Chapter 4.
Measurement of Unemployment in States and Local Areas

Estimation Methodology

Estimates for States
Monthly labor force data for all States, the District of Columbia, the Los Angeles-Long Beach metropolitan area, New York City, and the balances of California and New York are based on the time series approach to sample survey data (Scott and Smith, 1974; Bell and Hillmer, 1990). The purpose of this approach is to reduce the high variability in monthly CPS estimates for these geographic areas due to small sample sizes. The actual monthly CPS sample estimates are represented in signal plus noise form as the sum of a stochastic true labor force series (signal) and error (noise) generated by sampling only a portion of the total population.

where:

= CPS estimate

= true labor force value

= sampling error

The signal is represented by a time series model that incorporates historical relationships in the monthly CPS estimates along with auxiliary data from the Unemployment Insurance and Current Employment Statistics (CES) programs. This time series model is combined with a noise model that reflects key characteristics of the sampling error (SE) to produce estimates of the true labor force values. This estimator is optimal under the model assumptions and has been shown to be design-consistent under general conditions by Bell and Hillmer (1990).

Two models—one for the employment-to-population ratio and one for the unemployment rate—are developed for each State using over 15 years of data. The signals for both models are based on a core model of the following form:

where X_t is a single explanatory variable with coefficient and T_t, S_t, and I_t are the trend, seasonal and irregular components. The variable used in the employment model is the statewide monthly estimate of workers on payrolls in non-farm industries from the CES program divided by the intercensal estimate of the State's population of working age. The unemployment model uses the ratio of the number of State workers claiming unemployment insurance benefits to the payroll employment estimate. The regression coefficient is allowed to change over time to adapt to changing relationships between the CPS and the explanatory variable. The trend and seasonal components change smoothly over time to control for systematic variation in the CPS not accounted for by the explanatory variable. The irregular component accounts for transitory residual variation not captured by other components of the model.

The degree to which the regression coefficient and the time series components vary over time is determined empirically for each State. Occasionally, the trend is a constant, acting as a fixed intercept. In some cases, the seasonal component is estimated to have a fixed pattern from year-to-year. For most models, the irregular component is zero.

Occasionally, there are sudden changes, either temporary or permanent, in the CPS that are not predictable from past history. These effects, manifested as aberrant observations or outliers, are handled by intervention analysis techniques which introduce dummy variables into the model components. Shifts in level are incorporated into the trend component and transitory changes into the irregular component.

The second major component of the signal plus noise model deals with CPS standard errors. Because of this survey's complex design, the behavior of the observed sample estimates differ in important ways from the true values. Sampled households are rotated in and out of the CPS over a period of 16 months, such that 75 percent of the sample from month-to-month consists of the same households and 50 percent from year-to-year. (See chapter 1.) Also, redesigns and major fluctuations in the size of the labor force cause major changes in the variance of the standard errors. These two features of the CPS, an overlapping sample design and changes in reliability, induce strong positive autocorrelation and heteroscedasticity in the standard error. These characteristics can seriously contaminate estimates of the true labor force if the standard error is ignored in the estimation process. For this reason, it is important to specify a model of the standard error process and combine it with the model of the signal to estimate the unobserved components of the CPS. The standard error model is specified as follows:

with reflecting the autocovariance structure, assumed to follow an ARMA process and representing a changing variance over time. The parameters of the ARMA model are derived from standard error autocorrelations developed independently of the time series model from design based information. The standard error variances (equivalent to the square of the standard error described in chapter 1) are estimated using the method of generalized variance functions (Zimmerman and Robison, 1996).

The unknown hyperparameters of the signal are estimated by maximum likelihood using the Kalman filter algorithm. Given these estimated hyperparameters, the Kalman filter is used to decompose the observed CPS into its signal and noise components. This algorithm efficiently updates the model estimates as new data become available each month. For the latest month, the Kalman filter calculates estimates based on all available data, but does not revise estimates for the previous months with the latest data. Previous estimates are updated by a Kalman filter "smoother" which revises an estimate at time t using all available data before and following time t. Smoothing is performed at the end of each year.

Benchmarking. This process is a general statistical procedure used to adjust estimates to a control total. Each year, historical model estimates are benchmarked to the annual average CPS State estimates of employment and unemployment. The goal of benchmarking is twofold: (1) To insure that the annual average of the final benchmarked series equals the CPS annual average, and (2) to preserve the pattern of the model series as much as possible. In practice, these two goals are conflicting, and some changes to the pattern of the time series are made to meet the first goal. The Denton benchmarking method has been used since the introduction of model-based estimates in 1989. It avoids discontinuities between December and January estimates and, through a constrained quadratic minimization approach, minimizes the distortion to the original time-series estimates. The benchmarked series are seasonally adjusted with X-11 ARIMA.

Next: Estimates for Sub-State Areas—The Handbook Method