The information matrix test: Simplified calculation via a score test interpretation

In Chesher (1982) I show that the Information Matrix test introduced by White (1982) is a score test for parameter constancy. In this letter I show that this result leads to a simple computational procedure for calculating the Information Matrix test. The procedure involves computing, for a sample of n observations, n times the R² from the least squares regression of a column of ones on a matrix whose elements are functions of 1st and 2nd derivatives of the log density function.     

Individual stability in Delphi studies

This article carries forward the work discussed in a recent article in this journal by Dajani, Sincoff, and Talley (Technol. Forecast. Soc. Change 13, 83, 1979) by investigating individual stability as opposed to group stability for successive Delphi rounds. It is demonstrated that individual stability implies group stability but that the converse does not hold. A χ² test is proposed for testing the stability of individual responses between consecutive Delphi rounds. Also, an estimation procedure is presented for measuring the extent of individual stability.     

Stochastic unit root processes: Maximum likelihood estimation,and new Lagrange multiplier and likelihood ratio tests

We show in this study that the maximum likelihood estimators of stochastic unit root (STUR) processes are consistent and asymptotically normally distributed. We also present two new tests for STUR. We first propose a Lagrange multiplier test and show that it has a standard χ² distribution asymptotically. We also propose a likelihood ratio test and show that it has an asymptotic distribution of 50–50 mixture of χ² and a point mass at 0. As an empirical example, we test the existence of STUR in the Canadian real exchange rate and explore the implication of STUR on the validity of purchasing power parity.     

Bayes prediction density and regression estimation — A semiparametric approach

This paper is concerned with the Bayes estimation of an arbitrary multivariate density,f(x), x ? R ^k. Such anf(x) may be represented as a mixture of a given parametric family of densities {h (x¦θ)} with support inR ^k, whereθ (inR ^d) is chosen according to a mixing distributionG. We consider the semiparametric Bayes approach in whichG, in turn, is chosen according to a Dirichlet process prior with given parameterα. We then specialize these results whenf is expressed as a mixture of multivariate normal densitiesΦ (x¦Μ, λ) whereΜ is the mean vector and λ is the precision matrix. The results are finally applied to estimating a regression parameter.     

On least squares estimation with a particular linear function of the dependent variable

We consider the standard linear regression model where the endogenous variable y is substituted by Ty, T being a symmetric, idempotent matrix. Comparing the mean square error (MSE) matrices we show that a ‘naive’ LS-procedure may work better than a competing estimator usually proposed in the literature and may even perform better than the LS-estimator based on untransformed data. We derive necessary and sufficient conditions for MSE-dominance and outline some ideas for testing.     

Selden's model of inflation: A reexamination

Selden has presented a model which appears to offer strong support for a monetary explanation of inflation. Using bivariate regression to relate percentage changes in quarterly prices to percentage changes in M1 in previous quarters, he finds values of R² from 0.70 to 0.88 and t-statistics of 12 or more when using data for the U.S., Canada, Belgium, and the Netherlands. Unfortunately, the unreported DW statistics for the regressions range from 0.30 to 0.46. Reestimated with a correction for autocorrelation, the t-statistics collapse, values of rho are almost unity, and there is little evidence to support a monetary explanation of inflation.     

Conflict among testing procedures?

This paper demonstrates that under certain conditions the Lagrange Multiplier test provides a better approximation to the asymptotic X² distribution than the more familiar Wald and Likelihood Ratio tests.     

Saddle points of Hamiltonian systems in convex Lagrange problems having a nonzero discount rate

Problems are studied in which an integral of the form ∫₀^+∞L(k(t),k(t))e^?ptdt is minimized over a class of arcs k: [0, +∞) → Rⁿ. It is assumed that L is a convex function on Rⁿ × Rⁿ and that the discount rate ? is positive. Optimality conditions are expressed in terms of a perturbed Hamiltonian differential system involving a Hamiltonian function H(k, q) which is concave in k and convex in q, but not necessarily differentiable. Conditions are given ensuring that, for ? sufficiently small, the system has a stationary point, in a neighborhood of which one has classical “saddle point” behavior. The optimal arcs of interest then correspond to the solutions of the system which tend to the stationary point as t → +∞. These results are motivated by questions in theoretical economics and extend previous work of the author for the case ? = 0. The case ? < 0 is also covered in part.     

Calendar effects in structural time series models with trend and season

Calendar effects are analysed in the class of structural time series models one of the two main model based approaches in time series decomposition. While Bell and Hillmer (1983) modeled calendar variation in the ARIMA model based approach, we represent structural models in the generalized regression form which allows to apply classical estimation and test procedures. It turns out that the expected high computaional complexity 0(T ³) in the generalized regression model can be reduced to 0(T). As all parameters are estimated by maximizing the likelihood the Likelihood Ratio statistics can be used to test effects of the calendar composition.     

Measuring systemic risk with regime switching in tails

This paper motivates the importance of modeling nonlinearities in measuring systemic risk. I capitalize this motivation by generalizing the CoVaR approach proposed by Adrian and Brunnermeier (2016) to allow it switching between a high and a normal risk regime filtered from data.. Considering the U.S. large bank holding companies (BHCs), this paper shows that modeling regime changes in tails is capable of capturing both amplification and mean-reversion effects of an adverse shock to a bank's balance sheet on the banking system. Using the Kolmogorov–Smirnov test statistics with and without bootstrapping, I perform the significance test to identify systemically important financial institutions (SIFIs), and the stochastic dominance test to rank the identified SIFIs. The stochastic dominance test raises the concern that the CoVaR measure underestimates systemic risk contributions for SIFIs but overestimates for non-SIFIs. Finally, applying the BHCs' characteristics and housing market price to forecast the regime-switching systemic risk out-of-sample, I obtain from 4- and 8-quarter-ahead horizons a desirable countercyclical, forward-looking measure of systemic risk.     

Some results on “an income fluctuation problem”

A consumer at each period, given the income available, y, has to decide how much to consume and save. If he consumes c ? 0 units he gets u(c) units of satisfaction or utility, and if x = y ? c ? 0 is the amount saved then the available income in the next period is rx + ω^k, where ω^k is a random variable, and r is an interest factor that is assumed to be known with certainty. Infinite time horizon problems are considered, and it is shown that if 0 < δr < 1, where 0 < δ < 1 is a discount factor, then the limiting policy is optimal. Questions about the behavior of the stock level, such as boundness, are considered, and an example is given that shows that the stock level might converge almost surely to infinity. Finally an economic explanation is given.     

Unmediated communication in games with complete and incomplete information

We study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^U(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^U(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e., all profiles of types have positive probability), then S^U(G) is equal to the set of communication equilibria of G.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

A more general class of biased estimators of the co-efficients in linear regression

In a recent paper, Ullah and Ullah (1978) proposed a class of biased estimators, namely double k-class (k₁, k₂) for the coefficients in a linear regression model. Even though, this set of estimators contains James and Stein (1961) as a special case, in its present form, it does not contain the ridge type estimators. The aim of this note is to extend Ullah and Ullah set of estimators and then establish a relationship with the various operational ridge estimators. The conditions under which the extended set of estimators dominates the ordinary least squares estimator are analyzed.     

Testing spatial regression models under nonregular conditions

In time series context, estimation and testing issues with autoregressive and moving average (ARMA) models are well understood. Similar issues in the context of spatial ARMA models for the disturbance of the regression, however, remain largely unexplored. In this paper, we discuss the problems of testing no spatial dependence in the disturbances against the alternative of spatial ARMA process incorporating the possible presence of spatial dependence in the dependent variable. The problems of conducting such a test are twofold. First, under the null hypothesis, the nuisance parameter is not identified, resulting in a singular information matrix (IM), which is a nonregular case in statistical inference. To take account of singular IM, we follow Davies (Biometrika 64(2):247–254, 1977; Biometrika 74(1):33–43, 1987) and propose a test procedure based on the supremum of the Rao score test statistic. Second, the possible presence of spatial lag dependence will have adverse effect on the performance of the test. Using the general test procedure of Bera and Yoon (Econom Theory 9:649–658, 1993) under local misspecification, we avoid the explicit estimation of the spatial autoregressive parameter. Thus our suggested tests are entirely based on ordinary least squares estimation. Tests suggested here can be viewed as a generalization of Anselin et al. (Reg Sci Urban Econ 26:77–104, 1996). We conduct Monte Carlo simulations to investigate the finite sample properties of the proposed tests. Simulation results show that our tests have good finite sample properties both in terms of size and power, compared to other tests in the literature. We also illustrate the applications of our tests through several data sets.     

A comparison of commodity price forecasting by Box-Jenkins and regression-based techniques

This article describes a study of forecasting methods performed for the corporate purchasing function, which required monthly forecasts of high-volume rubber-commodity prices as an aid to formulating its future purchasing strategy. Four mathematical forecasting procedures are applied to the same set of rubber-commodity price-index data. The forecasting techniques used are the Box-Jenkins time-series method, multiple linear regression analysis, and two new regression-based techniques, referred to as minimum relative error regressionanalysis and dynamic regression analysis.The rationale behind each method is briefly described. The forecast results generated by each algorithm are presented in graphic and numerical form. The accuracy of each method is evaluated by comparing forecasted versus actual values of the rubber-commodity price index. For this data, the new minimum relative error regression technique compares quite favorably with the powerful Box-Jenkins method, followed by standard multiple regression. The dynamic regression method is the least accurate of the four in this application.     

A medium-N approach to macroeconomic forecasting

This paper considers methods for forecasting macroeconomic time series in a framework where the number of predictors, N, is too large to apply traditional regression models but not sufficiently large to resort to statistical inference based on double asymptotics. Our interest is motivated by a body of empirical research suggesting that popular data-rich prediction methods perform best when N ranges from 20 to 40. In order to accomplish our goal, we resort to partial least squares and principal component regression to consistently estimate a stable dynamic regression model with many predictors as only the number of observations, T, diverges. We show both by simulations and empirical applications that the considered methods, especially partial least squares, compare well to models that are widely used in macroeconomic forecasting.     

The meta-technology cost ratio: An indicator for judging the cost performance of CO2 reduction

This paper constructs an innovative ratio of meta-technology cost (MTCR) by combining the meta-frontier framework and shadow price model. This ratio is not only an indicator to account for country-specific technology heterogeneity or technology heterogeneity across a group of countries to measure improvement toward the meta-frontier. This ratio also computes the absolute value difference between the real and ideal minimum marginal abatement costs for the non-market good CO₂ to achieve specific emission reduction. An equally weighted combination of technological readiness and innovation is used to classify countries into groups. The value of the MTCR via the computation of the meta marginal abatement cost (MAC^meta) and group marginal abatement cost (MAC^group-k) allows us to identify the possible improvement of marginal abatement cost (MAC) in absolute value. That is, observation of the MTCR, along with its components the MAC^meta and MAC^group-k, provides more intuitive and comprehensive information for commanding the cost performance of CO₂ reduction than the traditional meta-technology ratio alone.     

Wald,likelihood ratio,and Lagrange multiplier tests as an F test

This expository note contains, in the case of a multiple regression model, a derivation of the Wald, Lagrange, and Likelihood Ratio tests as a function of the small sample F test. This also allows us to readily establish the well-known small sample inequalities among the first three large sample tests.     

Testing for Granger non-causality in heterogeneous panels

This paper proposes a very simple test of Granger (1969) non-causality for heterogeneous panel data models. Our test statistic is based on the individual Wald statistics of Granger non causality averaged across the cross-section units. First, this statistic is shown to converge sequentially to a standard normal distribution. Second, the semi-asymptotic distribution of the average statistic is characterized for a fixed T sample. A standardized statistic based on an approximation of the moments of Wald statistics is hence proposed. Third, Monte Carlo experiments show that our standardized panel statistics have very good small sample properties, even in the presence of cross-sectional dependence.