Despite the widespread use of key concepts of the Neyman–Pearson (N–P) statistical paradigm—type I and II errors, significance levels, power, confidence levels—they have been the subject of philosophical controversy and debate for over 60 years. Both current and long-standing problems of N–P tests stem from unclarity and confusion, even among N–P adherents, as to how a test’s (pre-data) error probabilities are to be used for (post-data) inductive inference as opposed to inductive behavior. We argue that the relevance of error probabilities is to ensure that only statistical hypotheses that have passed severe or probative tests are inferred from the data. The severity criterion supplies a meta-statistical principle for evaluating proposed statistical inferences, avoiding classic fallacies from tests that are overly sensitive, as well as those not sensitive enough to particular errors and discrepancies.

The curve-fitting problem is often viewed as an exemplar which encapsulates the multitude of dimensions and issues associated with inductive inference, including underdetermination and the reliability of inference. The prevailing view is that the ‘fittest ’ curve is one which provides the optimal trade-off between goodness-of-fit and simplicity, with the Akaike Information Criterion (AIC) the preferred method. The paper argues that the AIC-type procedures do not provide an adequate solution to the curve fitting problem because (a) they have no criterion to assess when a curve captures the regularities in the data inadequately, and (b) they are prone to unreliable inferences. The thesis advocated is that for more satisfactory answers one needs to view the curvefitting problem in the context of error-statistical approach where (i) statistical adequacy provides a criterion for selecting the fittest curve and (ii) the error probabilities can be used to calibrate the reliability of inductive inference. This thesis is illustrated by comparing the Kepler and Ptolemaic models in terms of statistical adequacy, showing that the latter does not ‘save the phenomena ’ as often claimed. This calls into question the view concerning the pervasiveness of the problem of underdetermination; statistically adequate ‘fittest ’ curves are rare, not common. ∗Thanks are due to Deborah Mayo and Clark Glymour for valuable suggestions and comments on an earlier draft of the paper. 1 1

The main aim of this paper is to revisit the curve-fitting problem using the reliability of inductive inference as a primary criterion for the ‘fittest ’ curve. Viewed from this perspective, it is argued that a crucial concern with the current framework for addressing the curve-fitting problem is, on the one hand, the undue influence of the mathematical approximation perspective, and on the other, the insufficient attention paid to the statistical modeling aspects of the problem. Using goodness-of-fit as the primary criterion for best, the mathematical approximation perspective undermines the reliability of inference objective by giving rise to selection rules which pay insufficient attention to ‘capturing the regularities in the data’. A more appropriate framework is offered by the error-statistical approach, where(i)statistical adequacy provides the criterion for assessing when a curve captures the regularities in the data adequately, and (ii) the relevant error probabilities canbeusedtoassessthereliabilityof inductive inference. Broadly speaking, the fittest curve (statistically adequate) is not determined by the smallness if its residuals, tempered by simplicity or other pragmatic criteria, but by the non-systematic (e.g. white-noise) nature of its residuals. The advocated error-statistical arguments are illustrated by comparing the Kepler and Ptolemaic models on empirical grounds. ∗ Forthcoming in Philosophy of Science, 2007. † I’m grateful to Deborah Mayo and Clark Glymour for many valuable suggestions and comments on an earlier draft of the paper; estimating the Ptolemaic model was the result of Glymour’s prompting and encouragement. 1 1

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Despite the impressive developments in panel data modeling, the statistical foundations of such models are rather weak in so far that they are inadequate for securing the reliability and precision of inference. In statistical induction we learn from data about phenomena of interest when we employ reliable and incisive inference procedures. When one invokes either untested probabilistic assumptions, or/and broad (including nonparametric) premises, one has to rely on crude approximations (asymptotic) for evaluating the relevant error probabilities. This invariably leads to imprecise inference of unknown reliability because one does not know how closely the actual error probabilities approximate the assumed nominal ones, or how effective the inference procedures are! The primary objective of the paper is to revisit these probabilistic foundations with a view to: (a) recast the error assumptions in terms of the probabilistic structure of the observable processes underlying the data, (b) provide a complete and internally consistent set of testable probabilistic assumptions for several statistical models for panel data, and (c) propose pertinent interpretations for the individual-specific (fixed or random) and time-specific effects. It is shown that the current interpretations of the individual-specific effects (fixed or random) need to be reconsidered in light of the implicit statistical parameterizations in terms of the observable stochastic processes involved. This provides a more appropriate framework for securing learning from panel data by bringing out the neglected facets of empirical modeling which include specification, misspecification testing and respecification. 1 1

Where do statistical models come from? Revisiting the problem of specification