Small area estimation for unemployment using latent Markov models
Section 6. Final remarks

• Previous
• Next

In this paper we develop a new area level SAE method that uses a Latent Markov Model (LMM) as the linking model. In LMMs (Bartolucci et al., 2013), the characteristic of interest, and its evolution in time, is represented by a latent process that follows a Markov chain, usually of first order. Under the assumption of normality for the conditional distribution of the response variables given the latent variables, the model is estimated using an augmented data Gibbs sampler. The proposed model has been applied to quarterly data from the Italian LFS from 2004 to 2014. The model-based method has been found to be effective for developing LMAs level estimates of unemployment incidence and the reduction in the coefficient of variation compared to the direct estimator is quite evident. The proposed approach is also more accurate than the direct and the time-series model-based estimator proposed by You et al. (2003) in reproducing census data. An advantage of this methodology is that it also provides a clustering of the small areas in homogeneous groups.

LMMs can be seen as an extension of latent class models to longitudinal data. In this regard, our approach represents an extension of the latent class SAE model proposed by Fabrizi et al. (2016). Moreover, LMMs may be seen as an extension of Markov chain models to control for measurement errors and can easily handle multivariate data, providing a very flexible modeling framework. The approach could be extended using spatial correlation information, and it could consider different distributions for the manifest variables, such as Poisson, Binomial, and Multinomial responses. In this scenario, we could fit unmatched sampling and linking models and handle departures from the normality assumption, but a Gibbs sampler cannot be used any longer, and Metropolis-Hastings sampling is an option. The proposed univariate model can account for measurement errors, but the extension to multivariate framework could be also possible, taking into account the conditional independence assumption.

In this application we have not accounted explicitly for the serial correlation induced by the rotating panel design. A natural way to take the different features of this design into account, such as the rotating group bias and the autocorrelation of the survey errors, is to use state space-model specifications, as in Pfeffermann (1991), Pfeffermann and Rubin-Bleuer (1993) and, more recently, Van den Brakel and Krieg (2015) and Boonstra and Van den Brakel (2016). In this context, it would also be interesting to extend to SAE the LMM with serial correlation in the measurement model proposed by Bartolucci and Farcomeni (2009). State space-model specifications can also be a useful tool to capture and model the strong trend and seasonality of this type of data.

Acknowledgements

The work of Bertarelli, Ranalli, D’Aló and Solari has been developed under the support of the project PRIN-2012F42NS8 “Household wealth and youth unemployment: new survey methods to meet current challenges”. The Authors are grateful to the Associated Editor and two anonymous Referees who provided very useful comments on earlier versions of this paper.

Appendix A

Model estimation

In the following we first illustrate Bayesian estimation and model selection based on a MCMC algorithm which is implemented in a data augmentation framework (Tanner and Wong, 1987).

A.1  Data augmentation method

In order to estimate the small area parameters $\Theta ,$ the measurement parameters ${\varphi }_{\text{obs}}\text{},$ and the latent parameters ${\varphi }_{\text{lat}}\text{},$ we follow a data augmentation approach. We recall that the observed data consist of the direct estimates ${\widehat{\theta }}_{it},$ the corresponding smoothed ${\widehat{\text{MSE}}}_{it},$ and the covariate vectors $x_{it},$ with $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m$ and $t\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}T.$ Moreover, the data augmentation approach explicitly introduces the latent variables $U_{it}$ treated as missing data, the values of which are updated during the MCMC algorithm that is, therefore, based on a complete data likelihood. In this context, the use of conjugate priors to the complete data likelihood allows us to sample from the conditional posterior of the latent states in a straightforward way. Since the state space is finite, sampling the latent states conditionally given the model parameters is also simple.

To generate samples from the joint posterior distribution of the model parameters and latent states, the proposed MCMC algorithm proceeds as follows. Let $\widehat{\Theta }$ be the matrix of realizations of the available direct estimates that is defined as in (4.1), with each ${\theta }_{it}$ replaced by ${\widehat{\theta }}_{it},$ and let $U$ be the matrix of the latent variable $U_{it},$ with elements organized as in $\widehat{\Theta }.$ Then the posterior distribution of all model parameters and latent variables, given the observed data, has the following expression:

$$p\left(U\mathrm{,}{\varphi }_{\text{lat}}\mathrm{,}{\varphi }_{\text{obs}}\mathrm{,}\Theta |\widehat{\Theta }\right)\propto p\left(U|{\varphi }_{\text{lat}}\right)\pi \left({\varphi }_{\text{lat}}\right)\pi \left({\varphi }_{\text{obs}}\right)p\left(\Theta \mathrm{<mo>|</mo>}U\mathrm{,}{\varphi }_{\text{obs}}\right)p\left(\widehat{\Theta }|\Theta \right)\mathrm{.}$$

The MCMC algorithm alternates between sampling the latent variables and the parameters from the corresponding full conditional distribution. This scheme is repeated for $R$ iterations. At the end of each iteration $r,$ $r\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}R,$ the sampled model parameters and latent variables are obtained and are denoted by $U^{\left(r\right)}\text{},$ ${\varphi }_{\text{lat}}^{\left(r\right)}\text{},$ ${\varphi }_{\text{obs}}^{\left(r\right)}\text{},$ and ${\Theta }^{\left(r\right)}\text{}.$ More precisely, each iteration consists in:

1. drawing $U^{\left(r\right)}$ from $p\left(U|{\varphi }_{\text{lat}}^{\left(r-1\right)}\text{}\mathrm{,}{\varphi }_{\text{obs}}^{\left(r-1\right)}\text{}\mathrm{,}{\Theta }^{\left(r-1\right)}\right);$
2. drawing ${\varphi }_{\text{lat}}^{\left(r\right)}$ from $p\left({\varphi }_{\text{lat}}|U^{\left(r\right)}\right);$
3. drawing ${\varphi }_{\text{obs}}^{\left(r\right)}$ from $p\left({\varphi }_{\text{obs}}|U^{\left(r\right)}\text{},{\Theta }^{\left(r-1\right)}\right);$
4. drawing ${\Theta }^{\left(r\right)}$ from $p\left(\Theta |U^{\left(r\right)}\text{},{\varphi }_{\text{obs}}^{\left(r\right)}\mathrm{,}\widehat{\Theta }\right).$

In the following we illustrate in details each of the above steps. In this regard, note that our illustration is referred to the case where all elements of $\widehat{\Theta }$ are available. However, in our application, some elements of this matrix are missing. This requires minor adjustments to the MCMC algorithm, consisting in imputing the missing values by a Gibbs sampler and sampling directly from its full conditional distribution.

A.1.1  Simulation of $U^{\left(r\right)}$

Each latent variable $U_{it}$ is drawn separately from the corresponding full conditional distribution, which is of multinomial type with specific parameters. In particular, we have that

$$U_{it}|U_{i\text{}\mathrm{,}t-1}^{\left(r\right)}\text{},U_{i\text{}\mathrm{,}t+1}^{\left(r-1\right)}\text{}\mathrm{,}{\varphi }_{\text{lat}}^{\left(r-1\right)}\text{}\mathrm{,}{\varphi }_{\text{obs}}^{\left(r-1\right)}\text{}\mathrm{,}{\Theta }^{\left(r-1\right)}\sim {\text{Multi}}_k\left(q_{it}\right)\mathrm{,}t\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}T\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m\mathrm{,}(\text{A}\text{.1})$$

where $U_{i\text{}\mathrm{,}t-1}^{\left(r\right)}$ disappears for $t\mathrm{<mo>=</mo>\; 1}$ and $U_{i\text{}\mathrm{,}t+1}^{\left(r\right)}$ disappears for $t\mathrm{<mo>=</mo>}T.$ Moreover, the probability vector $q_{it}$ is defined as follows:

• for $t\mathrm{<mo>=</mo>\; 1},$ $q_{it}$ has elements proportional to

$${\pi }_u^{\left(r-1\right)}{\pi }_{u|U_{i2}^{\left(r-1\right)}}^{\left(r-1\right)}\text{}\mathrm{,}u\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}k\mathrm{<mo>;</mo>}$$

• for $t\mathrm{<mo>=</mo>\; 2,}\dots \mathrm{,}T-1,$ $q_{it}$ has elements proportional to

$${\pi }_{u|U_{i\text{}\mathrm{,}t-1}^{\left(r\right)}}^{\left(r-1\right)}{\pi }_{U_{i\text{}\mathrm{,}t+1}^{\left(r-1\right)}|u}^{\left(r-1\right)}\text{}\mathrm{,}u\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}k\mathrm{<mo>;</mo>}$$

• for $t\mathrm{<mo>=</mo>}T,$ $q_{it}$ has elements proportional to

$${\pi }_{u|U_{i\text{}\mathrm{,}T-1}^{\left(r\right)}}^{\left(r-1\right)}\text{}\mathrm{,}u\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}k\mathrm{.}$$

A.1.2  Simulation of ${\varphi }_{\text{lat}}^{\left(r\right)}$

Recalling that ${\varphi }_{\text{lat}}\mathrm{<mo>=</mo>}\left\{\pi \mathrm{,}\Pi \right\},$ we first draw ${\pi }^{\left(r\right)}$ from the full conditional distribution:

$$\pi |U^{\left(r\right)}\sim \text{Dirichlet}\left(1_k+n_1\right)\mathrm{,}$$

where $n_1\mathrm{<mo>=</mo>}{\left(n_{11}\mathrm{,}\dots \mathrm{,}{n}_{1k}\right)}^{\prime }$ and $n_{1u}$ is the number of areas in state $u$ at time 1, with $u\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}k.$ Moreover, we draw each row of matrix $\Pi$ from the distribution

$${\pi }_{\overline{u}}|U^{\left(r\right)}\sim \text{Dirichlet}\left(1_k+n_{\overline{u}\text{}\mathrm{,}t}\right)\mathrm{,}t\mathrm{<mo>=</mo>\; 2,}\dots \mathrm{,}T\text{}\mathrm{,}$$

where $n_{\overline{u}\text{}\mathrm{,}t}\mathrm{<mo>=</mo>}{\left(n_{\overline{u}\text{}\mathrm{,}{t}_1}\text{}\mathrm{,}\dots \mathrm{,}{n}_{\overline{u}\text{}\mathrm{,}{t}_k}\right)}^{\prime }$ and $n_{\overline{u}\text{}\mathrm{,}{t}_u}$ is the number of areas moving from state $\overline{u}$ to state $u$ at time $t,$ with $t\mathrm{<mo>=</mo>\; 2,}\dots \mathrm{,}T$ and $u\mathrm{,}\overline{u}\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}k.$

A.1.3  Simulation of ${\varphi }_{\text{obs}}^{\left(r\right)}$

Considering that ${\varphi }_{\text{obs}}\mathrm{<mo>=</mo>}\left\{{\beta }_1\mathrm{,}\dots \mathrm{,}{\beta }_k\text{}\mathrm{,}{\sigma }_1^2\text{}\mathrm{,}\dots \mathrm{,}{\sigma }_k^2\right\}$ , we first draw each ${\beta }_u,$ $u\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}k,$ from the full conditional distribution:

$${\beta }_u|U^{\left(r\right)}\text{},{\Theta }^{\left(r\right)}\sim N_p\left({\eta }_{\mathrm{1,}u}\mathrm{,}{\Sigma }_{\mathrm{1,}u}\right)\mathrm{,}$$

where

$$\begin{array}{ll}{\eta }_{\mathrm{1,}u}\hfill & \mathrm{<mo>=</mo>}{\Lambda }_{\mathrm{1,}u}^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle \sum _{t\mathrm{<mo>=</mo>\; 1}}^T{\theta }_{it}I\left(U_{it}\mathrm{<mo>=</mo>}u\right)x_{it}\mathrm{,}}\hfill \\ {\Sigma }_{\mathrm{1,}u}\hfill & \mathrm{<mo>=</mo>}{\sigma }_u^2{\Lambda }_{\mathrm{1,}u}^{-1}\mathrm{,}\hfill \\ {\Lambda }_{\mathrm{1,}u}\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle \sum _{t\mathrm{<mo>=</mo>\; 1}}^T{x}_{it}{x}_{it}^{\prime }I\left(U_{it}\mathrm{<mo>=</mo>}u\right)}+{\Lambda }_0\mathrm{,}\hfill \end{array}$$

with $I(\cdot )$ denoting the indicator function equal to 1 if its argument is true and to 0 otherwise. Then, we draw each ${\sigma }_u^2$ from

$${\sigma }_u^2|U^{\left(r\right)}\text{},{\Theta }^{\left(r\right)}\sim \text{IG}\left(a_{\mathrm{1,}u}\text{}\mathrm{,}{b}_{\mathrm{1,}u}\right)\mathrm{,}$$

with

$$\begin{array}{ll}{a}_{\mathrm{1,}u}\hfill & \mathrm{<mo>=</mo>}{a}_0+\frac{n_{\mathrm{.}u}}{2}\mathrm{,}\hfill \\ b_{\mathrm{1,}u}\hfill & \mathrm{<mo>=</mo>}{b}_0+\frac{1}{2}\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle \sum _{t\mathrm{<mo>=</mo>\; 1}}^T{\theta }_{it}^{2}I\left(U_{it}\mathrm{<mo>=</mo>}u\right)}+{\eta }_0^{\prime }{\Lambda }_0{\eta }_0-{\eta }_{\mathrm{1,}u}^{\prime }{\Lambda }_{\mathrm{1,}u}{\eta }_{\mathrm{1,}u}\right)\mathrm{,}\hfill \end{array}$$

where $n_{\mathrm{.}u}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{t\mathrm{<mo>=</mo>\; 1}}^T}{n}_{tu}$ is the number of areas in state $u$ regardless of the specific time occasion.

A.1.4  Simulation of ${\Theta }^{\left(r\right)}$

The goal of SAE is to predict each ${\theta }_{it},$ $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m,$ $t\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}T,$ based on the model and the observed data. This amounts to draw these elements from

$${\theta }_{it}|U^{\left(r\right)}\text{},{\varphi }_{\text{obs}}^{\left(r\right)}\text{}\mathrm{,}{\widehat{\theta }}_{it}\sim N\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\theta }}_{it}^{\left(r\right)}\text{}\mathrm{,}{\gamma }_{it}^{\left(r\right)}{\psi }_{it}\mathrm{<mo\; stretchy="false">)</mo>\; ,}$$

where

$${\widehat{\theta }}_{it}^{\left(r\right)}\mathrm{<mo>=</mo>}{\gamma }_{it}^{\left(r\right)}{\widehat{\theta }}_{it}+\left(1-{\gamma }_{it}^{\left(r\right)}\right)x_{it}^{\prime }{\beta }_u^{\left(r\right)}\text{},(\text{A}\text{.2})$$

with ${\gamma }_{it}^{\left(r\right)}\mathrm{<mo>=</mo>}{\sigma }_u^{2\left(r\right)}/\left({\sigma }_u^{2\left(r\right)}+{\psi }_{it}\right).$

A.2  Model selection: The Chib estimator

The method proposed in Chib (1995) can be applied to perform model selection starting from the Gibbs sampler output. It is known that the posterior density can be written as the ratio of the product of the likelihood function and the priors divided by the marginal likelihood:

$$p\left(U\mathrm{,}{\varphi }_{\text{lat}}\text{}\mathrm{,}{\varphi }_{\text{obs}}\text{}\mathrm{,}\Theta |\widehat{\Theta }\right)\mathrm{<mo>=</mo>}\frac{p\left(U|{\varphi }_{\text{lat}}\right)\pi \left({\varphi }_{\text{lat}}\right)\pi \left({\varphi }_{\text{obs}}\right)p\left(\Theta |U\text{},{\varphi }_{\text{obs}}\right)p\left(\widehat{\Theta }|\Theta \right)}{p\left(\widehat{\Theta }\right)}\mathrm{.}(\text{A}\text{.3})$$

Therefore, it is possible to write the marginal likelihood of the data $\widehat{\Theta }$ as

$$p\left(\widehat{\Theta }\right)\mathrm{<mo>=</mo>}\frac{p\left(\widehat{\Theta }|\Theta \right)p\left(U|{\varphi }_{\text{lat}}\right)\pi \left({\varphi }_{\text{lat}}\right)\pi \left({\varphi }_{\text{obs}}\right)p\left(\Theta |U\mathrm{,}{\varphi }_{\text{obs}}\right)}{p\left(U\mathrm{,}{\varphi }_{\text{lat}}\text{}\mathrm{,}{\varphi }_{\text{obs}}\text{}\mathrm{,}\Theta |\widehat{\Theta }\right)}\mathrm{,}(\text{A}\text{.4})$$

for any $U,$ ${\varphi }_{\text{lat}}\text{},$ ${\varphi }_{\text{obs}}\text{},$ $\Theta$ and $\widehat{\Theta }.$ We drop the dependence on $k$ for ease of notation. This is the model selection criterion used in Section 5. Then, choosing specific values of the latent variables and model parameters, denoted by $\overline{U},$ ${\overline{\varphi }}_{\text{lat}}\text{},$ ${\overline{\varphi }}_{\text{obs}}\text{},$ and $\overline{\Theta },$ we can estimate $\log p\left(\widehat{\Theta }\right)$ through the following decomposition:

$$\begin{array}{ll}\log p\left(\widehat{\Theta }\right)=\hfill & \log p\left(\widehat{\Theta }|\overline{\Theta }\right)+\log p\left(\overline{U}|{\overline{\varphi }}_{\text{lat}}\right)+\log \pi \left({\overline{\varphi }}_{\text{lat}}\right)+\log \pi \left({\overline{\varphi }}_{\text{obs}}\right)\hfill \\ \hfill & +\log p\left(\overline{\Theta }|\overline{U},{\overline{\varphi }}_{\text{obs}}\right)-\log p\left(\overline{U}\mathrm{,}{\overline{\varphi }}_{\text{lat}}\text{}\mathrm{,}{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\overline{\Theta }|\widehat{\Theta }\right)\mathrm{.}(\text{A}\text{.5})\hfill \end{array}$$

The use of the log transformation is motivated by numerical stability (Chib, 1995).

The first five terms at the right hand side of (A.5) can be computed directly from the assumed distributions of the parameters and the data. On the other hand obtaining the last component is more challanging. By the law of total probability, $p\left(\overline{U}\mathrm{,}{\overline{\varphi }}_{\text{lat}}\text{}\mathrm{,}{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\overline{\Theta }|\widehat{\Theta }\right)$ may be decomposed as

$$p\left(\overline{U}\mathrm{,}{\overline{\varphi }}_{\text{lat}}\text{}\mathrm{,}{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\overline{\Theta }|\widehat{\Theta }\right)\mathrm{<mo>=</mo>}p\left(\overline{U}|{\overline{\varphi }}_{\text{lat}}\text{},{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)p\left({\overline{\varphi }}_{\text{lat}}|{\overline{\varphi }}_{\text{obs}}\text{},\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)p\left({\overline{\varphi }}_{\text{obs}}|\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)p\left(\overline{\Theta }|\widehat{\Theta }\right)\mathrm{.}(\text{A}\text{.6})$$

Following Chib (1995), we compute the first term of (A.6) following the Gibbs scheme outlined in Section A.1, whereas, the other three terms are estimated from the Gibbs output. In particular, we estimate

$$p\left({\overline{\varphi }}_{\text{lat}}|{\overline{\varphi }}_{\text{obs}}\text{},\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)\mathrm{<mo>=</mo>}{\displaystyle \int }p\left({\overline{\varphi }}_{\text{lat}}|\overline{U}\mathrm{,}{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)p\left(\overline{U}|{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)d\overline{U}$$

as $R^{-1}{\displaystyle {\sum }_{r\mathrm{<mo>=</mo>\; 1}}^R}p\left({\overline{\varphi }}_{\text{lat}}|U^{\left(r\right)}\text{}\mathrm{,}{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\overline{\Theta }\right)\mathrm{,}$ based on $R$ draws from a reduced Gibbs sampling where $U$ is not updated. In order to estimate

$$p\left({\overline{\varphi }}_{\text{obs}}|\overline{\Theta }\text{},\widehat{\Theta }\right)\mathrm{<mo>=</mo>}{\displaystyle \int }p\left({\overline{\varphi }}_{\text{obs}}|\overline{U},{\overline{\varphi }}_{\text{lat}}\text{},\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)p\left(\overline{U}\text{}\mathrm{,}{\overline{\varphi }}_{\text{lat}}|\overline{\Theta }\mathrm{,}\widehat{\Theta }\right)d\overline{U}d{\overline{\varphi }}_{\text{lat}}\text{}\mathrm{,}$$

we use $R^{-1}{\displaystyle {\sum }_{r\mathrm{<mo>=</mo>\; 1}}^R}p\left({\overline{\varphi }}_{\text{obs}}|U^{\left(r\text{},1\right)}\text{},{\overline{\varphi }}_{\text{lat}}^{\left(r\text{},1\right)}\text{}\mathrm{,}\overline{\Theta }\right).$ Finally, to estimate

$$p\left(\overline{\Theta }|\widehat{\Theta }\right)\mathrm{<mo>=</mo>}{\displaystyle \int }p\left(\overline{\Theta }|\overline{U}\text{},{\overline{\varphi }}_{\text{lat}}\text{}\mathrm{,}{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}\widehat{\Theta }\right)p\left(\overline{U}\mathrm{,}{\overline{\varphi }}_{\text{lat}}\text{}\mathrm{,}{\overline{\varphi }}_{\text{obs}}|\widehat{\Theta }\right)d\overline{U}d{\overline{\varphi }}_{\text{lat}}d{\overline{\varphi }}_{\text{obs}}\text{}\mathrm{,}$$

we use $R^{-1}{\displaystyle {\sum }_{r\mathrm{<mo>=</mo>\; 1}}^R}p\left(\overline{\Theta }|U^{\left(r\text{},2\right)}\text{},{\varphi }_{\text{lat}}^{\left(r\text{},2\right)}\text{}\mathrm{,}{\varphi }_{\text{obs}}^{\left(r\text{},2\right)}\right)\mathrm{,}$ with $R$ draws from a third reduced Gibbs sampling.

References

Bartolucci, F., and Farcomeni, A. (2009). A multivariate extension of the dynamic logit model for longitudinal data based on a latent Markov heterogeneity structure. Journal of the American Statistical Association, 104, 816-831.

Bartolucci, F., Farcomeni, A. and Pennoni, F. (2013). Latent Markov Models for Longitudinal Data. Boca Roton, FL: CRC Press.

Bartolucci, F., Lupparelli, M. and Montanari, G.E. (2009). Latent Markov model for longitudinal binary data: An application to the performance evaluation of nursing homes. The Annals of Applied Statistics, 3, 611-636.

Bartolucci, F., Pennoni, F. and Francis, B. (2007). A latent Markov model for detecting patterns of criminal activity. Journal of the Royal Statistical Society, Series A, 170, 115-132.

Boonstra, H.J. (2012). hbsae: Hierarchical Bayesian small area estimation. R Package Version 1.

Boonstra, H.J. (2014). Time-series small area estimation for unemployment based on a rotating panel survey. Technical report, CBS. Available at https://www.cbs.nl/nl-nl/achtergrond/2014/25/time-series-small-area-estimation-for-unemployment-based-on-a-rotating-panel-survey.

Boonstra, H.J., and van den Brakel, J.A. (2016). Estimation of Level and Change for Unemployment Using Multilevel and Structural Time Series Models. Discussion paper 2016-10. Statistics Netherlands, Heerlen.

Boys, R., and Henderson, D. (2003). Data augmentation and marginal updating schemes for inference in hidden Markov models. Technical report, Univ. Newcastle.

Carlin, B.P., and Chib, S. (1995). Bayesian model choice via Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B, 56, 473-484.

Chib, S. (1995). Marginal likelihood from the Gibbs output. Journal of the American Statistical Association, 90, 1313-1321.

D’Alò, M., Di Consiglio, L., Falorsi, S., Ranalli, M.G. and Solari, F. (2012). Use of spatial information in small area models for unemployment rate estimation at sub-provincial areas in Italy. Journal of the Indian Society of Agricultural Statistics, 66, 43-53.

Datta, G.S., Lahiri, P., Maiti, T. and Lu, K.L. (1999). Hierarchical Bayes estimation of unemployment rates for the states of the US. Journal of the American Statistical Association, 94, 1074-1082.

Fabrizi, E., Montanari, G.E. and Ranalli, M.G. (2016). A hierarchical latent class model for predicting disability small area counts from survey data. Journal of the Royal Statistical Society, Series A, 179, 103-132.

Fay, R.E., and Herriot, R.A. (1979). Estimates of income for small places: An application of James-Stein procedures to census data. Journal of the American Statistical Association, 74, 269-277.

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Analysis, 1(3), 515-534.

Gelman, A., Jakulin, A., Pittau, M.G. and Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 2, 1360-1383.

Germain, S.E. (2010). Bayesian Spatio-Temporal Modelling of Rainfall Through Non-Homogenous Hidden Markov Models. Ph.D. thesis, University of Newcastle Upon Tyne.

Ghosh, M., Nangia, N. and Kim, D.H. (1996). Estimation of median income of four-person families: A Bayesian time series approach. Journal of the American Statistical Association, 91, 1423-1431.

Harvey, A., and Chung, C.-H. (2000). Estimating the underlying change in unemployment in the UK. Journal of the Royal Statistical Society, Series A, 163, 303-309.

ISTAT (2014). I sistemi locali del lavoro 2011. Rapporto Annuale 2014.

Jasra, A., Holmes, C. and Stephens, D. (2005). Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Statistical Science, 20, 50-67.

Krieg, S., and van der Brakel, J.A. (2012). Estimation of the monthly unemployment rate for six domains through structural time series modelling with cointegrated trends. Computational Statistics & Data Analysis, 56, 2918-2933.

Lazarsfeld, P.F., Henry, N.W. and Anderson, T.W. (1968). Latent Structure Analysis. Houghton Mifflin Boston.

Liu, J.S., Wong, W.H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika, 81, 27-40.

MacDonald, I.L., and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-Valued Time Series. London: Series Chapman & Hall.

Marhuenda, Y., Molina, I. and Morales, D. (2013). Small area estimation with spatio-temporal Fay-Herriot models. Computational Statistics & Data Analysis, 58, 308-325.

Marin, J.-M., Mengersen, K. and Robert, C.P. (2005). Bayesian modelling and inference on mixtures of distributions. Handbook of Statistics, 25, 459-507.

Meng, X.-L. (1994). Posterior predictive p-values. The Annals of Statistics, 1142-1160.

Pfeffermann, D. (1991). Estimation and seasonal adjustment of population means using data from repeated surveys. Journal of Business & Economic Statistics, 9, 163-175.

Pfeffermann, D., and Burck, L. (1990). Robust small area estimation combining time series and cross-sectional data. Survey Methodology, 16, 2, 217-237. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/1990002/article/14534-eng.pdf.

Pfeffermann, D., and Rubin-Bleuer, S. (1993). Robust joint modelling of labour force series of small areas. Survey Methodology, 19, 2, 149-163. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/1993002/article/14458-eng.pdf.

Pfeffermann, D., and Tiller, R. (2006). Small-area estimation with state-space models subject to benchmark constraints. Journal of the American Statistical Association, 101, 1387-1397.

Polson, N.G., and Scott, J.G. (2012). On the half-cauchy prior for a global scale parameter. Bayesian Analysis, 7, 4, 887-902.

Rao, J.N.K. (2003). Small Area Estimation. Wiley Online Library.

Rao, J.N.K., and Yu, M. (1994). Small-area estimation by combining time-series and cross-sectional data. The Canadian Journal of Statistics, 22, 4, 511-528.

Spezia, L. (2010). Bayesian analysis of multivariate gaussian hidden Markov models with an unknown number of regimes. Journal of Time Series Analysis, 31, 1-11.

Statistics Canada (2016). Guide to the Labour Force Survey. Technical report, Statistics Canada, Catalogue 71-543-G, available at https://www150.statcan.gc.ca/n1/pub/71-543-g/71-543-g2016001-eng.pdf.

Tanner, M.A., and Wong, W.H. (1987). The calculation of posterior distributions by data augmentation. Journal of the American statistical Association, 82, 528-540.

Van den Brakel, J.A., and Krieg, S. (2015). Dealing with small sample sizes, rotation group bias and discontinuities in a rotating panel design. Survey Methodology, 41, 2, 267-296. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/2015002/article/14231-eng.pdf.

Van der Brakel, J.A., and Krieg, S. (2016). Small area estimation with state space common factor models for rotating panels. Journal of the Royal Statistical Society, Series A, 179, 763-791.

Van Dyk, D.A., and Meng, X.-L. (2001). The art of data augmentation. Journal of Computational and Graphical Statistics, 10(1), 1-50.

Vermunt, J.K., and Magidson, J. (2002). Latent class cluster analysis. Applied Latent Class Analysis, 11, 89-106.

Wiggins, L.M. (1973). Panel Analysis: Latent Probability Models for Attitude and Behavior Processes. Jossey-Bass.

Wolter, K. (2007). Introduction to Variance Estimation. New York: Springer Science & Business Media.

You, Y., Rao, J.N.K. and Gambino, J. (2003). Model-based unemployment rate estimation for the Canadian Labour Force Survey: A hierarchical Bayes approach. Survey Methodology, 29, 1, 25-32. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/2003001/article/6602-eng.pdf.

• Previous
• Article main page
• Next