# Research Reports

### The Timing and Magnitude Relationships Between Month-to-Month Changes and Year-to-Year Changes That Make Comparing Them Difficult

#### KEY WORDS:

Business cycle; Gain and phase shift functions; Growth rates; Moving averages.

### ABSTRACT

When a monthly economic indicator series contracts sharply for a few months and then starts to recover, the published annual and monthly growth rates can give conicting signals: the annual growth rate can indicate a decrease and the monthly growth rate an increase or vice versa. This is well known to the seasonal adjustment community, see, for example, Shiskin (1957). In this paper, we revisit, illustrate and then explain this potential for conict more analytically. For example, the annual differences lag the monthly differences by five and a half months because the same-month-year-ago difference is the sum of the current and eleven preceding monthly differences, and the annual sum has a phase shift of five and a half months. Illustrative examples are followed by an elementary formal mathematical derivation using the gain and phase functions of the annual sum.

#### CITATION:

Benoit Quenneville and David F. Findley. (2012). The Timing and Magnitude Relationships Between Month-to-Month Changes and Year-to-Year Changes That Make Comparing Them Difficult. Center for Statistical Research & Methodology Research Report Series (Statistics #2012-05). U.S. Census Bureau. Available online at <http://www.census.gov/srd/papers/pdf/rrs2012-05.pdf>.

Source: U.S. Census Bureau, Center for Statistical Research & Methodology, Research and Methodology Directorate

Published online: May 9, 2012
Last revised: May 9, 2012

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

This symbol 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 | Research and Methodology Directorate | Center for Statistical Research & Methodology | (301) 763-3215 (or chad.eric.russell@census.gov) |   Last Revised: September 11, 2013