Social media as a data source for official statistics; the Dutch Consumer Confidence Index
Section 3. Structural time series modelling of the CCI and the SMI

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In this section, univariate and bivariate structural time series models for the CCI and SMI are developed. With a structural time series model, a series is decomposed in a trend component, seasonal component, other cyclic components, regression component and an irregular component. For each component a stochastic model is assumed. This allows the trend, seasonal, and cyclic component but also the regression coefficients to be time dependent. If necessary autoregressive-moving-average (ARMA) components can be added to capture the autocorrelation in the series beyond these structural components. See Harvey (1989) or Durbin and Koopman (2012) for details about structural time series modelling.

The question addressed in this paper is to which extent the SMI follows a similar pattern as the CCI such that the SMI can be used in the estimation procedure of the CCI or, in the most extreme case, even can replace the CCI. This question is addressed by developing a bivariate structural time series model for the CCI and the SMI and modeling the correlation between the disturbance terms of the different components of the structural time series model for both series. The concept of cointegration is used to investigate to which extent the unobserved components of both series are driven by common factors. If e.g., the trends of both series are driven by one underlying common trend an argument can be made that the SMI represents similar evolution of sentiment feelings compared to the CCI. Alternatively, the SMI can be used as an auxiliary series in a model based estimation procedure for the CCI or in a nowcasting procedure to obtain more precise real time estimates.

3.1  Univariate model CCI series

As a first step, a univariate time series model for the CCI series is proposed. With the design-based approach described in Section 2.1, the sample information observed in each separate month is used to obtain an estimate for the CCI in that month. A drawback of this approach is that information observed in preceding periods is not used to obtain more accurate estimates for the CCI. In survey methodology, time series models are frequently applied to develop estimates for periodic surveys. Blight and Scott (1973) and Scott and Smith (1974) proposed to regard the unknown population parameters as a realization of a stochastic process that can be described with a time series model. This introduces relationships between the estimated population parameters at different time points in the case of non-overlapping as well as overlapping samples. The explicit modelling of this relationship between these survey estimates with a time series model can be used to combine sample information observed in the past to improve the precision of estimates obtained with periodic surveys. Some key references to authors that applied the time series approach to repeated survey data to improve the efficiency of survey estimates are Scott, Smith and Jones (1977), Tam (1987), Binder and Dick (1989, 1990), Bell and Hillmer (1990), Tiller (1992), Rao and Yu (1994), Pfeffermann and Burck (1990), Pfeffermann (1991), Pfeffermann and Rubin-Bleuer (1993), Pfeffermann, Feder and Signorelli (1998), Pfeffermann and Tiller (2006), Harvey and Chung (2000), Feder (2001), Lind (2005) and van den Brakel and Krieg (2009, 2015).

Developing a time series model for survey estimates observed with a periodic survey starts with a model, which states that the survey estimate can be decomposed in the value of the population variable and a sampling error:

$$I_t={\theta }_t+e_t,(3.1)$$

where ${\theta }_t$ denote the real CCI in month $t$ under a complete enumeration of the target population and $e_t$ the sampling error.

The CCI is observed at a monthly frequency. Therefore, as a first step, the series of the finite population parameter can be decomposed in a stochastic trend, seasonal component to model systematic deviations from the trend within a year, and a white noise component for the remaining unexplained variation. These considerations lead to the following model for the series of the finite population parameter:

$${\theta }_t=L_t+S_t+{\xi }_t,(3.2)$$

where $L_t$ denotes a stochastic trend, $S_t$ a stochastic seasonal component and ${\xi }_t$ the unexplained variation of the finite population parameter. Inserting (3.2) into measurement model (3.1) gives

$$I_t=L_t+S_t+{\xi }_t+e_t.(3.3)$$

In a cross-sectional survey it is difficult to separate the sampling error from the white noise of the population parameter. Therefore, both components are combined in one disturbance term

$${\upsilon }_t={\xi }_t+e_t.(3.4)$$

It is assumed that $E\left({\upsilon }_t\right)=0$ and $\text{Var}\left({\upsilon }_t\right)={\sigma }_{\upsilon }^2.$ To allow for nonhomogeneous variance in the sampling errors, Binder and Dick (1990) proposed a measurement error where the disturbance terms ${\upsilon }_t$ are proportional to the standard errors of $I_t,$ i.e.,

$${\upsilon }_t=\sqrt{\text{Var}\left(I_t\right)}{\tilde{\upsilon }}_t,(3.5)$$

with $E\left({\tilde{\upsilon }}_t\right)=0,$ $\text{Var}\left({\tilde{\upsilon }}_t\right)={\sigma }_{\upsilon }^2,$ and where $\text{Var}\left(I_t\right)$ is defined by (2.3) and is used as a priori information in the time series model. Such a model would be useful if the sampling error dominates the white noise in the population parameter. Initial analyses indicate that in this application the variance of the population white noise is substantial, invalidating (3.5) for this application. In addition, the variance of the sampling error in this application is constant over time. Therefore, it is decided to combine the sampling error with the population white noise and assume a constant variance over time. The question how to account for sampling variance is also an issue in seasonal adjustment variances (Pfeffermann and Sverchkov, 2014). Bell (2005) studied the contribution of the sampling variance in the variance of the estimation error of seasonally adjusted series and in the nonseasonal component. In the case of (rotating) panels, the sampling error can be separated from the population white noise. In cross-sectional repeated surveys, it is difficult to identify the separate components and therefore both terms are combined in one disturbance term that captures both the sampling variance and the unexplained variation of the population parameter.

An extensive model selection showed that a smooth trend model is the most appropriate model to capture the trend and the economic cycle in the CCI series. The smooth trend model is defined as (Durbin and Koopman, 2012):

$$L_t=L_{t-1}+R_t,$$

$$\begin{array}{l}{R}_t=R_{t-1}+{\eta }_t,\text{E}\left({\eta }_t\right)=0,(3.6)\hfill \\ \text{Cov}\left({\eta }_t,{\eta }_{t^{\prime }}\right)=\{\begin{array}{ll}{\sigma }_{\eta }^2\hfill & \text{if}t=t^{\prime }\hfill \\ 0\hfill & \text{if}t\ne t^{\prime }\hfill \end{array}.\hfill \end{array}$$

Adding a random component for the level in (3.6) improves the log-likelihood with five units but results in an overfit of the data in a sense that the smoothed signal almost exactly follows the observed series with a very small measurement error variance. A local level model (random level without a slope) improves the log-likelihood with three units but also intends to overfit the data.

The seasonal component is modelled with a trigonometric model, which is defined as (Durbin and Koopman, 2012):

$$S_t={\displaystyle \sum _{j=1}^6{\gamma }_{jt}},(3.7)$$

with

$$\begin{array}{ll}{\gamma }_{jt}\hfill & ={\gamma }_{jt-1}\cos \left({\lambda }_j\right)+{\tilde{\gamma }}_{jt-1}\sin \left({\lambda }_j\right)+{\omega }_{jt},\hfill \\ {\tilde{\gamma }}_{jt}\hfill & =-{\gamma }_{jt-1}\sin \left({\lambda }_j\right)+{\tilde{\gamma }}_{jt-1}\cos \left({\lambda }_j\right)+{\tilde{\omega }}_{jt}.\hfill \end{array}$$

Here ${\lambda }_j$ denotes the frequency of the different cycles in radians and is defined as

$${\lambda }_j=\frac{2\pi j}{12},\text{for}j=1,\dots ,6.$$

For the disturbance terms, it is assumed that

$$\begin{array}{c}E\left({\omega }_{jt}\right)=0,E\left({\tilde{\omega }}_{jt}\right)=0,\\ \text{Cov}\left({\omega }_t,{\omega }_{t^{\prime }}\right)=\{\begin{array}{cc}{\sigma }_{\omega }^2& \text{if}t=t^{\prime }\\ 0 & \text{if}t\ne t^{\prime }\end{array}.\end{array}$$

For reasons of parsimony, the same variance structure is assumed with the same hyperparameter for ${\tilde{\omega }}_{jt}.$ Furthermore, it is assumed that ${\omega }_t$ and ${\tilde{\omega }}_t$ are uncorrelated.

After including the stochastic trend component (3.6) and seasonal component (3.7), no additional cycle components are required. The model selection procedure indicated that two level interventions are needed to model sudden jumps in the series. The first one is due to the financial crisis in September 2008, and the second one is due to the economic downturn in September of 2011. Finally, an outlier is required for September 2007. Adding these three components increases the log-likelihood with 15 units. These considerations lead to the following model for the observed CCI series

$$I_t=L_t+S_t+{\beta }^{07}{\delta }_t^{07}+{\beta }^{08}{\delta }_t^{08}+{\beta }^{11}{\delta }_t^{11}+{\upsilon }_t,(3.8)$$

with

$${\delta }_t^{07}=\{\begin{array}{cc}1& \text{if}t=2007\left(9\right)\\ 0& \text{if}t\ne 2007\left(9\right)\end{array},{\delta }_t^{08}=\{\begin{array}{cc}1& \text{if}t\ge 2008\left(9\right)\\ 0& \text{if}t<2008\left(9\right)\end{array},{\delta }_t^{11}=\{\begin{array}{cc}1& \text{if}t\ge 2011\left(9\right)\\ 0& \text{if}t<2011\left(9\right)\end{array},$$

and ${\beta }^x$ the corresponding regression coefficients.

Finally, autoregressive (AR) and moving average (MA) components can be added to the structural time series model (3.8). In this application, there were no indications that such components are required, since there are no clear signs of remaining serial correlation in the standardized innovations. Adding an AR(1) or an MA(1) to (3.8) increases the log-likelihood with 5 and 4.5 units respectively. Adding second-order AR or MA models does not further improve the log-likelihood. Adding an ARMA(1,1) also does not further increases the log-likelihood. An AR(1) or MA(1) slightly improves the correlogram but also increases the standard error of the filtered smoothed signals. Therefore, model (3.8) was finally selected for the CCI series.

State space models assume that the disturbance terms are normally and independently distributed. These assumptions translate into the assumption that the innovations are normally and independently distributed. Table A.1 in the appendix contains an overview of goodness of fit statistics applied to the standardized innovations. The values for skewness, kurtosis and the Bowman-Shenton test do not indicate deviations from normality of the standardized innovations. The values for the Ljung-Box test and Durban-Watson test do not indicate serial correlations in the standardized innovations. This is also confirmed by a correlogram (not shown). In conclusion, these diagnostics indicate that (3.8) fits the series of the CCI reasonably well.

3.2  Bivariate model CCI and SMI series

The next step is to combine the univariate model for the CCI with the series for the SMI. Before combining CCI and SMI in a bivariate model, a univariate model for the SMI is developed with the purpose to better understand the behaviour of this series. A model selection procedure, similar to the one conducted for the CCI series in Subsection 3.1, indicated that the observed series for the SMI can be modelled with a smooth trend model and a white noise component for the unexplained variation. No significant seasonal component or business cycle is established. There are no signs for outliers or level shifts. AR(1) and MA(1) components are not included since there is no serial correlation in the standardized innovations. These considerations led to a bivariate model for the CCI and SMI where the CCI contains a trend and a seasonal component and the SMI a trend component.

Tables A.2 and A.3 in the appendix contain an overview of goodness of fit statistics for the standardized innovations of the CCI and SMI respectively. There are no indications that the standardized innovations of both series deviate from a normal distributions. The null hypothesis of no serial correlation in the standardized innovations could not be rejected. The correlogram of the innovations for the SMI, however, show a non-significant seasonal pattern (not shown). The innovations of the SMI, also contain heteroscedasticity.

The disturbance terms of the trend of both series are correlated. Since the series for the SMI is available from June 2010, the model for the CCI also contains the last intervention for September 2011, but not the outlier in September 2007 and the intervention in September 2008. As a result the following bivariate model is obtained:

$$\left(\begin{array}{c}{I}_t\\ X_t\end{array}\right)=\left(\begin{array}{c}{L}_t^I\\ L_t^X\end{array}\right)+\left(\begin{array}{c}{S}_t^I\\ 0\end{array}\right)+\left(\begin{array}{c}{\beta }^{11}{\delta }_t^{11}\\ 0\end{array}\right)+\left(\begin{array}{c}{\upsilon }_t^I\\ {\upsilon }_t^X\end{array}\right),(3.9)$$

with $L_t^I$ and $L_t^X$ the smooth trend model as defined in (3.6) with covariance structure

$$\begin{array}{ll}\text{Cov}\left({\eta }_t^I\text{},{\eta }_{t^{\prime }}^I\right)\hfill & =\{\begin{array}{lll}{\sigma }_{\eta I}^2\hfill & \text{if}\hfill & t=t^{\prime }\hfill \\ 0\hfill & \text{if}\hfill & t\ne t^{\prime }\hfill \end{array},\hfill \\ \text{Cov}\left({\eta }_t^X\text{},{\eta }_{t^{\prime }}^X\right)\hfill & =\{\begin{array}{lll}{\sigma }_{\eta X}^2\hfill & \text{if}\hfill & t=t^{\prime }\hfill \\ 0\hfill & \text{if}\hfill & t\ne t^{\prime }\hfill \end{array},\hfill \\ \text{Cov}\left({\eta }_t^I\text{},{\eta }_{t^{\prime }}^X\right)\hfill & =\{\begin{array}{lll}{\sigma }_{\eta I}{\sigma }_{\eta X}{\rho }_{\eta }\hfill & \text{if}\hfill & t=t^{\prime }\hfill \\ 0\hfill & \text{if}\hfill & t\ne t^{\prime }\hfill \end{array}.\hfill \end{array}$$

In the last expression ${\rho }_{\eta }$ denotes the correlation between the slope disturbances of the CCI and SMI. Furthermore, $S_t^I$ is the seasonal effect defined by (3.7) and ${\delta }_t^{11}$ the intervention for September 2011 with ${\beta }^{11}$ the corresponding regression coefficient. Finally, ${\upsilon }_t^I$ and ${\upsilon }_t^X$ are the disturbance terms for the CCI and SMI series and are defined as:

$$\begin{array}{ll}E({\upsilon }_t^I)\hfill & =E({\upsilon }_t^X)=0,\hfill \\ \text{Cov}\left({\upsilon }_t^I\text{},{\upsilon }_{t^{\prime }}^I\right)\hfill & =\{\begin{array}{c}\begin{array}{ccc}{\sigma }_{\upsilon I}^2 & \text{if}& t=t^{\prime }\end{array}\\ \begin{array}{ccc}0 & \text{if}& t\ne t^{\prime }\end{array}\end{array},\hfill \\ \text{Cov}\left({\upsilon }_t^X\text{},{\upsilon }_{t^{\prime }}^X\right)\hfill & =\{\begin{array}{c}\begin{array}{ccc}{\sigma }_{\upsilon X}^2& \text{if}& t=t^{\prime }\end{array}\\ \begin{array}{ccc}0 & \text{if}& t\ne t^{\prime }\end{array}\end{array},\hfill \\ \text{Cov}\left({\upsilon }_t^I\text{},{\upsilon }_{t^{\prime }}^X\right)\hfill & =0\text{for}\text{all}t\text{and}{t}^{\prime }.\hfill \end{array}$$

If the model detects a strong correlation between the trends of the CCI and the SMI, then the trends of both series will develop into the same direction more or less simultaneously. In this case, the additional information from the SMI series will result in an increased precision of the estimates of the CCI figures. In the case of strong correlation between the disturbances of the trends, i.e., if ${\rho }_{\eta }\to 1,$ the trends are said to be cointegrated. In that case, there is one underlying common trend that drives the evolution of the trends of the two observed series. To see this, it is noted that the covariance matrix of the slope disturbances is implemented as a singular value decomposition:

$$\text{cov}\left(\begin{array}{c}{\eta }_t^I\\ {\eta }_t^X\end{array}\right)=\left(\begin{array}{cc}{\sigma }_{\eta I}^2& {\sigma }_{\eta I}{\sigma }_{\eta X}{\rho }_{\eta }\\ {\sigma }_{\eta I}{\sigma }_{\eta X}{\rho }_{\eta }& {\sigma }_{\eta X}^2\end{array}\right)=\left(\begin{array}{cc}1& 0\\ a& 1\end{array}\right)\left(\begin{array}{cc}{d}_1& 0\\ 0& d_2\end{array}\right)\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right).(3.10)$$

Instead of estimating ${\sigma }_{\eta I}^2,$ ${\sigma }_{\eta X}^2,$ and ${\rho }_{\eta },$ parameters $d_1,$ $d_2,$ and $a$ are estimated. If $d_2\to 0,$ it follows that ${\rho }_{\eta }\to 1.$ In that case, the covariance matrix of the slope disturbances is of reduced rank and both trends are driven by one common trend. This implies that the slope disturbances of both series simultaneously move up or down and that the slope disturbances of the SMI can be perfectly predicted from slope disturbances of the CCI by ${\eta }_t^X=a{\eta }_t^I.$ Furthermore, the slope for the SMI series can be expressed as a linear combination of the slope for the CCI series as $R_t^X=a{R}_t^I+\overline{R}.$ Similarly, the trend for the SMI series can be expressed as a linear combination of the trend for the CCI series as $L_t^X=a{L}_t^I+\overline{L}+\overline{R}t.$ Note that $\overline{R}$ and $\overline{L}$ are constants that are derived from the estimated states at the last two time periods of the series.

Cointegration increases the precision of the estimated trend and signal of the CCI series, allows for formulating more parsimonious models, but could also be seen as an argument to replace the CCI series by the SMI series since both series are driven by and represent the same common trend. For a more detailed discussion about cointegration in the context of state space modelling, see Koopman, Harvey, Shephard and Doornik (2009, Sections 6.4 and 9.1).

3.3  Estimation of structural time series models

The general way to analyse a structural time series model is to express it in the so-called state space representation and apply the Kalman filter to obtain optimal estimates for the state variables, see e.g., Durbin and Koopman (2012). The software for the analysis and estimation of the time series models is developed in Ox in combination with the subroutines of SsfPack 3.0, see Doornik (2009) and Koopman, Shephard and Doornik (2008).

All state variables are non-stationary and initialised with a diffuse prior, i.e., the expectation of the initial states are equal to zero and the initial covariance matrix of the states is diagonal with large diagonal elements. In Ssfpack 3.0, an exact diffuse log-likelihood function is obtained with the procedure proposed by Koopman (1997). Maximum likelihood (ML) estimates for the hyperparameters, i.e., the variance components of the stochastic processes for the state variables are obtained using a numerical optimization procedure (Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, Doornik, 2009). To avoid negative variance estimates, the log-transformed variances are estimated. More technical details about the analysis of state space models can be found in Harvey (1989) or Durbin and Koopman (2012).

Under the assumption of normally distributed disturbance terms, the Kalman filter can be applied to obtain optimal estimates for the state variables, see e.g., Durbin and Koopman (2012). The Kalman filter assumes that the variance and covariance terms are known in advance and are often referred to as hyperparameters. In practise, these hyperparameters are not known and are therefore substituted with their ML estimates. Estimates for state variables for period $t$ based on the information available up to and including period $t$ are referred to as the filtered estimates. They are obtained with the Kalman filter where the ML estimates for the hyperparameters are based on the complete time series. The filtered estimates of past state vectors can be updated, if new data become available. This procedure is referred to as smoothing and results in smoothed estimates that are based on the complete time series.

Standard errors of the Kalman filter estimates do not reflect the additional uncertainty of using the ML estimates for the unknown hyperparameters. Therefore, the estimates of the standard errors are too optimistic.

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