Model Misspecification Errors Research Articles

Photometric redshifts and intrinsic alignments: Degeneracies and biases in the 3×2pt analysis

We present a systematic study of cosmological parameter bias in weak lensing and large-scale structure analyses for upcoming imaging surveys induced by the interplay of intrinsic alignments (IA) and photometric redshift (photo-z) model misspecification error. We first examine the degeneracies between the parameters of the tidal alignment–tidal torquing (TATT) model for IA and of a photo-z model including a mean shift (Δz¯) and variance (σz) for each tomographic bin of lenses and sources, under a variety of underlying true IA behaviors. We identify strong degeneracies between: (1) the redshift scaling of the tidal alignment amplitude and the mean shift and variances of source bins, (2) the redshift scaling of the tidal torquing amplitude and the variance of the lowest-z source bin, and (3) the IA source density weighting and the mean shift and variance of several source bins. We then use this information to guide our exploration of the level of cosmological parameter bias which can be induced given incorrect modeling of IA, photo-z, or both. We find that marginalizing over all the parameters of TATT is generally sufficient to preclude cosmological parameter bias in the scenarios we consider. However, this does not necessarily mean that IA and photo-z parameters are themselves unbiased, nor does it mean that the best-fit model is a good fit to the data. We also find scenarios where the inferred parameters produce χd.o.f.2 values indicative of a good fit but cosmological parameter bias is significant, particularly when the IA source density weighting parameter is not marginalized over. Published by the American Physical Society 2024

Modeling Model Misspecification in Structural Equation Models

Structural equation models constrain mean vectors and covariance matrices and are frequently applied in the social sciences. Frequently, the structural equation model is misspecified to some extent. In many cases, researchers nevertheless intend to work with a misspecified target model of interest. In this article, a simultaneous statistical inference for sampling errors and model misspecification errors is discussed. A modified formula for the variance matrix of the parameter estimate is obtained by imposing a stochastic model for model errors and applying M-estimation theory. The presence of model errors is quantified in increased standard errors in parameter estimates. The proposed inference is illustrated with several analytical examples and an empirical application.

Exploratory Bi-factor Analysis with Multiple General Factors

Exploratory bi-factor analysis (EBFA) is a very popular approach to estimate models where specific factors are concomitant to a single, general dimension. However, the models typically encountered in fields like personality, intelligence, and psychopathology involve more than one general factor. To address this circumstance, we developed an algorithm (GSLiD) based on partially specified targets to perform exploratory bi-factor analysis with multiple general factors (EBFA-MGF). In EBFA-MGF, researchers do not need to conduct independent bi-factor analyses anymore because several bi-factor models are estimated simultaneously in an exploratory manner, guarding against biased estimates and model misspecification errors due to unexpected cross-loadings and factor correlations. The results from an exhaustive Monte Carlo simulation manipulating nine variables of interest suggested that GSLiD outperforms the Schmid-Leiman approximation and is robust to challenging conditions involving cross-loadings and pure items of the general factors. Thereby, we supply an R package (bifactor) to make EBFA-MGF readily available for substantive research. Finally, we use GSLiD to assess the hierarchical structure of a reduced version of the Personality Inventory for DSM-5 Short Form (PID-5-SF).

Digital twinning of self-sensing structures using the statistical finite element method

Abstract The monitoring of infrastructure assets using sensor networks is becoming increasingly prevalent. A digital twin in the form of a finite element (FE) model, as commonly used in design and construction, can help make sense of the copious amount of collected sensor data. This paper demonstrates the application of the statistical finite element method (statFEM), which provides a principled means of synthesizing data and physics-based models, in developing a digital twin of a self-sensing structure. As a case study, an instrumented steel railway bridge of $ 27.34\hskip1.5pt \mathrm{m} $ length located along the West Coast Mainline near Staffordshire in the UK is considered. Using strain data captured from fiber Bragg grating sensors at 108 locations along the bridge superstructure, statFEM can predict the “true” system response while taking into account the uncertainties in sensor readings, applied loading, and FE model misspecification errors. Longitudinal strain distributions along the two main I-beams are both measured and modeled during the passage of a passenger train. The statFEM digital twin is able to generate reasonable strain distribution predictions at locations where no measurement data are available, including at several points along the main I-beams and on structural elements on which sensors are not even installed. The implications for long-term structural health monitoring and assessment include optimization of sensor placement and performing more reliable what-if analyses at locations and under loading scenarios for which no measurement data are available.

The surprising robustness of dynamic Mean-Variance portfolio optimization to model misspecification errors

In single-period portfolio optimization settings, Mean-Variance (MV) optimization can result in notoriously unstable asset allocations due to small changes in the underlying asset parameters. This has resulted in the widespread questioning of whether and how MV optimization should be implemented in practice, and has also resulted in a number of alternatives being proposed to the MV objective for asset allocation purposes. In contrast, in dynamic or multi-period MV portfolio optimization settings, preliminary numerical results show that MV investment outcomes can be remarkably robust to model misspecification errors, which arise when the investor derives an optimal investment strategy based on some chosen model for the underlying asset dynamics (the investor model), but implements this strategy in a market driven by potentially completely different dynamics (the true model). In this paper, we systematically investigate the causes of this surprising robustness of dynamic MV portfolio optimization to model misspecification errors under both the pre-commitment MV (PCMV) and time-consistent MV (TCMV) approaches. We identify particular combinations of parameters that play a key role in explaining the observed model misspecification errors. We investigate the impact of the chosen dynamic MV approach, underlying model formulation, portfolio rebalancing frequency and the application of multiple realistic investment constraints on the robustness of investment outcomes, as well as the implications for model calibration.

Design and sample size considerations for valuation studies of multi-attribute utility instruments.

The EQ-5D, a widely used multiattribute utility instrument, is commonly used in health economic evaluations where the goal is to decide on which treatments to reimburse. Like other instruments, value sets of the EQ-5D are constructed using valuation studies typically valuing a subset of the health states and using predicted values from a regression model for the unvalued health states. In current practice the prediction errors associated with the value sets are substantial. The goal of this work is 2-fold. First, derive a formula of the mean squared error (MSE) of a value set assuming that the value set is estimated using a linear mixed model with either an independent or a Gaussian spatial correlation on the model misspecification error. Second, explore the effect of the number of health states directly valued, the number of participants and the correlation structure on the MSE. Keeping the total number of participants and the total number of valuations fixed, valuing all 242 health states of the EQ-5D-3L was found to reduce the MSE considerably compared with the common practice of valuing only 42 health states. Furthermore, an independent correlation structure with 3773 participants valuing 42 health states produced the MSE that can be achieved with less than 600 participants valuing all 242 health states under a Gaussian spatial correlation structure. Based on the comparison of the MSE values of some of the well-known designs our suggestion is to value more health states and to use a model with spatially correlated misspecification errors.

Making causal inferences about treatment effect sizes from observational datasets

In the era of big data and cloud computing, analysts need statistical models to go beyond predicting outcomes to forecasting how outcomes change when decision-makers intervene to change one or more causal factors. This paper reviews methods to estimate the causal effects of treatment choices on patient health outcomes using observational datasets. Methods are limited to those that model choice of treatment (propensity scoring) and treatment outcomes (instrumental variable, difference in differences, control function). A regression framework was developed to show how unobserved confounding covariates and heterogeneous outcomes can introduce biases to effect size estimates. In response to criticisms that outcome approaches are not systematic and subject to model misspecification error, we extend the control function approach of Lu and White by applying Best Approximating Model technology (BAM-CF). Results from simulation experiments are presented to compare biases between BAM-CF and propensity scoring in the presence of an unobserved confounder. We conclude no one strategy is ‘optimal’ for all datasets, and analyst should consider multiple approaches to assess robustness. For both observational and randomized datasets, researchers should assess how moderating covariates impact estimates of treatment effect sizes so that clinicians can understand what is best for each individual patient.

Treatment Effect Deviation as an Alternative to Blinder–Oaxaca Decomposition for Studying Social Inequality

The Blinder–Oaxaca decomposition (BOD) is a popular method for studying the contributions of explanatory factors to social inequality. The results have often been given causal interpretations. While recent work and this article both show that some types of BOD are equivalent to a counterfactual-based treatment effect/selection bias decomposition, this equivalence does not hold in general. Given this lack of general equivalence, in this article based on the counterfactual framework, we propose a method of treatment effect deviation (TED) to study social inequality. Essentially, the TED assesses to what extent the omission of particular covariates (i.e., selection bias in the omitted variables) can alter the estimated treatment effect. The TED has a better causal interpretation and can be estimated nonparametrically (and hence is more robust to model misspecification errors). Therefore, the TED may serve as an alternative to or may be used in tandem with the BOD. We illustrate the new method through two case studies. In the first case study, we show that the TED provides a more credible estimate of the treatment effect on the treated than does the BOD. However, both the TED and the BOD highlight the importance of accounting for prior earning in estimating the training effect. In the second case study, the results of the two methods differ notably, but both agree that normative factors play a significant role in generating the rural–urban disparity in social policy preference in China.

Slope meets Lasso: Improved oracle bounds and optimality

We show that two polynomial time methods, a Lasso estimator with adaptively chosen tuning parameter and a Slope estimator, adaptively achieve the minimax prediction and $\ell_{2}$ estimation rate $(s/n)\log(p/s)$ in high-dimensional linear regression on the class of $s$-sparse vectors in $\mathbb{R}^{p}$. This is done under the Restricted Eigenvalue (RE) condition for the Lasso and under a slightly more constraining assumption on the design for the Slope. The main results have the form of sharp oracle inequalities accounting for the model misspecification error. The minimax optimal bounds are also obtained for the $\ell_{q}$ estimation errors with $1\le q\le2$ when the model is well specified. The results are nonasymptotic, and hold both in probability and in expectation. The assumptions that we impose on the design are satisfied with high probability for a large class of random matrices with independent and possibly anisotropically distributed rows. We give a comparative analysis of conditions, under which oracle bounds for the Lasso and Slope estimators can be obtained. In particular, we show that several known conditions, such as the RE condition and the sparse eigenvalue condition are equivalent if the $\ell_{2}$-norms of regressors are uniformly bounded.

Sharp oracle inequalities for Least Squares estimators in shape restricted regression

The performance of Least Squares (LS) estimators is studied in shape-constrained regression models under Gaussian and sub-Gaussian noise. General bounds on the performance of LS estimators over closed convex sets are provided. These results have the form of sharp oracle inequalities that account for the model misspecification error. In the presence of misspecification, these bounds imply that the LS estimator estimates the projection of the true parameter at the same rate as in the well-specified case. In isotonic and unimodal regression, the LS estimator achieves the nonparametric rate $n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order $q/n$ up to logarithmic factors, where $q$ is the number of affine pieces of the true regression function. This adaptive risk bound holds for any collection of design points. While Guntuboyina and Sen [Probab. Theory Related Fields 163 (2015) 379–411] established that the nonparametric rate of convex regression is of order $n^{-4/5}$ for equispaced design points, we show that the nonparametric rate of convex regression can be as slow as $n^{-2/3}$ for some worst-case design points. This phenomenon can be explained as follows: Although convexity brings more structure than unimodality, for some worst-case design points this extra structure is uninformative and the nonparametric rates of unimodal regression and convex regression are both $n^{-2/3}$. Higher order cones, such as the cone of $\beta $-monotone sequences, are also studied.

Fair value accounting in the absence of prudence in accounting standards: an illustration with exotic derivatives

ABSTRACTThe aim of this paper is to contribute to the current discussion about fair value accounting (FVA). We discuss the problem surrounding FVA by relying on the role played by prudence, its meaning, and how the treatment of prudence has changed in the accounting framework of standard setters due to its ‘apparent’ inconsistency with neutrality. To highlight the relevance of this issue, we provide (1) a brief analysis of the high impact that Level 2 fair value estimates have on large US and European banks’ financial positions; (2) a ‘case study’ by pricing two common exotic derivatives and comparing the valuation results of two different assumptions of volatility (local vs. stochastic); and (3) a discussion of potential solutions to the problem surrounding FVA. Our findings are consistent with the argument that neutrality is supported by the exercise of prudence in achieving a faithful representation, since a non-conservative use of FVA can lead bank managers towards model misspecification error in the valuation of complex financial instruments. We conclude by arguing that the problem surrounding FVA can be mitigated if prudence is reinstated by standards setters.

The Effect of Estimation Error on Risk-Adjusted Survival Time CUSUM Chart Performance

Research on risk-adjusted control charts has gained great interest in healthcare settings. Based on monitored variables (binary outcome or survival times), risk-adjusted cumulative sum (CUSUM) charts are divided into Bernoulli and survival time CUSUM charts. The effect of estimation error on control chart performance has been systematically studied for Bernoulli CUSUM but not for survival time CUSUM in continuous time. We investigate the effect of estimation error on the performance of risk-adjusted survival time CUSUM scheme in continuous time with the cardiac surgery data. The impact is studied with the use of the median run lengths (medRLs) and the standard deviation (SD) of medRLs for different sample sizes, specified in-control median run length, adverse event rate and patient variability. Results show that estimation error affects the performance of risk-adjusted survival time CUSUM chart significantly and the performance is more sensitive to the specified in-control median run length (medRL0) and adverse event rate. To take the estimation error into account, the practitioners can bootstrap many samples from Phase I data and then determine the threshold that can guarantee at least a medRL0 with certain probability under which false alarms occur less frequently and meanwhile out-of-control alarms don't signal too slow. Moreover, additional event occurrences can be used to update the estimation but should be from in-control process. Finally, non-parametric bootstrap can be applied to reduce model misspecification error. Copyright © 2015 John Wiley & Sons, Ltd.

Prediction of Functional Status for the Elderly Based on a New Ordinal Regression Model

The functional mobility of the elderly is a very important factor in aging research, and prognostic information is valuable in making clinical and health care policy decisions. We develop a predictive model for the functional status of the elderly based on data from the Second Longitudinal Study of Aging (LSOA II). The functional status is an ordinal response variable. The ordered probit model has been moderately successful in analyzing such data; however, its reliance on the normal distribution for its latent variable hinders its accuracy and potential. In this paper, we focus on the prediction of conditional quantiles of the functional status based on a more general transformation model. The proposed estimation procedure does not rely on any parametric specification of the conditional distribution functions, aiming to reduce model misspecification errors in the prediction. Cross-validation within the LSOA II data shows that our prediction intervals are more informative than those from the ordered probit model. Monte Carlo simulations also demonstrate the merits of our approach in the analysis of ordinal response variables.

A Robust Conditional Realized Extended 4-CAPM

In this paper we present and extend the approach of Bollerslev and Zhang (2003) for realized measures and co-measures of risk in some classical asset pricing models, such as the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and the Arbitrage Pricing Theory (APT) model by Ross (1976). These extensions include higher-moments asset pricing models (see Jurczenko and Maillet, 2006), conditional asset pricing models (see Bollerslev et al., 1988, and Jondeau and Rockinger, 2004). Estimations are conducted using several methodologies aiming to neutralize data measurement and model misspecification errors (see Ledoit and Wolf, 2003 and 2004), properly dealing with inter-relations between financial assets in term of returns (see Zellner, 1962), but also in terms of higher conditional moments (see Bollerslev, 1988).

Demographic Issues in Longevity Risk Analysis

AbstractFundamental to the modeling of longevity risk is the specification of the assumptions used in demographic forecasting models that are designed to project past experience into future years, with or without modifications based on expert opinion about influential factors not represented in the historical data. Stochastic forecasts are required to explicitly quantify the uncertainty of forecasted cohort survival functions, including uncertainty due to process variance, parameter errors, and model misspecification errors. Current applications typically ignore the latter two sources although the potential impact of model misspecification errors is substantial. Such errors arise from a lack of understanding of the nature and causes of historical changes in longevity and the implications of these factors for the future. This article reviews the literature on the nature and causes of historical changes in longevity and recent efforts at deterministic and stochastic forecasting based on these data. The review reveals that plausible alternative sets of forecasting assumptions have been derived from the same sets of historical data, implying that further methodological development will be needed to integrate the various assumptions into a single coherent forecasting model. Illustrative calculations based on existing forecasts indicate that the ranges of uncertainty for older cohorts' survival functions will be at a manageable level. Uncertainty ranges for younger cohorts will be larger and the need for greater precision will likely motivate further model development.

The power of tests for equivalent ARMA models: The implications for practitioners

Analysts frequently find it convenient to use the same ARMA model to make forecasts for multiple time series. The trick is to know when it is safe to assume that multiple series are generated by the same underlying process. Although several authors have developed statistical procedures for testing whether two models are equivalent, no one has shown how to determine the power of these tests. This paper shows how to determine the power of the most general test for equivalent ARMA models. It also shows how to quantify the effect of model misspecification errors on the accuracy of the forecast. An illustrative example and flowchart are then used to show how calculating the power of the test can enable the practitioner to safeguard against a serious degradation in the accuracy of the forecast.

Capturing Geographically Localized Misspecification Error in Retail Store Choice Models

No retail store choice model, no matter how many relevant variables it might include, can realistically expect to model all the variation in store choice. There are always some variables that are left out, because they are difficult to measure, they have not yet been conceptualized in theory, or their estimated parameter stability suffers when an excessive number of predictors are included. Because these omitted variables can be correlated with geographic location, model misspecification error may itself be correlated with location. Estimating the geographically localized misspecification errors therefore suggests itself as a method for estimating (and predicting) the effects of these omitted variables. The authors show that spatial nonstationarity of the model parameters may also be expressed as an instance of omitted variables and therefore be addressed using their method. They show, using both a simulation study and an empirical natural experiment, that estimating the geographically localized misspecification error can appreciably reduce prediction error, even when the predictor model is reasonably well specified.