Filters for Short Nonstationary Sequences

This paper describes a methodology for implementing bidirectional frequency-selective filters in cases where the data sequence is short and nonstationary. A simple method is proposed for dealing with the start-up problem. The method has a firm theoretical basis and it is computationally efficient.Signal extraction, Linear Filtering, Frequency-domain analysis, Trend estimation

Signal Extraction, Maximum Likelihood Estimation and the Start-up Problem

In this paper, we portray the essential features of the finite-sample signal extraction problem in both the stationary and the nonstationary cases. The computational procedures can be simplified in the light of our analysis. An important outcome of the analysis is a demonstration that the start-up problem can be handled far more easily that one might expect from a passing acquaintance with the usual practices.Signal extraction, Linear filtering, Trend estimation

Circulant Matrices and Time-series Analysis

This paper sets forth some of the salient results in the algebra of circulant matrices which can be used in time-series analysis. It provides easy derivations of some results that are central to the analysis of statistical periodograms and empirical spectral density functions. A statistical test for the stationarity or homogeneity of empirical processes is also presented.Time-series analysis, Circulant matrices, Discrete Fourier transforms, Periodograms

Deconstructing the Consumption Function: New Tools and Old Problems

In this paper, we analyse anew the relationship between aggregate income and consumption in the United Kingdom. Our analysis entails a close examination of the structure of the data, for which we employ a variety of spectral methods which depend on the concepts of Fourier analysis. We discover that fluctuations in the rate of growth of consumption tend to precede similar fluctuations in income, which contradicts a common supposition. We also highlight the difficulty of uncovering from the aggregate data a structural equation representing the behaviour of consumers.Consumption function, Trend estimation, Seasonal adjustment, Spectral analysis

Econometric Methods of Signal Extraction

The Wiener-Kolmogorov signal extraction filters, which are widely used in econometric analysis, are constructed on the basis of statistical models of the processes generating the data. In this paper, such models are used mainly as heuristic devices that are to be specified in whichever ways are appropriate to ensure that the filters have the desired characteristics. The digital Butterworth filters, which are described and illustrated in the paper, are specified in this way. The components of an econometric time series often give rise to spectral structures that fall within well-defined frequency bands that are isolated from each other by spectral dead spaces. We find that the finite-sample Wiener-Kolmogorov formulation lends itself readily to a specialisation that is appropriate for dealing with band-limited components.Signal extraction, Linear filtering, Frequency-domain analysis, Trend estimation

On Kronecker Products, Tensor Products And Matrix Differential Calculus

The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. The vectorisation operations and the commutation transformations that are common in multivariate statistical analysis alter the positional relationship of the matrix elements. These elements correspond to numbers that are liable to be stored in contiguous memory cells of a computer, which should remain undisturbed. It is suggested that, in the absence of an adequate index notation that enables the manipulations to be performed without disturbing the data, even the most clear-headed of computer programmers is liable to perform wholly unnecessary and time-wasting operations that shift data between memory cells.

Recursive Estimation in Econometrics

An account is given of recursive regression and of Kalman filtering which gathers the important results and the ideas that lie behind them within a small compass. It emphasises the areas in which econometricians have made contributions, which include the methods for handling the initial-value problem associated with nonstationary processes and the algorithms of fixed-interval smoothing.Recursive regression, Kalman filtering, Fixed-interval smoothing, The initial-value problem

Improved Frequency-selective Filters

This paper gives an account of some techniques for designing recursive frequency-selective filters which can be applied to data sequences of limited duration which may be nonstationary. The designs are based on the Wiener-Kolmogorov theory of signal extraction which employs a statistical model of the processes generating the data. The statistical model may be regarded as an heuristic device which is designed with a view to ensuring that the resulting signal-extraction filters have certain preconceived properties.Signal extraction, Linear filtering, Filter design, Trend estimation, Frequency-domain analysis

Alternative Methods of Seasonal Adjustment

Alternative methods for the seasonal adjustment of economic data are described that operate in the time domain and in the frequency domain. The time-domain method, which employs a classical comb filter, mimics the effects of the model-based procedures of the SEATS–TRAMO and STAMP programs. The frequency-domain method eliminates the sinusoidal elements of which, in the judgment of the user, the seasonal component is composed. It is proposed that, in some circumstances, seasonal adjustment is best achieved by eliminating all elements in excess of the frequency that marks the upper limit of the trend-cycle component of the data. It is argued that the choice of the method seasonal adjustment is liable to affect the determination of the turning points of the business cycle.Wiener–Kolmogorov Filtering; Frequency-Domain Methods; The Trend-Cycle Component