State space time series modelling of the Dutch Labour Force Survey: Model selection and mean squared errors estimation
Section 4. The DLFS-specific simulation setup

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The performance of the five MSE estimation methods is examined on series of the original length from the DLFS survey (114 monthly time points from 2001(1) until 2010(6)), as well as on shorter series of lengths 48 and 80 months, and on longer ones of length 200. For each of these series lengths, a Monte-Carlo experiment is set up where multiple series (1,000) are simulated on the basis of the DLFS model for the number of unemployed. MSEs for each of these series are estimated based on $B\mathrm{<mo>=</mo>\; 300}$ bootstrap series; for asymptotic approximation, however, at least $B\mathrm{<mo>=</mo>\; 500}$ draws turned out to be needed. This number has been found sufficient for the approximated MSEs to converge. MSEs delivered by the five methods and averaged over the 1,000 simulations are compared to $\text{MSE}-$ averages produced by the “naive” Kalman filter. However, for the latter MSE estimates to converge to a certain average value, at least 10,000 simulations are needed.

The above-mentioned artificial series $Y_t^s$ for simulations $s=1,\dots ,\text{1,000}$ (or 10,000) are generated parametrically in the following way. First, the hyperparameter ML estimates ${\widehat{\theta }}_{\sigma }$ are obtained from fitting the STS model to the original series. Thereafter, state disturbances (recall that survey errors are also modelled as state variables) are randomly drawn from their joint normal distribution $N\left(0,\Omega \left({\widehat{\theta }}_{\sigma }\right)\right),$ and series are generated using the Kalman filter recursion. Since the system is non-stationary, the generated series $Y_t^s$ may take on negative or implausibly large numbers of the unemployed. In order to avoid an excessively large number of series with negative values, the state variables recursion is launched from the states’ smoothed estimates at one of the highest points of the observed series. Further, the first 30 time points are discarded in order to prevent that the series start at the same time-point. With an assumption that unemployment in the Netherlands will not exceed 15 percent of the total labour force, the simulation data set is restricted to contain only series with values between 0 and 1 mln of unemployed (this value comprised about 15 percent of the Dutch labour force in 2010); other series are discarded. Keeping the artificial series below the upper bound is also done in order not to extrapolate outside of the original data range when simulating the design-based standard errors $z_t^j.$

Every series of simulated GREG point-estimates needs its own series of simulated design-based standard error estimates, $z_t^j’\text{s}\text{.}$ The original known design-based standard error estimates $\sqrt{\widehat{\text{Var}}\left(Y_t^j\right)}$ would not be suitable for this simulation because the sampling error variance is proportional to the corresponding point-estimate. The following variance function is used to generate design-based variances for the simulated series of point-estimates (see Appendix B in Bollineni-Balabay et al. (2016b) for details):

$$\begin{array}{l}\text{ln}\left[\widehat{\text{Var}}\left(Y_t^1\right)\right]\mathrm{<mo>=</mo>}\text{ln}\left[{\left(z_t^1\right)}^2\right]\text{=}c+{\beta }_1\text{ln}\left(l_t^1\right)+{\epsilon }_t^1\text{}\mathrm{,}{\epsilon }_t^t\sim N\left(0,{\left({\sigma }_{\epsilon }^1\right)}^2\right)\mathrm{<mo>;</mo>}\hfill \\ \text{ln}\left[\widehat{\text{Var}}\left(Y_t^j\right)\right]\mathrm{<mo>=</mo>}\text{ln}\left[{\left(z_t^j\right)}^2\right]\text{=}{\psi }_j\text{ln}\left[{\left(z_{t-3}^{j-1}\right)}^2\right]+{\beta }_j\text{ln}\left(l_t^j\right)+{\epsilon }_t^j\text{},{\epsilon }_t^j\sim N\left(0,{\left({\sigma }_{\epsilon }^j\right)}^2\right)\mathrm{,}j\mathrm{<mo>=</mo>}\left\{\mathrm{2,<mtext>\hspace{0.17em}</mtext>3,<mtext>\hspace{0.17em}</mtext>4,<mtext>\hspace{0.17em}</mtext>5}\right\},(4.1)\hfill \end{array}$$

where $l_t^j\mathrm{,}j\mathrm{<mo>=</mo>}\left\{\mathrm{1,<mtext>\hspace{0.17em}</mtext>2,<mtext>\hspace{0.17em}</mtext>3,<mtext>\hspace{0.17em}</mtext>4,<mtext>\hspace{0.17em}</mtext>5}\right\}$ is the wave-signal being the sum of the trend, seasonal and RGB components. The regression coefficients in (4.1) are time-invariant and are obtained by regressing $\text{ln}{\left(z_t^j\right)}^2$ on $\text{ln}\left(l_t^j\right)$ and $\text{ln}\left({\left(z_{t-3}^{j-1}\right)}^2\right)$ from the original DLFS series. The superscripts are used to denote the wave these coefficients belong to. The coefficient estimates are presented in Table 4.1, together with the adjusted $\text{R}-$ square goodness of fit measure.

The simulation proceeds as follows. For each series length considered and in each simulation $s,$ five simulated signals $l_{t\mathrm{,}s}^j\mathrm{,}j\mathrm{<mo>=</mo>}\left\{\mathrm{1,2,3,4,5}\right\}$ are used to generate five sets of the design-based standard errors $z_{t\mathrm{,}s}^j$ according to the process defined by (4.1) and using the regression coefficients from Table 4.1. As soon as an artificial data set is generated, ${\widehat{\rho }}_s$ estimate is obtained, whereafter the rest of the hyperparameters are estimated with the quasi-ML method. Note that the same set of design-based standard errors $z_{t,s}$ is used to generate all bootstrap series within a particular simulation.

In order to obtain the true MSEs, the DLFS model is simulated a large number of times $\left(M=\text{50,000}\right),$ with each of these replications being restricted to the same limits as before, i.e., between zero and 1 mln of the unemployed. The true MSE is calculated in the following way using the true state vector ${\alpha }_{m,t}$ values known for every simulation $m:$

$$MS{E}_t^{\text{true}}=\frac{1}{\text{M}}{\displaystyle \sum _{m=1}^M\left[\left({\widehat{\alpha }}_{m,t}\left({\widehat{\theta }}_m\right)-{\alpha }_{m,t}\right){\left({\widehat{\alpha }}_{m,t}\left({\widehat{\theta }}_m\right)-{\alpha }_{m,t}\right)}^{\prime }\right]}.(4.2)$$

The true MSE of the signal is calculated in the same way by using the wave-signal values $l_{m,t}.$

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