The Kronos Retail Labor Index
Background | Data | Seasonal Adjustment | X-12 ARIMA Model
Calculating the Index | References
Kronos Incorporated provides a variety of products and services used by mid-size to large retail operations in the United States. One particular area of focus is on employee selection and onboarding. Provisioned to clients using the Software as a Service (SaaS) model and acting as the single application service provider (ASP) presents an excellent opportunity for the study of large volumes of application and hiring data in near real time. The relationship between the number of application and hiring transactions when aggregated across a range of retail operations can be used to inform a better understanding of the underlying labor-related economic forces impacting the retail sector. This paper describes the development of the Kronos Retail Labor Index, a time-series based Index that measures the supply-and-demand relationship for labor in the consumer retail sector.
Data used in calculating the Kronos Retail Labor Index is extracted from the Kronos system and represents the application and hiring transactions from 69 U.S. retail firms. These firms represent a broad cross section of in-person terminal transaction goods and services operations, generally known as “bricks and mortar” businesses, to consumer sales. With few exceptions, all firms represented are multisite operations.
The jobs being applied for and filled are primarily frontline hourly positions, such as cashiers and clerks at a grocery store. This broad category of positions accounts for more than 80 percent of all operational positions in the consumer retail sector.
Application and hiring data from August 2006 through the present are used in calculating the Index. To be included in the sample, a client company must have been using the system three or more months prior to August 2006, and must also be using the system throughout the time period over which the Index is calculated. Data was only included for firms that had used the system for the full duration of this time period. Grocers make up 30 percent of the sample data, with specialty retailers representing another 60 percent. Entertainment and Services account for 8 percent of the sample with casual dining and hospitality each providing approximately 1 percent of the sample.
For each month over which the Index is calculated, the total number of applications and hires are aggregated across customers. Larger customers (by application count) have a greater impact on the final value of the Index than smaller ones.
Demand for labor within the U.S. retail market is highly seasonal.  Analysis of economically related time series requires specialized forms of seasonal adjustment.  Seasonality must be defined, given a precise mathematical definition and then statistical methods can be devised (or selected from many already available) to remove the seasonality. For purposes here, a time series is defined to have seasonality if its sample autocorrelation function (ACF) has significant values at multiples of the seasonal period (e.g., in the case of monthly data, as we have here, at lags 12, 24, etc.). An initial visual inspection of the normalized level of hiring presented in Figure 1 indicates that there is a significant seasonal effect in hiring that is most pronounced in January and July.
Figure 1. Normalized Hiring Levels Showing Seasonal Spikes
The ACF values in Table 1 indicate a significant seasonality impact. Application of the Kruskal-Wallis  test indicates the presence of seasonality at p<.01, while an F-Test for the presence of seasonality suggests its presence with greater than 99.9 percent likelihood. 
Modern econometric techniques for seasonal adjustment fall into several categories, the most widely accepted being those that fall into the Box-Jenkins approach . In this approach, seasonality is understood to be the dominant nonstationary factor, and because the period of the seasonality is often known, it is a fairly straightforward adaptation of the well-known autoregressive integrated moving average (ARIMA) model, itself a generalization of the autoregressive moving average (ARMA) model.
Given a time series of data where t is an integer Index and the are real numbers, then an ARMA(p,q) model  is given by
where L is the lag operator, are the parameters of the autoregressive part of the model, are the parameters of the moving average part, and are error terms. The error terms are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.
An ARIMA(p,d,q) process is obtained by integrating an ARMA(p,q) process. That is,
where d is a positive integer that controls the level of differencing (or, if d = 0, this model is equivalent to an ARMA model). Conversely, applying term-by-term differencing d times to an ARMA(p,q) process gives an ARIMA(p,d,q) process.
Fairly exhaustive comparisons of a number of related ARIMA models were conducted, with the selection criteria being the greatest reduction in ACF and the Kruskal-Wallis coefficient. Of the various forms of ARIMA models that I tested for use with the Index data, the X-12 ARIMA model of the U.S. Census Bureau proved to be most appropriate.
The X-12 ARIMA Model
The U.S. Census Bureau deals with vast quantities of time series. In 1965, a group of econometricians at the Census Bureau developed what came to be known as the X-11 Variant of Method II. This esoteric name was attached to a significant breakthrough in the practical use of ARIMA type models for large scale data analysis. 
In 1998, a subsequent update of the technique known as the X-12 ARIMA model was released.  Among the important capabilities of these advanced models is an inclusion of diagnostic tests and corrective factors for what are known as “trading day effects.” For the purposes here, these effects are tied to particular calendar days, generally holiday related. Thus the impact of Christmas, Thanksgiving, Easter, and other specific holidays can be accounted for. At first consideration, this seems especially useful for non-stationary holidays such as Thanksgiving and Easter. These particular holidays do not appear on the same calendar day from one year to the next. In the retail sector, Thanksgiving in particular is an “anchor” date from which seasonal hiring strategies are planned. The ability to account for a shifting Thanksgiving holiday date is a small but important improvement.
As a wonderful public service, the U.S. Census Bureau provides public domain software called the X-12-ARIMA Seasonal Adjustment Program for a variety of platforms that can be used to analyze very large time series.  Several commercial econometric analysis package vendors have also implemented the X-12 algorithm.  I implemented the X-12 model using Mathematica Version 7 and general guidance on efficient algorithm development.  While it would be possible to use the software provided by the Census Bureau, doing so would not allow for integration with a number of the other procedures developed for the analysis of the data and for the graphical display of the results. We have used the software provided by the Census Bureau and found it to be excellent, with very clear documentation. The output provided by the Census Bureau software presents highly detailed diagnostics.
We tested and validated the implementation of X-12 by comparing the output we generated with the output from the Census Bureau software for the same known data set. Figure 2. below presents the normalized hiring levels seen in Figure 1. after they have been seasonally adjusted using the X-12 algorithm.
Figure 2. Seasonal Adjusted Hiring Levels (Example Data)
Calculating the Index
The Kronos Retail Labor Index is calculated simply as the rato of hires to applications. The time series for applications and for hires have different seasonal characteristics. We had two choices: seasonally adjust each of the time series and then calculate the Index (as the simple ratio) or calculate the ratio and then seasonally adjust it. The results do differ somewhat and we chose to adjust the series and then calculate the ratios.
Figure 3. Seasonal Adjusted Retail Labor Index (September Release)
Currently, the index is calculated at the highest aggregated level. It is possible to calculate the index within subsector, region, client, or most any combination of these. Until we have a better appreciation for how the index is being used, we will wait before determining what dimensions we should use to break the index into meaningful subindices. There will be a trade-off between more specific index series and sample sizes.
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