Quasi-likelihood Research Articles

Regularized maximum likelihood estimation for the random coefficients model

. The random coefficients model Y i = β 0 i + β 1 i X 1 i + β 2 i X 2 i + … + β d i X d i , with 𝐗 i , Y i , 𝜷 i i.i.d, and 𝜷 i independent of 𝐗 i is often used to capture unobserved heterogeneity in a population. We propose a quasi-maximum likelihood method to estimate the joint density distribution of the random coefficient model. This method implicitly involves the inversion of the Radon transformation in order to reconstruct the joint distribution, and hence is an inverse problem. To add stability to the solution, we apply Tikhonov-type regularization methods. Nonparametric estimation for the joint density of β i = ( β 0 i , … , β d i ) based on kernel methods or Fourier inversion have been proposed in recent years. Most of these methods assume a heavy tailed design density f 𝐗 . We analyze the convergence of the quasi maximum likelihood method without assuming heavy tails for f 𝐗 and illustrate performance by applying the method on simulated and real data.

Quasi Maximum Likelihood Estimation for Large-Dimensional Matrix Factor Models

In this study, we introduce a novel approach, called the quasi maximum likelihood estimation (Q-MLE), for estimating large-dimensional matrix factor models. In contrast to the principal component analysis based approach, Q-MLE considers the heteroscedasticity of the idiosyncratic error term, the heteroscedasticity of which is simultaneously estimated with other parameters. Interestingly, under the homoscedasticity assumption of the idiosyncratic error, the Q-MLE estimator encompassed the projected estimator (PE) as a special case. We provide the convergence rates and asymptotic distributions of the Q-MLE estimators under mild conditions. Extensive numerical experiments demonstrate that the Q-MLE method performs better, especially when heteroscedasticity exists. Furthermore, two real examples in finance and macroeconomics reveal factor patterns across rows and columns, which coincide with financial, economic, or geographical interpretations.

Threshold spatial autoregressive model

In this paper, we consider the estimation and inferential issues of the threshold spatial autoregressive (TSAR) model, which is a hybrid of the threshold and spatial autoregressive models. We use the quasi maximum likelihood (QML) method to estimate the model. In addition, we prove the tightness and the Hájek–Rényi type inequality for a quadratic form and establish a full inferential theory of the QML estimator under the setup that threshold effect shrinks to zero as the sample size increases. We conduct hypothesis testing on the presence of the threshold effect, using three super-type statistics. Their asymptotic behaviors are studied under the Pitman local alternatives. A bootstrap procedure is applied to obtain the asymptotically correct critical value. We also consider hypothesis testing on the threshold value set equal to a prespecified one. We run Monte Carlo simulations to investigate the finite sample performance of the QML estimators and find that the estimators perform well. In an empirical application, we apply the proposed TSAR model to study the relationship between financial development and economic growth, and we find firm evidence to support the TSAR model.

Fractional stochastic volatility model

This article introduces a discrete‐time fractional stochastic volatility model (FSV) based on fractional Gaussian noise. The new model includes the standard stochastic volatility model as a special case and has the same limit as the fractional integrated stochastic volatility (FISV) model, which is the continuous‐time fractional Ornstein–Uhlenbeck process. A simulated maximum likelihood method, which maximizes the time‐domain log‐likelihood function calculated by the importance sampling technique, and a frequency‐domain quasi maximum likelihood method (or quasi Whittle) are employed to estimate the model parameters. Simulation studies suggest that, while both estimation methods can accurately estimate the model, the simulated maximum likelihood method outperforms the quasi Whittle method. As an illustration, we fit the FSV and FISV models with the proposed estimation techniques to the S&P 500 composite index over a sample period spanning 45 years.

A simulation study to assess the impact missing values on the performance of different statistical methods for analysis of binary repeated measures data with an additional hierarchical structure

The primary objective of the study was to assess the impact of missing values on the analy- sis of binary repeated measures data with an additional hierarchical structure. One motivat- ing example for the present study was records of high somatic cell counts in milk samples obtained by approximately monthly sampling throughout the lactations of cows in dairy herds. Random effects models with autocorrelated (ρ = 1, 0.9 or 0.5) subject-level ran- dom effects were behind the simulated data. In general, the settings of the simulation were chosen to reflect a real somatic cell count dataset (scc40), except that the within-cow time series length was set to 8-time points for each cow. The estimation procedures consid- ered were: Ordinary Logistic Regression (OLR), Alternating Logistic Regression (ALR), Weighted Generalized Estimating Equations (WGEE), Penalized Quasi Likelihood (PQL), Maximum likelihood via numerical integration (ML) and Bayesian Markov chain Monte Carlo (MCMC). Multiple scenarios of simulated incomplete datasets were considered and include: a scenario corresponded to a combination of missingness patterns present in the scc40 dataset (scc40 scenario) The remaining scenarios involved only drop-outs, and corre- sponded to either moderate or high percentages of values either missing at random (MAR) or not missing at random (NMAR), respectively. In the scc40 scenario, all estimation procedures except OLR performed well and produced estimates with small relative bias (generally less than 5%) for levels of missingness that roughly corresponded to the scc40 data. In MAR missingness scenarios, some biases were found for ALR, WGEE and PQL procedures, whereas the likelihood-based procedures were largely unaffected by the miss- ing values. In NMAR scenarios, all procedures experienced similar and strong biases in the time coefficient; however, fixed effects estimates at the subject and cluster levels were relatively unaffected. Journal of Statistical Research 2023, Vol 57, No.1-2, pp.35-67

A dynamic count process

The current paper aims to complement the recent development of the observation-driven models of dynamic counts with a parametric-driven one for a general case, particularly discrete two parameters exponential family distributions. The current paper proposes a finite semiparametric exponential mixture of SETAR processes of the conditional mean of counts to capture the nonlinearity and complexity. Because of the intrinsic latency of the conditional mean, the general additive state-space representation of dynamic counts is firstly proposed then stationarity and geometric ergodicity are established under a mild set of conditions. We also propose to estimate the unknown parameters by using quasi maximum likelihood estimation and establishes the asymptotic properties of the quasi maximum likelihood estimators (QMLEs), particularly T-consistency and normality under the relatively mild set of conditions. Furthermore, the finite sample properties of the QMLEs are investigated via simulation exercises and an illustration of the proposed process is presented by applying the proposed method to the intraday transaction counts per minute of AstraZeneca stock.

Analysis of gene expression dynamics and differential expression in viral infections using generalized linear models and quasi-likelihood methods.

Our study undertakes a detailed exploration of gene expression dynamics within human lung organ tissue equivalents (OTEs) in response to Influenza A virus (IAV), Human metapneumovirus (MPV), and Parainfluenza virus type 3 (PIV3) infections. Through the analysis of RNA-Seq data from 19,671 genes, we aim to identify differentially expressed genes under various infection conditions, elucidating the complexities of virus-host interactions. We employ Generalized Linear Models (GLMs) with Quasi-Likelihood (QL) F-tests (GLMQL) and introduce the novel Magnitude-Altitude Score (MAS) and Relaxed Magnitude-Altitude Score (RMAS) algorithms to navigate the intricate landscape of RNA-Seq data. This approach facilitates the precise identification of potential biomarkers, highlighting the host's reliance on innate immune mechanisms. Our comprehensive methodological framework includes RNA extraction, library preparation, sequencing, and Gene Ontology (GO) enrichment analysis to interpret the biological significance of our findings. The differential expression analysis unveils significant changes in gene expression triggered by IAV, MPV, and PIV3 infections. The MAS and RMAS algorithms enable focused identification of biomarkers, revealing a consistent activation of interferon-stimulated genes (e.g., IFIT1, IFIT2, IFIT3, OAS1) across all viruses. Our GO analysis provides deep insights into the host's defense mechanisms and viral strategies exploiting host cellular functions. Notably, changes in cellular structures, such as cilium assembly and mitochondrial ribosome assembly, indicate a strategic shift in cellular priorities. The precision of our methodology is validated by a 92% mean accuracy in classifying respiratory virus infections using multinomial logistic regression, demonstrating the superior efficacy of our approach over traditional methods. This study highlights the intricate interplay between viral infections and host gene expression, underscoring the need for targeted therapeutic interventions. The stability and reliability of the MAS/RMAS ranking method, even under stringent statistical corrections, and the critical importance of adequate sample size for biomarker reliability are significant findings. Our comprehensive analysis not only advances our understanding of the host's response to viral infections but also sets a new benchmark for the identification of biomarkers, paving the way for the development of effective diagnostic and therapeutic strategies.

Testing Homoskedasticity in Spatial Panel Data Models

A robust test statistic for testing homoskedasticity in spatial panel data models that have entity and time fixed effects is introduced in a quasi maximum likelihood estimation setting. A size-correction approach is introduced to ensure that the score functions have an asymptotic distribution centered around zero in the local presence of certain nuisance parameters. The outer-product-of-martingale-difference (OPMD) method is used to formulate an estimator for the asymptotic variance of the score functions. The OPMD estimator and the adjusted score functions are used to formulate a computationally simple robust test statistic. The suggested test statistic does not require knowing the presence of spatial dependence in the outcome variable and/or the disturbance terms. The asymptotic distribution of the test statistic is established under the null and local alternative hypotheses. Through Monte Carlo simulations, the finite sample size and power properties of the proposed test statistic are investigated. Finally, two empirical applications are provided to illustrate the practical use of the proposed test statistic.

The Use Penalized Quasi Likelihood (PQL) to Estimate Multilevel Binary Logistic Model to Identify the Factors of Anemia

Multilevel logistic regression analysis is used in many fields, including health, medical, geography, social and educational, where the researchers were interested in this analysis to identify and study the nature of the relationship between the behavior of the vocabulary or units of study, and the social, environmental and economic variables in different environments in which they live and belong, and in such hierarchical data. In this research, multilevel data was used, Data were taken on anemia in children, as the dependent variable represents anemia infections (infected - uninfected) from four hospitals affiliated to the Babylon Health Department, namely (Al-Hah General Teaching Hospital - Imam Al-Sadiq Hospital (PBUH) - Marjan General Teaching Hospital - Babel Women's and Children's Hospital ) which represents the third level in the analysis and according to the type of lobby (public - private), which represents the second level. (50) cases of anemia were taken from the general ward from the general ward and (20) cases from the private ward, and from Imam al-Sadiq Hospital ( P) Peace, (75) sick cases were taken from the general ward and (20) sick cases from the private ward. From Marjan General Teaching Hospital, (30) sick cases were taken from the private ward and (20) sick cases from the general ward, and (40) sick cases were taken from the general ward and (25) sick cases from the general ward, and these sick cases represent the level Third, in the multi-level analysis, so that the total number of disease cases is (290) cases, and the independent variables that can affect anemia were taken (sex, age, weight, occupation, marital status, smoking, academic achievement, place of residence, infection with other diseases, blood pressure). That is, the first level contains (10) independent variables. The logistic analysis of the multilevel binary was carried out using the NCSS 2022 program, using the method of possibility semi-penalty. has been reached Significance of the multi-level binary logistic regression model, as the p-value = 0.0015, which is less than the level of significance 1%, and achieved a high odds ratio of (0.7767). The variables (age - occupation - smoking - blood pressure) are not significant and the variables (weight - water source - place of residence - other diseases - wealth index) are significant.

A novel loss function for neural network models exploring stock realized volatility using Wasserstein Distance

This study proposes a novel loss function for neural network models that explores the topological structure of stock realized volatility (RV) data by adding Wasserstein Distance (WD). The study shows that the proposed loss statistically significantly improves the forecast accuracy of neural network models for magnitude-dependent error measures, for example, Root Mean Square Error (RMSE) and Mean Absolute Error (MAE), but not necessarily for relative error measures, such as Quasi-likelihood (QLIKE). Additionally, this research provides user-friendly open-source code for researchers and practitioners to implement the proposed loss function efficiently and quickly.

The Spatial Autoregressive Panel Data Model with Spatial Moving Average Errors

This paper advocates the wider use of the spatial autoregressive (AR) panel data model with spatial moving average (MA) errors, individual and time effects, and different spatial weight matrices for each spatial lag. We demonstrate the practical relevance of this model, derive and investigate the asymptotic properties of a simple quasi maximum likelihood within estimator when is large and is finite, and provide an empirical example explaining military expenditures in 144 countries over the period 1993–2007.

A random coefficient integer autoregressive model of order one with seasonal structure (RCINAR(1)s)

Seasonality is an inherent part of most of the epidemic data. The fixed coefficient INAR(1) models with seasonal structure have been studied by many authors. The varying immunity and susceptibility affect the chances of catching or escaping an infection. This brings in the randomness in the phenomenon of the spread of the diseases. The fixed coefficient INAR models assume that the chance of infection remains the same for every individual, which is not true practically and hence one needs to study the disease spread phenomenon using random coefficient INAR models. The parameters of the proposed model have been estimated using quasi maximum likelihood estimation. Various probabilistic and inferential properties of the model have been studied. A simulation study has been carried out for parameter estimation. Two data sets having seasonal structures have been analyzed using the model. The model fits well to the data sets compared to the existing models.

Extended Glivenko–Cantelli theorem and L 1 strong consistency of innovation density estimator for time-varying semiparametric ARCH model

ABSTRACT This paper extends the classical Glivenko–Cantelli theorem for the empirical cumulative distribution function based on the innovations in the ARCH model with a slowly time-varying trend. In this semiparametric time-varying model, strong consistency for the innovation density estimator via kernel smoothing method is established, given that the trend and ARCH parameter estimators meet some mild conditions. Besides, the strong consistency for the Gaussian quasi maximum likelihood estimator (QMLE) in the time-varying ARCH parameter is established as well. Moreover, in terms of the existence of the trend in the data, two major nonparametric trend estimators, B-spline and kernel estimators, are shown to be appropriate for the strong consistency results.

Semi-nonparametric estimation of secret reserve prices in auctions

This letter proposes an estimation method for the distribution of unobservable secret seller reserve prices in auctions. Within the symmetric affiliated private values model, I construct a density estimator using the semi-nonparametric quasi maximum likelihood approach developed by Gallant and Nychka (1987). I then use Monte Carlo experiments to demonstrate its performance in relatively small samples. A discussion on applying this method to structurally similar identification problems is provided. • This letter offers an approach to identify the distribution of secret reserve prices. • The model allows for affiliated private values among buyers and sellers. • Input data include the auction’s paying bid and whether the item sold at said bid. • A semi-nonparametric estimator identifies the distribution of secret reserve prices. • This method works well in relatively small samples ( ∼ 100 ).

Forecasting Model of Air Pollution Index using Generalized Autoregressive Conditional Heteroskedasticity Family (GARCH)

The Air Pollution Index (API) of Malaysia has increased consistently in recent decades, becoming a serious environment issue concern. In this paper, we analyzed daily integer value time series data for API in Sarawak from January to June in 2019 using generalized autoregressive conditional heteroskedasticity (GARCH) family for discrete case namely poisson integer value GARCH (INGARCH), negative binomial integer value GARCH (NBINGARCH) and integer value autoregressive conditional heteroskedasticity (INARCH) models. The parameters of the models will be estimated using quasi likelihood estimator (QLE) and we compare their Akaike information criterion (AIC) to determine the best model fitted the data. The results showed that INGARCH (1,1) and INARCH (1,0) performed inconsistent results since the conventional methods of NBINGARCH (1,1) outperformed the performance of INGARCH (1,1) and INARCH (1,0). However, consistet results were achieved as the NBINGARCH (1,1) gave the smallest forecasting error compared to INGARCH (1,1) and INARCH (1,0). The findings are very important for controlling the API results in future and taking protection measure for conservation of the air.

High-dimensional volatility matrix estimation with high-frequency financial data: The GARCH-Itô grouped factor model

In the research area of the high-frequency financial data analysis, estimation and prediction of the high-dimensional volatility matrix are challenging problems, especially when the assets of interest have a naturally given group structure. In order to address this issue, we propose a novel GARCH-Itôgrouped factor model, in which we present the log price series of grouped assets with common factors, group-specific factors, and an asset-specific error term. We then embed the discrete GARCH structure into the volatility of the eigenvalue processes to capture the volatility dynamics of the observed price data. We propose a quasi maximum likelihood method for parameter estimation, establish its asymptotic properties and illustrate its good finite-sample performance with simulation. In real data analysis, we compare our method with the ungrouped factor model and the nonparametric high-frequency volatility estimator MSRV using the data from the SSE Main Board and the SZSE ChiNext market, where the proposed method outperforms its competitors in terms of prediction of the volatility matrix.

Distribution of test statistics under parameter uncertainty for time series data: an application to testing skewness, kurtosis and normality

In this paper, we provide a general result under some high level assumptions that shows how to account for the parameter uncertainty problem in test statistics formulated with the quasi maximum likelihood (QML) estimator. We use our general result to develop various test statistics for testing skewness, kurtosis and normality for time series data. We show that the asymptotic distributions of our test statistics coincide with the asymptotic distributions of some tests suggested in the literature. Thus, our general result provides a unified approach for test statistics formulated with the QML estimator for time series data.

Berry–Esseen Bounds of the Quasi Maximum Likelihood Estimators for the Discretely Observed Diffusions

For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, this paper obtains the Berry–Esseen bounds on the rates of convergence to normality of the distributions of the quasi maximum likelihood estimators based on stochastic Taylor approximation, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval, the high-frequency regime. It shows that the higher-order stochastic Taylor approximation-based estimators perform better than the basic Euler approximation in the sense of having smaller asymptotic variance.

Consistency without compactness of the parameter space in spatial econometrics

When studying the consistency of an estimator without a closed-form solution for a spatial econometric model, we usually assume that the parameter space is compact. However, compactness assumptions are restrictive as we need to know the boundaries of parameter spaces. We establish a consistency theorem for concave objective functions. We apply this result to rebuild the consistency of the quasi maximum likelihood estimator (QMLE) of a spatial autoregressive (SAR) model and a SAR Tobit model. Their log-likelihood functions are not concave, but they can be concave after proper reparameterization as in Olsen (1978).

Determinants of Long Acting Family Planning Utilization Among Reproductive Age Women in Ethiopia Based on Ethiopian Demographic and Health Survey Data 2019: Generalized Estimated Equation (GEE) Modeling

Background: Obstetric morbidity and mortality is reduced by universal access of reproductive health services including family planning. Long acting family planning is highly effective with low failure rate while the utilization and its determinants is not well studied in the recent EDHS data. Identifying the determinants of long acting family planning is important to design appropriate interventions and programs that address for the specific population. Objective: To identify the determinants of long acting family planning utilization among reproductive age women in Ethiopia based on Ethiopian demographic and health survey-2019. Method: A secondary data analysis was done from 2019 EDHS data. A data were extracted from kid record file by STATA version 15. A total of, 4,782 reproductive age women were selected for the study. After weighting the data, generalized estimated equation modeling was analyzed by xtgee STATA command. The model selection was done by quasi likelihood criteria (QIC) and fitting the model was done by two proposed working correlation structure (exchangeable and independent). Generalized estimated equation modeling of the study parameter was assessed accordingly. Result: The magnitude of long acting family planning utilization in this study was 8.6[95% CI: 7.8, 9.4]. Determinants of long acting family planning utilization was age of women 40-49 year (OR=1.87 [95%CI: 1.7, 4.7]), rural residence (OR= 0.47[95%CI: 0.19, 0.89]), female household head (OR=1.67[95%CI:1.5,2.2]), family size ≥13 (OR=0.04[95CI:0.003,0.68]) and number of under five children ≥4 (OR=0.26[95%CI:0.09,0.68]). Conclusion: The magnitude of long acting family planning utilization in Ethiopian women is relatively low. Age, residence, household head, family size, and number of under five children was the identified determinants of long acting family planning utilization. The finding imply that the country need scaling up the use of long acting family planning through improving young and adolescent FP use, empowering women and working on large family size households. RH interventions including family planning could address the rural male headed young aged women who had more than four under five children in the house to increase the utilization of long acting family planning utilization. Funding Information: The authors did not receive any fund for this research Declaration of Interests: The authors declare that they have no competing interests. Ethics Approval Statement: The dataset used in this study was obtained from demographic and health survey (DHS) database at http://dhsprogram.com/data/ after getting the approval letter from DHS program office. The Institution Review Board approved procedures for DHS public use datasets do not in any way allow respondents, households or sample communities to be identified.