arXiv subjects

Gopal K. Basak

Publications and source records attributed to Gopal K. Basak.

11 recordsLinked to original sources

arXiv2026

Linear Regression: Inference Based on Cluster Estimates

This article proposes a novel estimator for regression coefficients in clustered data that explicitly accounts for within-cluster dependence. We study the asymptotic properties of the proposed estimator under both finite and infinite cluster sizes. The analysis is then extended to a standard random coefficient model, where we derive asymptotic results for the average (common) parameters and develop a Wald-type test for general linear hypotheses. We also investigate the performance of the conventional pooled ordinary least squares (POLS) estimator within the random coefficients framework and show that it can be unreliable across a wide range of empirically relevant settings. Furthermore, we introduce a new test for parameter stability at a higher (superblock; Tier 2, Tier 3,...) level, assuming that parameters are stable across clusters within that level. Extensive simulation studies demonstrate the effectiveness of the proposed tests, and an empirical application illustrates their practical relevance.

stat.ME↗

arXiv2024

Cross Sectional Regression with Cluster Dependence: Inference based on Averaging

We re-investigate the asymptotic properties of the traditional OLS (pooled) estimator, $\hat{\beta} _P$, in the context of cluster dependence. The present study considers various scenarios under various restrictions on the cluster sizes and number of clusters. It is shown that $\hat{\beta}_P$ could be inconsistent in many realistic situations. We propose a simple estimator, $\hat{\beta}_A$ based on data averaging. The asymptotic properties of $\hat{\beta}_A$ are studied. It is shown that $\hat{\beta}_A$ is consistent even when $\hat{\beta}_P$ is inconsistent. It is further shown that the proposed estimator $\hat{\beta}_A$ is more efficient than $\hat{\beta}_P$ in many practical scenarios. As a consequence of averaging, we show that $\hat{\beta}_A$ retains consistency, asymptotic normality under classical measurement error problem circumventing the use of Instrumental Variables (IV). A detailed simulation study shows the efficacy of $\hat{\beta}_A$. It is also seen that $\hat{\beta}_A$ yields better goodness of fit.

stat.ME↗

arXiv2021

Optimal Lockdown Strategy in a Pandemic: An Exploratory Analysis for Covid-19

The paper addresses the question of lives versus livelihood in an SIRD model augmented with a macroeconomic structure. The constraints on the availability of health facilities - both infrastructure and health workers determine the probability of receiving treatment which is found to be higher for the patients with severe infection than the patients with mild infection for the specific parametric configuration of the paper. Distinguishing between two types of direct intervention policy - hard lockdown and soft lockdown, the study derives alternative policy options available to the government. The study further indicates that the soft lockdown policy is optimal from a public policy perspective under the specific parametric configuration considered in this paper.

math.DS↗

arXiv2020

Understanding the origin of extreme events in El Niño-Southern Oscillation

We investigate a low-dimensional slow-fast model to understand the dynamical origin of El Niño-Southern Oscillation. A close inspection of the system dynamics using several bifurcation plots reveals that a sudden large expansion of the attractor occurs at a critical system parameter via a type of interior crisis. This interior crisis evolves through merging of a cascade of period-doubling and period-adding bifurcations that leads to the origin of occasional amplitude-modulated extremely large events. More categorically, a situation similar to homoclinic chaos arises near the critical point, however, atypical global instability evolves as a channel-like structure in phase space of the system that modulates variability of amplitude and return time of the occasional large events and makes a difference from the homoclinic chaos. The slow-fast timescale of the low-dimensional model plays an important role on the onset of occasional extremely large events. Such extreme events are characterized by their heights when they exceed a threshold level measured by a mean-excess function. The probability density of events' height displays multimodal distribution with an upper-bounded tail. We identify the dependence structure of interevent intervals to understand the predictability of return time of such extreme events using autoregressive integrated moving average model and box plot analysis.

nlin.CD↗

arXiv2016

A Mathematical Model of Foreign Capital Inflow

The paper models foreign capital inflow from the developed to the developing countries in a stochastic dynamic programming (SDP) framework. Under some regularity conditions, the existence of the solutions to the SDP problem is proved and they are then obtained by numerical technique because of the non-linearity of the related functions. A number of comparative dynamic analyses explore the impact of parameters of the model on dynamic paths of capital inflow, interest rate in the international loan market and the exchange rate.

econ.GN↗

arXiv2015

Langevin type limiting processes for Adaptive MCMC

Adaptive Markov Chain Monte Carlo (AMCMC) is a class of MCMC algorithms where the proposal distribution changes at every iteration of the chain. In this case it is important to verify that such a Markov Chain indeed has a stationary distribution. In this paper we discuss a diffusion approximation to a discrete time AMCMC. This diffusion approximation is different when compared to the diffusion approximation as in Gelman, Gilks and Roberts (1997) where the state space increases in dimension to infinity. In our approach the time parameter is sped up in such a way that the limiting distribution (as the mesh size goes to 0) is to a non-trivial continuous time diffusion process.

math.PR↗

arXiv2014

Diffusive Limits for Adaptive MCMC for Normal Target densities

In this paper we apply the Diffusion approximation procedure to a discrete time Adaptive Markov Chain Monte Carlo (AMCMC) method when the target distribution is standard Normal. We show that the limiting distribution of the diffusion admits a density which we identify as the standard Normal distribution.

math.ST↗

arXiv2013

An Ornstein-Uhlenbeck process associated to self-normalized sums

We consider an Ornstein-Uhleneck (OU) process associated to self-normalised sums in i.i.d. symmetric random variables from the domain of attraction of $N(0, 1)$ distribution. We proved the self-normalised sums converge to the OU process (in $C[0, \infty)$). Importance of this is that the OU process is a stationary process as opposed to the Brownian motion, which is a non-stationary distribution (see for example, the invariance principle proved by Csorgo et al (2003, Ann Probab) for self-normalised sums that converges to Brownian motion). The proof uses recursive equations similar to those that arise in the area of stochastic approximation and it shows (through examples) that one can simulate any functionals of any segment of the OU process. The similar things can be done for any diffusion process as well.

math.PR↗

arXiv2008

Asymptotic Properties of an Estimator of the Drift Coefficients of Multidimensional Ornstein-Uhlenbeck Processes that are not Necessarily Stable

In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator.

math.ST↗

arXiv2005

Stationarity of Switching VAR and Other Related Models

Switching ARMA models greatly enhance the standard linear models to the extent that different ARMA model is allowed in a different regime, and the regime switching is typically assumed a Markov chain on the finite states of potential regimes. Although statistical issues have been the subject of many recent papers, there is few systematic study of the probabilistic aspects of this new class of nonlinear models. This paper discusses some basic issues concerning this class of models including strict stationarity, influence of initial conditions, and second-order property by studying SVAR models. A number of examples are given to illustrate the theory and the variety of applications. Extensions to other models such as mean-shifting, and inhomogeneous transition probabilities are discussed.

math.ST↗

arXiv2005

Central limit theorems for a class of irreducible multicolor urn models

We take a unified approach to central limit theorems for a class of irreducible urn models with constant replacement matrix. Depending on the eigenvalue, we consider appropriate linear combinations of the number of balls of different colors. Then under appropriate norming the multivariate distribution of the weak limits of these linear combinations is obtained and independence and dependence issues are investigated.

math.PR↗