David F. Findley

New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program

X-12-ARIMA is the Census Bureaus new seasonal-adjustment program. It provides four types of enhancements to X-ll-ARIMA—(1) alternative seasonal, trading-day, and holiday effect adjustment capabilities that include adjustments for effects estimated with user-defined regressors; additional seasonal and trend filter options; and an alternative seasonal-trend-irregular decomposition; (2) new diagnostics of the quality and stability of the adjustments achieved under the options selected; (3) extensive time series modeling and model-selection capabilities for linear regression models with ARIMA errors, with optional robust estimation of coefficients; (4) a new user interface with features to facilitate batch processing large numbers of series.

Sliding-Spans Diagnostics for Seasonal and Related Adjustments

Abstract When are the results of a seasonal adjustment procedure (or another smoothing procedure) likely to be of little value? The diagnostic approach presented in this article offers an answer to this question and to other questions concerned with the comparison of competing adjustments. It is based on a straightforward idea. A minimal requirement of the output of any smoothing or adjustment procedure is stability: Appending or deleting a small number of series values should not substantially change the smoothed values—otherwise, what reliable interpretation can they have? An important related principle is that, for a given series, if only one of several plausible signal-extraction procedures has a stable output, then this procedure should be the preferred one for the series. To implement these principles successfully, the definition of stability must be made precise in an appropriate way. The implementation described in this article is focused on multiplicative adjustments produced by the widely used X...

New Techniques for Determining if a Time Series can be Seasonally Adjusted Reliably

Deciding when a series is a good candidate for seasonal adjustment can be difficult. There are situations where a series may show evidence of seasonality, but because of a dominating irregular component, for example, or a volatile seasonal component, many of its seasonal factors cannot be estimated reliably. In these circumstances, the estimates of a given month’s seasonal factor can change substantially when more data are added to the series and earlier data are deleted. Some seasonal adjustment programs, such as X-11 and X-11-ARIMA, provide diagnostics that can be used to help the analyst make this decision. However, the diagnostics provided by X-11 and X-11-ARIMA are sometimes inadequate. In this article, we will discuss two new sets of measures that help to determine when a series can be seasonally adjusted reliably by a proposed seasonal adjustment methodology.

Fitting Constrained Vector Autoregression Models

This paper expands the estimation theory for both quasi-maximum likelihood estimates (QMLEs) and Least Squares estimates (LSEs) for potentially misspecified constrained VAR(p) models. Our main result is a linear formula for the QMLE of a constrained VAR(p), which generalizes the Yule–Walker formula for the unconstrained case. We make connections with the known LSE formula and the determinant of the forecast mean square error matrix, showing that the QMLEs for a constrained VAR(p) minimize this determinant but not the component entries of the mean square forecast error matrix, as opposed to the unconstrained case. An application to computing mean square forecast errors from misspecified models is discussed, and numerical comparisons of the different methods are presented and explored.

Stock Series Holiday Regressors Generated from Flow Series Holiday Regressors

Stock economic time series, such as end-of-month inventories, arise as the cumulative sum of monthly inflows and outflows over time, i.e., as accumulations of monthly net flows. In this article, we derive holiday regressors for stock series from cumulative sums of flow-series holiday regressors. This is similar to how stock trading day regressors have been derived. The stock holiday regressors from this approach have a very simple and appealing form when the flow regressors have standard properties. The modeling, forecasting and graphical results we present, for Easter effects in U.S. manufacturing inventories and for Chinese New Year effects in economic indicator inventory series of Taiwan, confirm the utility of this first general approach tomodeling stock holiday effects. As with estimated holiday effects fromflowseries,we find that stock holiday effects are usually larger than trading day effects but smaller than seasonal effects.

OPTIMALITY OF GLS FOR ONE-STEP-AHEAD FORECASTING WITH REGARIMA AND RELATED MODELS WHEN THE REGRESSION IS MISSPECIFIED

We consider the modeling of a time series described by a linear regression component whose regressor sequence satisfies the generalized asymptotic sample second moment stationarity conditions of Grenander ( 1954 , Annals of Mathematical Statistics 25, 252–272). The associated disturbance process is only assumed to have sample second moments that converge with increasing series length, perhaps after a differencing operation. The models regression component, which can be stochastic, is taken to be underspecified, perhaps as a result of simplifications, approximations, or parsimony. Also, the autoregressive moving average (ARMA) or autoregressive integrated moving average (ARIMA) model used for the disturbances need not be correct. Both ordinary least squares (OLS) and generalized least squares (GLS) estimates of the mean function are considered. An optimality property of GLS relative to OLS is obtained for one-step-ahead forecasting. Asymptotic bias characteristics of the regression estimates are shown to distinguish the forecasting performance. The results provide theoretical support for a procedure used by Statistics Netherlands to impute the values of late reporters in some economic surveys. The author thanks two referees and the co-editor for comments and suggestions that led to substantial improvements in the exposition and also thanks John Aston and Tucker McElroy for helpful comments on an earlier draft. Any views expressed are the authors and not necessarily those of the U.S. Census Bureau.

Convergence of a Robbins-Monro Algorithm for Recursive Estimation with Non-Monotone Weights for a Function with a Restricted Domain and Multiple Zeros

Summary Convergence properties are established for the output of a deterministic Robbins- Monro recursion whose function can have singularities and multiple zeros. Our analysis is built largely on slight adaptations of some lemmas and proofs of Fradkov published only in an untranslated Russian monograph (Derevitzkii and Fradkov , 1981). A gap in Fradkovs proof of the final lemma is fixed but only for the scalar case. Our results can be applied to results of Cantor (2001) to establish the convergence of two well-known time series model recursive estimation schemes in the case of an incorrect moving average model. For such models, it is known that maximum likelihood estimates can converge w .p.1 to a set of values rather than to a single value. When the limit set is finite, our results show that , on a given realization of the time series, the (recursive) estimates will converge to single value. This is the first result establishing that estimates of a moving average coefficient do not oscillate forever among different limit set values when there are more than one.