# Dealing with small sample sizes, rotation group bias and discontinuities in a rotating panel design 6. Accounting for discontinuities in the time series model

The parallel run showed that the redesign resulted in discontinuities in the series of the monthly figures about the labour force. To avoid severe model misspecification, the intervention term ${\Delta}_{t}\beta $ has to be included in model (3.1). An additional question is how the available information about the discontinuities in the first panel, obtained with the parallel run, can be used efficiently in the time series model. Six different methods to use the available information from the parallel run in model (3.1) and (3.9) are discussed.

*Method 1: Model (3.1) with a diffuse prior for
all intervention variables.*

The time independent regression coefficients of the intervention variables for all five panels are included in the state vector and initialised with a diffuse prior, as described by Durbin and Koopman (2001), Subsection 6.2.2. The Kalman filter can be applied straightforwardly to obtain estimates for the regression coefficients. This approach ignores the information about the discontinuities that is available from the parallel run. In this application, this approach is interesting since comparing the time series model estimate for the discontinuity in the first panel with the direct estimates obtained with the parallel run illustrates how well discontinuities can be estimated with the intervention approach.

*Method 2: Model (3.1) with an exact prior for
the intervention variable of the first panel.*

The direct estimates of the discontinuities from the parallel run are incorporated into the model by using an informative prior for the initialization of ${\beta}^{1}.$ This can be done by using these estimates in the initial state vector for ${\beta}^{1}$ and their estimated variances as an uncertainty measure for ${\beta}^{1}$ in the covariance matrix of the initial state vector.

*Method 3: Model (3.1) where the regression coefficient
of the intervention variable for the first panel equals the average direct
estimate for the discontinuity obtained with the parallel run.*

Another possibility of using the direct estimate of the discontinuities in the first panel as a-priori information in model (3.1), is to assume that the regression coefficient for the intervention in the first panel is time independent and equal to the average value of the observed discontinuity in the parallel run, i.e.,

$${\overline{\beta}}^{1}=\frac{1}{6}{\displaystyle \sum _{t=\tilde{t}}^{\tilde{t}+5}\left({\widehat{Y}}_{t}^{t,\text{New}}-{\widehat{Y}}_{t}^{t,\text{Old}}\right)},$$

where $\tilde{t}$ denotes the start of the parallel run in January 2010. In this case the direct estimate for the discontinuity is treated as if it is a fixed value, known in advance. This approach ignores the uncertainty of using a survey estimate for the discontinuity.

*Method 4: As method 3, but with a time
dependent regression coefficient for the intervention variable of the first
panel. *

The direct estimates for the discontinuities fluctuate considerably over the six months of the parallel run, see Table 5.2. To have a smooth transition from the old to the new design, an alternative for method 3 is considered where during the parallel run, the regression coefficient of the first panel is time dependent and equals the observed monthly discontinuities. For the period after the parallel run, this regression coefficient is equal to the average value of the observed discontinuity in the parallel run, i.e.,

$${\beta}_{t}^{1}=\{\begin{array}{lll}{\widehat{Y}}_{t}^{t,\text{New}}-{\widehat{Y}}_{t}^{t,\text{Old}}\hfill & \text{if}\hfill & t\in \left[\tilde{t},\mathrm{...},\tilde{t}+5\right]\hfill \\ {\overline{\beta}}^{1}\hfill & \text{if}\hfill & t>\tilde{t}+5.\hfill \end{array}$$

This method comes down to replacing the observations under the new design by the observations under the old design during the parallel run and assumes that the results under the old design are more reliable during this period. Similar to method 3, the uncertainty of using a survey estimate for the discontinuity is ignored.

The four methods can be applied to model (3.9) that is extended with an auxiliary series about the number of people formally registered at the employment office. The following two methods are considered:

*Method 5: Equals Method 1 applied, to model (3.9).*

*Method 6: Equals Method 4 applied, to model (3.9).*

In practise, method 1 would be considered if no parallel run is available. In the case of a well conducted parallel run, method 2 is probably the most natural approach, because the sample estimate for the discontinuity together with its uncertainty are used as prior information in the model. The sample information that becomes available after the parallel run under the new design is still used to improve the estimate of the discontinuity. Methods 3 and 4 are considered as alternatives for method 2 for getting a smoother transition from the estimates obtained until June 2010 under the old design to the estimates under the new design, starting in July 2010. Method 3 might work well if the variation between the monthly estimates for the discontinuity during the parallel run is small. In the case of large fluctuations between the monthly discontinuities, method 4 might be considered because during the parallel run each monthly deviation of the estimate under the new design is nullified with the time dependent discontinuities. Method 4 will therefore result in the smoothest transition.

In the case of strong and reliable auxiliary information, each method can be combined with model (3.9). It is a requirement, however, that the evolution of this auxiliary series is not influenced by factors that are unrelated to the real developments of the labour market. Method 5 would be considered if no parallel run is available. The auxiliary series might result in more precise estimates for the discontinuity and the trend and signal of the unemployed labour force. In the case of a parallel run, method 2 in combination with model (3.9) is probably the most natural approach for similar reasons as mentioned before (results not presented). Method 6 can be used to get a smoother transition from the old to the new design and more precise estimates for the trend and the signal of the unemployed labour force by taking advantage of the available auxiliary information. For similar reasons method 3 can be combined with model (3.9) (results not presented).

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