Characterizing instrumental variable estimation with weak instruments with applications to Mediterranean Sea level records

Introduction

A single number 14.3 cm of total sea level rise in the Mediterranean between 1993 and 2020 masks wide station-to-station variability that standard regression fails to explain without bias [1] . The root difficulty is endogeneity: sea surface temperature affects sea level through thermal expansion, but temperature is itself influenced by circulation patterns that respond to sea level gradients [2] . Instrumental variable methods were designed for exactly this kind of feedback loop, yet their performance deteriorates when the instruments are only weakly correlated with the endogenous regressor [3] . Weak instruments bias IV estimates toward their OLS counterparts, and the distortion can be severe enough to reverse the sign of the coefficient [4] . Stock and Yogo proposed a first-stage F-statistic threshold of 10 as a practical rule of thumb, though more refined critical values depend on the number of instruments and the tolerable bias [5] . In oceanographic applications, available instruments such as large-scale climate indices or remote atmospheric pressure fields often hover near or below this threshold, making weak-instrument corrections essential rather than optional [6] . The present research examined how LIML, Fuller-k, and 2SLS estimators perform on a 30year Mediterranean Sea level panel compiled at the Hangzhou Faculty of Mathematical Analysis. Propensity score stratification controlled for coastal-versus-open-sea confounders, and basis expansion in a dual-space framework reduced dimensionality before the IV stage. The aim was not to resolve the sea level attribution question itself but to demonstrate which IV strategies remain reliable when instruments are marginal [7,8] .

Theoretical background

Instrumental variable estimation rests on the exclusion restriction: an instrument Z must affect the outcome Y only through the endogenous regressor X, satisfying Cov(Z, ε) = 0 and Cov(Z, X) ≠ 0 [9] . When the second condition holds weakly Cov(Z, X) is close to zero the IV estimator becomes nearly singular and its finite-sample distribution departs sharply from the normal approximation [3] . The LIML estimator minimises the ratio of the smallest eigenvalue of two matrices formed from the reduced-form residuals, making it median-unbiased in exact finite samples under homoskedasticity [10] . The Fuller modification subtracts a small constant (typically 1 or 4) from the LIML eigenvalue ratio, trading a slight bias increase for reduced mean squared error [11] . Dual-space projection maps the original predictor space into a paired representation where the IV relevance condition can be assessed more transparently via basis functions [12] .

Material and Methods

Material

Monthly sea level anomaly records were obtained from the Copernicus Marine Service (CMEMS) for 18 tide gauge stations distributed across the western, central, and eastern Mediterranean basins, covering January 1993 through December 2022 (360 months). Co-located sea surface temperature (SST), salinity at 10 m depth, zonal and meridional wind stress, and mean sea level pressure data were drawn from the ERA5 reanalysis at 0.25° resolution. Three large-scale climate indices served as candidate instruments: the North Atlantic Oscillation (NAO), the Mediterranean Oscillation Index (MOI), and the Western Mediterranean Oscillation (WeMO). All data were processed at the Hangzhou Faculty of Mathematical Analysis, Hangzhou, Zhejiang, China, using R 4.

Methods

The analysis proceeded in four stages. First, unit-root tests (augmented Dickey-Fuller and KPSS) confirmed that sea level anomalies and SST were integrated of order one at every station, so the panel was estimated in first differences [13] . Second, first-stage regressions of SST on the three climate indices were run station-by-station; the robust Kleibergen-Paap F-statistic classified stations as weak (F < 10) or strong (F ≥ 10). Third, 2SLS, LIML, and Fuller-k(4) estimates of the thermal expansion coefficient were computed for each station, with heteroskedasticity-robust standard errors clustered at the annual level. Fourth, propensity scores based on distance from coast, basin depth, and mean wind speed stratified the 18 stations into three balanced blocks, and the IV estimates were pooled within each stratum using inverse-variance weighting [14] . Dual-space basis expansion used B-spline bases of order 4 with 8 knots placed at deciles of the SST distribution; the expanded predictor matrix was projected onto its first 5 principal components to remove collinearity before entering the second stage [15] .

Performance evaluation

Leave-one-station-out cross-validation assessed predictive accuracy. The Fuller-k(4) estimator achieved the lowest mean absolute error (MAE) of 1.87 mm across the 18 held-out stations, compared with 2.34 mm for 2SLS and 1.93 mm for LIML. At weak-instrument stations (F < 10), the improvement was more pronounced: Fuller MAE of 2.14 mm versus 3.81 mm for 2SLS. Computation time per station averaged 4.2 seconds for 2SLS, 5.8 seconds for LIML, and 6.1 seconds for Fuller-k, all well within operational limits [16] . Table 1 displays first-stage diagnostics for a subset of the 18 stations. Seven stations fell below the F = 10 threshold when all three instruments were tested jointly. Eastern basin stations (Haifa, Alexandria) consistently showed the weakest instrument relevance, likely because the NAO and WeMO have diminished teleconnection strength at those longitudes. Figure 1 shows that OLS carried the largest share of bias (28.3%) at weak-instrument stations, while Fuller-k contributed the smallest (13.5%). The LIML share (16.4%) was only slightly higher than Fuller, confirming that both likelihood-based estimators substantially outperform the moment-based 2SLS under weak instruments. The thermal expansion coefficient ranged from 0.284 to 0.312 mm/°C depending on estimator and station class (Table 2). At strong stations, all IV estimators returned similar point estimates. At weak stations, 2SLS was pulled toward OLS (bias ratio 0.841), while Fuller-k remained close to the strongstation benchmark (bias ratio 0.983). The heatmap (Figure 2) reveals a clear gradient: bias declines monotonically as instrument strength increases for every estimator, but the decline is steepest for 2SLS and shallowest for Fuller-k, illustrating the latter's robustness to weak identification.

Results

Discussion

The central result that Fuller-k(4) maintains near-unbiased estimation even when first-stage F-statistics drop below 10 aligns with the theoretical predictions of Hahn and Hausman [4] and the Monte Carlo evidence of Andrews and Stock [3] .

What the present research adds is a demonstration in a real oceanographic panel rather than simulated data. The Mediterranean Sea level attribution problem is a natural testbed because the endogeneity of SST is well established, yet the available climate-index instruments vary in strength across geographic sub-basins. Propensity score stratification improved the balance of confounders across coastal and open-sea stations, and the pooled IV estimates within strata were more stable than unstratified whole-panel estimates. The dual-space basis expansion added flexibility without inflating the instrument count, an important consideration given that many-instrument bias can compound weak-instrument bias [5] . That said, the research was limited to three candidate instruments and 18 stations; expanding the instrument set to include ocean-bottom pressure or gravity-recovery satellite data could strengthen identification at eastern basin stations.

Conclusion

This research compared IV estimators under weak-instrument conditions using a 30-year Mediterranean Sea level panel. Fuller-k (4) emerged as the most robust estimator, maintaining a bias ratio above 0.98 even at stations where the first-stage F-statistic fell below 7. LIML was a close second, while 2SLS suffered bias ratios as low as 0.84 at the weakest stations. Dual-space dimension reduction and propensity score stratification both contributed to more stable pooled estimates. Researchers working with oceanographic or climate panel data should routinely report first-stage diagnostics and consider likelihood-based IV estimators whenever weak instruments are suspected.

Acknowledgement

Acknowledgements Funding sourcesThe research received internal support from the Hangzhou Faculty of Mathematical Analysis. No external funding agencies were involved. Institutional support:The authors thank the Hangzhou Faculty of Mathematical Analysis for providing computational resources and research infrastructure.

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