Jukka Nyblom

Tests against stationary and explosive alternatives in vector autoregressive models

The article proposes new tests for the number of unit roots in vector autoregressive models based on the eigenvalues of the companion matrix. Both stationary and explosive alternatives are considered. The limiting distributions of test statistics depend only on the number of unit roots. Size and power are investigated, and it is found that the new test against some stationary alternatives compares favourably with the widely used likelihood ratio test for the cointegrating rank. The powers are prominently higher against explosive than against stationary alternatives. Some empirical examples are provided to show how to use the new tests with real data. Copyright 2008 The Authors

A Model-Based Approach for the Recovery of Forest Attributes Using Airborne Laser Scanning Data

As three-dimensional wall-to-wall information on forest structure, ALS echoes provide information on the growing stock and canopy structure. Even though the ALS echo heights are associated with the dimensions of trees, a theoretical model to relate ALS data with interesting forest attributes is missing. The recorded observation of echo height can be viewed as an outcome of a complex process mixing several random sub-processes related to the forest and the atmosphere in a non-trivial way. The forest-related processes include those generating stand density, tree heights, tree locations, tree crown shapes, and the internal structure of tree crowns. This chapter presents our recent work on development of a theoretical model for ALS echo heights. Furthermore, extensions are presented to take into account randomness in tree crown shape, to incorporate the penetration of laser pulses into tree crowns, and to develop the model for mixed stands.

Permutation Tests in Linear Regression

Exact permutation tests are available only in rather simple linear models. The problem is that, although standard assumptions allow permuting the errors of the model, we cannot permute them in practice, because they are unobservable. Nevertheless, the residuals of the model can be permuted. A proof is given here which shows that it is possible to approximate the unobservable permutation distribution where the true errors are permuted by permuting the residuals. It is shown that approximation holds asymptotically and almost surely for certain quadratic statistics as well as for statistics which are expressible as the maximum of appropriate linear functions. The result is applied to testing the significance of predictors as well as to diagnostic checking of heteroscedasticity, autocorrelation, change-points, and changing regression function. Also a simulation experiment is made in order to evaluate the performance of the proposed tests.

Tests for real and complex unit roots in vector autoregressive models

The article proposes new tests for the number of real and complex unit roots in vector autoregressive models. The tests are based on the eigenvalues of the sample companion matrix. The limiting distributions of the eigenvalues converging to the unit eigenvalues turn out to be of a non-standard form and expressible in terms of Brownian motions. The tests are defined such that the null distributions related to eigenvalues +/-1 are the same. The tests for the unit eigenvalues with nonzero imaginary part are defined independently of the angular frequency. When the tests are adjusted for deterministic terms, the null distributions usually change. Critical values are tabulated via simulations. Also some simulation based finite sample properties are presented together with comparisons with corresponding likelihood ratio tests. The relation of the unit roots to cointegration is discussed. An empirical example is provided to show how to use the test with real data.