EU ETS Market Expectations and Rational Bubbles

Q1. What is the Fama Predictive Regression approach to testing rational bubbles, and what is its key limitation with a dynamic risk premium?

The Fama (1984) decomposition splits the futures price F_{n,t} into expected future spot price E_t[P_{t+n}] and a risk premium RP_{n,t}; from this, two predictive regressions (FPR 1 and FPR 2) have slope coefficients β₁,ₙ and β₂,ₙ that equal 1 and 0, respectively, in the absence of a rational bubble, and deviate from these values (β₁,ₙ < 1, β₂,ₙ > 0) when a bubble is present. The paper shows analytically (equations 27–33) that when a rational bubble is present, β₁,ₙ decreases monotonically as Var(B_t) increases and β₂,ₙ increases monotonically, with the direction of the bias confirmed under both zero and nonzero covariance between the bubble and the risk premium. The key limitation of standard OLS inference in this regression is that when the regressor (F_{n,t} − P_t) is highly persistent or mildly explosive, the OLS t-statistic has a non-standard distribution, and Stambaugh (1999) bias can lead to over-rejection of the no-bubble null. The paper addresses this by applying the IVX estimator, which replaces the persistent regressor with an instrument of controllable lower persistence, yielding a Wald statistic that converges to a standard chi-squared distribution regardless of the persistence or trending behavior of the risk premium.

Q2. Why does the KPSS test applied to the price-expectations differential fail in the presence of a nonstationary risk premium?

The KPSS test applied to P_{t+n} − F_{n,t} tests whether this differential is stationary; under the no-bubble null (equation 16 in the paper), the differential equals the negative risk premium RP_{n,t}, so the KPSS test has correct size when RP is stationary but incorrectly rejects the no-bubble null when RP is integrated of order one or explosive — because non-stationarity in the risk premium is incorrectly attributed to a bubble. The paper’s Monte Carlo simulations (Table 1) confirm that the KPSS test maintains nominal size of 5 percent only when the risk premium is stationary (λ ∈ {0, 0.5} under RP 1); when λ = 1 or λ = 1.01, the KPSS test rejects far more often than 5 percent under the null. The FPR approach with IVX inference, by contrast, maintains size close to 5 percent across all risk premium specifications including explosive ones. This is the central methodological motivation: prior EU ETS bubble tests that relied on KPSS may have detected non-stationarity in the risk premium rather than a genuine bubble component.

Q3. What does the SADF/GSADF test find for EU ETS spot and futures prices, and what is its role in the paper’s empirical strategy?

The paper applies SADF and GSADF tests (Phillips, Shi, and Yu, 2015a,b) to weekly EU ETS spot and futures price levels from 2013 to 2023 and finds evidence of explosive behavior at the 5 percent significance level in both series, with consistent timing of explosive phases across spot and all four futures contracts. The explosive episodes are date-stamped using the BSADF sequence with wild-bootstrapped critical values (999 repetitions): explosive periods are identified during the end of the third trading period (2018–2022) and at the commencement of the fourth trading period (2021–2023). These results confirm that the necessary condition for a rational bubble — an explosive price component — is satisfied. However, the paper emphasizes that explosiveness in levels is not sufficient for a rational bubble: a mildly explosive fundamental or an explosive risk premium would produce the same SADF/GSADF result without any bubble component. The FPR-IVX test is designed to distinguish between these cases and constitutes the primary bubble test.

Q4. What do the FPR-IVX tests find for the presence of a rational bubble?

Across the full sample (January 2018 to October 2023, T = 302), the IVX-AR Wald statistic (denoted W̃_β, adjusting for serial correlation in FPR 2’s error term using the Yang, Long, Peng, and Cai (2020) procedure) fails to reject the null β₂,ₙ = 0 against β₂,ₙ ≠ 0 for all four delivery horizons n ∈ {4, 8, 12, 16} weeks. The Bayesian Information Criterion selects models with lagged error terms for both the full sample and sub-samples, confirming the need for the IVX-AR procedure over the standard IVX Wald statistic. The sub-sample analysis separates the third trading period (January 2018 to December 2020, T = 156) and the fourth trading period (January 2021 to October 2023, T = 146); in both sub-samples the null is not rejected for all delivery horizons. The conventional OLS t-statistic (|t_β|) sometimes provides marginal evidence against the null, but the paper interprets this as reflecting the Stambaugh bias problem and defers to the IVX-AR inference. These results contradict both the collapsing bubble hypothesis (which would require β₂,ₙ < 0) and the ongoing bubble hypothesis (which would require β₂,ₙ > 0).

Q5. What do the tests on the differential between future spot rates and futures rates find?

Applying the SADF/GSADF test to the differential P_{t+n} − F_{n,t} for n ∈ {4, 8, 12, 16} weeks reveals no evidence of explosiveness in this differential across all horizons (Table 9 in the paper). This is consistent with the absence of a rational bubble: under the FPR framework, a rational bubble would generate an explosive component in the futures basis (F_{n,t} − P_t), which would in turn produce explosiveness in the differential between actual future spot prices and futures prices. The absence of explosiveness in this differential therefore provides an additional check corroborating the FPR-IVX finding.

Q6. What does the co-explosiveness test find, and why does a structural break affect the full-sample result?

The co-explosiveness test of Evripidou, Harvey, Leybourne, and Sollis (2022) tests whether spot and futures prices share a common explosive trend (null: co-explosive, no bubble) versus the alternative that they diverge by an explosive component (rational bubble or explosive risk premium). In the full sample from January 2018 to October 2023, the test rejects the null for all n ∈ {4, 8, 12, 16}, apparently indicating a non-stationary component separating spot and futures prices. However, sub-sample analysis dividing the sample at December 2021 reveals that neither the third trading period (January 2018 to December 2021) nor the fourth trading period (January 2022 to October 2023) sub-samples show rejection of the null — the co-explosive null cannot be rejected in either period alone. The paper interprets the full-sample rejection as reflecting a structural break in the risk premium at the boundary between the two trading periods (a mean shift in the risk premium) rather than a bubble, consistent with the KPSS size problem and with the lack of any significant positive serial correlation between the phases. The sub-sample co-explosiveness results align with the FPR-IVX findings.

Q7. What explains the EU ETS price surge if not a rational bubble, and what are the policy implications?

The paper interprets the consistent co-movement of spot and futures prices in an explosive common trend — without any divergence between them — as evidence that the fundamental value of allowances itself became explosive, driven by a regime shift in market expectations about future allowance scarcity. The scarcity shift is traced to two policy changes: (a) the progressive tightening of the EU ETS cap trajectory under the European Green Deal and the Fit-for-55 legislation, which reduced the total number of allowances available over time; and (b) reform of the Market Stability Reserve, which removed surplus allowances from circulation, making the cap effectively more binding than its nominal level. When market participants updated their expectations about how scarce allowances would become, the fundamental value — the present discounted value of allowance scarcity rents — rose along an explosive path without any bubble component. For policy, this distinction matters: if the price surge reflected a rational bubble, regulatory intervention to deflate it could be efficiency-improving (bubbles misallocate resources and their bursting creates financial instability); if the surge reflects genuine scarcity expectations, intervention would undermine the price signal that guides firms’ decarbonization investment decisions. The paper concludes there is no basis from historical data to justify bubble-prevention intervention in the EU ETS architecture.

Key concepts

rational bubble (in the EU ETS context): a component of the allowance price that exceeds the present discounted value of future allowance scarcity rents and grows at the discount rate; theoretically possible because allowances are storable and have positive returns from banking across periods; the paper finds no evidence of this component in EU ETS prices during 2018–2023.

Fama Predictive Regression (FPR): a regression of the basis (F_{n,t} − P_t) on itself or on subsequent spot-futures differentials, used here to test rational bubbles; FPR 2 (the regression of P_{t+n} − P_t on F_{n,t} − P_t) has slope β₂,ₙ = 0 under no bubble and β₂,ₙ > 0 under an ongoing bubble, with the direction of β₂,ₙ identifying both the presence and type (ongoing vs. collapsing) of the bubble.

IVX estimator: the instrumental-variable estimator of Kostakis, Magdalinos, and Stamatogiannis (2015) that instruments a mildly explosive or highly persistent regressor with an instrument of controllable lower persistence; produces a Wald statistic with a standard chi-squared limiting distribution regardless of the persistence or trending behavior of the risk premium, enabling valid inference on bubble hypotheses in the FPR when the risk premium is nonstationary.

IVX-AR procedure: the extension of the IVX estimator by Yang, Long, Peng, and Cai (2020) that additionally accounts for serial correlation in the error term of the predictive regression; the paper applies this as its primary inference procedure because BIC selects models with lagged errors in both full-sample and sub-sample analyses.

allowance scarcity expectations: market participants’ beliefs about the future tightness of the EU ETS cap relative to aggregate emissions; the paper finds that the price surge since 2018 is consistent with a shift in these expectations driven by cap trajectory tightening and Market Stability Reserve reform, rather than with a speculative bubble.

mildly explosive process: a time series with autoregressive root θ = 1 + c·T^{−α} for c > 0, α ∈ (0,1), converging to unity as T → ∞; used in the paper’s Monte Carlo and theoretical analysis to model the fundamental price process and the risk premium under the alternative hypothesis of ongoing rational bubble behavior, following Phillips and Magdalinos (2007).