Skip Main Navigation Skip To Navigation Content

X-13ARIMA-SEATS Seasonal Adjustment Program

You are here: Census.govSubjects A to ZX-13ARIMA-SEATSSeasonal Adjustment PapersPapers by Year › Abstract of RRS2012-09
Skip top of page navigation

Model Estimation, Prediction, and Signal Extraction for Nonstationary Stock and Flow Time Series Observed at Mixed Frequencies

Tucker S. McElroy (1) and Brian C. Monsell (2)


An important practical problem for statistical agencies and central banks that publish economic data is the seasonal adjustment of mixed frequency stock and flow time series. This may arise in practice due to changes in funding of a particular survey. Mathematically, the problem can be reduced to the need to compute imputations, forecasts, and backcasts from a given model of the highest available frequency data. The nonstationarity of the economic time series coupled with the alteration of sampling frequency makes the problem of model estimation and imputation challenging. For flow data the analysis cannot be recast as a missing value problem, and our methods are needed. We provide explicit formulas and algorithms that allow one to compute the log Gaussian likelihood of the mixed sample, as well as any imputations and forecasts. Formulas for the relevant mean squared error are also derived. We illustrate the techniques on two economic time series.


seasonal adjustment, calendarization, ARIMA model, missing values, sampling frequency

(1) Tucker S. McElroy is Mathematical Statistican, Center for Statistical Research and Methodology U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email :

(2) Brian C. Monsell is Mathematical Statistician, Center for Statistical Research and Methodology, U.S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email :

[PDF] or PDF denotes a file in Adobe’s Portable Document Format. To view the file, you will need the Adobe® Reader® Off Site available free from Adobe.

This symbol Off Site indicates a link to a non-government web site. Our linking to these sites does not constitute an endorsement of any products, services or the information found on them. Once you link to another site you are subject to the policies of the new site.

Source: U.S. Census Bureau | Statistical Research Division | (301) 763-1649 (or |  Last Revised: April 02, 2015