Limit predictions in the Tobit model: an alternative derivation approach

Purpose This study aims to contribute to the ongoing assessment of executive compensation by investigating the nexus between managerial entrenchment factors, adopting a multifaceted perspective encompassing both economic and non-economic dimensions. Design/methodology/approach This research employs pooled cross-sectional Ordinary Least Squares (OLS) regression and Least Squares with Dummy Variables (LSDV) models with fixed effects to examine the determinants of Chief Executive Officer (CEO) compensation. Findings This research identifies firm size, performance (via ROA and Tobin’s Q), and CEO characteristics (age, tenure, stock ownership, MBA degree) as significant determinants of executive compensation at the 0.05 level. In contrast, the prestige of educational institutions, doctoral degrees, and the MBA’s relevance to short-term performance, along with CEO tenure, do not significantly affect pay. Additionally, the study highlights the significance of industry type (manufacturing vs technology) in shaping compensation, emphasizing the role of firm metrics and CEO credentials in designing executive pay packages. Originality/value This research introduces an innovative approach to controlling unobserved heterogeneity and adjusting for the dynamic nature of CEO compensation attributes across diverse CEO characteristics. By integrating both pooled Ordinary Least Squares (OLS) and Least Squares Dummy Variable (LSDV) models, the study addresses the challenges posed by time-invariant variables and unobservable heterogeneity. Such issues have historically skewed the accuracy of traditional OLS models in identifying the comprehensive array of factors—both economic and non-economic—that influence CEO compensation. This novel methodological framework significantly advances the examination of unobservable variables that may vary not only across the firms selected for analysis but also over time periods, thereby offering a more detailed understanding of the determinants of CEO pay.

One of the problems of reliability engineering, survival analysis, and applied statistics is correct estimation of lifetime distributions. Relevancy of the inverse Rayleigh distribution, where non-monotonic failure rates are considered, has been especially noted, and classical estimation tools are frequently unable to give good estimates in nonlinear, complicated situations. This paper gives comparative research on four estimation methods namely Ordinary Least Squares (OLS), Maximum Likelihood Estimation (MLE), Support Vector Machines (SVM), and Long Short-Term Memory (LSTM) networks in the estimation of the parameters of the inverse Rayleigh process. The performance of these methods was compared in terms of RMSE and AIC criteria using both as well as real-world data which were obtained through simulation experiments. The findings show the superiority of smart computational methods over traditional methods, where SVM was the most accurate and the strongest in all the conditions. Estimation performance was also significantly higher in LSTM than in OLS and MLE, but slightly lower than that of SVM. These results indicate the applied benefits of machine-learning-based methods in dealing with the nonlinear and complex forms of reliability data. The key finding of the present study is that it offers a comparative analysis of traditional and intelligent estimation approaches in a systematic way and demonstrates that the application of machine learning to statistical modelling may contribute to the high performance of the parameter estimation to a significant degree. This article contributes to the research by showing how SVM is better in estimating the inverse Rayleigh parameter, as well as by highlighting the significance of the hybrid statistical-computational methods in the reliability of applications in the real world. Future directions include the experimental use of additional high-level machine learning architectures, hybrids estimation systems and Bayesian approaches in order to achieve further accuracy improvements and understanding.

Detecting Discrimination: Analyzing Racial Disparities in Public Contracting

ObjectivesHealth-related quality of life (HRQoL) measures have been increasingly used in economic evaluations for policy guidance. We investigate the impact of 11 self-reported long-standing health conditions on HRQoL using the EQ-5D in a UK sample.MethodsWe used data from 13,955 patients in the South Yorkshire Cohort study collected between 2010 and 2012 containing the EQ-5D, a preference-based measure. Ordinary least squares (OLS), Tobit and two-part regression analyses were undertaken to estimate the impact of 11 long-standing health conditions on HRQoL at the individual level.ResultsThe results varied significantly with the regression models employed. In the OLS and Tobit models, pain had the largest negative impact on HRQoL, followed by depression, osteoarthritis and anxiety/nerves, after controlling for all other conditions and sociodemographic characteristics. The magnitude of coefficients was higher in the Tobit model than in the OLS model. In the two-part model, these four long-standing health conditions were statistically significant, but the magnitude of coefficients decreased significantly compared to that in the OLS and Tobit models and was ranked from pain followed by depression, anxiety/nerves and osteoarthritis.ConclusionsPain, depression, osteoarthritis and anxiety/nerves are associated with the greatest losses of HRQoL in the UK population. The estimates presented in this article should be used to inform economic evaluations when assessing health care interventions, though improvements can be made in terms of diagnostic information and obtaining longitudinal data.

This research article primarily focuses on the estimation of parameters of a linear regression model by the method of ordinary least squares and depicts Gauss-Mark off theorem for linear estimation which is useful to find the BLUE of a linear parametric function of the classical linear regression model. A proof of generalized Gauss-Mark off theorem for linear estimation has been presented in this memoir. Ordinary Least Squares (OLS) regression is one of the major techniques applied to analyse data and forms the basics of many other techniques, e.g. ANOVA and generalized linear models [1]. The use of this method can be extended with the use of dummy variable coding to include grouped explanatory variables [2] and data transformation models [3]. OLS regression is particularly powerful as it relatively easy to check the model assumption such as linearity, constant, variance and the effect of outliers using simple graphical methods [4]. J.T. Kilmer et.al [5] applied OLS method to evolutionary and studies of algometry.

This study investigates the interconnections among sustainability reporting, social performance, and firm value across the BRICS nations (Brazil, Russia, India, China, and South Africa). Employing a quantitative research design, the study utilizes firm-level data from the Refinitiv database, covering 862 firms operating in the BRICS countries from 2017 to 2022. The analysis begins with Ordinary Least Squares (OLS) regression and extends to models incorporating year-fixed effects and firm-fixed effects to account for heterogeneity and omitted variable bias. Robustness checks are conducted using OLS regression with robust standard errors, fixed effects regression with Driscoll–Kraay standard errors, and an instrumental variable approach to address potential endogeneity concerns. To examine the moderating role of sustainability reporting, interaction terms are incorporated into the regression models and margin plots are used for visualization. The findings reveal that social performance positively impacts firm value, underscoring the role of social responsibility in driving financial performance. Furthermore, sustainability reporting strengthens this relationship, indicating that firms with well-established reporting frameworks can effectively leverage social initiatives to enhance market valuation. Therefore, this study contributes to the literature by providing empirical evidence on the moderating effect of sustainability reporting in emerging markets. The findings offer valuable implications for policymakers, investors, and corporate leaders seeking to optimize CSR strategies and enhance firm value in dynamic economic environments.

The inverse Mills ratio is $$R:=\varphi /\Psi $$ , where $$\varphi $$ and $$\Psi $$ are, respectively, the probability density function and the tail function of the standard normal distribution. Exact bounds on R(z) for complex z with $$\mathfrak {R}z\geqslant 0$$ are obtained, which then yield logarithmically exact upper bounds on high-order derivatives of R. These results complement the many known bounds on the (inverse) Mills ratio of the real argument. The main idea of the proof is a non-asymptotic version of the so-called stationary-phase method. This study was prompted by a recently discovered alternative to the Euler–Maclaurin formula.

This study examines how vertical and horizontal pay disparities influence corporate innovation in publicly listed Indonesian firms from 2018 to 2022. Using a dataset of 1,505 firm-year observations, we apply Social Comparison Theory to analyze how perceived compensation inequalities impact innovation performance, measured by patent filings and citations. To ensure robustness, we employ Ordinary Least Squares (OLS), Two-Stage Least Squares (2SLS), Propensity Score Matching (PSM), Difference-in-Differences (DID), Entropy Balancing, and Tobit Regression. The results indicate that vertical pay disparity (CV_MT-RDP) positively affects patent quantity but negatively impacts patent quality, implying that larger managerial pay gaps encourage more patents but may not enhance their impact. Conversely, horizontal pay disparity (CV_RDP-OE) consistently reduces both patent output and citation impact, demonstrating that excessive pay differences across departments undermine cross-functional collaboration and innovation efficiency. Further, state-owned enterprises (SOEs) experience stronger negative effects of pay disparities on innovation than private firms, reinforcing the role of fairness concerns in shaping employee motivation. These findings suggest that firms should strategically design compensation policies to balance tournament incentives and pay equity to sustain long-term innovation performance.

The purpose of this study is twofold, based on the macroeconomic and microeconomic analysis. On the one hand, it is attempted to interpret and analyze the effects of fiscal consolidation in the countries-members of Eurozone while at the same time consumer behavior regarding current economic policy is estimated. On the other hand the research’s focus is on consumer behavior regarding specific categories of consumption. The interpretation of the effects of austerity measures in Greece during the period 2010-2011 is one of a high interest. The question of the research hypothesis is based on two fundamental theories about fiscal policy; the Keynesian and Ricardian theory. According to the first theory, a tax cut or an increase in government spending, results to higher income and higher consumption. Keynesian model is also the base of the theory of twin deficits. A fiscal expansion leads simultaneously to the deterioration of the current account deficit (CA). On the contrary, the proponents of Ricardian Equivalence Theorem claim that the aforementioned economic policies lead to an increase in private savings (precautionary savings). Consumers predict the government’s future fiscal adjustment in order to reduce deficits; thus they do not perceive the reduction of the taxes as additional income.As regard to the macroeconomic analysis, the econometric methodology of fixed-effects estimators and Arrelano-Bond models was investigated and based on country data from 12 Eurozone countries (Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Netherlands, Portugal, Spain). According to the results, fiscal adjustment has led to a decline in private consumption. More specifically, empirical research found that a high fiscal consolidation (above 2.5% of structural deficit) leads to a higher drop in private consumption. Especially for Greece, the impact of austerity has led to a higher drop in private consumption. The results of this research support Keynesian theory, confirming additionally the negative relationship between the volatility of government revenues and the growth of individual consumption. Furthermore, unemployment rate is negatively related to growth of individual consumption.As far as microeconomic analysis is concerned, the aim of this thesis is to identify the factors that influence consumers’ behavior regarding to private saving, consumers’ response to the first package of austerity measures in Greece and the level of consumption for food, heating and entertainment. In order to achieve this goal, we used a sample of 800 consumers from Athens and Crete. A questionnaire was distributed during the period August-November 2011. Previous studies on consumer’s behavior were used for the preparation of the questionnaire and the methodological approach. Moreover, Ordinary Least Square (OLS), Logit models, Ordered-Logit models, Tobit models and Quantile regressions were used.Generally, a high drop in private consumption is revealed. This result is supported on a macro and micro-economic level. Simultaneously, austerity measures have led consumers to save less. The study of consumer behavior is a valuable basis for the adoption and implementation of an effective policy in order to stimulate economic recovery.

We use cross sectional data from the World Bank enterprise surveys gathered in 2007 in South Africa’s four cities (Johannesburg, Cape Town, Durban and Port Elizabeth) to assess the impact of business related crimes on firm performance proxied using firm sales. Using Ordinary Least Squares (OLS) and Tobit model, we find that crime in the form of theft, robbery, arson and vandalism has a negative effect on sales and hence firm performance. However the impact of domestic shipment crime is mixed and varies from city to city depending on the magnitude of losses incurred by firms in each city. Results also show that crime is regressive in nature because crime related losses are relatively higher among small firms than large firms. The prevalence of crime amongst small firms and its negative effect on firm performance suggest the need for government and the business community to come together and develop security systems that are effective and affordable to small businesses. This is because, supporting small businesses is important for growth and employment creation

If the ordinary least squares (OLS) diagnostics discussed in the previous chapter indicate the existence of spatial lag or spatial error dependence, the researcher will wish to model the type of dependence indicated by these diagnostics. If the OLS diagnostics indicate the presence of a diffusion process, the researcher will wish to estimate a spatial lag model via maximum likelihood (ML) estimation or an instrumental variables specification incorporating instruments for the spatially lagged dependent variable. Alternatively, if the OLS diagnostics indicate the existence of spatial error dependence, the researcher may choose to estimate a more fully specified OLS model to model the spatial dependence or may choose to employ a ML or generalized method of moments (GMM) approach incorporating the spatial dependence in the errors. The spatial dependence diagnosed via the diagnostics discussed in Chapter 5 may alternatively be produced by spatial heterogeneity in the effects of covariates. If this is the only source of spatial dependence, modeling this heterogeneity will be sufficient to capture the spatial dependence. As a consequence, any specification search should also consider the possibility of spatial heterogeneity, which is the focus of Chapter 7. This chapter will first, however, examine alternative approaches for modeling spatial dependence if spatial heterogeneity is not present. This chapter begins by examining ML estimation of spatial lag models that derives from Ord (1975). Next, I explore alternative instrumental variables and GMM estimators for spatial lag dependence. Next, I turn to approaches for estimating spatial error models. I conclude by considering areas of concern in the estimation of spatial models. These include estimators for large sample sizes and diagnostics for continued spatial dependence. MAXIMUM LIKELIHOOD SPATIAL LAG ESTIMATION The mixed regressive, spatial autoregressive model, or spatial lag model, extends the pure spatial autoregressive model considered in Section 3.2 to include also the set of covariates and associated parameters: y = ρ W y +Xβ+e where X is again an N by K matrix of observations on the covariates, β is a K by 1 vector of parameters, and the remaining notation is as discussed in Section 3.2.

A restricted maximum likelihood estimator for truncated height samples

The change point identification has played a vital role in process improvement for an attribute process. This identification is able to effectively help process personnel to quickly determine the corresponding root causes and significantly improve the underlying process. Although many studies have focused on identifying the change point of a process, a generic identification approach has not been developed. The typical maximum likelihood estimator (MLE) approach has limitations: particularly, the known prior process distribution and mathematical difficulties. These deficiencies are commonly encountered in practice. Accordingly, this study proposes an artificial neural network (ANN) mechanism to overcome the difficulties of typical MLE approach in determining the change point of an attribute process. Specifically, the performance among the statistical process control (SPC) chart alone, the typical MLE approach, and the proposed ANN mechanism are investigated for the following cases: (1) a known attribute process distribution with the associated MLE being available to be used, (2) an unknown attribute process distribution with the MLE being unable to be used, and (3) an unknown attribute process distribution with the MLE being misused. The superior results and the performance of the proposed approach are reported and discussed.

Abstract. Type I linear regression models, which allow for measurement errors only in the criterion variable, are frequently used in modeling research in psychology and the social sciences. Although there are frequently measurement errors and large natural variation both in the criterion and predictor variables, type II regression methods that account for these errors are seldom used in these fields of study. The consistency and efficiency of three type II regression methods (reduced major axis, Kendall's robust line-fit and Bartlett's three-group) were evaluated in comparison to ordinary least squares (OLS) and the maximum likelihood with known variance ratio used frequently in biometrics and econometrics. When predictors are measured with error, OLS slope estimates are biased toward zero, and the same bias was observed with both Kendall's and Bartlett's methods. Reduced major axis produced consistent estimates even for small sample sizes, whenever the measurement errors in X are similar in magnitude to measurement errors in Y, but there was a consistent bias when the measurement error in X was smaller/greater than in Y. Maximum likelihood estimates behaved erroneously for small sample sizes, but for larger sample sizes they converged to the expected values.

Purpose- The study investigates the effect of financial deepening on stock market performance in selected Sub-Saharan African countries by determining the relationship that exist between financial deepening and stock market performance. Expansion in the financial services to reach out to the underbanked or unbanked in our society enables these individuals to assess banking services, thereby boosting economic activities. Methodology- The study considers four selected countries in Sub-Saharan African over the period 2001 to 2019. Multiple regression analysis techniques were used with Seemingly Unrelated Regression (SUR) to analyse the data. SUR used in this analysis provides the lowest standard errors of the estimated parameters. Findings- Ordinary Least Square (OLS) gives consistent results. However, it is not as efficient as the SUR method, which amounts to feasible generalised least squares with a specific form of the variance matrix. It solves the problem of endogeneity. The study conducted Augmented Dicky Fuller (ADF) test, Hausman test, and Bruce Pagan test to avoid any challenges associated with data normality. Conclusion- The research finds out that broad money supply, a proxy for financial deepening, positively and statistically significantly impacted stock market performance in each of the four countries. It was recommended that all countries involved in this study and others implement policies that seek to enhance financial deepening in increasing broad money supply as a percentage of GDP. The increase in overall money supply allows for investment in productive sectors of the economy. Keywords: Sub-Saharan Africa, seemingly unrelated regression, Augmented Dicky Fuller, investigates, ordinary least squares. JEL Codes: E51, F21