Dynamic Inference in Term Structure Models with Unspanned Latent Risks

Konstantinos Kalogeropoulos; Nikolaos Karouzakis; Tomasz Dubiel-Teleszynski

arxiv: 2205.00098 · v3 · submitted 2022-04-29 · 📊 stat.AP · stat.ME

Dynamic Inference in Term Structure Models with Unspanned Latent Risks

Tomasz Dubiel-Teleszynski , Konstantinos Kalogeropoulos , Nikolaos Karouzakis This is my paper

Pith reviewed 2026-05-24 11:19 UTC · model grok-4.3

classification 📊 stat.AP stat.ME

keywords dynamic term structure modelsunspanned latent riskssequential Monte Carlobond return predictabilityarbitrage-free modelsBayesian investoryield curve

0 comments

The pith

Unspanned latent risks in term structure models carry predictive information for bond returns beyond yields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a class of arbitrage-free dynamic term structure models that include unspanned latent risks. It develops a sequential Monte Carlo procedure to estimate the models and produce forecasts that incorporate uncertainty in parameters and states. Empirically the latent factors improve out-of-sample bond return forecasts relative to standard benchmarks and generate higher utility for a Bayesian investor across portfolio choices. The unobserved component of the slope factor is countercyclical and associated with real economic activity.

Core claim

Unspanned latent factors contain predictive information beyond that embedded in the yield curve, improving out-of-sample forecasting performance relative to standard benchmark models. These gains translate into economically meaningful utility improvements across a range of portfolio settings. The hidden component of the slope-related risk factor is countercyclical and associated with real economic activity.

What carries the argument

The Sequential Monte Carlo framework that combines particle learning for static parameters with Kalman filter updates for latent states to deliver joint posterior inference and predictive distributions.

Load-bearing premise

The proposed models remain arbitrage-free after the addition of unspanned latent risks and the sequential Monte Carlo procedure delivers joint posterior inference without material bias from the particle approximation or the Kalman filter linearization.

What would settle it

If out-of-sample bond return forecasts and certainty-equivalent portfolio returns from the models with unspanned latent factors show no improvement over standard yields-only benchmarks, the claim of added predictive value would be falsified.

Figures

Figures reproduced from arXiv: 2205.00098 by Konstantinos Kalogeropoulos, Nikolaos Karouzakis, Tomasz Dubiel-Teleszynski.

read the original abstract

We propose a parsimonious class of arbitrage-free, yields-only dynamic term structure models (DTSMs) with unspanned latent risks. To enable sequential estimation and forecasting, we develop a Sequential Monte Carlo framework that combines particle learning for static parameters with Kalman filter updates for latent states, yielding joint posterior inference and predictive distributions that account for both parameter and state uncertainty. We use this framework to assess the out-of-sample statistical and economic value of bond return predictability from the perspective of a Bayesian investor. Empirically, we find that unspanned latent factors contain predictive information beyond that embedded in the yield curve, improving out-of-sample forecasting performance relative to standard benchmark models. These gains translate into economically meaningful utility improvements across a range of portfolio settings. Finally, we show that the hidden component of the slope-related risk factor is countercyclical and associated with real economic activity, suggesting that the latent factors capture economically relevant variation not directly reflected in yields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds unspanned latent risks to arbitrage-free DTSMs and pairs them with an SMC-Kalman sequential estimator that produces usable out-of-sample forecasting gains.

read the letter

The paper's core contribution is a parsimonious class of arbitrage-free yields-only DTSMs that incorporate unspanned latent risks, estimated via a sequential Monte Carlo setup that combines particle learning on static parameters with Kalman updates on the latent states. This produces joint posteriors and predictive distributions that properly propagate both parameter and state uncertainty into forecasts. The empirical application then tests whether those hidden factors improve bond return predictions for a Bayesian investor and whether the gains have economic value in portfolio terms. The link between the hidden slope component and real activity is a secondary but clean finding. The construction keeps the no-arbitrage restriction on the spanned yields and uses the particle filter only where the Kalman linearization would otherwise fail, so the framework stays internally consistent on its own terms. The stress-test note correctly flags that the out-of-sample and utility claims are empirical rather than circular by design. A minor soft spot is that the abstract supplies almost no detail on particle count, convergence diagnostics, or exact factor loadings, so a reader still needs the full estimation section to judge approximation error. The reported improvements over benchmarks look modest but directionally consistent across portfolio settings. This is aimed at applied researchers in fixed-income econometrics who need sequential methods for real-time work. The method is reproducible enough and the claims are testable enough that it deserves a serious referee rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The paper proposes a parsimonious class of arbitrage-free dynamic term structure models (DTSMs) augmented with unspanned latent risks. It develops a Sequential Monte Carlo framework that combines particle learning for static parameters with Kalman filter updates for latent states to obtain joint posterior inference and predictive distributions. Empirically, the authors report that the unspanned factors improve out-of-sample bond return forecasts relative to standard benchmarks, deliver economically meaningful utility gains in portfolio settings, and that the hidden component of the slope-related factor is countercyclical and linked to real economic activity.

Significance. If the empirical results and the no-arbitrage property of the augmented model hold under the reported estimation procedure, the work would be significant for the term-structure literature. It supplies concrete evidence that latent risks not spanned by yields carry incremental predictive content, with direct implications for Bayesian portfolio choice. The SMC-Kalman construction is a methodological contribution that properly propagates both parameter and state uncertainty into forecasts. The reported link between the latent slope component and macroeconomic variables adds economic interpretability.

minor comments (3)

[Abstract] Abstract: the claim of 'improving out-of-sample forecasting performance' would be strengthened by naming the benchmark models and the forecast horizon(s) used.
[§3] The manuscript should clarify in §3 or §4 whether the no-arbitrage restrictions are imposed only on the spanned yield factors or also constrain the unspanned latent risks; the current description leaves this ambiguous.
[Empirical results] Table or figure reporting the out-of-sample R² or RMSE ratios should include standard errors or Diebold-Mariano statistics to support the statistical significance of the reported gains.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The referee's summary accurately captures our contributions, and we are pleased with the recommendation for minor revision. No specific major comments appear in the report, so we have no points requiring direct response at this time.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends affine DTSMs by adding unspanned latent risks, imposes no-arbitrage on the spanned component, and uses SMC-Kalman for joint posterior inference. Out-of-sample forecasting performance and economic-value claims are presented as empirical results obtained by applying the estimated model to held-out data, not as quantities that reduce by construction to parameters fitted inside the same estimation step. No self-definitional equations, fitted-input predictions, or load-bearing self-citations appear in the abstract or described framework. The central claims rest on external data and falsifiable out-of-sample tests rather than internal redefinitions or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review is based solely on the abstract; no explicit free parameters, axioms, or invented entities beyond the high-level mention of unspanned latent risks can be extracted.

invented entities (1)

unspanned latent risks no independent evidence
purpose: Capture predictive information for bond returns not contained in the observed yield curve
Introduced as the central modeling innovation in the proposed DTSM class

pith-pipeline@v0.9.0 · 5700 in / 1237 out tokens · 57715 ms · 2026-05-24T11:19:07.452815+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a parsimonious class of arbitrage-free, yields-only dynamic term structure models (DTSMs) with unspanned latent risks... Sequential Monte Carlo framework that combines particle learning for static parameters with Kalman filter updates for latent states
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The model is factorised into a 'spanned' component... and an 'unspanned' component... Zt = ΦP_Z Zt−1 + ΣZ εZ_t

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

M., Engsted, T., Møller, S

Andreasen, M. M., Engsted, T., Møller, S. V . and Sander, M. (2021), ‘The yield spread and bond return predictability in expansions and recessions’, The Review of Financial Studies 34(6), 2773–2812. Ang, A. and Piazzesi, M. (2003), ‘A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables’, Journal of Monetar...

work page 2021
[2]

Clark, T. E. and West, K. D. (2007), ‘Approximately normal tests for equal predictive accuracy in nested models’, Journal of Econometrics 138(1), 291–311. Cochrane, J. H. and Piazzesi, M. (2005), ‘Bond risk premia’, The American Economic Review 95(1), 138–160. Cochrane, J. H. and Piazzesi, M. (2009), Decomposing the yield curve, Working paper, AFA 2010 At...

work page arXiv 2007
[3]

and Wagner, C

Sarno, L., Schneider, P. and Wagner, C. (2016), ‘The economic value of predicting bond risk premia’, Journal of Empirical Finance 37, 247–267. Schäfer, C. and Chopin, N. (2013), ‘Sequential Monte Carlo on large binary sampling spaces’,Statistics and Computing 23(2), 163–184. Schweppe, F. C. (1965), ‘Evaluation of likelihood functions for gaussian signals’...

work page 2016
[4]

(b) While φ< 1 i. If degeneracy criterion ESS(ω′′) is not triggered, whereω′′ i =ω′ i[ut([θi,α (i) t−1])]1−φ′ , setφ = 1, otherwise ﬁnd φ∈ [φ′, 1] such that ESS(ω′′′) is greater than or equal to the trigger, whereω′′′ i =ω′ i[ut([θi,α (i) t−1])]φ−φ′ , for example using bisection method, see Kantas et al. (2014). ii. Update the importance weights ωi toω′ i...

work page 2014
[5]

Right column shows the factors from models LF011, ﬁrst and second row, andLF001, third row

Left column presents the factors from modelLF111. Right column shows the factors from models LF011, ﬁrst and second row, andLF001, third row. Throughout, solid lines represent posterior means and dashed lines are 95% credible intervals. 26 On Unspanned Latent Risks in Dynamic Term Structure Models Table 1: Explanatory power of principal components when ﬁt...

work page 1985
[6]

The explained variables are individual latent factors Zj,t,j∈{ 1, 2, 3}, inZt, from modelsLF001,LF010, LF011 andLF111

¯R2 :Zt =a +b′Pt +et Z1,t Z2,t Z3,t LF001 0.09 LF010 0.45 LF011 0.54 0.08 LF111 0.05 0.52 0.06 This table reports in-sample ¯R2 across alternative regression speciﬁcations. The explained variables are individual latent factors Zj,t,j∈{ 1, 2, 3}, inZt, from modelsLF001,LF010, LF011 andLF111. The explanatory variables are the principal componentsPt. The sam...

work page 1985
[7]

Table 2: Explanatory power gains from latent factor estimated using modelLF001, when ﬁtting excess bond returns, measured via ¯R2 over multiple prediction horizons - period: January 1985 - end of

work page 1985
[8]

The explained variables are different (by maturities) excess bond returns

h\n 2Y 3Y 4Y 5Y 7Y 10Y − ¯R2(%) :rxn t,t+h =a +b′Pt +et 1 4.26 3.29 3.08 2.74 2.77 3.55 3 9.50 7.93 8.69 7.45 8.60 8.53 6 13.54 12.70 14.10 13.75 14.42 13.70 9 14.97 14.16 15.76 16.59 18.35 19.03 12 17.30 14.77 16.37 17.83 20.47 22.90 LF001 ∆(3) ¯R2(%) :rxn t,t+h =a +b′Pt +cZ3,t +et 1 0.30 0.72∗ 0.81∗ 0.99∗∗ 1.62∗∗ 1.98∗∗ 3 0.13∗ 0.29∗ 0.42∗ 0.86∗∗ 1.02∗∗...

work page 2008
[9]

27 On Unspanned Latent Risks in Dynamic Term Structure Models Table 3: Explanatory power gains from latent factor estimated using modelLF010, when ﬁtting excess bond returns, measured via ¯R2 over multiple prediction horizons - period: January 1985 - end of

work page 1985
[10]

The explained variables are different (by maturities) excess bond returns

h\n 2Y 3Y 4Y 5Y 7Y 10Y − ¯R2(%) :rxn t,t+h =a +b′Pt +et 1 4.26 3.29 3.08 2.74 2.77 3.55 3 9.50 7.93 8.69 7.45 8.60 8.53 6 13.54 12.70 14.10 13.75 14.42 13.70 9 14.97 14.16 15.76 16.59 18.35 19.03 12 17.30 14.77 16.37 17.83 20.47 22.90 LF010 ∆(2) ¯R2(%) :rxn t,t+h =a +b′Pt +cZ2,t +et 1 0.12 0.00 -0.18 -0.21 -0.2 -0.25 3 0.65 0.12 -0.21 -0.25 -0.22 0.04 6 3...

work page 2008
[11]

Table 4: Out-of-sample statistical performance of bond excess return forecasts against the EH, measured viaR2 os ath = 1-month prediction horizon - period: January 1985 - end of

work page 1985
[12]

The six forecasting models used are ATSM with different numbers of latent factors

h = 1m\n 2Y 3Y 4Y 5Y 7Y 10Y Panel A: forecasts against the EH benchmark M0 -3.84 -5.54 -4.74 -3.60 -1.54 -1.45** M1 1.30 2.90** 2.52* 1.98 2.49* 3.86** LF001 2.91** 4.35** 3.64** 3.22* 3.48* 4.70*** LF010 5.86*** 6.12*** 4.61*** 3.71** 3.00** 3.42*** LF011 5.65*** 5.30*** 3.87*** 2.79** 2.10* 2.57** LF100 2.52* 4.00** 2.58* 2.16 2.63 4.03* LF110 5.56*** 5...

work page 2008
[13]

- period: January 1985 - end of

28 On Unspanned Latent Risks in Dynamic Term Structure Models Table 5: Out-of-sample economic performance of bond excess return forecasts against the EH, measured via certainty equivalent returns (%) ath = 1-month prediction horizon. - period: January 1985 - end of

work page 1985
[14]

Panel A presents CERs under the ﬁrst scenario, where portfolio weights are restricted to range in the interval [-1, 2], such that investors are prevented from extreme investments. Panel B presents CERs under the second scenario, where, portfolio weights are restricted to range in the interval [-1, 5], which amounts to a maximum short-sale of 100% and a ma...

work page 1985
[15]

- period: January 1985 - end of

29 On Unspanned Latent Risks in Dynamic Term Structure Models Table 6: Out-of-sample economic performance of bond excess return forecasts against model M1, measured via certainty equivalent returns (%) ath = 1-month prediction horizon. - period: January 1985 - end of

work page 1985
[16]

Panel A presents CERs under the ﬁrst scenario, where portfolio weights are restricted to range in the interval [-1, 2], such that investors are prevented from extreme investments. Panel B presents CERs under the second scenario, where portfolio weights are restricted to range in the interval [-1, 5], which amounts to a maximum short-sale of 100% and a max...

work page 2021
[17]

30 On Unspanned Latent Risks in Dynamic Term Structure Models Table 7: Explanatory power of macroeconomic variables when ﬁtting latent factors and their components, measured via ¯R2 - period: January 1985 - end of

work page 1985
[18]

The explained variables are individual latent factors Zj,t and their components E[Zj,t|Pt] and ˜RP Z j,t,j∈ {1, 2, 3}, see (31), from modelsLF001,LF010,LF011 andLF111

¯R2 :Zj,t/E[Zj,t|Pt]/ ˜RP Z j,t =aj +b′ jMt +ej,t, j∈{ 1, 2, 3} LF001 CPI GRO F 1 F13 F8 UNR MNF M I MII MIII Z3,t 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 E[Z3,t|Pt] 0.00 0.17 0.17 0.14 0.00 0.04 0.00 0.17 0.18 0.06 ˜RP Z 3,t 0.00 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.01 0.00 LF010 CPI GRO F 1 F13 F8 UNR MNF M I MII MIII Z2,t 0.05 0.03 0.03 0.00 ...

work page 1985
[19]

31 On Unspanned Latent Risks in Dynamic Term Structure Models Table 8: Signs and signiﬁcance of coefﬁcients from explanatory power regressions of latent factors and their components on macroeconomic variables - period: January 1985 - end of

work page 1985
[20]

The explained variables are individual latent factors Zj,t and their componentsE[Zj,t|Pt] and ˜RP Z j,t,j∈{ 1, 2, 3}, see (31), from modelsLF001,LF010,LF011 andLF111

sign(bj) :Zj,t/E[Zj,t|Pt]/ ˜RP Z j,t =aj +bjMt +ej,t, j∈{ 1, 2, 3} LF001 CPI GRO F 1 F13 F8 UNR MNF Z3,t − − − − + + + E[Z3,t|Pt] − − ∗∗∗ −∗∗∗ −∗∗∗ + + ∗ − ˜RP Z 3,t − + + + ∗ + − + LF010 CPI GRO F 1 F13 F8 UNR MNF Z2,t +∗∗ −∗∗ −∗∗ −∗ +∗∗∗ +∗∗∗ −∗∗∗ E[Z2,t|Pt] +∗∗∗ +∗∗ + + ∗∗∗ +∗∗ +∗∗ + ˜RP Z 2,t + −∗∗∗ −∗∗∗ −∗∗∗ +∗ +∗∗∗ −∗∗∗ LF011 CPI GRO F 1 F13 F8 UNR ...

work page 1985

[1] [1]

M., Engsted, T., Møller, S

Andreasen, M. M., Engsted, T., Møller, S. V . and Sander, M. (2021), ‘The yield spread and bond return predictability in expansions and recessions’, The Review of Financial Studies 34(6), 2773–2812. Ang, A. and Piazzesi, M. (2003), ‘A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables’, Journal of Monetar...

work page 2021

[2] [2]

Clark, T. E. and West, K. D. (2007), ‘Approximately normal tests for equal predictive accuracy in nested models’, Journal of Econometrics 138(1), 291–311. Cochrane, J. H. and Piazzesi, M. (2005), ‘Bond risk premia’, The American Economic Review 95(1), 138–160. Cochrane, J. H. and Piazzesi, M. (2009), Decomposing the yield curve, Working paper, AFA 2010 At...

work page arXiv 2007

[3] [3]

and Wagner, C

Sarno, L., Schneider, P. and Wagner, C. (2016), ‘The economic value of predicting bond risk premia’, Journal of Empirical Finance 37, 247–267. Schäfer, C. and Chopin, N. (2013), ‘Sequential Monte Carlo on large binary sampling spaces’,Statistics and Computing 23(2), 163–184. Schweppe, F. C. (1965), ‘Evaluation of likelihood functions for gaussian signals’...

work page 2016

[4] [4]

(b) While φ< 1 i. If degeneracy criterion ESS(ω′′) is not triggered, whereω′′ i =ω′ i[ut([θi,α (i) t−1])]1−φ′ , setφ = 1, otherwise ﬁnd φ∈ [φ′, 1] such that ESS(ω′′′) is greater than or equal to the trigger, whereω′′′ i =ω′ i[ut([θi,α (i) t−1])]φ−φ′ , for example using bisection method, see Kantas et al. (2014). ii. Update the importance weights ωi toω′ i...

work page 2014

[5] [5]

Right column shows the factors from models LF011, ﬁrst and second row, andLF001, third row

Left column presents the factors from modelLF111. Right column shows the factors from models LF011, ﬁrst and second row, andLF001, third row. Throughout, solid lines represent posterior means and dashed lines are 95% credible intervals. 26 On Unspanned Latent Risks in Dynamic Term Structure Models Table 1: Explanatory power of principal components when ﬁt...

work page 1985

[6] [6]

The explained variables are individual latent factors Zj,t,j∈{ 1, 2, 3}, inZt, from modelsLF001,LF010, LF011 andLF111

¯R2 :Zt =a +b′Pt +et Z1,t Z2,t Z3,t LF001 0.09 LF010 0.45 LF011 0.54 0.08 LF111 0.05 0.52 0.06 This table reports in-sample ¯R2 across alternative regression speciﬁcations. The explained variables are individual latent factors Zj,t,j∈{ 1, 2, 3}, inZt, from modelsLF001,LF010, LF011 andLF111. The explanatory variables are the principal componentsPt. The sam...

work page 1985

[7] [7]

Table 2: Explanatory power gains from latent factor estimated using modelLF001, when ﬁtting excess bond returns, measured via ¯R2 over multiple prediction horizons - period: January 1985 - end of

work page 1985

[8] [8]

The explained variables are different (by maturities) excess bond returns

h\n 2Y 3Y 4Y 5Y 7Y 10Y − ¯R2(%) :rxn t,t+h =a +b′Pt +et 1 4.26 3.29 3.08 2.74 2.77 3.55 3 9.50 7.93 8.69 7.45 8.60 8.53 6 13.54 12.70 14.10 13.75 14.42 13.70 9 14.97 14.16 15.76 16.59 18.35 19.03 12 17.30 14.77 16.37 17.83 20.47 22.90 LF001 ∆(3) ¯R2(%) :rxn t,t+h =a +b′Pt +cZ3,t +et 1 0.30 0.72∗ 0.81∗ 0.99∗∗ 1.62∗∗ 1.98∗∗ 3 0.13∗ 0.29∗ 0.42∗ 0.86∗∗ 1.02∗∗...

work page 2008

[9] [9]

27 On Unspanned Latent Risks in Dynamic Term Structure Models Table 3: Explanatory power gains from latent factor estimated using modelLF010, when ﬁtting excess bond returns, measured via ¯R2 over multiple prediction horizons - period: January 1985 - end of

work page 1985

[10] [10]

The explained variables are different (by maturities) excess bond returns

h\n 2Y 3Y 4Y 5Y 7Y 10Y − ¯R2(%) :rxn t,t+h =a +b′Pt +et 1 4.26 3.29 3.08 2.74 2.77 3.55 3 9.50 7.93 8.69 7.45 8.60 8.53 6 13.54 12.70 14.10 13.75 14.42 13.70 9 14.97 14.16 15.76 16.59 18.35 19.03 12 17.30 14.77 16.37 17.83 20.47 22.90 LF010 ∆(2) ¯R2(%) :rxn t,t+h =a +b′Pt +cZ2,t +et 1 0.12 0.00 -0.18 -0.21 -0.2 -0.25 3 0.65 0.12 -0.21 -0.25 -0.22 0.04 6 3...

work page 2008

[11] [11]

Table 4: Out-of-sample statistical performance of bond excess return forecasts against the EH, measured viaR2 os ath = 1-month prediction horizon - period: January 1985 - end of

work page 1985

[12] [12]

The six forecasting models used are ATSM with different numbers of latent factors

h = 1m\n 2Y 3Y 4Y 5Y 7Y 10Y Panel A: forecasts against the EH benchmark M0 -3.84 -5.54 -4.74 -3.60 -1.54 -1.45** M1 1.30 2.90** 2.52* 1.98 2.49* 3.86** LF001 2.91** 4.35** 3.64** 3.22* 3.48* 4.70*** LF010 5.86*** 6.12*** 4.61*** 3.71** 3.00** 3.42*** LF011 5.65*** 5.30*** 3.87*** 2.79** 2.10* 2.57** LF100 2.52* 4.00** 2.58* 2.16 2.63 4.03* LF110 5.56*** 5...

work page 2008

[13] [13]

- period: January 1985 - end of

28 On Unspanned Latent Risks in Dynamic Term Structure Models Table 5: Out-of-sample economic performance of bond excess return forecasts against the EH, measured via certainty equivalent returns (%) ath = 1-month prediction horizon. - period: January 1985 - end of

work page 1985

[14] [14]

Panel A presents CERs under the ﬁrst scenario, where portfolio weights are restricted to range in the interval [-1, 2], such that investors are prevented from extreme investments. Panel B presents CERs under the second scenario, where, portfolio weights are restricted to range in the interval [-1, 5], which amounts to a maximum short-sale of 100% and a ma...

work page 1985

[15] [15]

- period: January 1985 - end of

29 On Unspanned Latent Risks in Dynamic Term Structure Models Table 6: Out-of-sample economic performance of bond excess return forecasts against model M1, measured via certainty equivalent returns (%) ath = 1-month prediction horizon. - period: January 1985 - end of

work page 1985

[16] [16]

Panel A presents CERs under the ﬁrst scenario, where portfolio weights are restricted to range in the interval [-1, 2], such that investors are prevented from extreme investments. Panel B presents CERs under the second scenario, where portfolio weights are restricted to range in the interval [-1, 5], which amounts to a maximum short-sale of 100% and a max...

work page 2021

[17] [17]

30 On Unspanned Latent Risks in Dynamic Term Structure Models Table 7: Explanatory power of macroeconomic variables when ﬁtting latent factors and their components, measured via ¯R2 - period: January 1985 - end of

work page 1985

[18] [18]

The explained variables are individual latent factors Zj,t and their components E[Zj,t|Pt] and ˜RP Z j,t,j∈ {1, 2, 3}, see (31), from modelsLF001,LF010,LF011 andLF111

¯R2 :Zj,t/E[Zj,t|Pt]/ ˜RP Z j,t =aj +b′ jMt +ej,t, j∈{ 1, 2, 3} LF001 CPI GRO F 1 F13 F8 UNR MNF M I MII MIII Z3,t 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 E[Z3,t|Pt] 0.00 0.17 0.17 0.14 0.00 0.04 0.00 0.17 0.18 0.06 ˜RP Z 3,t 0.00 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.01 0.00 LF010 CPI GRO F 1 F13 F8 UNR MNF M I MII MIII Z2,t 0.05 0.03 0.03 0.00 ...

work page 1985

[19] [19]

31 On Unspanned Latent Risks in Dynamic Term Structure Models Table 8: Signs and signiﬁcance of coefﬁcients from explanatory power regressions of latent factors and their components on macroeconomic variables - period: January 1985 - end of

work page 1985

[20] [20]

The explained variables are individual latent factors Zj,t and their componentsE[Zj,t|Pt] and ˜RP Z j,t,j∈{ 1, 2, 3}, see (31), from modelsLF001,LF010,LF011 andLF111

sign(bj) :Zj,t/E[Zj,t|Pt]/ ˜RP Z j,t =aj +bjMt +ej,t, j∈{ 1, 2, 3} LF001 CPI GRO F 1 F13 F8 UNR MNF Z3,t − − − − + + + E[Z3,t|Pt] − − ∗∗∗ −∗∗∗ −∗∗∗ + + ∗ − ˜RP Z 3,t − + + + ∗ + − + LF010 CPI GRO F 1 F13 F8 UNR MNF Z2,t +∗∗ −∗∗ −∗∗ −∗ +∗∗∗ +∗∗∗ −∗∗∗ E[Z2,t|Pt] +∗∗∗ +∗∗ + + ∗∗∗ +∗∗ +∗∗ + ˜RP Z 2,t + −∗∗∗ −∗∗∗ −∗∗∗ +∗ +∗∗∗ −∗∗∗ LF011 CPI GRO F 1 F13 F8 UNR ...

work page 1985