## cp

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### Newey-West versus Hansen-Hodrick covariance corrections
- The Hansen-Hodrick (HH, 12 lags) covariance matrix in the sample is positive definite, but the χ2(5) statistic is 811.3, which suggests near-singularity and that some linear combinations of parameters are suspiciously well measured.
- A χ2 statistic calculated using only diagonal elements (sum of squared individual t-statistics) is 113.
- Replacing HH with Newey-West with 18 lags (NW, 18 lags) produces a positive definite covariance matrix that underweights higher covariances; the χ2(5) statistic falls from 811.3 to 105.5.
- The 105.5 value remains well above the 1-percent critical value of 15.1, so the expectations hypothesis is still decisively rejected.
- The individual (unrestricted) bond regressions of Table 1B using NW, 18 lags reject zero coefficients with χ2 values near 100.
- The paper reports HH, 12 lag standard errors in subsequent tables but uses the NW, 18 lag calculation for joint test statistics.

### Alternative asymptotic and simplified calculations
- A “Simplified HH” that ignores conditional heteroskedasticity and treats correlation as only due to overlapping homoskedastic forecast errors raises standard errors by about one-third and lowers the χ2(5) statistic to 42.4 (still above the 1-percent critical value).
- Using regressions with nonoverlapping data (averaging 12 month-to-month possibilities) and correcting for heteroskedasticity yields larger standard errors and a χ2(5) statistic of 22.6.
- Conclusion: even when using larger covariance matrices (more conservative), the expectations hypothesis is rejected.

### Small-sample distributions and bootstrap simulations
- Three small-sample distributions are computed:
  - “12 lag VAR” (unrestricted 12 monthly lag VAR of all 5 yields) with bootstrapped residuals: χ2(5) ≈ 40.2; small-sample standard errors about one-third larger than asymptotic.
  - Cointegrated VAR (12 lag VAR imposing a single unit root and four cointegrating vectors): χ2(5) ≈ 38.1; similar small-sample standard errors as the 12 lag VAR.
  - Simulation under the expectations hypothesis (unrestricted 12 lag autoregression of the one-year yield, bootstrap residuals, other yields constructed as expected values): generates a conventional small-sample distribution for the χ2 test statistics; under this distribution the NW, 18 lags χ2 value of 105.5 occurs so infrequently that rejection is at the 0-percent level.
- Small-sample joint Wald tests use the covariance matrix of parameter estimates across 50,000 simulations.

### Key results from Table 1 — Single-factor model estimates
- OLS estimates of return-forecasting factor coefficients (β0 … β5, R2, χ2(5)):
  - OLS estimates: 3.24, -2.14, 0.81, 3.00, 0.80, -2.08, R2 = 0.35
  - HH, 12 lags asymptotic standard errors in parentheses: (1.45)(0.36)(0.74)(0.50)(0.45)(0.34); χ2(5) = 811.3
  - NW, 18 lags asymptotic standard errors: (1.31)(0.34)(0.69)(0.55)(0.46)(0.41); χ2(5) = 105.5
  - Simplified HH standard errors: (1.80)(0.59)(1.04)(0.78)(0.62)(0.55); χ2(5) = 42.4
  - No overlap standard errors: (1.83)(0.84)(1.69)(1.69)(1.21)(1.06); χ2(5) = 22.6
- Small-sample (Small T) distributions:
  - 12 lag VAR standard errors: (1.72)(0.60)(1.00)(0.80)(0.60)(0.58); R2 confidence interval [0.22, 0.56]; χ2(5) = 40.2
  - Cointegrated VAR standard errors: (1.88)(0.63)(1.05)(0.80)(0.60)(0.58); R2 confidence interval [0.18, 0.51]; χ2(5) = 38.1
  - “Exp. Hypo.” R2 interval: [0.00, 0.17]
- Notes: 10-percent, 5-percent and 1-percent critical values for χ2(5) are 9.2, 11.1, and 15.1 respectively. All p-values are less than 0.005. Monthly observations of annual returns, 1964–2003.

### Individual-bond regressions (Table 1B highlights)
- Reported rows (restricted and unrestricted regressions) for maturities n = 2, 3, 4, 5:
  - n=2: coefficient 0.47 with standard errors (0.03)(0.02), R2 small-sample [0.18, 0.52], Small T R2 = 0.32, Exp. Hypo. R2 [0, 0.17], χ2(5) = 121.8
  - n=3: coefficient 0.87 with standard errors (0.02)(0.02), R2 small-sample [0.21, 0.54], Small T R2 = 0.34, Exp. Hypo. R2 [0, 0.17], χ2(5) = 113.8
  - n=4: coefficient 1.24 with standard errors (0.01)(0.02), R2 small-sample [0.24, 0.57], Small T R2 = 0.37, Exp. Hypo. R2 [0, 0.17], χ2(5) = 115.7
  - n=5: coefficient 1.43 with standard errors (0.04)(0.03), R2 small-sample [0.21, 0.55], Small T R2 = 0.34, Exp. Hypo. R2 [0, 0.17], χ2(5) = 88.2
- Interpretation: Individual-bond regressions using NW, 18 lag correction also strongly reject the expectations hypothesis.

### R2 sampling error, overfitting, and robustness checks
- Sample R2 is biased upward; small-sample 95-percent confidence intervals for unadjusted R2 are reported.
- The 0.32–0.37 unrestricted R2 in Table 1B lie well above the 0.17 upper end of the 95-percent R2 confidence interval calculated under the expectations hypothesis.
- Regressions repeated using actual excess returns (levels) e r t+1 (n) − e y t (1) produce nearly identical coefficients; the “Level R2” column reports slightly higher R2 than the log regressions.
- Regressions account for 1/2 β2 terms and conditional heteroskedasticity.

### Fama-Bliss regressions (Table 2 summary)
- Updated Fama and Bliss (1987) regressions of excess returns on same-maturity forward spread:
  - Reported slope coefficients are all within one standard error of 1.0; expected excess returns move essentially one-for-one with forward spreads.
  - Table 2 entries (selected):
    - n=2: 0.99 (0.33), Small T R2 0.16, χ2(1)=18.4
    - n=3: 1.35 (0.41), Small T R2 0.17, χ2(1)=19.2
    - n=4: 1.61 (0.48), Small T R2 0.18, χ2(1)=16.4
    - n=5: 1.27 (0.64), Small T R2 0.09, χ2(1)=5.7
  - Notes: The 5-percent and 1-percent critical values for χ2(1) are 3.8 and 6.6.
- Comparison with multiple regressions: Table 1 produces stronger χ2 rejections for all maturities and more than double Fama and Bliss’s R2. The return-forecasting factor drives out Fama-Bliss spreads in multiple regressions.

### Forecasting stock returns (Table 3 summary)
- Regressions of one-year excess stock returns (value-weighted NYSE less 1-year bond yield) on the return-forecasting factor and other predictors:
  - Regression 1: coefficient on ˆf is 1.73 (t-stat 2.20), R2 = 0.07.
  - Regression 2: dividend yield (d/p) coefficient 3.30 (t-stat 1.68), R2 = 0.05.
  - Regression 3: term spread coefficient 2.84 (t-stat 1.14), R2 = 0.02.
  - Regression 4: d/p and term together: d/p 3.56 (1.80), term 3.29 (1.48), R2 = 0.08.
  - Regression 5: ˆf and term: ˆf 1.87 (2.38), term −0.58 (−0.20), R2 = 0.07.
  - Regression 6: ˆf and d/p: ˆf 1.49 (2.17), d/p 2.64 (1.39), R2 = 0.10.
  - Regression 7: all forward rates together (All f) produce R2 = 0.10.
  - Regression 8: moving average of ˆf: coefficient 2.11 (3.39), R2 = 0.12.
  - Regression 9: MA ˆf, term, d/p: ˆf 2.23 (3.86), term 1.95 (1.02), d/p −1.41 (−0.63), R2 = 0.15.
- Notes: Standard errors use the Hansen-Hodrick correction. The return-forecasting factor has a statistically significant coefficient and corresponds to a somewhat longer duration than a five-year bond (five-year bond coefficient = 1.43 on ˆf in Table 1). The R2 for stock returns (0.07) is lower than for bond returns, consistent with additional cash-flow shocks in stock returns.

### Factor models and relation to yield-curve principal components
- The return-forecasting factor ˆf is a symmetric (tent-shaped) function of forward rates and is not simply a pure slope movement; it is not fully captured by conventional yield-curve factors (level, slope, curvature).
- Expressing ˆf as a function of yields produces coefficients (denoted ˆβ* ) that show large swings at long maturities because the tent-shaped forward coefficients are multiplied by maturity in conversion to yields.
- First five principal components of yields explain 98.6, 1.4, 0.03, 0.02, and 0.01 percent of yield variance respectively.
- Those same principal components explain 9.1, 58.7, 7.6, 24.3, and 0.3 percent of the variance of ˆf respectively.
  - Interpretation: the slope factor explains 58.7 percent of ˆf variance, but the fourth factor (loading heavily on the four- to five-year yield spread) explains 24.3 percent of ˆf variance and is essential for forecasting expected returns despite being unimportant for yield variance.
- Table 4 findings:
  - Level + slope factors achieve R2 = 0.22.
  - Adding curvature raises R2 to 0.26.
  - These values remain substantially below the R2 = 0.35 achieved by ˆf, which in practice requires including the last two principal components.

### Tests of slope-only (restricted) versus unrestricted regressions
- Restricted regression specification used:
  - rx_{t+1} = a + b · slope_t + ... with q_2 generating the slope factor from yields.
  - To test joint equality of restricted coefficients a, (b q_2) to unrestricted coefficients, the authors add 3 yields to the right-hand side so the regression becomes unconstrained and exactly equal to the unrestricted regression.
  - Unconstrained form: rx_{t+1} = a + b · slope_t + c_2 y_t^{(2)} + c_3 y_t^{(3)} + c_4 y_t^{(4)} + c_5 y_t^{(5)} + ε_{t+1}.
- Hypothesis tested: c_2 through c_5 jointly equal zero.
- Main empirical conclusion:
  - The hypothesis that slope, or any combination of level, slope, and curvature, are enough to forecast excess returns is decisively rejected.
  - For all three covariance-matrix computations, χ^2 values are well above 5-percent critical values and p-values are well below 1 percent.
  - The difference between 22-percent and 35-percent R^2 is statistically significant.

### Table 4 — Excess-return forecasts using yield factors and individual yields (selected numeric outcomes)
- Reported patterns:
  - Slope: R^2 0.226 0.6; p-values 0.00; test-statistics include 22.6, 24.9, 9.5.
  - Level, slope: R^2 0.243 7.0; p-values 0.00; test-statistics 20.5, 18.6, 7.8.
  - Level, slope, curve: R^2 0.263 1.9; p-values 0.00; test-statistics 17.3, 16.7, 6.0.
  - y^{(5)} − y^{(1)}: R^2 0.158 5.5; p-values 0.00; test-statistics 30.2, 33.2, 9.5.
  - y^{(1)}, y^{(5)}: R^2 0.224 5.7; p-values 0.00; test-statistics 24.6, 22.2, 7.8.
  - y^{(1)}, y^{(4)}, y^{(5)}: R^2 0.339 9.1; p-values 0.01; test-statistics 4.6, 0.10 with 4.9, 0.09 with 6.0.
- Notes: The χ^2 test is c′0 in regressions rx_{t+1} = a + b x_t + c z_t + ε_{t+1} where x_t are indicated right-hand variables and z_t are yields such that {x_t, z_t} span all five yields.
- Interpretation:
  - Although level, slope, and curvature together explain 99.97 percent of the variance of yields, these factors alone are insufficient to forecast excess returns.
  - The return-forecasting factor is curved at the long end; slope and curvature curved at the short end do a poor job matching the unrestricted regression.
  - Tiny four- to five-year yield spread is important for forecasting all-maturity bond returns.

### Simple spreads versus the tent-shaped factor
- Simple 5–1 spread (y^{(5)} − y^{(1)}) produces R^2 = 0.15, typical of Fama-Bliss / Campbell-Shiller.
- Allowing one- and five-year yields separately raises R^2 to 0.22, but model is still rejected.
- Best three-yield combination found: one-, four-, and five-year yields:
  - Produces R^2 = 0.33.
  - Not rejected with two of the three parameter covariance matrix calculations.
  - Produces the correct pattern of one-, four-, and five-year yields consistent with the tent-shaped factor.

### Implications about factor structure and hidden state variables
- If yields followed an exact factor structure, all state variables including the return-forecasting factor f would be functions of those factors.
- Because yields do not follow an exact factor structure, an important state variable like f can be hidden in small factors often dismissed as specification errors.
- Recommendation for researchers:
  - To see expected returns, first look at expected returns and then investigate reduced-factor representations for yields.
  - A reduced-factor representation that captures expected-return facts must include the return-forecasting factor f along with yield curve factors (level, slope, curvature), even though including f barely affects fit to yields.

### Affine models: constructing a model consistent with the return regressions
- It is trivial to construct market prices of risk that generate exactly the return regressions in an affine model.
- Discount factor specified: M_{t+1} = exp(−y_t′ φ − 1/2 φ′ Σ_t φ − φ′ Σ_t^{1/2} ε_{t+1}) with ε_{t+1} normally distributed.
- One-period log excess returns obey relation linking E_t(rx_{t+1}) to φ_t and covariances.
- Choice ensuring model reproduces one-period return regression makes φ_t proportional to the return-forecasting factor f_t.
- VAR for prices corresponding to return regression: p_{t+1} = A p_t + v_{t+1}, cov(v_{t+1}) = V.
- Key points:
  - The constructed affine model exactly generates the return regression.
  - Prices generated by the model equal the state variables p_t(n).
  - This yields a discrete-time affine term structure model that reproduces the pattern of bond return predictability at an annual horizon.
- Caveats:
  - The model is a candidate pricing kernel; it is not claimed to be the true marginal utility growth process.
  - The construction does not specify higher-frequency filling-in, monthly/instantaneous market prices of risk, or continuous-time dynamics.

### Lags and measurement error — single-lag regressions
- Motivation: measurement error in prices at time t can induce spurious predictability when the same noisy price appears on both sides of regressions; using a one-month lag of forward rates can avoid this shared-noise bias if measurement error is serially uncorrelated.
- Empirical findings for single-lag regressions:
  - The tent-shaped coefficient pattern is unaltered at the first and second lags.
  - As right-hand variables are moved backward in time, the tent shape shifts slightly to the right; pattern continues through first year of lags.
  - R^2 declines slowly with horizon: from 35 percent at lag 0 to 35, 32, and 31 percent at lags 1, 2, and 3.
  - Coefficients remain highly jointly significant.
- Conclusion:
  - Serially uncorrelated measurement errors (or mild serial correlation) are not the reason for the large R^2, single-factor structure, and tent-shaped coefficients.
  - The most natural interpretation: f reflects a slow-moving underlying risk premium nearly as well revealed by f_{t−i/12} as by f_t.

### Lags and measurement error — why measurement error alone is insufficient
- Pure measurement-error models produce step-function coefficient patterns, not tents.
- In measurement-error-only world, the n-year bond excess return loads on n-year yield and the one-year yield; coefficient patterns differ across maturity.
- To produce a single-factor prediction via measurement error would require measurement errors at t and t+1 to be correlated in a very particular cross-section way—deemed far-fetched.
- Therefore measurement error conceals rather than creates the single-factor structure unless mitigated (e.g., looking at annual horizons).

### Multiple-lag regressions and single-factor across time
- Unrestricted multiple-lag regressions display similar tent shapes across maturities.
- Restricted single-factor across time specification: rx_{t+1}(n) = b_n · [ Σ_{j=0}^k δ_j f_{t−j/12} ] + ε_{t+1}(n), with Σ_{j=0}^k δ_j = 1.
- Table 5 results (selected outcomes):
  - R^2 rises from 35 to 44 percent after adding 3 additional lags.
  - δ estimates show geometric decay pattern; marginal forecast power of additional lags statistically significant out to 4 additional lags.
  - b loadings across bonds essentially unchanged as lags are added.
- Stock returns:
  - Including 3 additional lags raises R^2 for stocks from 7 percent to 12 percent.
  - Slope coefficient rises to 2.11 (corresponds to a bond substantially longer than five years).
- Conclusion:
  - Multiple lags add useful forecasting power and preserve single-factor structure across maturities.

### Failure of monthly Markovian (VAR(1)) models to capture annual predictability
- Fitting an unconstrained monthly yield VAR: y_{t+1/12} = A y_t + η_{t+1/12}, then deriving implied annual VAR y_{t+1} = A^{12} y_t + u_{t+1}.
- Coefficients implied by monthly VAR(1) are very different from direct annual regressions:
  - They resemble measurement-error step functions rather than tents.
  - No single combination of yields/forwards summarizes return forecastability across maturities.
  - Implied R^2 are lower: 22–26 percent instead of 32–37 percent.
- Interpretation:
  - Looking directly at annual horizon raises signal-to-noise and uncovers the single-factor tent-shaped pattern.
  - High-frequency Markovian specifications that omit lags understate predictability and miss single-factor structure.

### Summary and interpretation on lags
- Additional lags matter empirically, but lags are awkward as forecasting variables because bond prices are time-t expectations of future discount factors; in principle, time-t bond yields should drive out lagged yields.
- Most natural interpretation:
  - Time-t yields are sufficient state variables but observed with small, poorly time-correlated measurement errors.
  - Best estimate of the true state given noisy observations is a geometrically weighted moving average (Kalman-filter intuition), consistent with δ pattern in Table 5.
- Measurement error is not the main driver of predictability; rather, it conceals the single-factor model and reduces observed R^2 unless addressed.

### Historical evidence and pattern robustness
- Forecast using three lags: E_t(rx_{t+1}) = Σ_{j=0}^3 δ_j f_{t−j/12}; forecast using no lags nearly indistinguishable.
- Ex-post return line shifted left one year to align forecast and realization.
- Return-forecasting factor f has clear business-cycle pattern: high in troughs, low at peaks.
- f often outperforms slope-based (Fama-Bliss) forecasts in episodes including late 1960s, early 1980s, mid-1990s, 2002, 2003.
- Episode examples:
  - December 1981: flat yield curve, Fama-Bliss forecasts nothing, tent-shaped forward curve yields strong forecast via f (correct ex post).
  - April 1983: upward slope but no tent shape; f gives little expected excess return and proves correct as yields rise producing negative returns to long bonds.
  - August 1985: forward curve curves down; f predicts nearly 10-percent expected excess return (realized).
  - April 1992 vs August 1993: similar slope but April 1992 has tent shape and leads to good returns; August 1993 loses tent shape and f correctly signals to bail out.
- Conclusion:
  - The tent-shaped operator (f) often produces better forecasts than slope-only measures across key historical episodes.

### Tests of the single-factor model (GMM, Wald, J_T)
- Moments underlying unrestricted regressions: regression forecast errors multiplied by forward rates.
- Two-step estimate of single-factor model sets moments to zero; GMM standard errors computed from these moments.
- Wald test computed for joint parameter restrictions vec(b f) = vec(Π) using cov(vec(Π)).
- Table 6 — selected results:
  - Lag 0:
    - J_T = 1269; p-value 0.00 (NW, 18)
    - Wald = 3460; p-value 0.00
  - Lag 1:
    - J_T = 157; p-value 0.00 (NW, 18) but some S constructions give lower χ^2 (19.8 with p-value 0.18; 22.1 p-value 0.11).
  - Lag 2:
    - J_T = 134; p-value 0.00 (NW, 18) but other S constructions show reduced χ^2 (20.4 p-value 0.16; 22.7 p-value 0.09).
- Interpretation:
  - With no lags, tests decisively reject the single-factor model under most covariance constructions.
  - With one-month lag, evidence against single-factor model is much weaker; several tests no longer reject.

### Diagnosing the rejection — regressions of residual portfolios
- Failures characterized by forecasting portfolios rx_{t+1}(n) − b_n · rx̄_{t+1}:
  - Regression: rx_{t+1}(n) − b_n rx̄_{t+1} = ɸ̃_n′ f_t + ω_{t+1}(n).
  - Coefficients ɸ̃ are differences between unrestricted and constrained regression coefficients.
- Table 7 findings:
  - Clear diagonal pattern: when a specific mid-term yield is high relative to others, the corresponding bond return does better than the single-factor model predicts.
  - Example diagonal coefficients:
    - Two-year yield coefficient 0.80.
    - Three-year yield coefficient 2.36.
    - Four-year yield coefficient 1.84.
    - Five-year yield coefficient 0.72.
- Interpretation:
  - These idiosyncratic forecasts are statistically significant with large t-statistics and cause statistical rejection of the single-factor model with no lags.
  - The forecasts resemble the pattern induced by measurement error: each bond is forecast by a different function of yields rather than by a single factor.

### Main empirical findings on factor dominance and magnitudes
- Models using the single return-forecast factor replicate most of the variation in expected bond returns: the single-factor model explains 99.5 percent of the variance of expected returns.
- Deviations from the single-factor model are tiny:
  - The standard deviation of expected portfolio returns (˜y) ranges from 17 to 21 basis points.
  - The standard deviation of the ex post portfolio returns (lhs) ranges from 30 to 61 basis points.
  - The standard deviation of expected excess returns (bn ˜y) ranges from 1.12 to 3.45 percentage points.
  - The standard deviation of ex post excess returns ranges from 1.93 to 6 percentage points.
- Tiny returns are forecast by tiny yield spreads, but with good R2 and statistical significance.
- Portfolios isolating failures of the single-factor model are predictable and idiosyncratic; for most portfolios the first factor overwhelms smaller additional factors and forecasts are close to f_t.

### Principal components analysis of expected returns
- Eigenvalue decomposition of cov(ft, ft′):
  - Standard deviations of the principal components are 5.16, 0.26, 0.20, and 0.16 percentage points.
  - As fractions of variance, the principal components account for 99.51, 0.25, 0.15, and 0.09 percent of the total variance of expected returns.
- Conclusion: the first principal component is almost exactly the return-forecast factor f̂ and dominates the variance of expected returns; level, slope, and curvature dominate many yield-related decompositions, but the dominant component for expected excess returns is tent-shaped and poorly related to level, slope, and curvature.

### Character and significance of additional forecasting factors
- Additional factors are very small and idiosyncratic:
  - Represent small movements of bond yields and forecast small returns to the corresponding portfolios.
  - Phenomenon appears to last only one month; evidence against the single-factor model with lagged forecasting variables is much weaker.
  - Failures seem to represent one-month serially uncorrelated pricing discrepancies.
- Economic significance caveat:
  - Portfolios are zero-cost; large leveraged positions could extract substantial profits from small pricing discrepancies even with a 35-percent R2.
  - Transaction costs may eliminate practical arbitrage; the interpolated zero-coupon bond data cannot answer whether 20-basis-point pricing discrepancies survive real-world transaction costs.
- Measurement error interpretation is favored:
  - Serially uncorrelated measurement error would cause multiple lags to help forecast bond returns; would cause rejection of the single-factor model with zero lags and failure to reject with lagged right-hand variables; and would cause the idiosyncratic nature of observed forecasting failures.
  - If the failures were true pricing anomalies, one would expect non-idiosyncratic structures (factors moving bonds of all maturities).

### Evidence from portfolio regressions (Table 7 summary)
- Panel A reports coefficients and t-statistics for regressions forecasting rx_{t+1} for specific portfolio weight vectors b.
  - Example coefficient entries include: -0.11, -0.20, 0.80, -0.30, -0.66, 0.40 with t-stats (−0.75)(−1.43)(2.19)(−0.90)(−1.94)(1.68).
- Panel B regression statistics:
  - rx_{t+1}(2): R2 = 0.1541, 2(5) = 0.170, (˜y) = 0.43, (lhs) = 1.12, (bn ˜y) = 1.93
  - rx_{t+1}(3): R2 = 0.3715, 2(5) = 10.210, (˜y) = 0.34, (lhs) = 2.09, (bn ˜y) = 3.53
  - rx_{t+1}(4): R2 = 0.3319, 2(5) = 30.180, (˜y) = 0.30, (lhs) = 2.98, (bn ˜y) = 4.90
  - rx_{t+1}(5): R2 = 0.1232, 2(5) = 0.210, (˜y) = 0.61, (lhs) = 3.45, (bn ˜y) = 6.00
  - Notes: 2(5) gives the 2 statistic for joint significance of all right-hand variables (excluding constant). The 5-percent critical value is 11.07; p-values are all below 1 percent.

### Two-step estimation versus efficient GMM
- Rationale: the single-factor model is statistically rejected; goal is to estimate a good approximate model rather than an efficient estimator under the null.
- Efficient GMM properties:
  - Under the null that the single-factor model is true, efficient GMM is asymptotically efficient.
  - When the single-factor model is only approximately true, efficient GMM may overweight well-measured but economically uninteresting linear combinations of moments, producing a single factor that fits tiny, well-measured moments perfectly but misses the dominant economically interesting factor.
- Two-step procedure rationale:
  - The two-step estimation constrains GMM to pay attention to a portfolio such as 1′ rx_{t+1}, ensuring the estimated single factor explains most of the variance of expected returns.
  - The return-forecasting factor is so dominant that the exact linear combination matters little as long as GMM is prevented from overweighting special linear combinations that produce tiny but statistically significant additional factors.

### Concluding remarks and extensions
- Appendix robustness checks and additional results:
  - Return forecasts are stable across subsamples, survive in real-time estimates, generate trading rule profits, and are verified in a different dataset.
  - Appendix shows how f̂ forecasts the short rate; presents individual-bond forecasting regressions; and details the principal components analysis of expected excess returns.
  - Appendix documents computations and presents the affine model.
- Limitations and suggested extensions:
  - Analysis limited to one- to five-year maturity bonds at a one-year horizon.
  - Understanding variation across investment horizon and extending to all maturities require subtle time-series analysis, probably including explicit treatment of measurement errors.
  - Extensions require data beyond zero coupon bonds at one-year spaced maturities and integration with an explicit term structure model when returns cannot be measured directly.
  - The paper does not model time-varying second moments, so it does not address Sharpe ratios or optimal portfolios.
  - The paper does not tie time-varying premia to macroeconomic or monetary fundamentals and offers no deep interpretation of the tent-shaped function of forward rates or the four- to five-year yield spread beyond a suggestive correlation with business cycles.

*Source: excerpt from the provided PDF chapter/section.*

### 15. This value is suspiciously large. The Han-

### 15. This value is suspiciously large. The Han-

### Newey-West versus Hansen-Hodrick covariance corrections
- The Hansen-Hodrick (HH, 12 lags) covariance matrix in the sample is positive definite, but the χ2(5) statistic is 811.3, which suggests near-singularity and that some linear combinations of parameters are suspiciously well measured.
- A χ2 statistic calculated using only diagonal elements (sum of squared individual t-statistics) is 113.
- Replacing HH with Newey-West with 18 lags (NW, 18 lags) produces a positive definite covariance matrix that underweights higher covariances; the χ2(5) statistic falls from 811.3 to 105.5.
- The 105.5 value remains well above the 1-percent critical value of 15.1, so the expectations hypothesis is still decisively rejected.
- The individual (unrestricted) bond regressions of Table 1B using NW, 18 lags reject zero coefficients with χ2 values near 100.
- The paper reports HH, 12 lag standard errors in subsequent tables but uses the NW, 18 lag calculation for joint test statistics.

### Alternative asymptotic and simplified calculations
- A “Simplified HH” that ignores conditional heteroskedasticity and treats correlation as only due to overlapping homoskedastic forecast errors raises standard errors by about one-third and lowers the χ2(5) statistic to 42.4 (still above the 1-percent critical value).
- Using regressions with nonoverlapping data (averaging 12 month-to-month possibilities) and correcting for heteroskedasticity yields larger standard errors and a χ2(5) statistic of 22.6.
- Conclusion: even when using larger covariance matrices (more conservative), the expectations hypothesis is rejected.

### Small-sample distributions and bootstrap simulations
- Three small-sample distributions are computed:
  - “12 lag VAR” (unrestricted 12 monthly lag VAR of all 5 yields) with bootstrapped residuals: χ2(5) ≈ 40.2; small-sample standard errors about one-third larger than asymptotic.
  - Cointegrated VAR (12 lag VAR imposing a single unit root and four cointegrating vectors): χ2(5) ≈ 38.1; similar small-sample standard errors as the 12 lag VAR.
  - Simulation under the expectations hypothesis (unrestricted 12 lag autoregression of the one-year yield, bootstrap residuals, other yields constructed as expected values): generates a conventional small-sample distribution for the χ2 test statistics; under this distribution the NW, 18 lags χ2 value of 105.5 occurs so infrequently that rejection is at the 0-percent level.
- Small-sample joint Wald tests use the covariance matrix of parameter estimates across 50,000 simulations.

### Key results from Table 1 — Single-factor model estimates (summary of reported numbers)
- OLS estimates of return-forecasting factor coefficients (β0 … β5, R2, χ2(5)):
  - OLS estimates: 3.24, -2.14, 0.81, 3.00, 0.80, -2.08, R2 = 0.35
  - HH, 12 lags asymptotic standard errors in parentheses: (1.45)(0.36)(0.74)(0.50)(0.45)(0.34); χ2(5) = 811.3
  - NW, 18 lags asymptotic standard errors: (1.31)(0.34)(0.69)(0.55)(0.46)(0.41); χ2(5) = 105.5
  - Simplified HH standard errors: (1.80)(0.59)(1.04)(0.78)(0.62)(0.55); χ2(5) = 42.4
  - No overlap standard errors: (1.83)(0.84)(1.69)(1.69)(1.21)(1.06); χ2(5) = 22.6
- Small-sample (Small T) distributions:
  - 12 lag VAR standard errors: (1.72)(0.60)(1.00)(0.80)(0.60)(0.58); R2 confidence interval [0.22, 0.56]; χ2(5) = 40.2
  - Cointegrated VAR standard errors: (1.88)(0.63)(1.05)(0.80)(0.60)(0.58); R2 confidence interval [0.18, 0.51]; χ2(5) = 38.1
  - “Exp. Hypo.” R2 interval: [0.00, 0.17]
- Notes: 10-percent, 5-percent and 1-percent critical values for χ2(5) are 9.2, 11.1, and 15.1 respectively. All p-values are less than 0.005. Monthly observations of annual returns, 1964–2003.

### Individual-bond regressions (Table 1B highlights)
- Reported rows (restricted and unrestricted regressions) for maturities n = 2, 3, 4, 5 (summary entries):
  - For n=2: coefficient 0.47 with standard errors (0.03)(0.02), R2 small-sample [0.18, 0.52], Small T R2 = 0.32, Exp. Hypo. R2 [0, 0.17], χ2(5) = 121.8
  - For n=3: coefficient 0.87 with standard errors (0.02)(0.02), R2 small-sample [0.21, 0.54], Small T R2 = 0.34, Exp. Hypo. R2 [0, 0.17], χ2(5) = 113.8
  - For n=4: coefficient 1.24 with standard errors (0.01)(0.02), R2 small-sample [0.24, 0.57], Small T R2 = 0.37, Exp. Hypo. R2 [0, 0.17], χ2(5) = 115.7
  - For n=5: coefficient 1.43 with standard errors (0.04)(0.03), R2 small-sample [0.21, 0.55], Small T R2 = 0.34, Exp. Hypo. R2 [0, 0.17], χ2(5) = 88.2
- Interpretation: Individual-bond regressions using NW, 18 lag correction also strongly reject the expectations hypothesis.

### R2 sampling error, overfitting, and robustness checks
- Sample R2 is biased upward; small-sample 95-percent confidence intervals for unadjusted R2 are reported.
- The 0.32–0.37 unrestricted R2 in Table 1B lie well above the 0.17 upper end of the 95-percent R2 confidence interval calculated under the expectations hypothesis.
- Regressions repeated using actual excess returns (levels) e r t+1 (n) − e y t (1) produce nearly identical coefficients; the “Level R2” column reports slightly higher R2 than the log regressions.
- Regressions account for 1/2 β2 terms and conditional heteroskedasticity.

### Fama-Bliss regressions (Table 2 summary)
- Updated Fama and Bliss (1987) regressions of excess returns on same-maturity forward spread:
  - Reported slope coefficients are all within one standard error of 1.0; expected excess returns move essentially one-for-one with forward spreads.
  - Table 2 entries (selected): for maturities n = 2, 3, 4, 5 the regression coefficients and standard errors (in parentheses) and χ2(1) statistics: 
    - n=2: 0.99 (0.33), Small T R2 0.16, χ2(1)=18.4
    - n=3: 1.35 (0.41), Small T R2 0.17, χ2(1)=19.2
    - n=4: 1.61 (0.48), Small T R2 0.18, χ2(1)=16.4
    - n=5: 1.27 (0.64), Small T R2 0.09, χ2(1)=5.7
  - Notes: Standard errors in parentheses; probability values in angled brackets. The 5-percent and 1-percent critical values for χ2(1) are 3.8 and 6.6.
- Comparison with multiple regressions: Table 1 produces stronger χ2 rejections for all maturities and more than double Fama and Bliss’s R2. The return-forecasting factor drives out Fama-Bliss spreads in multiple regressions.

### Forecasting stock returns (Table 3 summary)
- Regressions of one-year excess stock returns (value-weighted NYSE less 1-year bond yield) on the return-forecasting factor and other predictors:
  - Regression 1: coefficient on ˆf is 1.73 (t-stat 2.20), R2 = 0.07.
  - Regression 2: dividend yield (d/p) coefficient 3.30 (t-stat 1.68), R2 = 0.05.
  - Regression 3: term spread coefficient 2.84 (t-stat 1.14), R2 = 0.02.
  - Regression 4: d/p and term together: d/p 3.56 (1.80), term 3.29 (1.48), R2 = 0.08.
  - Regression 5: ˆf and term: ˆf 1.87 (2.38), term −0.58 (−0.20), R2 = 0.07.
  - Regression 6: ˆf and d/p: ˆf 1.49 (2.17), d/p 2.64 (1.39), R2 = 0.10.
  - Regression 7: all forward rates together (All f) produce R2 = 0.10.
  - Regression 8: moving average of ˆf: coefficient 2.11 (3.39), R2 = 0.12.
  - Regression 9: MA ˆf, term, d/p: ˆf 2.23 (3.86), term 1.95 (1.02), d/p −1.41 (−0.63), R2 = 0.15.
- Notes: Standard errors use the Hansen-Hodrick correction. The return-forecasting factor has a statistically significant coefficient and corresponds to a somewhat longer duration than a five-year bond (five-year bond coefficient = 1.43 on ˆf in Table 1). The R2 for stock returns (0.07) is lower than for bond returns, consistent with additional cash-flow shocks in stock returns.

### Factor models and relation to yield-curve principal components
- The return-forecasting factor ˆf is a symmetric (tent-shaped) function of forward rates and is not simply a pure slope movement; it is not fully captured by conventional yield-curve factors (level, slope, curvature).
- Expressing ˆf as a function of yields produces coefficients (denoted ˆβ* ) that show large swings at long maturities because the tent-shaped forward coefficients are multiplied by maturity in conversion to yields.
- First five principal components of yields explain 98.6, 1.4, 0.03, 0.02, and 0.01 percent of yield variance respectively.
- Those same principal components explain 9.1, 58.7, 7.6, 24.3, and 0.3 percent of the variance of ˆf respectively.
  - Interpretation: the slope factor explains 58.7 percent of ˆf variance (hence slope forecasts returns in single regressions), but the fourth factor (loading heavily on the four- to five-year yield spread) explains 24.3 percent of ˆf variance and is essential for forecasting expected returns despite being unimportant for yield variance.
- Table 4 (referenced) asks whether yield-curve factors in place of ˆf can forecast bond excess returns:
  - Level + slope factors achieve R2 = 0.22.
  - Adding curvature raises R2 to 0.26.
  - These values remain substantially below the R2 = 0.35 achieved by ˆf, which in practice requires including the last two principal components.

*Source: excerpt from the provided PDF chapter/section.*

### 4. We start with the slope factor alone. We run

### cp - 4. We start with the slope factor alone. We run

### Tests of slope-only (restricted) versus unrestricted regressions
- Restricted regression specification used:
  - rx_{t+1} = a + b · slope_t + ... with q_2 generating the slope factor from yields.
  - To test joint equality of restricted coefficients a, (b q_2) to unrestricted coefficients, the authors add 3 yields to the right-hand side so the regression becomes unconstrained and exactly equal to the unrestricted regression.
  - Unconstrained form shown (equation (4)):
    - rx_{t+1} = a + b · slope_t + c_2 y_t^{(2)} + c_3 y_t^{(3)} + c_4 y_t^{(4)} + c_5 y_t^{(5)} + ε_{t+1}.
- Hypothesis tested: c_2 through c_5 jointly equal zero (i.e., restricted coefficients suffice).
- Main empirical conclusion:
  - The hypothesis that slope, or any combination of level, slope, and curvature, are enough to forecast excess returns is decisively rejected.
  - For all three covariance-matrix computations, χ^2 values are well above 5-percent critical values and p-values are well below 1 percent.
  - The difference between 22-percent and 35-percent R^2 is statistically significant.

### Table 4 — Excess-return forecasts using yield factors and individual yields (selected numeric outcomes and patterns)
- Reported R^2 and test-statistic / p-value patterns (as presented):
  - Slope: R^2 0.226 0.6; p-values reported as 0.00 with test-statistics including 22.6, 24.9, 9.5 (as in the table cell formatting).
  - Level, slope: R^2 0.243 7.0; p-values 0.00 with test-statistics 20.5, 18.6, 7.8.
  - Level, slope, curve: R^2 0.263 1.9; p-values 0.00 with test-statistics 17.3, 16.7, 6.0.
  - y^{(5)} − y^{(1)}: R^2 0.158 5.5; p-values 0.00 with test-statistics 30.2, 33.2, 9.5.
  - y^{(1)}, y^{(5)}: R^2 0.224 5.7; p-values 0.00 with test-statistics 24.6, 22.2, 7.8.
  - y^{(1)}, y^{(4)}, y^{(5)}: R^2 0.339 9.1; p-values 0.01 with test-statistics 4.6, 0.10 with 4.9, 0.09 with 6.0.
- Notes associated with Table 4:
  - The χ^2 test is c′0 in regressions rx_{t+1} = a + b x_t + c z_t + ε_{t+1} where x_t are indicated right-hand variables and z_t are yields such that {x_t, z_t} span all five yields.
- Interpretation:
  - Although level, slope, and curvature together explain 99.97 percent of the variance of yields, these factors alone are insufficient to forecast excess returns.
  - The return-forecasting factor is curved at the long end; slope and curvature curved at the short end do a poor job matching the unrestricted regression.
  - Tiny four- to five-year yield spread is important for forecasting all-maturity bond returns.

### Simple spreads versus the tent-shaped factor
- Simple 5–1 spread (y^{(5)} − y^{(1)}) produces R^2 = 0.15, typical of Fama-Bliss / Campbell-Shiller.
- Allowing one- and five-year yields separately raises R^2 to 0.22, but model is still rejected.
- Best three-yield combination found: one-, four-, and five-year yields:
  - Produces R^2 = 0.33.
  - Not rejected with two of the three parameter covariance matrix calculations.
  - Produces the correct pattern of one-, four-, and five-year yields consistent with the tent-shaped factor.

### Implications about factor structure and hidden state variables
- If yields followed an exact factor structure, all state variables including the return-forecasting factor f would be functions of those factors.
- Because yields do not follow an exact factor structure, an important state variable like f can be hidden in small factors often dismissed as specification errors.
- Recommendation for researchers:
  - To see expected returns, first look at expected returns and then investigate reduced-factor representations for yields.
  - A reduced-factor representation that captures expected-return facts must include the return-forecasting factor f along with yield curve factors (level, slope, curvature), even though including f barely affects fit to yields.

### Affine models: constructing a model consistent with the return regressions
- It is trivial to construct market prices of risk that generate exactly the return regressions in an affine model.
- Discount factor specified (equation (6)):
  - M_{t+1} = exp(−y_t′ φ − 1/2 φ′ Σ_t φ − φ′ Σ_t^{1/2} ε_{t+1}) with ε_{t+1} normally distributed (notation as in text).
- One-period log excess returns obey relation (7), linking E_t(rx_{t+1}) to φ_t and covariances.
- Choice ensuring model reproduces one-period return regression (equation (8)):
  - φ_t = −1 − 1/2 diag(·) + (terms making φ_t proportional to the return-forecasting factor f_t).
- VAR for prices corresponding to return regression (equation (9)):
  - p_{t+1} = A p_t + v_{t+1}, cov(v_{t+1}) = V.
- Key points:
  - The constructed affine model exactly generates the return regression (5).
  - Prices generated by the model equal the state variables p_t(n).
  - This yields a discrete-time affine term structure model that reproduces the pattern of bond return predictability at an annual horizon.
- Caveats:
  - The model is a candidate pricing kernel; it is not claimed to be the true marginal utility growth process.
  - The construction does not specify higher-frequency filling-in, monthly/instantaneous market prices of risk, or continuous-time dynamics; those remain open/model-specific choices.

### Lags and measurement error — single-lag regressions
- Motivation: measurement error in prices at time t can induce spurious predictability when the same noisy price appears on both sides of regressions; using a one-month lag of forward rates can avoid this shared-noise bias if measurement error is serially uncorrelated.
- Empirical findings for single-lag regressions:
  - The tent-shaped coefficient pattern is unaltered at the first and second lags.
  - As right-hand variables are moved backward in time, the tent shape shifts slightly to the right; pattern continues through first year of lags.
  - R^2 declines slowly with horizon: from 35 percent at lag 0 to 35, 32, and 31 percent at lags 1, 2, and 3.
  - Coefficients remain highly jointly significant.
- Conclusion:
  - Serially uncorrelated measurement errors (or mild serial correlation) are not the reason for the large R^2, single-factor structure, and tent-shaped coefficients.
  - The most natural interpretation: f reflects a slow-moving underlying risk premium nearly as well revealed by f_{t−i/12} as by f_t.

### Lags and measurement error — why measurement error alone is insufficient
- Pure measurement-error models produce step-function coefficient patterns, not tents (Figure 4).
  - In measurement-error-only world, the n-year bond excess return loads on n-year yield and the one-year yield; coefficient patterns differ across maturity.
- To produce a single-factor prediction via measurement error would require measurement errors at t and t+1 to be correlated in a very particular cross-section way—deemed far-fetched.
- Therefore measurement error conceals rather than creates the single-factor structure unless mitigated (e.g., looking at annual horizons).

### Multiple-lag regressions and single-factor across time
- Unrestricted multiple-lag regressions display similar tent shapes across maturities.
- Restricted single-factor across time specification (equation (10) and (11)):
  - rx_{t+1}(n) = b_n · [ Σ_{j=0}^k δ_j f_{t−j/12} ] + ε_{t+1}(n), with Σ_{j=0}^k δ_j = 1.
- Table 5 results (selected numeric outcomes):
  - R^2 rises from 35 to 44 percent after adding 3 additional lags.
  - δ estimates show geometric decay pattern; marginal forecast power of additional lags statistically significant out to 4 additional lags.
  - b loadings across bonds essentially unchanged as lags are added.
- Stock returns:
  - Including 3 additional lags raises R^2 for stocks from 7 percent to 12 percent (Table 3, row 8).
  - Slope coefficient rises to 2.11 (interpretation: corresponds to a bond substantially longer than five years).
- Conclusion:
  - Multiple lags add useful forecasting power and preserve single-factor structure across maturities.

### Failure of monthly Markovian (VAR(1)) models to capture annual predictability
- Fitting an unconstrained monthly yield VAR: y_{t+1/12} = A y_t + η_{t+1/12}, then deriving implied annual VAR y_{t+1} = A^{12} y_t + u_{t+1}.
- Coefficients implied by monthly VAR(1) are very different from direct annual regressions:
  - They resemble measurement-error step functions rather than tents (Figure 5).
  - No single combination of yields/forwards summarizes return forecastability across maturities.
  - Implied R^2 are lower: 22–26 percent instead of 32–37 percent.
- Interpretation:
  - Looking directly at annual horizon raises signal-to-noise and uncovers the single-factor tent-shaped pattern.
  - High-frequency Markovian specifications that omit lags understate predictability and miss single-factor structure.

### Summary and interpretation on lags
- Additional lags matter empirically, but lags are awkward as forecasting variables because bond prices are time-t expectations of future discount factors; in principle, time-t bond yields should drive out lagged yields.
- Most natural interpretation:
  - Time-t yields are sufficient state variables but observed with small, poorly time-correlated measurement errors.
  - Best estimate of the true state given noisy observations is a geometrically weighted moving average (Kalman-filter intuition), consistent with δ pattern in Table 5.
- Measurement error is not the main driver of predictability; rather, it conceals the single-factor model and reduces observed R^2 unless addressed.

### Historical evidence and pattern robustness
- Figure 6 shows forecasted and actual average (across maturity) excess returns, with:
  - Forecast using three lags: E_t(rx_{t+1}) = Σ_{j=0}^3 δ_j f_{t−j/12}; forecast using no lags nearly indistinguishable.
  - Ex-post return line shifted left one year to align forecast and realization.
- Findings from historical episodes:
  - Return-forecasting factor f has clear business-cycle pattern: high in troughs, low at peaks.
  - f often outperforms slope-based (Fama-Bliss) forecasts in episodes including late 1960s, early 1980s, mid-1990s, 2002, 2003.
  - Examples:
    - December 1981: flat yield curve, Fama-Bliss forecasts nothing, but tent-shaped forward curve yields strong forecast via f (correct ex post).
    - April 1983: upward slope but no tent shape; f gives little expected excess return and proves correct as yields rise producing negative returns to long bonds.
    - August 1985: similar level and slope to April 1983, but forward curve curves down; f predicts nearly 10-percent expected excess return (realized).
    - April 1992 vs August 1993: similar slope but April 1992 has tent shape and leads to good returns; August 1993 loses tent shape and f correctly signals to bail out.
- Conclusion:
  - The tent-shaped operator (f) often produces better forecasts than slope-only measures across key historical episodes.

### Tests of the single-factor model (GMM, Wald, J_T)
- Moments underlying unrestricted regressions (equation (13)): regression forecast errors multiplied by forward rates.
- Two-step estimate of single-factor model sets moments (14)-(15) to zero; GMM standard errors computed from these moments.
- Wald test computed for joint parameter restrictions vec(b f) = vec(Π) using cov(vec(Π)).
- Table 6 — tests of single-factor model rx_{t+1} = b Σ_{j=0}^k δ_j f_{t−j/12} + ε_{t+1} against unrestricted model. Selected results (as presented):
  - Lag 0:
    - J_T = 1269; p-value 0.00 (NW, 18 construction)
    - Wald = 3460; p-value 0.00
    - Other S-matrix constructions also give large χ^2 and p-value 0.00 (e.g., 110, 87, 174 reported in cells).
  - Lag 1:
    - J_T = 157; p-value 0.00 (NW, 18) but some S constructions give lower χ^2 (19.8 with p-value 0.18; 22.1 p-value 0.11); small-sample and no-overlap columns still show rejection in some cases.
  - Lag 2:
    - J_T = 134; p-value 0.00 (NW, 18) but other S constructions show reduced χ^2 (20.4 p-value 0.16; 22.7 p-value 0.09).
- Interpretation:
  - With no lags, tests decisively reject the single-factor model under most covariance constructions.
  - With one-month lag, evidence against single-factor model is much weaker; several tests no longer reject.

### Diagnosing the rejection — regressions of residual portfolios
- The authors characterize failures of the single-factor model by forecasting portfolios rx_{t+1}(n) − b_n · rx̄_{t+1}:
  - Regression form (16): rx_{t+1}(n) − b_n rx̄_{t+1} = ɸ̃_n′ f_t + ω_{t+1}(n).
  - Coefficients ɸ̃ are differences between unrestricted and constrained regression coefficients.
- Table 7 (discussion of results):
  - A clear diagonal pattern emerges: when a specific mid-term yield (e.g., two-, three-, four-, five-year) is high relative to others, the corresponding bond return does better than the single-factor model predicts.
  - Example diagonal coefficients (as reported in text):
    - Two-year yield coefficient 0.80 (first row diagonal emphasized).
    - Three-year yield coefficient 2.36.
    - Four-year yield coefficient 1.84.
    - Five-year yield coefficient 0.72.
- Interpretation:
  - These idiosyncratic forecasts are statistically significant with large t-statistics.
  - They cause the statistical rejection of the single-factor model with no lags.
  - The forecasts resemble the pattern induced by measurement error: each bond is forecast by a different function of yields rather than by a single factor.

*Source: cp - 4. We start with the slope factor alone. We run*

### 0.37  are  at  least  as  good  as  the  Fama-Bliss

### cp - 0.37  are  at  least  as  good  as  the  Fama-Bliss

### Main empirical findings
- Models using the single return-forecast factor replicate most of the variation in expected bond returns: the single-factor model explains 99.5 percent of the variance of expected returns.
- Deviations from the single-factor model are tiny:
  - The standard deviation of expected portfolio returns (˜y) ranges from 17 to 21 basis points.
  - The standard deviation of the ex post portfolio returns (lhs) ranges from 30 to 61 basis points.
  - By contrast, the standard deviation of expected excess returns (bn ˜y) ranges from 1.12 to 3.45 percentage points.
  - The standard deviation of ex post excess returns ranges from 1.93 to 6 percentage points.
- Tiny returns are forecast by tiny yield spreads, but with good R2 and statistical significance.
- Portfolios constructed to isolate failures of the single-factor model are predictable and exhibit patterns nothing like the dominant factor; however, for most other portfolios the first factor overwhelms smaller additional factors and forecasts are close to f_t.

### Principal components analysis of expected returns
- Eigenvalue decomposition of cov(ft, ft′) shows the first principal component is almost exactly the return-forecast factor f̂, with bond returns loading on it nearly exactly as in earlier b loadings.
- Standard deviations of the principal components are 5.16, 0.26, 0.20, and 0.16 percentage points.
- As fractions of variance, the principal components account for 99.51, 0.25, 0.15, and 0.09 percent of the total variance of expected returns.
- Conclusion: the first factor, f̂, dominates the variance of expected returns; level, slope, and curvature dominate many yield-related covariance decompositions, but the dominant component for expected excess returns is a tent-shaped f̂ poorly related to level, slope, and curvature.

### Character and significance of additional forecasting factors
- The additional factors are very small and idiosyncratic:
  - They represent small movements of bond yields and forecast small returns to the corresponding portfolios.
  - The phenomenon appears to last only one month; evidence against the single-factor model with lagged forecasting variables is much weaker.
  - Failures seem to represent one-month serially uncorrelated pricing discrepancies.
- Economic significance caveat:
  - Portfolios are zero-cost; large leveraged positions could extract substantial profits from small pricing discrepancies even with a 35-percent R2.
  - Transaction costs may eliminate practical arbitrage; the interpolated zero-coupon bond data cannot answer whether 20-basis-point pricing discrepancies survive real-world transactions costs.
- Measurement error interpretation is favored:
  - Serially uncorrelated measurement error would cause multiple lags to help forecast bond returns; would cause rejection of the single-factor model with zero lags and failure to reject with lagged right-hand variables; and would cause the idiosyncratic nature of the observed forecasting failures.
  - If the failures were true pricing anomalies, one would expect non-idiosyncratic structures (factors moving bonds of all maturities).
  - Additional factors in expected returns cannot induce the non-Markovian structure that additional lags appear to help forecast returns.

### Evidence from portfolio regressions (Table 7 summary)
- Panel A reports coefficients and t-statistics for regressions forecasting rx_{t+1} for specific portfolio weight vectors b (rows correspond to rx_{t+1}(2) through rx_{t+1}(5) regressions shown).
  - Example coefficient entries include: -0.11, -0.20, 0.80, -0.30, -0.66, 0.40 with t-stats (−0.75)(−1.43)(2.19)(−0.90)(−1.94)(1.68).
  - Other reported coefficients and t-stats appear in the table for rx_{t+1}(3), rx_{t+1}(4), rx_{t+1}(5).
- Panel B regression statistics:
  - R2 and 2(5) statistics for the left-hand variables (rx_{t+1}(2) through rx_{t+1}(5)):
    - rx_{t+1}(2)  R2 = 0.1541, 2(5) = 0.170, (˜y) = 0.43, (lhs) = 1.12, (bn ˜y) = 1.93
    - rx_{t+1}(3)  R2 = 0.3715, 2(5) = 10.210, (˜y) = 0.34, (lhs) = 2.09, (bn ˜y) = 3.53
    - rx_{t+1}(4)  R2 = 0.3319, 2(5) = 30.180, (˜y) = 0.30, (lhs) = 2.98, (bn ˜y) = 4.90
    - rx_{t+1}(5)  R2 = 0.1232, 2(5) = 0.210, (˜y) = 0.61, (lhs) = 3.45, (bn ˜y) = 6.00
  - Notes: 2(5) gives the 2 statistic for joint significance of all right-hand variables (excluding constant). The 5-percent critical value is 11.07; p-values are all below 1 percent.
  - The (˜y) column gives the standard deviation of the right-hand side variables; (lhs) gives the standard deviation of the left-hand variables; (bn ˜y) gives the standard deviation of the single-factor model forecast of rx_{t+1}(n); (rx_{t+1}(n)) gives the standard deviation of that left-hand variable.

### Two-step estimation versus efficient GMM
- Rationale: the single-factor model is statistically rejected; the goal is to estimate a good approximate model rather than an efficient estimator under the null of an exact model.
- Efficient GMM properties:
  - Under the null that the single-factor model is true, efficient GMM (min_{b,φ} g_T(b,φ)′ S^{-1} g_T(b,φ)) produces asymptotically efficient estimates.
  - When the single-factor model is only approximately true, efficient GMM may focus on well-measured but economically uninteresting linear combinations of moments (guided by S), thereby producing a single factor that fits tiny, well-measured moments perfectly but misses the dominant economically interesting factor.
  - Example: if a particular linear combination of returns is exactly measured (100 percent R2), efficient GMM will fit that moment exactly and may ignore the factor that explains 99.5 percent of expected-return variance.
- Two-step procedure rationale:
  - The two-step estimation constrains GMM to pay attention to a portfolio such as 1′ rx_{t+1}, ensuring the estimated single factor explains most of the variance of expected returns.
  - The return-forecasting factor is so dominant that the exact linear combination matters little as long as GMM is prevented from overweighting special linear combinations with b′ ≠ 0 that produce tiny but statistically significant additional factors.

### Concluding remarks and extensions
- Appendix contains many robustness checks and additional results:
  - Return forecasts are stable across subsamples, survive in real-time estimates, generate trading rule profits, and are verified in a different dataset.
  - Appendix shows how f̂ forecasts the short rate; presents individual-bond forecasting regressions; and details the principal components analysis of expected excess returns.
  - Appendix documents computations and presents the affine model.
- Limitations and suggested extensions:
  - Analysis limited to one- to five-year maturity bonds at a one-year horizon.
  - Understanding variation across investment horizon and extending to all maturities require subtle time-series analysis, probably including explicit treatment of measurement errors.
  - Extensions require data beyond zero coupon bonds at one-year spaced maturities and integration with an explicit term structure model when returns cannot be measured directly.
  - The paper does not model time-varying second moments (variances or covariances), so it does not address Sharpe ratios or optimal portfolios.
  - The paper does not tie time-varying premia to macroeconomic or monetary fundamentals and offers no deep interpretation of the tent-shaped function of forward rates or the four- to five-year yield spread beyond a suggestive correlation with business cycles.

*Source: cp - 0.37  are  at  least  as  good  as  the  Fama-Bliss (PDF chapter/section).*

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_Source: http://www.stanford.edu/~piazzesi/cp.pdf_