Computation of Standard Errors for Geary–Khamis Parities and international Prices: A Stochastic Approach

One of the areas receiving little attention in the past in index-number theory is providing standard errors for the index number estimates. Recently, Clements and Izan and Selvanathan used the stochastic approach to index numbers to derive standard errors for the rate of inflation and Laspeyres and Paasche index numbers. In this article, we use this approach to compute standard errors associated with purchasing power parities computed using Geary-Khamis aggregation procedure in the International Comparisons Project of the United Nations. We assess the quality of the standard errors using Efron's bootstrap technique.     

A Class of Multiplicative Estimators of Laspeyres Price Indexes

Publication of indexes measuring changes in prices of retail, wholesale, export, and import items is an important part of many governmental statistics programs. One form of price index that is often used is the fixed-base Laspeyres, in which a fixed market basket of goods is priced over time. This article introduces a new class of multiplicative estimators of Laspeyres indexes. The optimum within the class is derived for long-term price change and compared with two other members of the class when used for estimating both long-term and short-term change. Theoretical properties are derived under a model in which long-term relative price changes for individual items have common within-stratum means and are correlated over time. Theory for long-term and short-term change estimators is tested in a simulation study in which a large number of stratified probability samples is selected from a population extracted from items priced for the U.S. consumer price index.     

Measuring price-induced changes in standard of living in least developed countries: economic validity of laspeyres price indices and some further suggestions

Usually, also in developing countries, price-induced changes in standard of living are measured by means of Laspeyres price indices. Looking at the special economic situation characterizing least developed countries, the economic validit of such indices is doubtful. The main theoretical results of this paper are derived on the basis of a special utility concept which may be considered typical for least developed countries. It is shown that, under certain conditions, a Laspeyres price index over- or underestimates respectively changes in standard of living. Furthermore, it does not sufficiently evaluate households’ supply positions. These theoretical results are motivated and illustrated by a special least developed country, the Republic of Niger. The paper closes with some further suggestions for measuring changes in standard of living in developing countries.     

Variance Estimation for Price Indexes From a Two-Stage Sample With Rotating Panels

Estimation of price indexes in the United States is generally based on complex rotating panel surveys. The sample for the Consumer Price Index, for example, is selected in three stages—geographic areas, establishments, and individual items—with 20% of the sample being replaced by rotation each year. At each period, a time series of data is available for use in estimation. This article examines how to best combine data for estimation of long-term and short-term changes and how to estimate the variances of the index estimators in the context of two-stage sampling. I extend the class of estimators, introduced by Valliant and Miller, of Laspeyres indexes formed using sample data collected from the current period back to a previous base period. Linearization estimators of variance for indexes of long-term and short-term change are derived. The theory is supported by an empirical simulation study using two-stage sampling of establishments and items from a population derived from U.S. Bureau of Labor Statistics data.     

Approximation of the Fisher price index by using Lowe,Young, and AG Mean indices

The Consumer Price Index (CPI) approximates changes in the costs of household consumption assuming the constant utility (COLI, Cost of Living Index). In practice, the Laspeyres price index is used to measure the CPI despite the fact that many economists consider the superlative indices to be the best approximation of COLI. The Fisher index is one of the superlative indices and additionally it satisfies most of tests from the axiomatic price index theory. Nevertheless, the Fisher price index makes use of current-period expenditure data and its usefulness in CPI measurement is limited. In this article, we verify the utility of using the Lowe, Young, and AG Mean indices for Fisher price index approximation. We confirm this utility in a simulation study and we provide an empirical proof.     

Nonresponse Assessment of a Consumer Price Index

A household budget survey often suffers from a high nonresponse rate and a selective response. The bias that may be introduced in the estimation of budget shares because of this nonresponse can affect the estimate of a consumer price index, which is a weighted sum of partial price index numbers (weighted with the estimated budget shares). The bias is especially important when related to the standard error of the estimate. Because of the impossibility of subsampling nonrespondents to the budget survey, no exact information on the bias can be obtained. To evaluate the nonresponse bias, bounds for this bias are calculated using linear programming methods for several assumptions. The impact on a price index of a high nonresponse rate among people with a high income can also be assessed by using the elasticity with respect to total expenditure. Attention is also given to the possible nonresponse bias in a time series of price index numbers. The possible nonresponse bias is much larger than the standard error of the estimate.     

Construction of confidence intervals for the Laspeyres price index

In the paper, we present and discuss several methods of the construction of confidence intervals for the Laspeyres price index. We assume that prices of commodities are normally distributed and we consider both independent and dependent prices. Using Monte Carlo simulation, the paper compares the confidence interval computed from a simple econometric model with those obtained based on the Laspeyres density function. Our conclusions can be generalized to other price index formulas.     

Efficient run orders for a two-factor response surface experiment on a correlated process

The standard assumption in the design and analysis of response surface experiments is that the errors are uncorrelated with common variance. In practice however there may be serial dependence between the errors, in which case the efficiency of the design depends on the order in which the runs are carried out. We investigate the effect of run order on a two-factor response surface experiment when there is positive serial correlation, and recommend some particular run orders which are efficient under a wide range of conditions.     

Tests for serial correlation in mean and variance of a sequence of time series objects

In this era of Big Data, large-scale data storage provides the motivation for statisticians to analyse new types of data. The proposed work concerns testing serial correlation in a sequence of sets of time series, here referred to as time series objects. An example is serial correlation of monthly stock returns when daily stock returns are observed. One could consider a representative or summarized value of each object to measure the serial correlation, but this approach would ignore information about the variation in the observed data. We develop Kolmogorov–Smirnov-type tests with the standard bootstrap and wild bootstrap Ljung–Box test statistics for serial correlation in mean and variance of time series objects, which take the variation within a time series object into account. We study the asymptotic property of the proposed tests and present their finite sample performance using simulated and real examples.     

Small-area estimation by combining time-series and cross-sectional data

A model involving autocorrelated random effects and sampling errors is proposed for small-area estimation, using both time-series and cross-sectional data. The sampling errors are assumed to have a known block-diagonal covariance matrix. This model is an extension of a well-known model, due to Fay and Herriot (1979), for cross-sectional data. A two-stage estimator of a small-area mean for the current period is obtained under the proposed model with known autocorrelation, by first deriving the best linear unbiased prediction estimator assuming known variance components, and then replacing them with their consistent estimators. Extending the approach of Prasad and Rao (1986, 1990) for the Fay-Herriot model, an estimator of mean squared error (MSE) of the two-stage estimator, correct to a second-order approximation for a small or moderate number of time points, T, and a large number of small areas, m, is obtained. The case of unknown autocorrelation is also considered. Limited simulation results on the efficiency of two-stage estimators and the accuracy of the proposed estimator of MSE are présentés.     

APPROXIMATING VOLATILITIES BY ASYMMETRIC POWER GARCH FUNCTIONS

ARCH/GARCH representations of financial series usually attempt to model the serial correlation structure of squared returns. Although it is undoubtedly true that squared returns are correlated, there is increasing empirical evidence of stronger correlation in the absolute returns than in squared returns. Rather than assuming an explicit form for volatility, we adopt an approximation approach; we approximate the γth power of volatility by an asymmetric GARCH function with the power index γ chosen so that the approximation is optimum. Asymptotic normality is established for both the quasi-maximum likelihood estimator (qMLE) and the least absolute deviations estimator (LADE) in our approximation setting. A consequence of our approach is a relaxation of the usual stationarity condition for GARCH models. In an application to real financial datasets, the estimated values for γ are found to be close to one, consistent with the stylized fact that the strongest autocorrelation is found in the absolute returns. A simulation study illustrates that the qMLE is inefficient for models with heavy-tailed errors, whereas the LADE is more robust.     

Adaptive Elastic Net GMM Estimation With Many Invalid Moment Conditions: Simultaneous Model and Moment Selection

This article develops the adaptive elastic net generalized method of moments (GMM) estimator in large-dimensional models with potentially (locally) invalid moment conditions, where both the number of structural parameters and the number of moment conditions may increase with the sample size. The basic idea is to conduct the standard GMM estimation combined with two penalty terms: the adaptively weighted lasso shrinkage and the quadratic regularization. It is a one-step procedure of valid moment condition selection, nonzero structural parameter selection (i.e., model selection), and consistent estimation of the nonzero parameters. The procedure achieves the standard GMM efficiency bound as if we know the valid moment conditions ex ante, for which the quadratic regularization is important. We also study the tuning parameter choice, with which we show that selection consistency still holds without assuming Gaussianity. We apply the new estimation procedure to dynamic panel data models, where both the time and cross-section dimensions are large. The new estimator is robust to possible serial correlations in the regression error terms.     

Estimation of Laspeyres Price Indexes Using the Prediction Approach for Finite Population Sampling

Estimators of chain and fixed-base Laspeyres price indexes are studied using the prediction approach to finite population sampling. The estimators include some that are based on those used in several U.S. government index programs and others derived from prediction models. Biases and variances of the estimators are studied for a case in which the reference period index weights are unknown for nonsample items. Under a model for a one-period price change in which items have common within-stratum means, unbiased estimators can be constructed, but under a more general regression model, special sample balance conditions are needed for unbiasedness of those estimators. The theory for the estimators of fixed-base indexes is illustrated in an empirical study using a population of items priced for the U.S. Consumer Price Index.     

Variance function in regression analysis of longitudinal data using the generalized estimating equation approach

Longitudinal or clustered response data arise in many applications such as biostatistics, epidemiology and environmental studies. The repeated responses cannot in general be assumed to be independent. One method of analysing such data is by using the generalized estimating equations (GEE) approach. The current GEE method for estimating regression effects in longitudinal data focuses on the modelling of the working correlation matrix assuming a known variance function. However, correct choice of the correlation structure may not necessarily improve estimation efficiency for the regression parameters if the variance function is misspecified [Wang YG, Lin X. Effects of variance-function misspecification in analysis of longitudinal data. Biometrics. 2005;61:413–421]. In this connection two problems arise: finding a correct variance function and estimating the parameters of the chosen variance function. In this paper, we study the problem of estimating the parameters of the variance function assuming that the form of the variance function is known and then the effect of a misspecified variance function on the estimates of the regression parameters. We propose a GEE approach to estimate the parameters of the variance function. This estimation approach borrows the idea of Davidian and Carroll [Variance function estimation. J Amer Statist Assoc. 1987;82:1079–1091] by solving a nonlinear regression problem where residuals are regarded as the responses and the variance function is regarded as the regression function. A limited simulation study shows that the proposed method performs at least as well as the modified pseudo-likelihood approach developed by Wang and Zhao [A modified pseudolikelihood approach for analysis of longitudinal data. Biometrics. 2007;63:681–689]. Both these methods perform better than the GEE approach.     

A generalized estimating equation method for fitting autocorrelated ordinal score data with an application in horticultural research

Summary.  Generalized estimating equations for correlated repeated ordinal score data are developed assuming a proportional odds model and a working correlation structure based on a first-order autoregressive process. Repeated ordinal scores on the same experimental units, not necessarily with equally spaced time intervals, are assumed and a new algorithm for the joint estimation of the model regression parameters and the correlation coefficient is developed. Approximate standard errors for the estimated correlation coefficient are developed and a simulation study is used to compare the new methodology with existing methodology. The work was part of a project on post-harvest quality of pot-plants and the generalized estimating equation model is used to analyse data on poinsettia and begonia pot-plant quality deterioration over time. The relationship between the key attributes of plant quality and the quality and longevity of ornamental pot-plants during shelf and after-sales life is explored.     

Analysis of longitudinal data unbalanced over time

Summary.  The paper considers modelling, estimating and diagnostically verifying the response process generating longitudinal data, with emphasis on association between repeated meas-ures from unbalanced longitudinal designs. Our model is based on separate specifications of the moments for the mean, standard deviation and correlation, with different components possibly sharing common parameters. We propose a general class of correlation structures that comprise random effects, measurement errors and a serially correlated process. These three elements are combined via flexible time-varying weights, whereas the serial correlation can depend flexibly on the mean time and lag. When the measurement schedule is independent of the response process, our estimation procedure yields consistent and asymptotically normal estimates for the mean parameters even when the standard deviation and correlation are misspecified, and for the standard deviation parameters even when the correlation is misspecified. A generic diagnostic method is developed for verifying the models for the mean, standard deviation and, in particular, the correlation, which is applicable even when the data are severely unbalanced. The methodology is illustrated by an analysis of data from a longitudinal study that was designed to characterize pulmonary growth in girls.     

Faktorenanalyse als ein Instrument zur Konstruktion von Preisniveauindizes

This paper proposes a solution to the problem of calculating price index numbers based on factor analysis. We present that the construction of price index numbers can be identical with determination of factors within the factor analysis. The price material for FRG during the period 1970 to 1979 (private consumption) shows that a definite identification and estimation for price index numbers in the two and multiple situation case is possible.     

Linear Regression With Nested Errors Using Probability‐Linked Data

Probabilistic matching of records is widely used to create linked data sets for use in health science, epidemiological, economic, demographic and sociological research. Clearly, this type of matching can lead to linkage errors, which in turn can lead to bias and increased variability when standard statistical estimation techniques are used with the linked data. In this paper we develop unbiased regression parameter estimates to be used when fitting a linear model with nested errors to probabilistically linked data. Since estimation of variance components is typically an important objective when fitting such a model, we also develop appropriate modifications to standard methods of variance components estimation in order to account for linkage error. In particular, we focus on three widely used methods of variance components estimation: analysis of variance, maximum likelihood and restricted maximum likelihood. Simulation results show that our estimators perform reasonably well when compared to standard estimation methods that ignore linkage errors.     

A parametric model for heterogeneity in paired Poisson counts

We present a model for data in the form of matched pairs of counts. Our work is motivated by a problem in fission-track analysis, where the determination of a crystal's age is based on the ratio of counts of spontaneous and induced tracks. It is often reasonable to assume that the counts follow a Poisson distribution, but typically they are overdispersed and there exists a positive correlation between the numbers of spontaneous and induced tracks in the same crystal. We propose a model that allows for both overdispersion and correlation by assuming that the mean densities follow a bivariate Wishart distribution. Our model is quite general, having the usual negative-binomial and Poisson models as special cases. We propose a maximum-likelihood estimation method based on a stochastic implementation of the EM algorithm, and we derive the asymptotic standard errors of the parameter estimates. We illustrate the method with a data set of fission-track counts in matched areas of zircon crystals.     

关于物价综合指数性质的探析

通过对拉氏和帕氏物价指数的讨论，明确了物价综合指数之间的相互关系和物价综合指数的数学性质，找到了物价综合指数的变化特点。在保持原有指数特性的前提下，提出了新的综合指数改进方案，并对其进行了检验。检验表明，新的综合指数方案计算简便，能够准确地反映物量和价格的变化，并且误差较小。