Kernel density estimation for multivariate data is an important technique that has a wide range of applications. However, it has received significantly less attention than its univariate counterpart. The lower level of interest in multivariate kernel density estimation is mainly due to the increased difficulty in deriving an optimal data-driven bandwidth as the dimension of the data increases. We provide Markov chain Monte Carlo (MCMC) algorithms for estimating optimal bandwidth matrices for multivariate kernel density estimation. Our approach is based on treating the elements of the bandwidth matrix as parameters whose posterior density can be obtained through the likelihood cross-validation criterion. Numerical studies for bivariate data show that the MCMC algorithm generally performs better than the plug-in algorithm under the Kullback-Leibler information criterion, and is as good as the plug-in algorithm under the mean integrated squared error (MISE) criterion. Numerical studies for five dimensional data show that our algorithm is superior to the normal reference rule. Our MCMC algorithm is the first datadriven bandwidth selector for multivariate kernel density estimation that is applicable to data of any dimension.

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Inference on the autocorrelation coefficient p of a linear regression model with first-order autoregressive normal disturbances is studied. Both stationary and nonstationary processes are considered. Locally best and point-optimal invariant tests for any given value of p are derived. Special cases of these tests include tests for independence and tests for unit root hypotheses. The powers of alternative tests are compared numerically for a number of selected testing problems and for a range of design matrices. The results suggest that point-optimal tests are usually preferable to locally best tests, especially for testing values of p greater than or equal to one.

Applications of exponential smoothing to forecast time series usually rely on three basic methods: simple exponential smoothing, trend corrected exponential smoothing and a seasonal variation thereof. A common approach to select the method appropriate to a particular time series is based on prediction validation on a withheld part of the sample using criteria such as the mean absolute percentage error. A second approach is to rely on the most appropriate general case of the three methods. For annual series this is trend corrected exponential smoothing: for sub-annual series it is the seasonal adaptation of trend corrected exponential smoothing. The rationale for this approach is that a general method automatically collapses to its nested counterparts when the pertinent conditions pertain in the data. A third approach may be based on an information criterion when maximum likelihood methods are used in conjunction with exponential smoothing to estimate the smoothing parameters. In this paper, such approaches for selecting the appropriate forecasting method are compared in a simulation study. They are also compared on real time series from the M3 forecasting competition. The results indicate that the information criterion approach appears to provide the best basis for an automated approach to method selection, provided that it is based on Akaike's information criterion.

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This paper puts the case for the inclusion of point optimal tests in the econometrician's repertoire. They do not suit every testing situation but the current evidence, which is reviewed here, indicates that they can have extremely useful small-sample power properties. As well as being most powerful at a nominated point in the alternative hypothesis parameter space, they may also have optimum power at a number of other points and indeed be uniformly most powerful when such a test exists. Point optimal tests can also be used to trace out the maximum attainable power envelope for a given testing problem, thus providing a benchmark against which test procedures can be evaluated. In some cases, point optimal tests can be constructed from tests of a simple null hypothesis against a simple alternative. For a wide range Of models of interest to econometricians, this paper shows how one can check whether a point optimal test can be constructed in this way. When it cannot, one may wish to consider approximately point optimal tests. As an illustration, the approach is applied to the non-nested problem of testing for AR(1) disturbances against MA(1) disturbances in the linear regression model.

This paper considers a nonparametric time series regression model with a nonstationary regressor. We construct a nonparametric test for whether the regression is of a known parametric form indexed by a vector of unknown parameters. We establish the asymptotic distribution of the proposed test statistic. Both the setting and the results differ from earlier work on nonparametric time series regression with stationarity. In addition, we develop a bootstrap simulation scheme for the selection of suitable bandwidth parameters involved in the kernel test as well as the choice of simulated critical values. An example of implementation is given to show that the proposed test works in practice.

This paper outlines several difficulties with testing economic theories, particularly that the theories may be vague, may relate to a decision interval different from the observation period and may need construction of a metric to convert a complicated testing situation to an easier one. We argue that it is better to use model selection procedures rather than formal hypothesis testing when asking the data to decide on model specification. This is because testing favors the null hypothesis, typically uses an arbitrary choice of significance level and researchers working with the same data could easily end up with different final models, which would make policy recommendations difficult.