The power of four tests of autocorrelation in the linear regression model

The power of each of four tests of first-order autocorrelation in the linear regression model is determined for a simple and multiple regression model whose parameters are presumed to be known. The tests are: Durbin-Watson bounds test, a test based on Theil's best linear unbiased scalar estimator, a test devised by Abrahamse, Koerts and Louter, and an exact test devised by Durbin.For positive values of the coefficient of autocorrelation the Durbin-Watson bounds test is generally better than the tests based on the estimator proposed by Abrahamse, Koerts and Louter, the best linear unbiased scalar estimator, and the Durbin exact test. For negative values of the coefficient of autocorrelation, the pattern of results is mixed for all four test procedures. A byproduct of these experiments is the demonstrated feasibility of enumerating the distribution of the Durbin-Watson test statistic for any regression matrix and thus eliminating the region of indeterminacy from the Durbin-Watson test procedure.     

Index

We study two Durbin-Watson type tests for serial correlation of errors inregression models when observations are missing. We derive them by applying standard methods used in time series and linear models to deal with missing observations. The first test may be viewed as a regular Durbin-Watson test in the context of an extended model. We discuss appropriate adjustments that allow one to use all available bounds tables. We show that the test is locally most powerful invariant against the same alternative error distribution as the Durbin-Watson test. The second test is based on a modified Durbin-Watson statistic suggested by King (1981a) and is locally most powerful invariant against a first-order autoregressive process.     

Fourth-order autocorrelation: Further significance points for the Wallis test

This note provides tables of 0.5, 1.0 and 2.5 percent significance points of the bounds of Wallis's statistic for testing for fourth-order autocorrelation in regressions either including or excluding quarterly dummy variables.     

A consistent nonparametric test for nonlinear causality—Specification in time series regression

Since the pioneering work by Granger (1969), many authors have proposed tests of causality between economic time series. Most of them are concerned only with “linear causality in mean”, or if a series linearly affects the (conditional) mean of the other series. It is no doubt of primary interest, but dependence between series may be nonlinear, and/or not only through the conditional mean. Indeed conditional heteroskedastic models are widely studied recently. The purpose of this paper is to propose a nonparametric test for possibly nonlinear causality. Taking into account that dependence in higher order moments are becoming an important issue especially in financial time series, we also consider a test for causality up to the Kth conditional moment. Statistically, we can also view this test as a nonparametric omitted variable test in time series regression. A desirable property of the test is that it has nontrivial power against T^1/2-local alternatives, where T is the sample size. Also, we can form a test statistic accordingly if we have some knowledge on the alternative hypothesis. Furthermore, we show that the test statistic includes most of the omitted variable test statistics as special cases asymptotically. The null asymptotic distribution is not normal, but we can easily calculate the critical regions by simulation. Monte Carlo experiments show that the proposed test has good size and power properties.     

The small sample bias of Durbin's tests for serial correlation: When one of the regressors is the lagged dependent variable and the null hypothesis is true

Durbin (1970) has recently proposed two asymptotically equivalent statistics which can be used to test for the presence of serial correlation when some of the regressors are lagged dependent variables. This study reports on simulation experiments designed to compare the two statistics, in use with small samples, in terms of their tendencies to detect serial correlation when none exists. Of the two test statistics, it is found that the one based on estimated residuals detects the absence of serial correlation in the expected proportion of trials; the other statistic (Durbin's h), which involves the application of a correction factor to the Durbin-Watson statistic, gives evidence of serious small sample bias which varies with both the sample size and the assumed size of the coefficient attaching to the lagged dependent variable.     

A directional test of exponentiality based on maximum correlations

In Fortiana and Grané (J Stat Plann Infer 108:85–97), we study a scale-free statistic, based on Hoeffding’s maximum correlation, for testing exponentiality. This statistic admits an expansion along a countable set of orthogonal axes, originating a sequence of statistics. Linear combinations of a given number p of terms in this sequence can be written as a quotient of L-statistics. In this paper, we propose a scale-free adaptive statistic for testing exponentiality with optimal power against a specific alternative and obtain its exact distribution. An empirical power study shows that the test based on this new statistic has the same level of performance than the best tests in the statistical literature.     

A new test for fourth-order autoregressive disturbances

This paper proposes a new test for simple fourth-order autoregressive disturbances in the linear regression model. The test is shown to be most powerful invariant in a given neighourhood of the alternative hypothesis for all design matrices. An empirical power comparison suggests that the test is generally more powerful than the Wallis test, the difference in power probably being slight for most economic applications, although for certain design matrices, the power advantage of the new test is very real. Selected bounds for the test's significance points are tabulated.     

A point optimal test for autoregressive disturbances

This paper is concerned with testing for first-order autoregressive disturbances in the linear regression model and recommends an alternative test to the Durbin-Watson test. The new test is most powerful invariant in a given neighbourhood of the alternative hypothesis parameter space. An empirical power comparison indicates that the test is generally more powerful than the Durbin-Watson test. The comparison also suggests that for many economic applications, the difference in power will be small, although circumstances do exist in which the power advantage of the new test is very real. Selected bounds for the test's significance points are tabulated.     

A robust Hotelling test

Hotelling's T ² statistic is an important tool for inference about the center of a multivariate normal population. However, hypothesis tests and confidence intervals based on this statistic can be adversely affected by outliers. Therefore, we construct an alternative inference technique based on a statistic which uses the highly robust MCD estimator [9] instead of the classical mean and covariance matrix. Recently, a fast algorithm was constructed to compute the MCD [10]. In our test statistic we use the reweighted MCD, which has a higher efficiency. The distribution of this new statistic differs from the classical one. Therefore, the key problem is to find a good approximation for this distribution. Similarly to the classical T ² distribution, we obtain a multiple of a certain F-distribution. A Monte Carlo study shows that this distribution is an accurate approximation of the true distribution. Finally, the power and the robustness of the one-sample test based on our robust T ² are investigated through simulation.     

A non-parametric spatial independence test using symbolic entropy

In the present paper, we construct a new, simple, consistent and powerful test for spatial independence, called the SG test, by using the new concept of symbolic entropy as a measure of spatial dependence. The standard asymptotic distribution of the test is an affine transformation of the symbolic entropy under the null hypothesis. The test statistic, with the proposed symbolization procedure, and its standard limit distribution have appealing theoretical properties that guarantee the general applicability of the test. An important aspect is that the test does not require specification of the W matrix and is free of a priori assumptions. We include a Monte Carlo study of our test, in comparison with the well-known Moran's I, the SBDS (de Graaff et al., 2001) and τ test (Brett and Pinkse, 1997) that are two non-parametric tests, to better appreciate the properties and the behaviour of the new test. Apart from being competitive compared to other tests, results underline the outstanding power of the new test for non-linear dependent spatial processes.     

Optimizing high school curriculum assignments

A quantitative model is described for the optimal assignment of students to courses for their complete high school careers. Values of curriculum, assigned by counselor-judges for types of students, are found to be highly reliable (r_tt = 0.65). Model-generated student schedules are found to be unfailing in their conformity to all of the constraints of the test high school's curriculum. ACPP (the Automatic Curriculum Planning Program) incorporates a two-stage dynamic programming algorithm with curricular sequence constraints resolved by Boolean reduction techniques.     

Smooth preferences and the regularity of equilibria

To study equilibria we describe an economy by its distribution of consumers' preferences and endowments. All preferences are smooth and weakly convex. Demand of an economy need not be single valued, but there is an open dense set of economies for which demand is a C¹-function in a neighborhood of the equilibrium prices. We call an economy regular if its excess demand is transversal to zero. A regular economy has locally unique equilibria. It is shown that regular economies form an open dense set on which the equilibrium price correspondence varies continuously and the number of equilibria is locally constant.     

On the impact of the tests for serial correlation upon the test of significance for the regression coefficient

Monte Carlo methods are used to investigate the relationship between the power of different pretests for autocorrelation, and the Type I error and power of the significance test for a resulting two-stage estimate of the slope parameter in a simple regression. Our results suggest it may be preferable to always transform without pretesting. Moreover we find little room for improvement in the Type I errors and power of two-stage estimators using existing pretests for autocorrelation, compared with the results obtained given perfect knowledge about when to transform (i.e., given a perfect pretest). Rather, researchers should seek better estimators of the transformation parameter itself.     

Goodness of fit for seemingly unrelated regressions: Glahn's R2y.x and Hooper's r̄2

This paper presents a measure of goodness of fit for Zellner's seemingly unrelated regressions. The measure is a monotonic transform of the appropriate (asymptotic) F-statistic, is bounded on [0, 1] and is maximized by Zellner's estimation technique. Glahn's composite correlation coefficient is shown to be a special case of this measure. It is also compared to Hooper's squared trace correlation coefficient. All three measures as well as some additional asymptotic summary test statistics are calculated for the two-equation example of Zellner. The applicability of the latter test statistics seems not to be recognized in applied work.     

Detecting a shift in location: Some robust tests

The test statistic W, suggested by Hawkins (1977) and Worsley (1979) for testing a sequence of observations for a shift in location, is sensitive to the assumption that the distribution of the population being sampled is normal. A modification of this statistic is proposed which is robust against ‘heavy tailed’ distributions. We also study the performance of several alternative testing procedures by means of a simulation experiment. The Farley-Hinich test (1975) and the Homogeneity test discussed in Brown, Durbin and Evans (1975) perform well in the Simulation experiment against ‘heavy tailed’ distributions. Some applications to economic data are also discussed in the paper.     

Simple and powerful GMM over-identification tests with accurate size

Based on the series long run variance estimator, we propose a new class of over-identification tests that are robust to heteroscedasticity and autocorrelation of unknown forms. We show that when the number of terms used in the series long run variance estimator is fixed, the conventional J statistic, after a simple correction, is asymptotically F-distributed. We apply the idea of the F-approximation to the conventional kernel-based J tests. Simulations show that the J^∗ tests based on the finite sample corrected J statistic and the F-approximation have virtually no size distortion, and yet are as powerful as the standard J tests.     

Testing rational expectations by the use of overidentifying restrictions

A simple econometric test for rational expectations in the case in which unobservable, rationally expected variables appear in a structural equation is presented. Using McCallum's instrumental variable estimator as a base, a test for rational expectations per se and a joint test of rational expectations and hypotheses about the structural equation are presented. The new test is shown to be a new interpretation of Basmann's test of overidentifying restrictions. As an illustration, the hypothesis that the forward exchange rate is the rationally expected future spot exchange rate is tested and rejected.     

Semiparametric tests of conditional moment restrictions under weak or partial identification

We propose two new semiparametric specification tests which test whether a vector of conditional moment conditions is satisfied for any vector of parameter values θ₀. Unlike most existing tests, our tests are asymptotically valid under weak and/or partial identification and can accommodate discontinuities in the conditional moment functions. Our tests are moreover consistent provided that identification is not too weak. We do not require the availability of a consistent first step estimator. Like Robinson [Robinson, Peter M., 1987. Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55, 875–891] and many others in similar problems subsequently, we use k-nearest neighbor (knn) weights instead of kernel weights. The advantage of using knn weights is that local power is invariant to transformations of the instruments and that under strong point identification computation of the test statistic yields an efficient estimator of θ₀ as a byproduct.     

The Bierens test for certain nonstationary models

We adapt the Bierens (1990) test to the I-regular models of Park and Phillips (2001). Bierens (1990) defines the test hypothesis in terms of a conditional moment condition. Under the null hypothesis, the moment condition holds with probability one. The probability measure used is that induced by the variables in the model, that are assumed to be strictly stationary. Our framework is nonstationary and this approach is not always applicable. We show that the Lebesgue measure can be used instead in a meaningful way. The resultant test is consistent against all I-regular alternatives.