arXiv subjects

Deepankar Basu

Publications and source records attributed to Deepankar Basu.

6 recordsLinked to original sources

arXiv2023

The Yule-Frisch-Waugh-Lovell Theorem for Linear Instrumental Variables Estimation

In this paper, I discuss three aspects of the Frisch-Waugh-Lovell theorem. First, I show that the theorem holds for linear instrumental variables estimation of a multiple regression model that is either exactly or overidentified. I show that with linear instrumental variables estimation: (a) coefficients on endogenous variables are identical in full and partial (or residualized) regressions; (b) residual vectors are identical for full and partial regressions; and (c) estimated covariance matrices of the coefficient vectors from full and partial regressions are equal (up to a degree of freedom correction) if the estimator of the error vector is a function only of the residual vectors and does not use any information about the covariate matrix other than its dimensions. While estimation of the full model uses the full set of instrumental variables, estimation of the partial model uses the residualized version of the same set of instrumental variables, with residualization carried out with respect to the set of exogenous variables. Second, I show that: (a) the theorem applies in large samples to the K-class of estimators, including the limited information maximum likelihood (LIML) estimator, and (b) the theorem does not apply in general to linear GMM estimators, but it does apply to the two step optimal linear GMM estimator. Third, I trace the historical and analytical development of the theorem and suggest that it be renamed as the Yule-Frisch-Waugh-Lovell (YFWL) theorem to recognize the pioneering contribution of the statistician G. Udny Yule in its development.

econ.EM↗

arXiv2023

The Yule-Frisch-Waugh-Lovell Theorem

This paper traces the historical and analytical development of what is known in the econometrics literature as the Frisch-Waugh-Lovell theorem. This theorem demonstrates that the coefficients on any subset of covariates in a multiple regression is equal to the coefficients in a regression of the residualized outcome variable on the residualized subset of covariates, where residualization uses the complement of the subset of covariates of interest. In this paper, I suggest that the theorem should be renamed as the Yule-Frisch-Waugh-Lovell (YFWL) theorem to recognize the pioneering contribution of the statistician G. Udny Yule in its development. Second, I highlight recent work by the statistician, P. Ding, which has extended the YFWL theorem to a comparison of estimated covariance matrices of coefficients from multiple and partial, i.e. residualized regressions. Third, I show that, in cases where Ding's results do not apply, one can still resort to a computational method to conduct statistical inference about coefficients in multiple regressions using information from partial regressions.

econ.EM↗

arXiv2023

Formal Covariate Benchmarking to Bound Omitted Variable Bias

Covariate benchmarking is an important part of sensitivity analysis about omitted variable bias and can be used to bound the strength of the unobserved confounder using information and judgments about observed covariates. It is common to carry out formal covariate benchmarking after residualizing the unobserved confounder on the set of observed covariates. In this paper, I explain the rationale and details of this procedure. I clarify some important details of the process of formal covariate benchmarking and highlight some of the difficulties of interpretation that researchers face in reasoning about the residualized part of unobserved confounders. I explain all the points with several empirical examples.

econ.EM↗

arXiv2022

Singularity of Input-Output Matrices: Structural Linkages and Spectral Properties

Analyzing 5 historical and 670 contemporary input-output (IO) tables, we show that all of these IO matrices are singular. We map this phenomenon onto underlying production network topologies. While singularity trivially emerges from sectors lacking intermediate forward or backward linkages, it could frequently persist due to deeper economic properties: proportional cost structures across sectors, isolated economic sub-networks, and supply-chain hierarchies lacking intra-industry transactions. As a subset of these matrices is non-diagonalizable, we leverage this singularity to derive simple, rank and eigenvalue based diagnostic criteria for diagonalizability. This provides an efficient computational test for network shock propagation models using eigendecomposition.

econ.TH↗

arXiv2022

Marx after Okishio: Falling Rate of Profit with Constant Rate of Exploitation

Can cost-reducing technical change lead to a fall in the long run rate of profit if class struggle manages to keep the rate of exploitation constant? In a general circulating capital model, we derive sufficient conditions for cost-reducing technical change to both keep the rate of exploitation constant and lead to a fall in the equilibrium rate of profit. Further, if the real wage bundle is such that the maximum price-value ratio is larger than 1 plus the rate of exploitation, then starting from any configuration of technology and real wage, we can always find a viable, CU-LS technical change that satisfies the sufficient conditions for the previous result. Taken together, these results vindicate Marx's claim in Volume III of Capital, that if the rate of exploitation remains unchanged then viable, CU-LS technical change in capitalist economies can lead to a fall in the long run rate of profit.

econ.TH↗

arXiv2022

Bounds for Bias-Adjusted Treatment Effect in Linear Econometric Models

In linear econometric models with proportional selection on unobservables, omitted variable bias in estimated treatment effects are real roots of a cubic equation involving estimated parameters from a short and intermediate regression. The roots of the cubic are functions of $\delta$, the degree of selection on unobservables, and $R_{max}$, the R-squared in a hypothetical long regression that includes the unobservable confounder and all observable controls. In this paper I propose and implement a novel algorithm to compute roots of the cubic equation over relevant regions of the $\delta$-$R_{max}$ plane and use the roots to construct bounding sets for the true treatment effect. The algorithm is based on two well-known mathematical results: (a) the discriminant of the cubic equation can be used to demarcate regions of unique real roots from regions of three real roots, and (b) a small change in the coefficients of a polynomial equation will lead to small change in its roots because the latter are continuous functions of the former. I illustrate my method by applying it to the analysis of maternal behavior on child outcomes.

econ.EM↗