Benoit Quenneville

A note on Musgrave asymmetrical trend-cycle filters

Abstract In this paper, we derive the implicit forecasts in the asymmetrical trend-cycle averages used in the X-11 seasonal adjustment method. We give an algorithm to calculate them, and we study their statistical properties. We express the forecasts as Stein estimators. We derive expressions for their bias, variance, covariances and prediction mean squared errors. We show that the prediction mean squared errors of the implied predictors are always smaller or equal to those obtained using the least squares predictors. Finally, we derive the prior distributions under which the implied predictors are Bayes estimators.

Bayesian Prediction Mean Squared Error for State Space Models with Estimated Parameters

Hamilton (A standard error for the estimated state vector of a state-space model. J. Economet. 33 (1986), 387–97) and Ansley and Kohn (Prediction mean squared error for state space models with estimated parameters. Biometrika 73 (1986), 467–73) have both proposed corrections to the naive approximation (obtained via substitution of the maximum likelihood estimates for the unknown parameters) of the Bayesian prediction mean squared error (MSE) for state space models, when the models parameters are estimated from the data. Our work extends theirs in that we propose enhancements by identifying missing terms of the same order as that in their corrections. Because the approximations to the MSE are often subject to a frequentist interpretation, we compare our proposed enhancements with their original versions and with the naive approximation through a simulation study. For simplicity, we use the random walk plus noise model to develop the theory and to get our empirical results in the main body of the text. We also illustrate the differences between the various approximations with the Purse Snatching in Chicago series. Our empirical results show that (i) as expected, the underestimation in the naive approximation decreases as the sample size increases; (ii) the improved Ansley–Kohn approximation is the best compromise considering theoretical exactness, bias, precision and computational requirements, though the original Ansley–Kohn method performs quite well; finally, (iii) both the original and the improved Hamilton methods marginally improve the naive approximation. These conclusions also hold true with the Purse Snatching series.

Dynamic linear models for time series components

Abstract This paper describes a general state space approach for the modelling of the unobserved components; trend cycle, seasonality, trading-day variations, and irregulars of a time series and the calculation of the mean square errors of the estimated unobserved components. The unobserved components models are presented in a state space form. The Kalman filter and fixed interval smoother are applied on the observed series to obtain the estimates of the unobserved components and their corresponding variances. Implementation problems related to the estimation of the initial conditions and the initial values for the variances of both the observation noise and noise processes of the unobserved components are solved using the estimates of the unobserved components from X-11-ARIMA [Dagum (1980)]. The estimation of the signal to noise ratio is made by maximum likelihood using the method of scoring. The MLE of the noise variance is obtained analytically, conditional on the estimates of the signal to noise ratios. Statistical tests to distinguish between alternative models are also provided.

A non-parametric iterative smoothing method for benchmarking and temporal distribution

This article considers the problem of benchmarking and temporal distribution and presents a non-parametric method based on iterative smoothing using the Henderson moving averages. The properties of the method are discussed and two examples are provided to illustrate the application.

Outline of the X-11 Method

The X-11 method is based on an iterative principle of estimation of the different components, this estimation being done at each step using appropriate moving averages. The method is designed for the decomposition and seasonal adjustment of monthly and quarterly series.

Modelling of the Easter Effect

X-11-ARIMA and X-12-ARIMA propose different models for correcting the Easter effect based on an estimate of the irregular component. The proposed models and the methods used are sometimes quite different in the two softwares, which is why we have not integrated them directly into the seasonal adjustment example of Chapter 4.

The Various Tables

This chapter presents a complete and detailed example of seasonal adjustment with the X-11 method. The series that is used in this example is a monthly series; it is in such cases that the softwares’ options are most numerous and complex. The series studied X t is the monthly index of industrial production in France between October 1985 and March 19951. The series is represented in the top panel of Figure 4.1, which gives a decomposition plot. The seasonal factors S t (Table D10) are graphed in the third panel, and, in the case at hand, the trading-day factors D t (Table C18) are provided in the fourth panel. These two sets of factors are used to compute the seasonally adjusted series A t (Table D11), which is shown with the original series in the top panel, and with the trend-cycle C t (Table D12) in the second panel. Finally, the irregular component I t (Table D13) is graphed in the bottom panel. It is obtained by removing the trend-cycle from the seasonally adjusted series.

Brief History of Seasonal Adjustment

It is common today to decompose an observed time series X t into several components, themselves unobserved, according to a model such as:

Benchmarking by State Space Models

X_t = T_t + C_t + S_t + I_t ,