Multilevel Modeling for Economists: Why, When and How
The paper deals with multilevel regression modelling (MLM) as a method preferred to the ordinary least-squares regression in the analysis of comparative data with hierarchical data structure. We present substantive reasons (contextual sources of heterogeneity, causal heterogeneity, and generalisability of results) and statistical reasons (obtaining more precise and reliable estimates) for multilevel modelling. We also provide an overview of MLM implementation in several statistical packages. Using the cross-national World Values Survey (WVS) data, we outline a step-by-step procedure for building and fitting a two-level linear regression model of generalized trust on educational attainment levels (the “null” model, the fixed-intercept model, the random-intercept model, the random-intercept random-slope model, the model with a country-level predictor, and the cross-level interaction model). Then we describe and compare existing goodness-of-fit measures for MLM (AIC, BIC, maximum likelihood functions, and pseudo-R2). We also demonstrate robustness check techniques for multilevel models (visualization, Cook’s distance, and DFBETAs). In the final section, we overview alternative approaches to multilevel modelling when dealing with hierarchical data (cluster robust standard errors, generalized estimating equations, country fixed effects, country means, and aggregation) as currently practiced in comparative cross-national social science research. The replicable R code is attached.
What factors best explain the low incidence of skills training in a late industrial society like Russia? This research undertakes a multilevel analysis of the role of occupational structure against the probability of training. The explanatory power of occupation-specific determinants and skills polarisation are evaluated, using a representative 2012 sample from the Russian Longitudinal Monitoring Survey. Applying a two-level Bayesian logistic regression model, we show that the incidence of training in Russia is significantly contextualised within the structure of occupations and the inequalities between them. The study shows that extremely high wage gaps within managerial class jobs can discourage training, an unusual finding. Markets accumulating interchangeable and disposable labour best explain the low incidence of training; workers within generic labour are less likely to develop their skills formally, except in urban markets. Although we did not find strong evidence of skills polarisation, Russians are yet to live in a knowledge economy.
This paper explores the relationship between the characteristics of a street protest and its effectiveness. We propose a new approach to solving the problem through the use of a combination of several statistical techniques: the logistic regression models with mixed effects from the Frequentist approach, the hierarchical modeling from the Bayesian approach, the propensity score matching from the Quasiexperimental approach
Some Internet stores manage to charge prices that are significantly higher than market averages, therefore, obtaining some sort of price premium. This paper is dedicated to building a model that can be used to explain and predict a typical price premium that an Internet store charges for a specific product based on the information about the characteristics of the store and the features of the market for this product. Such models can provide support for pricing and assortment decisions: in particular, they allow detecting products that a store is likely to sell with the highest or the lowest markup based on price premia that are charged by stores with similar characteristics on similar markets.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.