Stochastic trends and seasonality in economic time series: new evidence from Bayesian stochastic model specification search

Abstract

An important issue in modelling economic time series is whether key unobserved components representing trends, seasonality and calendar components, are deterministic or evolutive. We address it by applying a recently proposed Bayesian variable selection methodology to an encompassing linear mixed model that features, along with deterministic effects, additional random explanatory variables that account for the evolution of the underlying level, slope, seasonality and trading days. Variable selection is performed by estimating the posterior model probabilities using a suitable Gibbs sampling scheme. The paper conducts an extensive empirical application on a large and representative set of monthly time series concerning industrial production and retail turnover. We find strong support for the presence of stochastic trends in the series, either in the form of a time-varying level, or, less frequently, of a stochastic slope, or both. Seasonality is a more stable component, although in at least 60 % of the cases we were able to select one or more stochastic trigonometric cycles. Most frequently the time variation is found in correspondence with the fundamental and the first harmonic cycles. An interesting and intuitively plausible finding is that the probability of estimating time-varying components increases with the sample size available. However, even for very large sample sizes we were unable to find stochastically varying calendar effects.

Appendix A: markov chain monte carlo estimation

This appendix provides details of the Gibbs sampling scheme outlined at the end of Sect. 3. Recall that the linear mixed model was defined as $y = Z_\varUpsilon \psi _\varUpsilon + \epsilon$, where $y$ and $\epsilon$ are vectors stacking the values $\{y_t\}$ and $\{\epsilon _t\}$, respectively, and the generic row of matrix $Z_\varUpsilon$ contains the relevant subset of the explanatory variables.

Step a. is carried out by sampling the indicators with probabilities proportional to the conditional likelihood of the regression model, as

$$\begin{aligned} \begin{array}{lll} \pi (\varUpsilon | \alpha , y)&\propto \pi (\varUpsilon )\pi (y|\varUpsilon , \alpha ) \end{array} \end{aligned}$$

which is available in closed form (see below). All the $2^{10}$ combinations of indicators are sampled jointly in a multimove way as in FS-W, conditional on the latent process $\alpha$.

Under the normal-inverse Gamma conjugate prior for $(\psi _\varUpsilon , \sigma _\epsilon ^2)$

$$\begin{aligned} \sigma ^{2}_{\epsilon }\sim \text{ IG }(c_{0}, C_{0}), \;\;\;\; \psi _{\varUpsilon }|\sigma ^{2}_{\epsilon } \sim \text{ N }\left( 0,\sigma _\epsilon ^2 D_{\varUpsilon }\right) , \end{aligned}$$

where $D_{\varUpsilon }$ is a diagonal matrix with elements $\kappa _\mu , \kappa _A,$ etc., steps b. and c. are carried out by sampling from the posteriors

$$\begin{aligned} \begin{array}{lll} \sigma _\epsilon ^2|\varUpsilon , \psi _\varUpsilon , \alpha , y &{}\sim &{}\text{ IG }(c_{T}, C_{T}) \\ \psi _\varUpsilon |\varUpsilon ,\sigma _\epsilon ^2, \alpha ,y &{}\sim &{}\text{ N }(m, \sigma _\epsilon ^2 S) \\ \end{array} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{llllll} S &{}=&{} \left( Z_{\varUpsilon }'Z_{\varUpsilon }+D_{\varUpsilon }^{-1}\right) ^{-1}, \quad &{} m &{} = &{} SZ_{\varUpsilon }'y, \\ c_{T} &{} =&{} c_0+T/2, &{} C_{T} &{} =&{} C_0 + \frac{1}{2} \left( y'y- m'S^{-1} m\right) . \\ \end{array} \end{aligned}$$

Finally,

$$\begin{aligned} \pi (y|\varUpsilon , \alpha ) \propto \frac{\left| S\right| ^{0.5}}{\left| D_{\varUpsilon }\right| ^{0.5}} \frac{\Gamma (c_{T})}{\Gamma (c_{0})} \frac{C_{0}^{c_{0}}}{C_{T }^{c_{T }}}, \end{aligned}$$

see e.g. Geweke (2005), where $\Gamma (\cdot )$ denotes the Gamma function.

The sample from the posterior distribution of the latent states, conditional on the model and its parameters, in step d., is obtained by the conditional simulation smoother proposed by Durbin and Koopman (2002).

The draws of the parameters $\beta _\mu$, $\beta _A$, $\beta _{sj}, j = 1, \ldots , 6$, $\beta _{TD}$ are obtained by performing a final random sign permutation. This is achieved by drawing independently Bernoulli random variables $\mathsf{B}_\mu$, $\mathsf{B}_A$, $\mathsf{B}_{sj}, j = 1, \ldots , 6, \mathsf{B}_{TD}$ with probability 0.5, and recording $(-1)^{\mathsf{B}_{\mu }} (\sigma _\eta , \tilde{\mu }_{t})$, $(-1)^{\mathsf{B}_{A}} (\sigma _\zeta , \tilde{A}_{t}, a_t)$, etc.

The starting values are obtained by iterating for the full model (with all the indicators being equal to 1) the above GS scheme 1000 times, with initial values of the hyperparameters in $\psi _\varUpsilon$ set equal to zero.

Under a fractional prior the marginal likelihood is evaluated as follows:

$$\begin{aligned} \begin{array}{llll} &{}c_{T}&{} = c_{0} + \frac{1-b}{2}T, C_{T } = C_{0} + \frac{(1-b)}{2} \left( y'y - m'\left( Z_{\varUpsilon }'Z_{\varUpsilon }\right) ^{-1}m\right) ,\\ &{}\pi &{}(y\mid \varUpsilon , \alpha ) \propto \frac{b^{q/2} }{{2 \pi }^{T (1-b)/2} }\frac{\Gamma (c_{T})}{\Gamma (c_{0})} \frac{C_{0}^{c_{0}}}{C_{T }^{c_{T }}}, \end{array} \end{aligned}$$

(13)

where $q$ is the dimension of $\psi _{\varUpsilon }$, and $m$ is as given above. The free parameter in the fractional prior specification is $b$ that in our study is set to be $10^{-4}$ and $10^{-5}$.

A key assumption is that $\sigma ^2_\epsilon$ is strictly greater than zero, i.e. the irregular component is always present.