The Employment Situation Summary of February 5, 2010, released last Friday, has managed to turn heads and raise eyebrows. Namely, how is it that the unemployment rate declined, while payroll employment simultaneously declined? Preliminary estimate on payroll employment, pulling from The Establishment Survey, projected -20,000 jobs lost in January. Differingly, the unemployment rate, pulling from The Household Survey, estimated a drop from 10% in December to 9.7% in January.

As a response, in this article we explore the relationship between observations on the unemployment rate and changes in payroll employment data. Results are interesting and counterintuitive. Correlation between the two variables appears to be present more on a lagged, time-offset basis — specifically, 7/8 months lagged.

First, here is some language from the BLS release, regarding payroll employment:

Total nonfarm payroll employment was essentially unchanged in January (-20,000). Job losses continued in construction and in transportation and warehousing, while employment increased in temporary help services and retail trade. Since the start of the recession in December 2007, payroll employment has fallen by 8.4 million. Over the last 3 months, however, employment has shown little net change.

And, some information on the marginally attached:

Among the marginally attached, there were 1.1 million discouraged workers in January, up from 734,000 a year earlier. (The data are not seasonally adjusted.) Discouraged workers are persons not currently looking for work because they believe no jobs are available for them. The remaining 1.5 million people marginally attached to the labor force had not searched for work in the 4 weeks preceding the survey for reasons such as school attendance or family responsibilities.

For the analysis, let’s begin with a scatter plot glance at the data, arranged concurrently. Here (see below), compared and plotted are variables: change in the unemployment rate and change in total non-farm jobs; again, all monthly data, with observations spanning several decades.

Notably surprising, the most recent January 2010 coordinate fits snuggly into the lower lefthand quadrant, i.e., [-0.3, -20,000]. Such an observation apparently is not that rare. For example, from just looking at the data you can see that there are many instances where there is a break in intuition. In other words, January employment data does not represent a Black Swan event. But how prevalent is the break in theory?

For a review of the Cartesian coordinate systems and quadrants, read this primer — Wikipedia’s Entry on Descartes.

The summary table and graphic below display just how often the breakdown occurs. For comparison, the table includes only observations exhibiting change in measured variables; whereas, the graphic incorporates all observations, including those where zero change occurs in the variables.

Notice the number of rejections of the null hypothesis, i.e., 358 rejections, where an increase in the unemployment rate associates with a decrease in payroll employment. Observations in quadrants I and III reject the hypothesis, while the remaining coordinates in quadrants II and IV confirm the hypothesis.

On a more subtle note, if you solely consider tail observations where change in the unemployment rate is greater than or equal to 4% or less than or equal to -4%, confirmations are more present, statistically significant and highly reliable. There exists here, 56 observations confirming and 4 observations rejecting, i.e., a 95% level of confirmation of the null hypothesis. Interpretation would read that if the magnitude of movement is large enough in one variable, the other will concurrently budge in the expected direction.

Now consider some basic diagnostics on lagged intervals and correlation relationships. The process begins with eyeing the real data (see below):

After specifying some OLS linear regressions of unemployment rate on change in payroll employment (11 lags explored), and specifying a corresponding lag matrix, we reflect the unemployment rate across the x-axis and shift it back 7 lags, as follows:

Clearly, one discovers a heightened degree of correlation at lag 7 and lag 8:

From my perspective, these are interesting results.

Obviously, they are not end-game in terms of forecasting usefulness, afterall R^2 explanation hovers around 28% in one reading (see the first graph above); but, they do display a more clarified relationship between two much watched macro-variables. It is worthwhile to remind oneself that the data is derived from ultimately two different surveys, as stated earlier. Multiple linear regression has a tendency to trump simple linear regression in terms of providing a more robust overall analysis.

The Employment Situation Summary released last Friday by the BLS, should no longer turn heads and raise eyebrows.