Expectations hypothesis — cross-market evidence
The pure expectations hypothesis (PEH) of the term structure fails on the Polish sovereign curve, but the way it fails is very different from the US and euro-area benchmarks. The figures below reproduce the headline empirical results of the EH paper.
Read the short — "The PEH dies in three different ways"How to read this page (methodology and glossary)
The pure expectations hypothesis (PEH) says that long yields are unbiased averages of expected future short rates, so any term premium is zero in expectation. Three classical regression batteries test this on a common monthly panel: the Fama-Bliss type-1 cross-sectional slope, the Campbell-Shiller log-yield-spread slope, and the Cochrane-Piazzesi forward-rate single-factor projection. We run each one on PL, US and EA on the same nine forecast horizons by ten maturities grid so the cross-market comparison is on identical footing. Inference is reported under both asymptotic Newey-West HAC and a wild block bootstrap (Bauer-Hamilton 2018) so the reader can see how much each rejection rests on the asymptotic approximation.
Glossary
| PEH | Pure expectations hypothesis: long yields equal averages of expected short rates plus a constant, so the term premium is zero in expectation. |
| FB1 | Fama-Bliss type-1 regression. Slope β from yn(t+h) − yn(t) = α + β · [n/(n−h) · (yn(t) − yh(t))] + ε. Under PEH, β = 1. |
| CS | Campbell-Shiller log-yield-spread regression. Tests the same restriction in spread-of-yields space. |
| CP | Cochrane-Piazzesi single-factor projection of excess returns on a linear combination of forward rates. |
| NW HAC | Newey-West heteroskedasticity- and autocorrelation-consistent standard errors. Asymptotic but biased downward in small samples with persistent regressors. |
| Bootstrap | Wild block bootstrap of the regression residuals; block length tuned via Politis-White. Less anti-conservative under near-unit-root regressors. |
| MCS | Model confidence set (Hansen-Lunde-Nason). Used in the systems comparison to identify the set of forecasting systems statistically indistinguishable from the best. |
| Reject | A cell is marked "reject" when both the NW HAC t-test and the bootstrap p-value reject H0: β = 1 at 5 percent. |
What each panel shows
FB1 cross-market heatmaps. Slope coefficient β on the (h, n) grid for the three markets. Negative cells in blue, positive in red, zero white. PL is uniformly close to zero or mildly negative, US mildly positive at long horizons, EA strongly positive across the long-end grid.
CS / CP companion plots. Same construction in the spread-of-yields and the forward-curve specifications.
Rejection tables. Per-market count of cells rejecting PEH at 5 percent under NW HAC and under the wild block bootstrap. The gap between the two columns is the asymptotic-vs-finite-sample wedge.
The headline asymmetry: EA looks like a strong PEH rejection under asymptotic NW but only marginal under bootstrap (Bauer-Hamilton concern), PL drifts mildly anti-PEH on both, US sits roughly consistent with PEH at long horizons.
Fama-Bliss type-1 slope coefficients across markets
Slope coefficient β from the FB1 regression y_n(t+h) − y_n(t) = α + β · [n/(n−h) · (y_n(t) − y_h(t))] + ε on the (h, n) grid for the three markets, 2005:01 to 2026:04. Negative cells are blue, positive cells are red, zero is white. The Polish panel is uniformly close to zero or mildly negative, the US panel mildly positive at long horizons, the euro-area panel strongly positive at long horizons.
PL
US
EA
Under the pure expectations hypothesis the population value of β is 1.0. The Polish panel sits closer to zero than to one on every cell, the US panel approaches one only at horizons of 36 months and longer, and the euro-area panel overshoots one across the whole long-end grid. The geometry is the cross-market evidence against PEH.
Macro-spanning regressions, median R² by horizon
For each forecasting horizon h, the figure shows the median in-sample R² across the 32 (h, n) pairs for three nested specifications run on Polish data, 2010:01 to 2026:04. The grey bar is the Fama-Bliss type-1 regressor only, the blue bar is the four-element macro vector (CPI YoY, registered unemployment, EUR/PLN exchange-rate volatility, change in NBP reference rate), the green bar is the joint specification.
Three patterns are visible. (1) The Fama-Bliss regressor alone delivers a median R² below 5% at every horizon — the forward-spot spread carries essentially no predictive content for Polish excess returns over this sample. (2) The macro-only R² rises monotonically from 13% at h=6 months to 84% at h=60 months. (3) The joint specification adds little beyond macro-only at long horizons. The marginal contribution of FB above macro is at most six percentage points (at h=36 months) and below one percentage point at h=60 months.
Cochrane-Piazzesi single-factor regressions
Second-stage one-year-ahead excess-return regressions on the CP single factor (one-year-ahead forward-rate combination from the first-stage tent). Coefficients should rise monotonically with maturity if a single factor prices the term structure.
Out-of-sample STV — pure-EH path versus random walk
Diebold-Mariano test of the pure-expectations-hypothesis short-rate path against a random-walk benchmark, h months ahead, three markets. Negative DM with low p-value means the EH path beats the random walk; positive DM means the random walk wins.
FB1 — PL detailed table
Top-20 (n, h) cells from the PL FB1 panel sorted by |β − 1|. Highlighted rows are PEH rejections at 5% Newey-West.
| h (months) | n (months) | n_obs | α | SE(α) | p(α) | β | SE(β) | p(β=1) | R² | NW reject |
|---|
Methodology: FB1 = Fama-Bliss (1987) yield-change regression; CP = Cochrane-Piazzesi (2005) single-factor regression; OOS-STV = expanding-window short-rate prediction comparison à la Sarno-Thornton-Valente (2007); macro-spanning is the Fama-Bliss regressor augmented with a four-element macro vector. All standard errors are Newey-West with bandwidth = h months. See the EH short above for a 2-page interpretation, or the about page for the full working paper.