Cont.
The growing level of confidence regarding attribution of climate change to GHG’s expressed by the IPCC and others over the past two decades rests principally on the many studies that employ the AT99 method, including the RC test. The methodology is still in wide use, albeit with a couple of minor changes that don’t address the flaws identified in my critique. (Total Least Squares or TLS, for instance, introduces new biases and problems which I analyze elsewhere; and regularization methods to obtain a matrix inverse do not fix the underlying theoretical flaws). There have been a small number of attribution papers using other methods, including ones which the TAR mentioned. “Temporal” or time series analyses have their own flaws which I will address separately (put briefly, regressing I(0) temperatures on I(1) forcings creates obvious problems of interpretation).

The Gauss-Markov (GM) Theorem

As with regression methods generally, everything in this discussion centres on the GM Theorem. There are two GM conditions that a regression model needs to satisfy to be BLUE. The first, called homoskedasticity, is that the error variances must be constant across the sample. The second, called conditional independence, is that the expected values of the error terms must be independent of the explanatory variables. If homoskedasticity fails, least squares coefficients will still be unbiased but their variance estimates will be biased. If conditional independence fails, least squares coefficients and their variances will be biased and inconsistent, and the regression model output is unreliable. (“Inconsistent” means the coefficient distribution does not converge on the right answer even as the sample size goes to infinite.)

I teach the GM theorem every year in introductory econometrics. (As an aside, that means I am aware of the ways I have oversimplified the presentation, but you can refer to the paper and its sources for the formal version). It comes up near the beginning of an introductory course in regression analysis. It is not an obscure or advanced concept, it is the foundation of regression modeling techniques. Much of econometrics consists of testing for and remedying violations of the GM conditions.

The AT99 Method

(It is not essential to understand this paragraph, but it helps for what follows.) Optimal Fingerprinting works by regressing observed climate data onto simulated analogues from climate models which are constructed to include or omit specific forcings. The regression coefficients thus provide the basis for causal inference regarding the forcing, and estimation of the magnitude of each factor’s influence. Authors prior to AT99 argued that failure of the homoskedasticity condition might thwart signal detection, so they proposed transforming the observations by premultiplying them by a matrix P which is constructed as the matrix root of the inverse of a “climate noise” matrix C, itself computed using the covariances from preindustrial control runs of climate models. But because C is not of full rank its inverse does not exist, so P can instead be computed using a Moore-Penrose pseudo inverse, selecting a rank which in practice is far smaller than the number of observations in the regression model itself.

The Main Error in AT99

AT99 asserted that the signal detection regression model applying the P matrix weights is homoscedastic by construction, therefore it satisfies the GM conditions, therefore its estimates are unbiased and efficient (BLUE). Even if their model yields homoscedastic errors (which is not guaranteed) their statement is obviously incorrect: they left out the conditional independence assumption. Neither AT99 nor—as far as I have seen—anyone in the climate detection field has ever mentioned the conditional independence assumption nor discussed how to test it nor the consequences should it fail.

And fail it does—routinely in regression modeling; and when it fails the results can be spectacularly wrong, including wrong signs and meaningless magnitudes. But you won’t know that unless you test for specific violations. In the first version of my paper (written in summer 2019) I criticized the AT99 derivation and then ran a suite of AT99-style optimal fingerprinting regressions using 9 different climate models and showed they routinely fail standard conditional independence tests. And when I implemented some standard remedies, the greenhouse gas signal was no longer detectable. I sent that draft to Allen and Tett in late summer 2019 and asked for their comments, which they undertook to provide. But hearing none after several months I submitted it to the Journal of Climate, requesting Allen and Tett be asked to review it. Tett provided a constructive (signed) review, as did two other anonymous reviewers, one of whom was clearly an econometrician (another might have been Allen but it was anonymous so I don’t know). After several rounds the paper was rejected. Although Tett and the econometrician supported publication the other reviewer and the editor did not like my proposed alternative methodology. But none of the reviewers disputed my critique of AT99’s handling of the GM theorem. So I carved that part out and sent it in winter 2021 to Climate Dynamics, which accepted it after 3 rounds of review.

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