 1.23 8.68 0.99a 0.92a,* (0.00) (0.01) (0.00) (0.10) (1.00) (0.00) 2.77 (0.00) 2.43 (0.00) (1.00) (0.00) (0.00) (0.00) (0.00) 0.78a,* 0.80 1.89 1.00 0.89 (1.00) (0.04) (0.00) (0.00) (0.00) 0.85a 0.93 1.58 1.02 0.82a,* (0.65) (0.02) (0.00) (0.00) (1.00) (0.13) 4.84a (0.17) 5.23a (0.20) (0.02) (0.06) (0.00) (0.26) (1.00) 4.71 1.05 5.56 0.99a 0.95a,* (0.01) (0.06) (0.00) (0.63) (1.00) 9.20 1.15 6.21 0.99a 0.93a,* (0.00) (0.09) (0.00) (0.64) (1.00) (0.00) 1.63 (0.04) 1.39a (0.52) (0.00) (0.00) (0.00) (0.00) (1.00) 1.28 1.74 15.12 1.01 0.95a,* (0.00) (0.00) (0.00) (0.04) (1.00) 1.63 2.20 24.91 0.99a 0.99a,* (0.00) (0.00) (0.00) (0.95) (1.00) Note: This table reports the relative MSFEs with respect to the benchmark model, which is the recursive OLS. The notation is the same as in table 2.1. ‘Recursive OLS’ refers to the OLS model with an expanding estimation window; ‘Rolling 5 years and 10 years’ refer to OLS models with window lengths set at 5 years and 10 years; ‘TVP’ stands for the time varying parameter model; ‘AR_LS’ is the AR(1) model allowing for level shifts in the constant term; ‘AR_LS_SV’ incorporates stochastic volatility into the error term. ‘AR_K1t,K2t’ allows for both the constant term and the AR coefficient to follow a level shift process with two different latent variables and mean reversion. Figure 1: Mixture Kalman Filtered Estimates and True Parameter Process 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0 100 200 300 400 500 600 700 800 900 1000 Note: The true process for βt is generated using equations (1) and (2) with mean reversion and time varying probability with the parameters (r0 , r1 , σe , ση , ρ) = (−1.96, 4, 0.2, 0.2, −0.1). The covariate wt is set to be 1 every 50 observations, 0 otherwise. T=1000. The red step function is the true βt parameter process; the blue solid line is the corresponding filtered estimates of βt using the mixture Kalman filter. Figure 2: Mixture Kalman Filtered Estimates and True Parameter and Stochastic Volatility Processes Panel A: True and Filtered Estimates of the Parameter Process 0.2 true  filtered estimates 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0 100 200 300 400 500 600 700 800 900 1000 Panel B: True and Filtered Estimates of the Stochastic Volatility Process 3 true volatility filtered estimates 2.5 2 1.5 1 0.5 0 0 100 200 300 400 500 600 700 800 900 1000 Note: The true βt and the stochastic volatility processes are generated using equations (1)-(3) with mean reversion and time varying probability with the parameters (r0 , r1 , ϕ, σv , ση , ρ) = (−1.96, 4, 0.95, 0.2, 0.2, −0.1). The covariate wt is set to be 1 every 50 observations, 0 otherwise. T=1000. Figure 3: In-Sample Confidence Bands for the Parameters Process and Forecast Confidence Bands Panel A: In-Sample Confidence Bands for the Parameter Process Note: The true βt process is generated using (1) and (2) with mean reversion and time varying probability and parameters (r0 , r1 , σe , ση , ρ) = (−1.96, 4, 0.2, 1, −0.5). The covariate wt is set to be 1 every 20 observations, 0 otherwise. T=500. The red solid step line is the true βt parameter process; while the blue solid line is path of the filtered estimates of βt . The two dashed lines represent the 2.5% and 97.5% percentiles of the simulated parameter paths. The computation of the in-sample bands are based on M=1000 and S=1000 simulations. Panel B: Out-of-Sample Forecast Bands for the Parameter Process Note: βt is generated using (1) and (2) with mean reversion and time varying probability and parameters (r0 , r1 , σe , ση , ρ) = (−1.96, 4, 0.2, 1, −0.5). The covariate 𝑤𝑡 is set to be 1 every 20 observations, 0 otherwise. T=500. For the upper graph, we choose to forecast from the 115th observation, which is in a high regime; while for the lower graph, we choose to forecast from the 325th observation, which is in a low regime. The forecasting horizon is set to be 100 steps. The red solid line is the true βt process. The blue solid line is the forecasted βt . The two dashed lines represent the 2.5% and 97.5% percentiles of the extrapolated paths. The computation of the out-of-sample bands are based on M=1000 and S=1000 simulations.