Víctor Gómez

Estimation, Prediction, and Interpolation for Nonstationary Series with the Kalman Filter

Abstract We show how our definition of the likelihood of an autoregressive integrated moving average (ARIMA) model with missing observations, alternative to that of Kohn and Ansley and based on the usual assumptions made in estimation of and forecasting with ARIMA models, permits a direct and standard state-space representation of the nonstationary (original) data, so that the ordinary Kalman filter and fixed point smoother can be efficiently used for estimation, forecasting, and interpolation. In this way, the problem of estimating missing values in nonstationary series is considerably simplified. The results are extended to regression models with ARIMA errors, and a computer program is available from the authors.

Wiener–Kolmogorov Filtering and Smoothing for Multivariate Series With State–Space Structure

Wiener-Kolmogorov filtering and smoothing usually deal with projection problems for stochastic processes that are observed over semi-infinite and doubly infinite intervals. For multivariate stationary series, there exist closed formulae based on covariance generating functions that were first given independently by N. Wiener and A.N. Kolmogorov around 1940. In this article, we consider multivariate series with a state-space structure and, using a new purely algebraic approach to the problem, we prove the equivalence between Wiener-Kolmogorov filtering and Kalman filtering. Up to now, this equivalence has only been partially shown. In addition, we get some new recursions for smoothing and some new recursions to compute the filter weights and the covariance generating functions of the errors. The results are extended to nonstationary series. Copyright 2007 The Author Journal compilation 2007 Blackwell Publishing Ltd.

A new state–space methodology to disaggregate multivariate time series

This article addresses the problem of disaggregating multivariate time series sampled at different frequencies using state-space models. In particular, we consider the relation between the high-frequency and low-frequency models, the possible loss of observability and identifiability in the latter with respect to the former, the estimation of the parameters of the low-frequency model by maximum likelihood, and the prediction and interpolation of high-frequency figures when only low-frequency data are available. Since vector autoregressive moving average models are a special case of state-space models, our results are also valid for those models, but they include other models as well, like structural models. We provide a rigorous theoretical development of the aforementioned issues, including a comparison with the classical model-based approaches, and we propose a practical methodology to disaggregate multivariate time series that is both efficient and easy to implement. Copyright 2009 The Authors. Journal compilation 2009 Blackwell Publishing Ltd

The Beveridge–Nelson Decomposition: A Different Perspective with New Results

We show in the paper that the decomposition proposed by Beveridge and Nelson (1981) for models that are integrated of order one can be generalized to seasonal Arima models by means of a partial fraction decomposition. Two equivalent algorithms are proposed to optimally (in the mean squared sense) compute the estimates of the components in the generalized decomposition. While the first algorithm is very fast and easy to implement, the second can also provide the standard errors of the estimated components. The properties of the implied filters are investigated and compared with those obtained using the model-based TRAMO/SEATS software package. The alternative methods are applied to the German unemployment series.

New Methods for Quantitative Analysis of Short-Term Economic Activity

We concern ourselves with statistical treatment of economic time-series data used in short-term economic policy, control and monitoring. Although other frequencies are possible, our attention centers on monthly (also quarterly) series. The statistical treatment we have in mind includes short-term forecasting, seasonal adjustment, estimation of the trend, estimation of the business cycle, estimation of special effects and removal of outliers, perhaps for a large number of series.

A Strongly Consistent Criterion to Decide Between I(1) and I(0) Processes Based on Different Convergence Rates

The usual procedure to determine whether a univariate time series is stationary or first-difference stationary is to perform some unit root test. In this article, an alternative methodology is presented that leads to a strongly consistent two-step criterion to estimate the number of unit roots. The criterion is based on estimating some autoregressive polynomials using regression procedures and exploiting the fact that the nonstationary roots converge at a faster rate than the stationary ones. The proposed procedure requires at most four regressions and is easy to implement. A simulation study demonstrates that it can perform significantly better in practice than the Dickey–Fuller and the generalized least squares (GLS)-detrended Dickey–Fuller tests.

An Alternative to Transfer Function Forecasting Based on Subspace Methods

In the time series literature, recent interest has focused on the so-called subspace methods. These techniques use canonical correlations and linear regressions to estimate the system matrices of an ARMAX model expressed in state space form. In this article, we use subspace methods to forecast two series with the help of some exogenous variables related to them. We compare the results with those obtained using traditional transfer function models and find that the forecasts obtained with both methods are similar. This result is very encouraging because, in contrast to transfer function models, subspace methods can be considered as almost automatic.