State space time series modelling of the Dutch Labour Force Survey: Model selection and mean squared errors estimation
Section 3. MSE estimation approaches

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Linear structural time series models with unobserved components are usually fitted with the help of the Kalman filter after putting them into a state space form. See Bollineni-Balabay, van den Brakel and Palm (2016b) for the state space representation of the STS model for the DLFS. The state vector ${\alpha }_t$ contains the state variables defined in the previous section, i.e., the trend, slope, seasonal harmonics, RGB, the population white noise and survey errors. All the non-stationary state variables are initialised with a diffuse prior (i.e., with a zero-mean and a very large variance). The five survey error components ${\tilde{e}}_t^j\mathrm{,}j\mathrm{<mo>=</mo>}\left\{\mathrm{1,<mtext>\hspace{0.17em}</mtext>2,<mtext>\hspace{0.17em}</mtext>3,<mtext>\hspace{0.17em}</mtext>4,<mtext>\hspace{0.17em}</mtext>5}\right\}$ and the population white noise ${\epsilon }_t$ are stationary state variables that are initialised with zeros. The initial variance of the first-wave sampling errors is taken equal to unity, whereas the one of the other waves is taken equal to $\left(1-{\rho }^2\right).$ One could try a small value for the initial variance of ${\epsilon }_t.$

Filtered estimates of the state vector ${\alpha }_t$ and its covariance matrix $P_{t|t}$ are usually extracted with the Kalman filter (see Harvey 1989). $P_{t|t}$ thus contains MSEs extracted by the filter conditionally on the information up to and including time $t:$

$$P_{t|t}={\text{E}}_t\left[\left({\widehat{\alpha }}_{t|t}\left(\theta \right)-{\alpha }_t\right){\left({\widehat{\alpha }}_{t|t}\left(\theta \right)-{\alpha }_t\right)}^{\prime }\right],(3.1)$$

where $\theta$ is assumed to be the true hyperparameter value, and the expectation is taken with respect to the joint distribution of the state vector and the $Y-$ values at time $t.$ In practice, the true hyperparameter vector is replaced by its estimate $\widehat{\theta }$ in the Kalman filter recursions. Then, the MSE in (3.1) is no longer the true MSE and is called “naive” as it does not incorporate the uncertainty around the $\widehat{\theta }-$ estimates. The true MSE then becomes:

$$MS{E}_{t|t}={\text{E}}_t\left[\left({\widehat{\alpha }}_{t|t}\left(\widehat{\theta }\right)-{\alpha }_t\right){\left({\widehat{\alpha }}_{t|t}\left(\widehat{\theta }\right)-{\alpha }_t\right)}^{\prime }\right],$$

which is larger than the MSE in (3.1) and can be decomposed as the sum of the filter uncertainty and parameter uncertainty, provided the error terms are normal:

$$MS{E}_{t|t}={\text{E}}_t\left[\left({\widehat{\alpha }}_{t|t}\left(\theta \right)-{\alpha }_t\right){\left({\widehat{\alpha }}_{t|t}\left(\theta \right)-{\alpha }_t\right)}^{\prime }\right]+{\text{E}}_t\left[\left({\widehat{\alpha }}_{t|t}\left(\widehat{\theta }\right)-{\widehat{\alpha }}_{t|t}\left(\theta \right)\right){\left({\widehat{\alpha }}_{t|t}\left(\widehat{\theta }\right)-{\widehat{\alpha }}_{t|t}\left(\theta \right)\right)}^{\prime }\right]\mathrm{.}(3.2)$$

The first term, the filter uncertainty, is what is estimated by the naive $\text{MSE}-$ estimates $P_{t|t}$ delivered by the Kalman filter. Estimation of the second term, the parameter uncertainty, requires some additional effort. The literature on MSE estimation proposes two main approaches: asymptotic approximation and bootstrapping. Bootstrapping can be performed in a parametric or non-parametric way. A few remarks have to be mentioned about these methods in the context of STS models and of the DLFS model specifically.

For the parametric bootstrap, the state disturbances, say, ${\eta }_t,$ are drawn from their estimated joint conditional multivariate normal density ${\eta }_t\stackrel{\text{iid}}{\mathrm{~}}\text{MN}\left(0,\widehat{\Omega }\right),\widehat{\Omega }$ being evaluated at the hyperparameter estimate $\widehat{\theta },$ and are used in the Kalman filter state recursions to generate the state variables. Non-parametric bootstrap, in turn, has an advantage of not depending on any particular assumption about this joint distribution. Unlike in the parametric case where state disturbances are drawn from their estimated distribution, in the non-parametric case, standardised innovations are resampled with replacement from the standardized innovations that are based on the original hyperparameter estimates. The resampled standardized innovations are further used to generate bootstrap series $\left\{Y_1^b,\dots ,Y_T^b\right\}$ by running the so-called innovation form of the Kalman filter, see Harvey (1989) or Bollineni-Balabay et al. (2016b) for details. In the DLFS model, the first 13 time points of standardised innovations are not subject to resampling, as they constitute the so-called diffuse sample (this is the time needed to construct a proper distribution for the non-stationary state variables; see Koopman (1997) for initialisation of non-stationary state variables).

If an STS model contains non-stationary components, as is the case with the DLFS model, the generated series are likely to diverge from the original dataset they have been bootstrapped from, i.e., from $\left\{Y_1,\dots ,Y_T\right\}.$ Therefore, a special procedure is required for bootstrap samples to be brought in accordance with the pattern of the given dataset. This can be done with the help of the simulation smoother algorithm developed by Durbin and Koopman (2002). Technical details for implementation can be found in Koopman et al. (2008), Chapter 8.4.2. The survey errors, generated as described in either parametric or non-parametric unconditional state recursion, do not need any adjustments as they constitute (autocorrelated) noise.

The following sections contain a brief presentation of the asymptotic approach, as well as of the recent Rodriguez and Ruiz (2012) bootstrap approaches (hereafter referred to as the Rodriguez and Ruiz (RR) bootstrap) and of Pfeffermann and Tiller (2005) (hereafter the Pfeffermann and Tiller (PT) bootstrap) bootstrap approaches.

3.1 Rodriguez and Ruiz bootstrapping approach

Rodriguez and Ruiz (2012) developed their bootstrap method for MSE estimation conditional on the data, which means that bootstrap hyperparameters are further applied to the original data series for obtaining bootstrap estimates of the state variables. Bootstrapping can be done both parametrically and non-parametrically, following the steps below:

1. Estimate the model and obtain the hyperparameter estimates $\widehat{\theta }.$
2. Generate a bootstrap sample $\left\{Y_1^b,\dots ,Y_T^b\right\}$ using $\widehat{\theta },$ either parametrically or non-parametrically, as described in the introduction to this section. If the model is non-stationary, the bootstrap sample has to be corrected with the help of the simulation smoother.
3. The bootstrap dataset $\left\{Y_1^b,\dots ,Y_T^b\right\}$ is used to obtain both the survey error autocorrelation parameter estimates ${\widehat{\rho }}^b$ and bootstrap ML estimates ${\widehat{\theta }}_{\sigma }^b.$ Thereafter, the Kalman filter is launched using the original series $\left\{Y_1,\dots ,Y_T\right\}$ and the newly-estimated ${\widehat{\theta }}^b,$ which produces ${\widehat{\alpha }}_{t|t}\left({\widehat{\theta }}^b\right)$ and $P_{t|t}\left({\widehat{\theta }}^b\right).$
4. Steps 2-3 are repeated $B$ times. Then, the MSE are estimated in the following way:

$${\widehat{MSE}}_{t|t}^{\text{RR}}=\frac{1}{\text{B}}{\displaystyle \sum _{b=1}^B{P}_{t|t}}\left({\widehat{\theta }}^b\right)+\frac{1}{\text{B}}{\displaystyle \sum _{b=1}^B\left[{\widehat{\alpha }}_{t|t}\left({\widehat{\theta }}^b\right)-{\overline{\alpha }}_{t|t}\right]{\left[{\widehat{\alpha }}_{t|t}\left({\widehat{\theta }}^b\right)-{\overline{\alpha }}_{t|t}\right]}^{\prime },}(3.3)$$

• where ${\overline{\alpha }}_{t|t}=\frac{1}{\text{B}}{{\displaystyle {\sum }_{b=1}^B\widehat{\alpha }}}_{t|t}\left({\widehat{\theta }}^b\right).$

Equation (3.3) is applied both for the parametric and non-parametric bootstrap $\text{MSE}-$ estimators (denoted hereafter as ${\text{MSE}}^{\text{RR1}}$ and ${\text{MSE}}^{\text{RR2}},$ respectively).

3.2 Pfeffermann and Tiller bootstrapping approach

The bootstrap developed by Pfeffermann and Tiller (2005) is an unconditional bootstrap. This implies that bootstrap state variables are derived from the bootstrap dataset $\left\{Y_1^b,\dots ,Y_T^b\right\},$ i.e., not from the original data $\left\{Y_1,\dots ,Y_T\right\}$ as in Rodriguez and Ruiz (2012). Pfeffermann and Tiller (2005) prove that they approximate the true MSE up to the order of $O\left(1/T^2\right)$ (Pfeffermann and Tiller 2005, Appendix C):

$${\widehat{MSE}}_{t|t}^{\text{PT}}=2P_{t|t}\left(\widehat{\theta }\right)-\frac{1}{\text{B}}{\displaystyle \sum _{b=1}^B{P}_{t|t}\left({\widehat{\theta }}^b\right)}+\frac{1}{\text{B}}{\displaystyle \sum _{b=1}^B\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)\right]{\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)\right]}^{\prime }}.(3.4)$$

Equation (3.4) is applied both for the parametric and non-parametric bootstrap $\text{MSE}-$ estimators (denoted further as ${\text{MSE}}^{\text{PT1}}$ and ${\text{MSE}}^{\text{PT2}},$ respectively). $\text{MSE}-$ calculation in (3.4) requires two Kalman filter runs for every bootstrap series. In the first run, ${\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)$ is estimated from the bootstrap data set $\left\{Y_1^b,\dots ,Y_T^b\right\}$ and the bootstrap parameters ${\widehat{\theta }}^b.$ In this run, $P_{t|t}\left({\widehat{\theta }}^b\right)$ can also be obtained based on ${\widehat{\theta }}^b,$ since matrix $P_{t|t}$ does not depend on the data. The second Kalman filter run is needed to produce the state estimates ${\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)$ based on $\left\{Y_1^b,\dots ,Y_T^b\right\}$ and $\widehat{\theta }-$ estimates that were obtained from the original dataset. The bootstrap procedure is summarized below:

1. Estimate the model using the original dataset and obtain the hyperparameter vector estimates $\widehat{\theta }.$ Apart from that, save the “naive” MSE estimates $P_{t|t}\left(\widehat{\theta }\right)$ for future use in (3.4).
2. Use the parametric or non-parametric method to generate a bootstrap sample $\left\{Y_1^b,\dots ,Y_T^b\right\}.$ Apply the simulation smoother correction to it if the model is non-stationary.
3. Estimate bootstrap hyperparameter estimates ${\widehat{\theta }}^b$ from the newly generated bootstrap dataset. Run the Kalman filter once to get ${\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)$ and $P_{t|t}\left({\widehat{\theta }}^b\right),$ and another time to obtain ${\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right),$ as described under (3.4).
4. Repeat steps 2-3 $B$ times. Then, estimate the MSE using (3.4).

Pfeffermann and Tiller (2005) note that, in the case of the parametric bootstrap, the second Kalman filter run can be avoided because the true state vector is generated (and thus known) for every bootstrap series. Thus, the state estimates ${\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)$ in (3.4) can be replaced by the true vector ${\alpha }_t^b$ to obtain the following MSE estimator:

$${\widehat{MSE}}_{t|t}^{\text{PT}1}=P_{t|t}\left(\widehat{\theta }\right)-\frac{1}{\text{B}}{\displaystyle \sum _{b=1}^B{P}_{t|t}\left({\widehat{\theta }}^b\right)+\frac{1}{\text{B}}{\displaystyle \sum _{b=1}^B\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\alpha }_t^b\right]{\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\alpha }_t^b\right]}^{\prime }.}}(3.5)$$

There is only one $P_{t|t}\left(\widehat{\theta }\right)$ in the right-hand side of (3.5). This is due to the fact that the new term ${\text{E}}_B\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\alpha }_t^b\right]{\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\alpha }_t^b\right]}^{\prime },$ corresponding to the last term on the right-hand side of (3.5), can itself be decomposed, in the same fashion as in (3.2), into the measure of parameter uncertainty ${\text{E}}_B\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)\right]{\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)\right]}^{\prime }$ and the filter uncertainty term $P_{t|t}^b\left(\widehat{\theta }\right)=\text{E}\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]{\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]}^{\prime }\text{},\widehat{\theta }$ being the true parameter vector the bootstrap state variables ${\alpha }_t^b$ are generated with. However, the bootstrap average term $\frac{1}{\text{B}}{\displaystyle {\sum }_{b=1}^B\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]{\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]}^{\prime }}$ replacing $P_{t|t}\left(\widehat{\theta }\right)$ may need much more bootstrap iterations to converge. Further, this simplified method may result in an additional bias if the normality assumption about the model error terms is violated. Then, the decomposition of the term ${\text{E}}_B\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\alpha }_t^b\right]{\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\alpha }_t^b\right]}^{\prime }$ according to (3.2) will also contain a non-zero cross-term: $\text{E}\left\{\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)\right]\right\}.$ In this application, the non-zero cross-term bootstrap averages have turned out to be negligible, but the bootstrap average $\frac{1}{\text{B}}{\displaystyle {\sum }_{b=1}^B\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]{\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]}^{\prime }}$ exhibited large departures (in both directions) from the term it was meant to replace. This may be explained by the fact that the true Kalman filter MSE in (3.1) can be obtained from simulated series if the distribution of the state-vector is sufficiently dispersed. When bootstrapping non-stationary models, however, the bootstrap series are forced to follow the pattern of the underlying original series, as it has been mentioned in the description of the simulation smoother algorithm. Therefore, the term $\frac{1}{\text{B}}{\displaystyle {\sum }_{b=1}^B\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]{\left[{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)-{\alpha }_t^b\right]}^{\prime }}$ that replaces $P_{t|t}\left(\widehat{\theta }\right)$ in (3.5) may not be sufficiently close to it. For this reason, both parametric (denoted as PT1) and non-parametric (PT2) bootstraps in this application rely on the estimator in (3.4).

A few words have to be said about the role of the simulation smoother of Durbin and Koopman (2002), mentioned at the end of the introduction to this section. We suggest that it should be used at the bootstrap series generation step. Without it, the bootstrap hyperparameter distribution obtained from uncorrected series for a non-stationary model could be very different from what it should be for a particular realisation of the data at hand. At least in the case of the DLFS, omitting the simulation smoother step resulted in bootstrap hyperparameter distributions having a much wider range than the range of distributions obtained with the simulation smoother. Moreover, such bootstrap hyperparameter distributions obtained from uncorrected series in the DLFS are centred around values that are much larger than the hyperparameter values the series have been generated with. This results in an excessively large bootstrap average $\frac{1}{\text{B}}{\displaystyle {\sum }_{b=1}^B{P}_{t|t}\left({\widehat{\theta }}^b\right)}$ (relatively to $P_{t|t}(\widehat{\theta })$ ) and, subsequently, in $\text{MSE}-$ estimates that are even lower than the naive ones. The term $\frac{1}{\text{B}}{\displaystyle {\sum }_{b=1}^B\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)\right]{\left[{\widehat{\alpha }}_{t|t}^b\left({\widehat{\theta }}^b\right)-{\widehat{\alpha }}_{t|t}^b\left(\widehat{\theta }\right)\right]}^{\prime }}$ also becomes very unstable over time and excessively large compared to when the simulation smoother is used, but that does not compensate for the negative bias obtained from (3.4) without the simulation smoother.

3.3 Asymptotic approximation

An asymptotic approximation (AA) to the true MSE in equation (3.2) was developed by Hamilton (1986) and can be expressed as an expectation over the hyperparameter joint asymptotic distribution $\pi \left(\widehat{\theta }|Y\right),$ conditional on the given dataset $Y\equiv \left\{Y_1,\dots ,Y_T\right\}.$ In the present application, the part of the hyperparameter vector estimated by the $\text{ML}-$ method, $\left({\widehat{\theta }}_{\sigma }\right),$ depends on the estimated value of the autoregressive parameter $\widehat{\rho }.$ Therefore, the joint asymptotic distribution of the hyperparameter estimator has the following form: $\pi \left(\widehat{\theta }|Y\right)=\pi \left(\widehat{\rho }|Y\right)\pi \left({\widehat{\theta }}_{\sigma }|\widehat{\rho },Y\right).$ The MSE is approximated as follows:

$$MS{E}_{t|t}={\text{E}}_{\pi \left(\widehat{\theta }|Y\right)}\left[P_{t|t}\left(\widehat{\theta },Y\right)\right]+{\text{E}}_{\pi \left(\widehat{\theta }|Y\right)}\left[\left({\widehat{\alpha }}_{t|t}\left(\widehat{\theta },Y\right)-{\widehat{\alpha }}_{t|t}\left(Y\right)\right){\left({\widehat{\alpha }}_{t|t}\left(\widehat{\theta },Y\right)-{\widehat{\alpha }}_{t|t}\left(Y\right)\right)}^{\prime }\right],(3.6)$$

where ${\text{E}}_{\pi \left(\widehat{\theta }|Y\right)}$ is an expectation taken over the hyperparameter estimator joint asymptotic distribution $\pi \left(\widehat{\theta }|Y\right),$ and ${\widehat{\alpha }}_{t|t}\left(Y\right)$ are the state vector estimates when the hyperparameters are not known (i.e., ${\text{E}}_{\pi \left(\widehat{\theta }|Y\right)}[{\widehat{\alpha }}_{t|t}(\widehat{\theta },Y)]).$

Distribution $N\left(\widehat{\rho }\mathrm{,}\text{Var}\left(\widehat{\rho }\right)\right)$ is chosen as $\widehat{\rho }’\text{s}$ asymptotic distribution $\pi \left(\widehat{\rho }|Y\right),$ from which random $\widehat{\rho }-$ realisations are drawn. Generally, the sampling distribution of the correlation coefficient has a complex form, but it may be well approximated by a normal distribution, which was the case in this application (the normal distribution fitted both the simulated and the bootstrap distribution of $\widehat{\rho }$ very well). From equation (3) in Bartlett (1946), and using the fact that the autoregressive coefficient in an AR(1) process is equal to the correlation for lag 1, the variance estimator of $\widehat{\rho }$ becomes: $\text{Var}\left(\widehat{\rho }\right)\approx \left(1-{\widehat{\rho }}^2\right)/T.$ In the case of the DLFS, where $\widehat{\rho }\mathrm{<mo>=</mo>}\text{0}\text{.208},$ this means that $\widehat{\text{Var}}\left(\widehat{\rho }\right)\approx 0.96\left(1/T\right).$ Taking into account the fact that $\widehat{\rho }’\text{s}$ standard error is used for making draws from the asymptotic distribution, and that the square root is a concave function, the sample standard deviation would be an underestimate. Therefore, making $\widehat{\rho }-$ draws by means of $1/\sqrt{T}$ as the asymptotic distribution’s standard deviation would be a reasonable choice.

A sample of $B$ draws from the hyperparameter asymptotic distribution is obtained in the following way. After a value, say, ${\widehat{\rho }}^a,$ is drawn from $\pi \left(\widehat{\rho }|Y\right),$ the other hyperparameters are re-estimated from the original data to obtain ${\widehat{\theta }}_{\sigma }^{\text{ML}}|{\widehat{\rho }}^a,Y$ and the information matrix $\widehat{I}\left({\widehat{\theta }}_{\sigma }^{\text{ML}}|{\widehat{\rho }}^a,Y\right).$ Finally, a ${\widehat{\theta }}_{\sigma }^a\text{}-$ draw is made from distribution $\text{MN}\left({\widehat{\theta }}_{\sigma }^{\text{ML}},{\widehat{I}}^{-1}\left({\widehat{\theta }}_{\sigma }^{\text{ML}}|{\widehat{\rho }}^a,Y\right)\right).$ The Kalman filter is run again using ${\widehat{\rho }}^a-$ and ${\widehat{\theta }}_{\sigma }^a\text{}-$ realisations to obtain the state estimates ${\widehat{\alpha }}_{t|t}\left({\widehat{\theta }}^a,Y\right)$ and their MSEs ${\widehat{P}}_{t|t}\left({\widehat{\theta }}^a\right).$ The procedure is repeated until $B$ ${\widehat{\theta }}^a\text{}-$ draws are obtained, whereafter (3.6) is obtained by averaging the necessary quantities over $B$ iterations. If all the hyperparameters of the model are estimated within the $\text{ML}-$ procedure, $B$ draws can be made directly from $\text{MN}\left({\widehat{\theta }}^{\text{ML}},{\widehat{I}}^{-1}\left({\widehat{\theta }}^{\text{ML}}\right)\right).$

The first term in (3.6) can be approximated by the average value of the Kalman filter variance $P_{t|t}$ across $B$ realizations of the hyperparameter vector, and the second term by the variance of the state vector estimates across the same $B$ realisations. An asymptotic approximation for the MSE could therefore be obtained in the following way:

$${\widehat{MSE}}_{t|t}^{\text{AA}}=\frac{1}{\text{B}}{\displaystyle \sum _{a=1}^B{P}_{t|t}}\left({\widehat{\theta }}^a\right)+\frac{1}{\text{B}}{\displaystyle \sum _{a=1}^B\left[{\widehat{\alpha }}_{t|t}\left({\widehat{\theta }}^a,Y\right)-{\overline{\widehat{\alpha }}}_{t|t}\right]{\left[{\widehat{\alpha }}_{t|t}\left({\widehat{\theta }}^a,Y\right)-{\overline{\widehat{\alpha }}}_{t|t}\right]}^{\prime },}(3.7)$$

where ${\widehat{\theta }}^a$ is the $a^{\text{th}}$ draw from the $\pi \left(\widehat{\theta }|Y\right)$ asymptotic distribution. As Hamilton (1986) suggests, the sample average ${\overline{\widehat{\alpha }}}_{t|t}=\frac{1}{\text{B}}{\displaystyle {\sum }_{a=1}^B{\widehat{\alpha }}_{t|t}\left({\widehat{\theta }}^a,Y\right)}$ can replace ${\widehat{\alpha }}_{t|t}\left(Y\right)$ in (3.6). Further, he states, such a decomposition of the total uncertainty into the filter and parameter uncertainty resembles the well-known decomposition: $\text{var}\left(X\right)\mathrm{<mo>=</mo>}E\left[\text{var}\left(X|Y\right)\right]+\text{var}\left[E\left(X|Y\right)\right].$ Obviously, this $\text{MSE}-$ estimator is entirely based on the assumption of asymptotic normality of the hyperparameter vector estimator. Apart from that, this approach usually produces significant biases if the series is not of a sufficient length, in which case the assumed asymptotic normal distribution would fail to approximate the finite (usually skewed) distribution of maximum-likelihood estimates.

Another problem with the asymptotic approach can occur if some of the hyperparameters are estimated to be close to zero. This can happen to the initial model estimates or during the procedure itself, e.g., due to certain extreme $\widehat{\rho }-$ draws. In these cases, the asymptotic variance of such hyperparameters will be very large, which will inflate the $\text{MSE}-$ estimates of the signal and its unobserved components. It may as well lead to a failure in inverting the information matrix for the hyperparameter vector.

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