LR Statistic Research Articles

Use of the linear regression method to evaluate population accuracy of predictions from non-linear models.

To address the limitations of commonly used cross-validation methods, the linear regression method (LR) was proposed to estimate population accuracy of predictions based on the implicit assumption that the fitted model is correct. This method also provides two statistics to determine the adequacy of the fitted model. The validity and behavior of the LR method have been provided and studied for linear predictions but not for nonlinear predictions. The objectives of this study were to 1) provide a mathematical proof for the validity of the LR method when predictions are based on conditional means, regardless of whether the predictions are linear or non-linear 2) investigate the ability of the LR method to detect whether the fitted model is adequate or inadequate, and 3) provide guidelines on how to appropriately partition the data into training and validation such that the LR method can identify an inadequate model. We present a mathematical proof for the validity of the LR method to estimate population accuracy and to determine whether the fitted model is adequate or inadequate when the predictor is the conditional mean, which may be a non-linear function of the phenotype. Using three partitioning scenarios of simulated data, we show that the one of the LR statistics can detect an inadequate model only when the data are partitioned such that the values of relevant predictor variables differ between the training and validation sets. In contrast, we observed that the other LR statistic was able to detect an inadequate model for all three scenarios. The LR method has been proposed to address some limitations of the traditional approach of cross-validation in genetic evaluation. In this paper, we showed that the LR method is valid when the model is adequate and the conditional mean is the predictor, even when it is a non-linear function of the phenotype. We found one of the two LR statistics is superior because it was able to detect an inadequate model for all three partitioning scenarios (i.e., between animals, by age within animals, and between animals and by age) that were studied.

GROWTH OPPORTUNITY, FIRM SIZE, FINANCIAL DISTRESS AND HEDGING DECISIONS

This study aims to determine the effect of Growth Opportunity, Firm Size, and Financial Distress on Hedging Decisions in Property and Real Estate Sector companies listed on the Indonesia Stock Exchange in 2018-2022. The type of research used is quantitative associative using secondary data. The sample selection was based on purposive sampling method, so that the samples obtained were 48 (fortyeight) companies. The data analysis technique used is logistic regression analysis, with data processing using the Eviews 12 for student program. Hypothesis testing in this study used the Likelihood Ratio Statistics (LR Statistics) test and the z-statistics test. The results of the Likelihood Ratio Statistics (LR Statistics) test show that the variables of Growth Opportunity, Firm Size, and Financial Distress together have an effect on hedging decisions. The results of the z-statistics test show that Firm Size has an effect on hedging decisions, while the Growth Opportunity and Financial Distress variables have no effect on hedging decisions.

Women's Empowerment and Food Security in Cameroon

The zero-hunger objective of the Sustainable Development Goals is still far from reach as food insecurity continues to be a problem. Literature largely associates women empowerment with food security but its empirical evidence is still sparse, especially in Cameroon. To fill this gap, the study investigates the effect of women empowerment on food security in Cameroon. The methodology employed eight indicators of women empowerment drawn from the Cameroon Consumption Household Survey IV to construct an index of women empowerment using the Multiple Correspondence Analysis. Likewise, Household Food Insecurity Access Scale is used in combination with other food security measures to construct a composite index of food security. The study then employs two stage least square instrumental variable model for analysis. Pre-estimation and post-estimation tests were employed for robustness; such as the Anderson canon LR Statistics to test for the strength of the instruments used and the endogeneity test to validate the need for the instrumental variable regression. The key result posits that women empowerment has a positive and significant effect on food security. A unit increase in women empowerment increases food security by 0.474 and it is significant at 5% significant level. This aligns with apriori expectation. Furthermore, household size, the square of household size, age and the square of age are all not significant. Though, household size and age are negative while their squares are positive. Also, the results show that households located in urban areas are significantly more food secure than those in rural areas at a 1% level of significance. Respondents who were married had significantly lower food security than the unmarried and Christian women had significantly lower food security than their counterparts. The study recommends that there is need to set up programs and institutions that empower women in all aspects so as to ensure food security consequently. Furthermore, the empowerment of women should be incorporated in food poverty reduction strategies that touches the household directly. Conclusively, the study insinuates a positive relationship between women empowerment and food security, therefore all women empowerment measures must be promoted and institutionalized if food security must be improved.

Small Sample Adjustment for Hypotheses Testing on Cointegrating Vectors

Abstract Johansen’s (2000. “A Bartlett Correction Factor for Tests of on the Cointegrating Relations.” Econometric Theory 16: 740–78) Bartlett correction factor for the LR test of linear restrictions on cointegrated vectors is derived under the i.i.d. Gaussian assumption for the innovation terms. However, the distribution of most data relating to financial variables is fat-tailed and often skewed; there is therefore a need to examine small sample inference procedures that require weaker assumptions for the innovation term. This paper suggests that using the non-parametric bootstrap to approximate a Bartlett-type correction provides a statistic that does not require specification of the innovation distribution and can be used by applied econometricians to perform a small sample inference procedure that is less computationally demanding than it’s analytical counterpart. The procedure involves calculating a number of bootstrap values of the LR test statistic and estimating the expected value of the test statistic by the average value of the bootstrapped LR statistic. Simulation results suggest that the inference procedure has good finite sample property and is less dependent on the parameter space of the data generating process.

Residual bootstrap tests in linear models with many regressors

This paper is concerned with bootstrap hypothesis testing in linear regression models with many regressors. I show that bootstrap F, LR and LM tests are asymptotically valid even when the numbers of estimated parameters and tested restrictions are not asymptotically negligible fractions of the sample size. One of the conditions for these results is that the regressors come from an asymptotically balanced design. Depending on the number of restrictions tested and on the errors’ distribution, violation of that condition might render the bootstrap tests asymptotically invalid. In that case, I propose bootstrapping Calhoun’s (2011) G statistic or modified versions of the LR and LM statistics, and show that these procedures remain asymptotically valid.Monte Carlo simulations indicate that the bootstrap tests often outperform asymptotic ones. However, analyzing the approximate third cumulant of the F statistic reveals that the bootstrap test does not generally provide the usual higher order asymptotic refinements. Nevertheless, it is found that the bootstrap third cumulant partially matches the population third cumulant, which might explain the bootstrap’s good finite sample performances, especially when the errors come from a symmetric distribution.

Computable Error Bounds for High-Dimensional Approximations of an LR Statistic for Additional Information in Canonical Correlation Analysis

Let $\lambda$ be the LR criterion for testing an additional information hypothesis on a subvector of $p$-variate random vector ${x}$ and a subvector of $q$-variate random vector ${y}$, based on a sample of size $N=n+1$. Using the fact that the null distribution of $-(2/N)\log \lambda$ can be expressed as a product of two independent $\Lambda$ distributions, we first derive an asymptotic expansion as well as the limiting distribution of the standardized statistic $T$ of $-(2/N)\log \lambda$ under a high-dimensional framework when the sample size and the dimensions are large. Next, we derive computable error bounds for the high-dimensional approximations. Through numerical experiments it is noted that our error bounds are useful in a wide range of $p$, $q$, and $n$.

Computable error bounds for high-dimensional approximations of an LR statistic for additional information in canonical correlation analysis

Пусть $\lambda$ есть статистика отношения правдоподобия, построенная по выборке размера $N=n+1$, для проверки гипотезы об избыточности информации в подвекторе $p$-мерного случайного вектора $\mathbf{x}$ и в подвекторе $q$-мерного случайного вектора $\mathbf{y}$. Используя тот факт, что при выполнении нулевой гипотезы распределение статистики $-(2/N)\ln \lambda$ выражается в виде произведения двух независимых $\Lambda$ распределений, сначала находится предельное распределение, а также строятся асимптотические разложения для стандартизованной статистики $T$ величины $-(2/N)\ln \lambda$ в условиях наблюдений высокой размерности, когда и размер выборки, и размерности велики. Далее выводятся вычислимые оценки для приближений в случае данных высокой размерности. Численные эксперименты показывают, что найденные оценки полезны в широком диапазоне значений величин $p$, $q$ и $n$.

Inference and testing on the boundary in extended constant conditional correlation GARCH models

We consider inference and testing in extended constant conditional correlation GARCH models in the case where the true parameter vector is a boundary point of the parameter space. This is of particular importance when testing for volatility spillovers in the model. The large-sample properties of the QMLE are derived together with the limiting distributions of the related LR, Wald, and score statistics. Due to the boundary problem, these large-sample properties become nonstandard. The size and power properties of the tests are investigated in a simulation study. As an empirical illustration we test for (no) volatility spillovers between foreign exchange rates.

MCMC Con dence Sets for Identi ed Sets

In complicated/nonlinear parametric models, it is generally hard to determine whether the model parameters are (globally) point identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of parameters in econometric models defined through a likelihood or a vector of moments. The CSs for the identified set or for a function of the identified set (such as a subvector) are based on inverting an optimal sample criterion (such as likelihood or continuously updated GMM), where the cutoff values are computed via Monte Carlo simulations directly from a quasi posterior distribution of the criterion. We establish new Bernstein-von Mises type theorems for the posterior distributions of the quasi-likelihood ratio (QLR) and profile QLR statistics in partially identified models, allowing for singularities. These results imply that the Monte Carlo criterion-based CSs have correct frequentist coverage for the identified set as the sample size increases, and that they coincide with Bayesian credible sets based on inverting a LR statistic for point-identified likelihood models. We also show that our Monte Carlo optimal criterion-based CSs are uniformly valid over a class of data generating processes that include both partially- and point-identified models. We demonstrate good finite sample coverage properties of our proposed methods in four non-trivial simulation experiments: missing data, entry game with correlated payoff shocks, Euler equation and finite mixture models. Finally, our proposed procedures are applied in two empirical examples.

Inference and Testing on the Boundary in Extended Constant Conditional Correlation GARCH Models

We consider inference and testing in extended constant conditional correlation GARCH models in the case where the true parameter vector is a boundary point of the parameter space. This is of particular importance when testing for volatility spillovers in the model. The large-sample properties of the QMLE are derived together with the limiting distributions of the related LR, Wald, and LM statistics. Due to the boundary problem, these large-sample properties become nonstandard. The size and power properties of the tests are investigated in a simulation study. As an empirical illustration we test for (no) volatility spillovers between foreign exchange rates.

Nonnull asymptotic distributions of the LR, Wald, score and gradient statistics in generalized linear models with dispersion covariates

The class of generalized linear models with dispersion covariates, which allows us to jointly model the mean and dispersion parameters, is a natural extension to the classical generalized linear models. In this paper, we derive the asymptotic expansions under a sequence of Pitman alternatives (up to order n −1/2) for the nonnull distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics in this class of models. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing a subset of dispersion parameters. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, we consider Monte Carlo simulations in order to compare the finite-sample performance of these tests in this class of models. We present two empirical applications to two real data sets for illustrative purposes.

Interim analysis (IA) results of COU-AA-302, a randomized, phase III study of abiraterone acetate (AA) in chemotherapy-naive patients (pts) with metastatic castration-resistant prostate cancer (mCRPC).

LBA4518 Background: AA is an androgen biosynthesis inhibitor that inhibits CYP17 and improves overall survival (OS) in post-docetaxel mCRPC. The primary objective of COU-AA-302 was to compare clinical benefit of AA + prednisone (P) vs placebo (PL) + P in chemo-naive, asymptomatic/mildly symptomatic mCRPC pts. Methods: 1088 pts (151 centers; 12 countries) were randomized 1:1 to AA (1 g) + P (5 mg BID) or PL + P. Co-primary endpoints: radiographic progression-free survival (rPFS) and OS. Median times estimated using K-M method including LR statistic for inference. The Lan-DeMets α-spending function was used for OS. Results:The Independent Data Monitoring Committee concluded that the OS, rPFS and secondary endpoints (Table) all favored the AA arm and unanimously recommended unblinding the study and crossing pts from PL to AA at IA (43% of total events). Median follow up = 22.2 mos. Grade 3/4 AEs (AA + P, PL + P) (%): hypertension 3.9 vs 3.0; hypokalemia 2.4 vs 1.9; ALT↑ 5.4 vs 0.7; AST↑ 3.0 vs 0.9. Conclusions:AA + P produced a statistically significant improvement in rPFS and a strong trend for increased OS at this IA. AA resulted in clinically and statistically significant effects on all secondary endpoints. IA results confirmed the acceptable tolerability/safety profile of AA. This is the first randomized trial to demonstrate both OS and rPFS benefits in chemo-naive mCRPC and that inhibition of persistent extragonadal androgen synthesis significantly delays initiation of cytotoxic chemo. While median OS (AA arm) has not been reached, median PL arm OS (27.2 mos) is the longest measured in any phase III mCRPC study. [Table: see text]

Note on asymptotic null distributions of LR statistics for testing covariance matrix under growth curve model when the number of the observation points is large

This paper is concerned with the testing problem about the covariance matrix under growth curve model. The testing problems treated in this paper are the following problems: (i) the problem of testing that a covariance matrix is equal to a specified positive definite matrix, (ii) the problem of testing for sphericity of the covariance matrix, and (iii) the problem of intraclass model for the covariance matrix. We give asymptotic distributions of the null distributions for the likelihood ratio statistics under an asymptotic framework that the total sample size n go to infinity, the number of the observation points p go to infinity, and p/n go to a constant c∈(0,1). Simulation reveals that the proposed approximations have good accuracies compared with the classical chi-square approximations.

Dynamic value at risk estimation for BELEX15

This paper uses RiskMetrics, GARCH and IGARCH models to calculate daily VaR for Belgrade Stock Exchange index BELEX15 returns based on the normal and Student t innovation distribution. In the case of GARCH and IGARCH models VaR values are obtained applying Extreme Value Theory on the standardized residuals. The Kupiec's LR statistics was used to test the accuracy of risk measurement models. The main conclusions are: (1) when modelling value-at-risk it is very important to have a good model for volatility of stock returns; (2) both stationary and integrated GARCH models outperform RiskMetrics in estimating VaR; (3) although long memory volatility is present in the BELEX15 index, IGARCH models cannot outperform GARCH type models in VaR evaluations for this index.

Testing for structural breaks in dynamic factor models

In this paper we investigate the consequences of structural breaks in the factor loadings for the specification and estimation of factor models based on principal components and suggest procedures for testing for structural breaks. It is shown that structural breaks severely inflate the number of factors identified by the usual information criteria. The hypothesis of a structural break is tested by using LR, LM and Wald statistics. The LM test (which performs best in our Monte Carlo simulations) is generalized to test for structural breaks in factor models where the break date is unknown and the common factors and idiosyncratic components are serially correlated. The proposed test procedures are applied to datasets from the US and the euro area.

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z 2(g 1), Z 2(g 2), D and K 2 for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z 2(g 1) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ1≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.

Genome-wide SNP-based linkage analysis of tuberculosis in Thais

Tuberculosis, a potentially fatal infectious disease, affects millions of individuals annually worldwide. Human protective immunity that contains tuberculosis after infection has not been clearly defined. To gain insight into host genetic factors, nonparametric linkage analysis was performed using high-throughput microarray-based single nucleotide polymorphism (SNP) genotyping platform, a GeneChip array comprised 59 860 bi-allelic markers, in 93 Thai families with multiple siblings, 195 individuals affected with tuberculosis. Genotyping revealed a region on chromosome 5q showing suggestive evidence of linkage with tuberculosis (Z(lr) statistics=3.01, logarithm of odds (LOD) score=2.29, empirical P-value=0.0005), and two candidate regions on chromosomes 17p and 20p by an ordered subset analysis using minimum age at onset of tuberculosis as the covariate (maximum LOD score=2.57 and 3.33, permutation P-value=0.0187 and 0.0183, respectively). These results imply a new evidence of genetic risk factors for tuberculosis in the Asian population. The significance of these ordered subset results supports a clinicopathological concept that immunological impairment in the disease differs between young and old tuberculosis patients. The linkage information from a specific ethnicity may provide unique candidate regions for the identification of the susceptibility genes and further help elucidate the immunopathogenesis of tuberculosis.

Large sample approximations for the LR statistic for equality of the smallest eigenvalues of a covariance matrix under elliptical population

This paper is concerned with large sample approximations of the LR statistic for testing the hypothesis that the smallest eigenvalues of a covariance matrix are equal. Under a normal population Lawley [1956. Tests of significance for the latent roots of covariance and correlation matrices. Biometrika 43, 128–136.] and Fujikoshi [1977. An asymptotic expansion for the distributions of the latent roots of the Wishart matrix with multiple population roots. Ann. Inst. Statist. Math. 29, 379–387.] obtained a Bartlett-correction factor and an asymptotic expansion for the LR statistic, respectively, when the sample size is large. In this paper we extend the Bartlett correction factor to an elliptical population. The accuracy of our approximations is examined through simulation experiments.

Asymptotic distribution of the LR statistic for equality of the smallest eigenvalues in high-dimensional principal component analysis

This paper deals with the distribution of the LR statistic for testing the hypothesis that the smallest eigenvalues of a covariance matrix are equal. We derive an asymptotic null distribution of the LR statistic when the dimension p and the sample size N approach infinity, while the ratio p / N converging on a finite nonzero limit c ∈ ( 0 , 1 ) . Numerical simulations revealed that our approximation is more accurate than the classical chi-square-type approximation as p increases in value.

Classification of microarray data with factor mixture models

The classification of few tissue samples on a very large number of genes represents a non-standard problem in statistics but a usual one in microarray expression data analysis. In fact, the dimension of the feature space (the number of genes) is typically much greater than the number of tissues. We consider high-density oligonucleotide microarray data, where the expression level is associated to an 'absolute call', which represents a qualitative indication of whether or not a transcript is detected within a sample. The 'absolute call' is generally not taken in consideration in analyses. In contrast to frequently used cluster analysis methods to analyze gene expression data, we consider a problem of classification of tissues and of the variables selection. We adopted methodologies formulated by Ghahramani and Hinton and Rocci and Vichi for simultaneous dimensional reduction of genes and classification of tissues; trying to identify genes (denominated 'markers') that are able to distinguish between two known different classes of tissue samples. In this respect, we propose a generalization of the approach proposed by McLachlan et al. by advising to estimate the distribution of log LR statistic for testing one versus two component hypothesis in the mixture model for each gene considered individually, using a parametric bootstrap approach. We compare conditional (on 'absolute call') and unconditional analyses performed on dataset described in Golub et al. We show that the proposed techniques improve the results of classification of tissue samples with respect to known results on the same benchmark dataset. The software of Ghahramani and Hinton is written in Matlab and available in 'Mixture of Factor Analyzers' on http://www.gatsby.ucl.ac.uk/~zoubin/software.html while the software of Rocci and Vichi is available upon request from the authors.