Correcting for truncation bias caused by a latent truncation variable

We discuss estimation of the model Y_i=X_ib_Y+e_Yi, T_i=X_ib_T+ e_Ti, when data on the continuous dependent variable Y and on the independent variables X are observed iff the ‘truncation variable’ T>0 and when T is latent. This case is distinct from both (i) the‘censored sample’ case, in which Y data are available iff T>0, T is latent and X data are available for all observations, and (ii) the ‘observed truncation variable’ case, in which both Y and X are observed iff T>0 and in which the actual value of T is observed whenever T>0. We derive a maximum-likelihood procedure for estimating this model and discuss identification and estimation.     

Cointegration and distance between differenced processes

In this note, it is argued that cointegration augments the distance between the differenced series. If two series, x _t and y _t , are integrated of order one and cointegrated and v _t and w _t are integrated of order one but not cointegrated then, under certain conditions, the distance between ??x _t and ??y _t is more than the distance between ??v _t and ??w _t .     

Continua of stochastic dominance relations for bounded probability distributions

Let P={F,G,…} be the set of probability distribution functions on [0,b]. For each αε[1, ∞), F_αG means that ∫^x_o(x−y^α−1dF(y)∫^x_o(x−y)^α−1dG(y) for all xε[0, b], and F>_αG means that F_αG and F≠G. Each _α is reflexive and transitive and each>_α is asymmetric and transitive. Both _α and>_α increase as α increases but their limits are not complete. A class U_α of utility functions is defined to give F>_αG iff ∫udF>∫udG for all uεU_α. These classes decrease as α increases, and their limit is empty. Similar decreasing classes are defined for each _α, and their limit is essentially the constant functions on (0, b].     

Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems

Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are X_t=X?t | X>t and Y_t=Y?t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time‐dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ):=P(X_t>Y_t) which enables one to obtain, at time t, the probability that the residual lifetime X_t is greater than the residual lifetime Y_t. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22 , 1973) that the probability of type R(t) is important for designing as long‐lived a product as possible. Several properties of R(t) and its connection with well‐known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

Social Distance,Respondent Cooperation and Item Nonresponse in Sex Survey

This paper examines the ways in which the number of item nonresponses is determined by social distance and/or interview rapport, with a focus on responses of “refusal” and “don’t know”, implying the respondent’s lack of willingness and ability to provide substantive responses to sensitive questions. The data analyzed were from 39 self- administered questions concerning sexual attitudes and behaviors in the 2002 Taiwan Social Change Survey for module “Family and Changing Gender Role”. Poisson Regression in 2-level Hierarchical Linear Model was employed to enhance the accuracy of the analysis of the accumulation of “don’t know” and “refusal” responses. The results showed that respondent cooperation significantly decreased the number of both “don’t know” and “refusal” replies. The decrease was not conditioned by any kind of social distance. Age and education distances have respectively negative and positive effect on the number of “don’t know” and “refusal” answers. The married–married interview produced more “don’t know” and “refusal” than other paired interview types. The larger the ethnicity distance is, the more “refusal” appears. The substantial findings imply that the effects of social-distance and rapport (respondent cooperation) on the number of item nonresponses deserve more attention in research on survey methodology. The divergent findings on gender-distance effect and marital-status effect, however, call for replication studies in the future.     

Improved stein-rule estimator for regression problems

Stein-Rule estimator for regression problems has been studied by several authors including Sclove (1968) and others listed in Vinod's (1978) survey. Ullah and Ullah (1978) provide the expressions for the mean squared error (MSE) of a double k-class (KK) estimator with parameters k₁ and k₂. When k₂=1 the Stein-Rule estimator becomes a special case of KK and an optimal choice of k₁ is known. This paper explores optimal theoretical choice of k₁ and k₂. We note that negative choices of k₂ are permissible, and that thereis a large range of choices for K₁ and k₂ where the MSE of the Stein-Rule estimator can be reduced for regression problems based on multicollinear data. A simulation experiment is included.     

Pythagorean generalization of testing the equality of two symmetric positive definite matrices

We provide a new test for equality of two symmetric positive-definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality or the sandwich asymptotic covariance matrix of the GMM estimator. The test relies on a new characterization of equality between two $k$ dimensional symmetric positive-definite matrices $A$ and $B$: the traces of $A{B}^{?1}$ and $B{A}^{?1}$ are equal to $k$ if and only if $A=B$. Using this simple criterion, we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions. A preferred test in the class with good omni-directional power is recommended for practical work. Monte Carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives. The test is applicable in many settings, including GMM estimation, SVAR models and high dimensional variance matrix settings.     

Multivariate L1 mean

The center of a univariate data set {x ₁,…,x _n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x ₁−μ|,…,|x _n−μ|). As the median and the mean are the minimizers of respectively the L ₁- and the L ₂-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x ₁,…,x _n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L ₁-distances y′₁=(∥x ₁- μ∥₁,…,∥x _n- μ∥₁) and the vector of L ₂-distances y′₂=(∥x ₁- μ∥₂,…,∥x _n-μ∥₂). We define the L ₁-median and the L ₁-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₁; and then the L ₂-median and the L ₂-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₂. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L ₁-mean appears to be a new concept. Received January 1999     

Uncorrelated residuals from linear models

In the usual linear model $\text{y}\text{=}\text{Xβ}\text{+}\text{u}$, the error vector u is not observable and the vector r of least squares residuals has a singular covariance matrix that depends on the design matrix X. We approximate u by a vector$\text{r}{}^1\text{=}\text{G}\text{(}\text{JA'y}\text{+}\text{Kz}\text{)}$ of uncorrelated ‘residuals’, where G and $\text{(}\text{J, K}\text{)}$ are orthogonal matrices, $\text{A'X}\text{=}\text{0}$ and $\text{A'A}\text{=}\text{I}$, while z is either 0 or a random vector uncorrelated with u satisfying $\text{E(}\text{z}\text{) = E(}\text{J'u}\text{) =}\text{0}$, $\text{V(}\text{z}\text{) = V(}\text{J'u}\text{) = σ}{}^2\text{I}$. We prove that $\text{r}{}^1\text{-}\text{r}$ is uncorrelated with $\text{r}\text{-}\text{u}$, for any such r¹, extending the results of Neudecker (1969). Building on results of Hildreth (1971) and Tiao and Guttman (1967), we show that the BAUS residual vector $\text{r}{}_{\mathrm{h}}\text{=}\text{r}\text{+}\text{P}{}_{\mathrm{<mtext>1</mtext>}}\text{z}$, where P₁ is an orthonormal basis for X, minimizes each characteristic root of $\text{V(}\text{r}{}^1\text{-}\text{u}\text{)}$, while the vector r_b of Theil's BLUS residuals minimizes each characteristic root of $\text{V(}\text{Jr}{}_{\mathrm{a}}\text{-}\text{r}\text{)}$, cf. Grossman and Styan (1972). We find that $\text{tr V(}\text{r}{}_{\mathrm{h}}\text{-}\text{u}\text{) < tr V(}\text{Jr}{}_{\mathrm{b}}\text{-}\text{u}\text{)}$ if and only if the average of the singular values of $\text{P}\text{′}{}_1\text{K}$ is less than $\text{1}\text{2}$, and give examples to show that BAUS is often better than BLUS in this sense.     

Henry George and British Socialism

Abstract . Henrys George, the American social reformer and Single Tax advocate, had a decisive impact on native British socialism considered apart from the Marxist and revolutionary types imported from the continent. Karl Marx and Frederick Engels were hostile critics but the typically English Fabian Society was influenced by George's seminal ideas. The Fabians were especially attracted to two notions: the conception that George gave to the thought of his time, that poverty was an evil preventable by political intervention—by State action; and that the disparity in incomes could be explained by the theory of unearned increment. In turn Sydney Olivier, George Bernard Shaw, Sidney Webb, Annie Besant, H. G. Wells and E, R. Pease came under the influence of George. Soon to affect the Fabians, however, was the development of the economist, P. H. Wicksteed, beyond George to Jenons and Marginalism. Key figures in the Parliamentary and Independent Labour parties almost achieved land value taxation.     

Forecasting contemporal aggregates of multiple time series

The paper considers forecasting a contemporal linear aggregate y_t of a vector time series Z^'_t =(Z_1t,...,Z_kt). We first disciss the case where Z_t follows a stationary multiple moving average process and propose a measure of the efficiency of aggregation. A necessary and sufficient condition is given for the case of no gain by employing the component series. Extension of the results to stationary multiple autoregressive process and some non-stationarity processes is discussed, and an illustrative example is given.     

Nonparametric identification of a binary random factor in cross section data

Suppose V and U are two independent mean zero random variables, where V has an asymmetric distribution with two mass points and U has some zero odd moments (having a symmetric distribution suffices). We show that the distributions of V and U are nonparametrically identified just from observing the sum V+U, and provide a pointwise rate root n estimator. This can permit point identification of average treatment effects when the econometrician does not observe who was treated. We extend our results to include covariates X, showing that we can nonparametrically identify and estimate cross section regression models of the form Y=g(X,D^∗)+U, where D^∗ is an unobserved binary regressor.     

Determination of uniformly most powerful tests in discrete sample spaces

Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H ₀ and the alternative hypothesis H ₁ are given by (i) H ₀ : θ≤θ₀, H ₁ : θ>θ₀, and (ii) H ₀ : θ≤θ₁ or θ≥θ₂, H ₁ : θ₁<θ<θ₂, where θ₁<θ₂.  Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H ₀ and H ₁ take the form (iii) H ₀ : θ₁≤θ≤θ₂, H ₁ : θ<θ₁ or θ>θ₂, where θ₁<θ₂, and (iv) H ₀ : θ=θ₀, H ₁:θ≠θ₀.  To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values c ₁, c ₂ and two randomization constants γ₁, γ₂. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases is rather difficult, unless distributions are symmetric. No straightforward method to determine two-sided UMP tests in discrete sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination of UMP tests in cases (ii) through (iv). Received: March 1999     

Existence of a convex extension of a preference relation

In this paper we say that a preference (i.e. irreflexive) relationP isregular (or aweak order) if both it and its non-comparability relation are transitive; we also say that a preference relationP ^* is aconvex extension of another preference relationP ifP?P ^* holds andP ^* is regular and convex-valued. We prove that a convex extension ofP exists if and only if every non-empty and finite set of alternativesA is not included in the convex hull of ∪ _x∈A P(x).     

Interest rate and bank stock returns asymmetry: Evidence from U.S. banks

We use the multivariate extension of Exponential Generalized Autoregressive Conditionally Heteroscedastic (EGARCH) of Nelson, Econometrica, 59: 347–370, 1991 to test for spillover effects and examine the extent of asymmetries between short- and long-term interest rates and portfolios of money center, large, and medium-size banks in the U.S. Our results indicate the existence of price and volatility spillovers from short- and long-term interest rates to the three bank portfolios. We also provide evidence of response asymmetries for the portfolios of money center and large banks, suggesting that money center and large banks are more sensitive to negative than positive short- and long-term interest rate changes.

Semiparametric estimation of a censored regression model with an unknown transformation of the dependent variable

This paper presents a method for estimating the model Λ(Y)=min(β′X+U, C), where Y is a scalar, Λ is an unknown increasing function, X is a vector of explanatory variables, β is a vector of unknown parameters, U has unknown cumulative distribution function F, and C is a censoring threshold. It is not assumed that Λ and F belong to known parametric families; they are estimated nonparametrically. This model includes many widely used models as special cases, including the proportional hazards model with unobserved heterogeneity. The paper develops n^1/2-consistent, asymptotically normal estimators of Λ and F. Estimators of β that are n^1/2-consistent and asymptotically normal already exist. The results of Monte Carlo experiments illustrate the finite-sample behavior of the estimators.