Empirical Economics (2023) 64:2467–2474
https://doi.org/10.1007/s00181-023-02438-2

Introduction
Subal C. Kumbhakar1 · Robin Sickles2 · Hung-Jen Wang3
Published online: 16 June 2023
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023

This Special Issue in Honor of Peter Schmidt was put in motion by Subal Kumbhakar who as a co-editor of Empirical Economics enlisted Hung-Jen Wang and Robin
Sickles to join him as co-editors of the Special Issue. When we sent requests for
submissions, based in large part on Peter’s suggestions, the responses were quick
and overwhelming. From the VERY best scholars in econometrics came commentsthat would soften the heart of the most cynical academics. These scholars grew theircareers and accomplished such remarkable professional achievements while admiringand benefiting from Peter’s writings, insights, and friendships. We are so very lucky tohave been able to put together this special of Empirical Economics in Honor of Peter
Schmidt and we thank the editors and Springer Nature for making this possible.
In our call for papers to selective potential submitters in late 2021, we pointed outthat Peter Schmidt stepped down as an Associate Editor of Empirical Economics after
serving in that position for over 24 years. We noted his distinguished service at the
Journal and that among his lifelong accomplishments in academics were important
contributions to many areas of econometric research, including time series econometrics, panel data econometrics, and stochastic frontier analysis. We also noted that
his research always had a fine balance between theoretical innovation and empiricalrelevance, which is what is valued at Empirical Economics.
This Special Issue contains contributions on state-of-the-art topics in econometrics
related to the three broad research areas mentioned above as well as a few others that,based on submissions, we felt deserved their own topic, namely his contributions to
the literature on applied econometrics, copulas, and nonparametric methods, as wellas on limited dependent variables.

B Subal C. Kumbhakar

kkar@binghamton.edu
Robin Sickles
rsickles@rice.edu
Hung-Jen Wangwangh@ntu.edu.tw

1

Department of Economics, Binghamton University - SUNY, Binghamton, NY, USA

2

Department of Economics, Rice University, Houston, TX, USA

3

Department of Economics, National Taiwan University, Taipei 10617, Taiwan

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The Special Issue is composed of 25 manuscripts. We will briefly discuss each paper
in order of the number of contributions on the topics, which give an order to our summary that begins with panel data, then stochastic frontiers and efficiency/productivity
measurement, time series, applied econometrics, copulas, and nonparametric methods,
and finishes with limited dependent variables.

1 Panel data
In “Robust dynamic space–time panel data models using ε-contamination: An application to crop yields and climate change,” Baltagi, Bresson, Chaturvedi, and Lacroix
extend the Baltagi et al. (2018, 2021) static and dynamic ε -contamination papers todynamic space–time models. They investigate the robustness of Bayesian panel data
models to possible misspecification of the prior distribution. Using an extensive Monte
Carlo simulation study, they compare the finite sample properties of their proposed
estimator to those of standard classical estimators. They obtain short-run as well as
long-run effects of climate change on corn producers in the USA.
In “Unbiased estimation of the OLS covariance matrix when the errors are clustered,” Boot, Niccodemi, and Wansbeek derive an estimator that is unbiased when the
random-effects model holds. They do the same for two more general structures andstudy the usefulness of these estimators against others by simulation, the size of the ttest being the criterion.
In “Refined GMM estimators for simultaneous equations models with network
interactions,” Egger and Prucha propose a refinement of the generalized spatial twostage and three-stage least squares estimators for simultaneous systems of equations
with network interdependence, introduced in Drukker et al. (2022). The refinement
proposed involves the weighting of the moment conditions underlying those estimators.
In “Identification and estimation of categorical random coefficient models,” Gao andPesaran propose a linear categorical random coefficient model, in which the randomcoefficients follow parametric categorical distributions. The distributional parameters
are identified based on a linear recurrence structure of moments of the random coefficients. A generalized method of moments estimation procedure is proposed to address
heterogeneity in time effects in panel data models (Ahn and Schmidt 1995; Ahn et al.
2013). The utility of the proposed estimator is illustrated by estimating the distributionof returns to education in the USA by gender and educational levels.
Han and Kim, in “Dynamic panel GMM estimators with improved finite sample
properties using parametric restrictions for dimension reduction,” propose reducing
finite sample bias by imposing parametric restrictions on the expected first derivative
matrix and the covariance matrix of the sample moment functions. The resulting
estimator is consistent under regularity irrespective of the correctness of the extra
restrictions and is first-order efficient if they are indeed correct. The method is applied
to a dynamic cigarette consumption model.
In “Testing for correlation between the regressors and factor loadings in heterogeneous panels with interactive effects,” Kapetanios, Serlenga, and Shin address whether
the regressors are correlated with factor loadings or not. They explore this issue and123Introduction

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propose a Hausman-type test to address the matter. Further, they develop two nonparametric variance estimators for the FE and PC estimators as well as their difference
that are robust to the presence of heteroscedasticity, autocorrelation, and slope heterogeneity.
Li, Li, and Hsiao, in their contribution “Assessing the impacts of pandemic andthe increase of minimum down payment rate on Shanghai housing prices,” study thetreatment effect of a major policy change on Shanghai’s housing market by employing
panel data from March 2009 to December 2021. They use the panel data approachsuggested by Hsiao et al. (2012) to estimate the treatment effects and a time series
approach to disentangle the treatment effects and the effects of the pandemic. Fortime periods after the outbreak of the pandemic, they find no significant impact of thepandemic on the real estate price indices between 2020 and 2021.
Papke and Wooldridge, in their paper “A simple, robust test for choosing the level of
fixed effects in linear panel data models,” propose a test that allows one to determine
whether controlling for fixed effects at the more aggregate level is sufficient. The
alternative is that one should allow for fixed effects at the unit level. The regressionbased test is simple to carry out, even for unbalanced panels. In addition, the test iseasily made robust to arbitrary heteroskedasticity, serial correlation across time, andeven cluster correlation at the group level.

2 Stochastic frontiers analysis and eﬃciency/productivity
measurement
Although not a formal contributor to the Special Issue, C. A. Knox Lovell was at thecore of the original contributions by Peter on the topic of stochastic frontier analysis(SFA) and efficiency/productivity measurement. He has provided us with a bit ofperspective in private correspondences on the development of SFA, for which of course
he was also responsible. Knox has pointed out that Peter was more than an important
contributor to the field; he was a creator of SFA, with Aigner et al. (1977) (Meeusen
and van den Broeck, 1977, published a similar and nearly simultaneous paper in theInternational Economic Review, unknown at the time to Aigner et al. and similarly
for Meeusen and van den Broeck) and directs us to the Journal of Econometrics
2023 50th Anniversary Jubilee Issue (2023) where the paper is reproduced along withthe authors’ recollections of the origins and subsequent developments of SFA. Knox
noted as well Peter Schmidt’s role in subsequent developments of SFA, in particular,
the conversion of the production frontier to the cost frontier (Schmidt and Lovell
1979, 1980), methods to estimate the efficiency of individual producers (Jondrow
et al. 1982), and extensions to panel models wherein efficiency is allowed to vary bothacross producers and through time (Schmidt and Sickles 1984; Cornwell et al. 1990).
Among the many additional contributions Peter has made in this field of study, Lovell
points out two additional seminal works that focus on “measurable and policy-sensitive
determinants of efficiency,” Wang and Schmidt (2002) and Alvarez et al. (2006).
We now turn to brief discussions of each formal contribution to the special issue
that relate to this aspect of Peter Schmidt’s work.

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Chetty and Heckman, in “Internal adjustment costs of firm-specific factors and theneoclassical theory of the firm,” consider the predictions for factor demand of a twosector vertically integrated model of firms producing output using firm-specific capitalalong with a second sector producing firm-specific capital that adapts raw capital purchased in the market. They find that aggregating over both sectors produces short-run
and long-run factor demand functions that appear to be perverse, but when disaggregated they obey standard neoclassical properties. They conclude that adjustment costs
create the appearance of static inefficiency in the presence of dynamic efficiency.
In “Proportional incremental cost probability functions and their frontiers,” Féve,
Florens, and Simar suggest an alternative semi-parametric model that avoids the drawbacks of the two-stage methods of estimating frontier models. Their approach is basedon a class of model called the proportional incremental cost functions, which is adaptedfrom the Cox proportional hazard model. Use of this approach avoids the first-stage
nonparametric estimation of the frontier and avoids the curse of dimensionality keeping the parametric rates of convergence for the parameters of interest.
Koenker, in “Hotelling tubes, confidence bands and conformal inference,” proposes
using Hotelling’s tube methods for constructing nonparametric quantile regression
confidence bands. Hotelling’s methods strengthen the performance of such bands.
Koenker’s innovation is based on recent developments in conformal inference—considered to be a new approach to nonparametric inference for stochastic frontier models.
“In Indirect inference estimation of stochastic production frontier models withskew-normal noise,” Lai and Kumbhakar consider a stochastic frontier model in which
both the noise and inefficiency components are asymmetric, viz., the noise term is skew
normal and the inefficiency term is half normal. This formulation avoids the criticism
that skewness of the composite error term (sum of the noise and inefficiency) cannot
be an indicator of inefficiency because skewness can also arise from the noise term.
They further generalize the model by introducing determinants of skewness of thenoise term as well as determinants of inefficiency, and provide both simulation andempirical results using the indirect inference estimation approach.
In “The noise error component in stochastic frontier analysis,” Papadopoulos
examines the relation between predicted noise and predicted inefficiency. For theNormal-Half Normal and the Normal-Exponential error specification, he provides its
conditional expectation as a predictor and examines its distribution in relation to themarginal law. He also derives the conditional distribution of the noise and computes
confidence intervals and the probability of over-predicting it.
In “An alternative corrected ordinary least squares estimator for the stochastic frontier model,” Parmeter and Zhao consider an extension of the corrected ordinary leastsquares (COLS) estimator for the stochastic frontier model. They propose a novel modification to COLS by using the first moment of the absolute value of the composite
error term in place of the third moment for both the Normal-Half Normal and NormalExponential specifications. They demonstrate via simulations that this modificationconsiderably reduces the occurrence of both Type I and Type II failures.

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3 Time series
Ahn’s “Likelihood based inference for dynamic panel data models” examines theasymptotic and finite sample distributions of maximum likelihood (ML) estimatorsfor both stationary and nonstationary time series processes. He links the identification
criteria developed by Sagan (1983) to the singularity of the information matrix for
ML estimators under nonstationary. A major finding is that when data follow unit
root processes without or with drift, the ML estimators are consistent, but they havenonstandard asymptotic distributions and their convergence rates are slower than n 1/2 .
Additionally, Monte Carlo experiments show that the modified LR tests are much better
sized than the corresponding Wald tests. Although LR tests tend to slightly over-reject
the unit root hypothesis in small samples they maintain good finite sample powerproperties.
In “Approximating long memory processes with low order autoregressions with
implications for modeling realized volatility,” Baillie, Cho, and Rho show that forrealistic ranges of the long memory parameter in fractionally integrated times series,
the ordinary least squares estimators of an AR(p) model will have nonstandard rates
of convergence to nonstandard distributions. The AR parameter and impulse response
function estimators will be of questionable value to researchers as well as models that
use these estimators to represent realized volatility (RV) in financial markets.
In the topical empirical treatment “Does climate change affect economic data?”,
Choi derives seasonal factors based on US temperature, gasoline price, and fresh foodprice data sets using Kalman state smoothers and principal component analysis to showthat seasonal volatilities have increased over the last four decades. Climate change
is undoubtedly reflected in the temperature data. The three data sets show similar
patterns from the 1990s, which suggests that climate change may have affected theprice volatility behavior.
Phillips and Yu integrate flat trading features into an efficient price process intheir modeling of asset price determination in their contribution “Information loss
in volatility measurement with flat price trading.” They develop a limit theory forthe usual measure of integrated volatility, the conventional realized volatility (RV),and show that estimated RV, as well as estimated quarticity, has inflated asymptotic
variances that depend on the probability of flat trading. Extensions to models withmicrostructure noise are also provided and the effect of flat trading using tick-by-tickdata is empirically evaluated.
In “Forecasting in the presence of in and out of sample breaks,” Xu and Perron use
a frequentist-based paradigm, modified by modeling the probability of shifts, basedon covariates that can also be forecasted and by a built-in mean reversion mechanism
for the evolution of the parameters. Estimation is based on a mixture Kalman filter anda Monte Carlo expectation maximization algorithm and simulation results show that
their approach is superior to standard forecasting models that are robust to model misspecifications. Empirical applications also are provided and illustrate the substantial
gains in forecasting accuracy based on their new methods.

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4 Applied econometrics, copulas, and nonparametric methods
The dependency structure of US commodity futures across sectors between 2004 and2022 is the focus of “Multivariate models of commodity futures markets: a dynamic
copula approach” by Chen, Li, Wang, and Zhang. During their study period, both the2008 financial crisis and the COVID-19 pandemic caused substantial disruptions inmarkets. Their copula-based models address these major shocks by allowing for timevarying nonlinear and asymmetric dependence using flexible integrating elliptical andskewed copulas. Their empirical findings point to increasing connectedness among
commodities during both of these events. Risk management strategies in commodity
markets are also analyzed and they find that strategies using portfolio weights basedon their dynamic copulas dominate those based on an equal-weighted portfolio.
Dang and Ullah use kernel regularized least squares (KRLS) in developing their
two-step estimator of a nonparametric regression function in “Generalized kernel regularized least squares estimator with parametric error covariance.” The KRLS canmodel a very general covariance structure, similar to the HAC heteroskedastic andautocorrelation consistent covariance estimator, and the authors detail the construction of the estimator and also provide derivations of bias, variance, and asymptotics.
Simulations and an empirical illustration provide both finite sample evidence of theappeal for the KRLS two-step estimator as well as its feasibility and relative ease ofinterpretability in terms of average partial effects of the inputs in their airline coststudy, which illustrates the usefulness and feasibility of their new techniques.
Lahiri and Yang develop a new ensemble econometric model to predict binary
outcomes in their paper, “Predicting binary outcomes based on the pair-copula construction.” Pair-copula construction (PCC) is used to optimally combine diverse
information while allowing the conditional copula to depend on the conditioning variable non-parametrically. The authors use their new methods to predict US businesscycle peaks, using Conference Board leading indicators. The predictive accuracy of
their estimates using the receiver operating characteristic curve criteria is found to
perform well in comparison with other widely used combination models. The appealof their new ensemble predictor is highlighted by a number of diagnostic measuresthat point to different aspects of its advantages.
Varaku and Sickles utilize relatively new nonparametric methods and machinelearning techniques in their paper “Public subsidies and innovation: a doubly robust
machine learning approach leveraging deep neural networks.” They use Eurostat firmlevel data to assess the effects of public subsidies on firms’ R&D input and output.
Average treatment effects are estimated based on the selection on observables assumption as well as selection on unobservables that may also result in nonrandom subsidy
assignment. Instrumental variables (IVs) are used to identify the local instrumental
variable (LIV) curve. Identification of the LIV is obtained via double machine learning, combining IV estimation, neural networks, and deep neural networks to learn
functional forms from the data. Major findings point to the positive and significant
role that public subsidies have in increasing both R&D intensity and R&D output.

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5 Limited dependent variables
Hirukawa, Liu, Murtazashvili, and Prokhorov focus on the Heckman (1979) sample selection model and utilize lasso-based criteria to select the control variables intheir contribution, “DS-HECK: double-lasso estimation of Heckman selection model.”They point out that the usual lasso under-selects, adding the additional problem ofomitted variable bias to the selection problem, and address the shortcoming of thelasso in this context by using a double lasso in the selection equation and in the calculation of the variance matrix. Simulations and a study of drivers of female US labormarket participation and earnings are provided as well as a dedicated Stata procedure,dsheckman.
In “Simultaneity in binary outcome models with an application to employment forcouples,” Honoré, Hu, Kyriazidou, and Weidner generalize the dynamic panel data
model of Ahn and Schmidt (1995) to allow not only for lagged dependent variable
and fixed effects in short panels, but also for bivariate outcomes studied by Schmidt
and Strauss (1975) in their simultaneous logit model. Their estimators combine bothconditional likelihood and method of moments methods and are used to examine theintra-household relationship in employment. Their major finding points to significant
variations in within-household dependence in employment due to a couple’s ethnicity
after controlling for unobserved household specific heterogeneity.

6 Closing Remarks
We owe a debt of gratitude to the contributing authors for trusting us to handle theirresearch at the Journal rather than submit elsewhere, and patience for our many pleas
to stay on top of revisions. On a final note, we would like to thank the many very
capable referees who provided excellent criticism and feedback on the contributed
papers. We thank them for their dedicated service in helping us put this special issue
together and ensuring its high quality. We also owe a great debt of appreciation to
the editorial board of Empirical Economics for approving the initial proposal for thisissue.

References
Ahn SC, Schmidt P (1995) Efficient estimation of models for dynamic panel data. J Econom 68(1):5–27
Ahn SC, Lee YH, Schmidt P (2013) Panel data models with multiple time-varying individual effects. J
Econom 174(1):1–14
Aigner D, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production
function models. J Econom 6(1):21–37
Alvarez A, Amsler C, Orea L, Schmidt P (2006) Interpreting and testing the scaling property in models
where inefficiency depends on firm characteristics. J Prod Anal 25(3):201–212
Baltagi BH, Bresson G, Chaturvedi A, Lacroix G (2018) Robust linear static panel data models using-contamination. J Econom 202:108–123
Baltagi BH, Bresson G, Chaturvedi A, Lacroix G (2021) Robust dynamic panel data models using –contamination. In: Chudik A, Hsiao C, Timmermann A (eds) Advances in econometrics, essays in honorof M. Hashem Pesaran. Emerald Publishing (forthcoming)

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Cornwell C, Schmidt P, Sickles RC (1990) Production frontiers with cross-sectional and time-series variationin efficiency levels. J Econom 46(1):185–200
Drukker DM, Egger PH, Prucha IR (2022) Simultaneous equations models with higher-order spatial orsocial network interactions. Econom Theory (forthcoming)
Heckman JJ (1979) Sample selection bias as a specification error. Econometric 47:153–161
Hsiao C, Ching HS, Wan SK (2012) A panel data approach for program evaluation: measuring the benefits ofpolitical and economic integration of Hong Kong with mainland China. J Appl Econom 27(5):705–740
Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in thestochastic frontier production function model. J Econom 19(2–3):233–238
Journal of Econometrics (2023) Journal of econometrics jubilee issue—celebrating our fiftieth anniversary:
1973—2023. Ng S, Tamer E (Eds.) 234(Supplement):1–120
Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb-Douglas production functions withcomposed error. Int Econ Rev 18(2):435–444
Sagan JD (1983) Identification and lack of identification. Econometrica 51:1605–1634
Schmidt P, Lovell CAK (1979) Estimating technical and allocative inefficiency relative to stochastic production and cost frontiers. J Econom 9(3):343–366
Schmidt P, Lovell CAK (1980) Estimating stochastic production and cost frontiers when technical andallocative inefficiency are correlated. J Econom 3(1):83–100
Schmidt P, Sickles RC (1984) Production frontiers and panel data. J Bus Economic Stat 2(4):367–374
Schmidt P, Strauss RP (1975) Estimation of models with jointly dependent qualitative variables: a simultaneous logit approach. Econometrica 43(4):745–755
Wang H-J, Schmidt P (2002) One-step and two-step estimation of the effects of exogenous variables ontechnical efficiency levels. J Prod Anal 18(2):129–144
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