A mixed integer linear program to compress transition probability matrices in Markov chain bootstrapping

Abstract

Bootstrapping time series is one of the most acknowledged tools to study the statistical properties of an evolutive phenomenon. An important class of bootstrapping methods is based on the assumption that the sampled phenomenon evolves according to a Markov chain. This assumption does not apply when the process takes values in a continuous set, as it frequently happens with time series related to economic and financial phenomena. In this paper we apply the Markov chain theory for bootstrapping continuous-valued processes, starting from a suitable discretization of the support that provides the state space of a Markov chain of order $k \ge 1$. Even for small k, the number of rows of the transition probability matrix is generally too large and, in many practical cases, it may incorporate much more information than it is really required to replicate the phenomenon satisfactorily. The paper aims to study the problem of compressing the transition probability matrix while preserving the “law” characterising the process that generates the observed time series, in order to obtain bootstrapped series that maintain the typical features of the observed time series. For this purpose, we formulate a partitioning problem of the set of rows of such a matrix and propose a mixed integer linear program specifically tailored for this particular problem. We also provide an empirical analysis by applying our model to the time series of Spanish and German electricity prices, and we show that, in these medium size real-life instances, bootstrapped time series reproduce the typical features of the ones under observation.

Appendices

Appendix 1: Trend and weekly seasonality removal

The estimation of the exponential trend and weekly seasonality is based on the following model:

$$\begin{aligned} e_{t}^{(c)}=\exp (rt+\eta _{1}{\mathbb {I}}_{1}(t)+\eta _{2}{\mathbb {I}} _{2}(t)+\eta _{3}{\mathbb {I}}_{3}(t)+\eta _{4}{\mathbb {I}}_{4}(t)+\eta _{5} {\mathbb {I}}_{5}(t)+\eta _{6}{\mathbb {I}}_{6}(t)+\eta _{7}{\mathbb {I}} _{7}(t)+\varepsilon _{t})\text {,} \end{aligned}$$

(14)

where $e_{t}^{(c)}$ are the raw original prices, ${\mathbb {I}}_{j}(t)$ is the dummy variable signalling whether t is the jth day of the week, with $j=1,\dots ,7$, r is the growth rate, $\eta _{j}$ is the coefficient of dummy variable ${\mathbb {I}}_{j}(t)$, with $j=1,\dots ,7$, and $\varepsilon _{t}$ are the errors. If we take the natural logarithm on both sides of formula (14), we obtain the following formula:

$$\begin{aligned} z_{t}=rt+\eta _{1}{\mathbb {I}}_{1}(t)+\eta _{2}{\mathbb {I}}_{2}(t)+\eta _{3} {\mathbb {I}}_{3}(t)+\eta _{4}{\mathbb {I}}_{4}(t)+\eta _{5}{\mathbb {I}}_{5}(t)+\eta _{6}{\mathbb {I}}_{6}(t)+\eta _{7}{\mathbb {I}}_{7}(t)+\varepsilon _{t}\text {,} \end{aligned}$$

where $z_{t}=\ln e_{t}^{(c)}$.

For estimation purposes, we assume that the usual hypotheses of linear regression on the errors $\varepsilon _{t}$ hold. We obtain the OLS estimates of r and $\eta _{j}$, $j=1,\dots ,7$, and they are significant at a level of $5\,\%$ (see Table 4).

Table 4 Coefficients estimates of an exponential regression model of trend and weekly seasonality applied to the series of electricity prices of Spain and Germany

To the purpose of removing the exponential trend and weekly seasonality from our original series, we define the series of prices $e(T)=(e_{1},\dots ,e_t, \dots , e_{T})$, where:

$$\begin{aligned} e_{t}= & {} \exp [z_{t}-(\hat{r}t+\hat{\eta }_{1}{\mathbb {I}}_{1}(t)+\hat{\eta }_{2} {\mathbb {I}}_{2}(t)+\hat{\eta }_{3}{\mathbb {I}}_{3}(t)+\hat{\eta }_{4}{\mathbb {I}} _{4}(t)+\hat{\eta }_{5}{\mathbb {I}}_{5}(t)+\hat{\eta }_{6}{\mathbb {I}}_{6}(t)\\&+\,\hat{ \eta }_{7}{\mathbb {I}}_{7}(t))]\text {, }t=1,\dots ,T\text {.} \end{aligned}$$

Set e(T) is an input of the bootstrapping method, while the output is the bootstrapped series $x(\ell )=(x_{1},\ldots ,x_{\ell })$. To re-introduce the exponential trend and weekly seasonality in $x(\ell )$, we multiply each point $x_{j}$ by $e^{(\hat{r}j+\hat{\eta }_{1}{\mathbb {I}} _{1}(j)+\hat{\eta }_{2}{\mathbb {I}}_{2}(j)+\hat{\eta }_{3}{\mathbb {I}}_{3}(j)+\hat{ \eta }_{4}{\mathbb {I}}_{4}(j)+\hat{\eta }_{5}{\mathbb {I}}_{5}(j)+\hat{\eta }_{6} {\mathbb {I}}_{6}(j)+\hat{\eta }_{7}{\mathbb {I}}_{7}(j))}$,$j=1,\dots ,\ell$.

Appendix 2: Initial states, or intervals

Table 5 reports the 12 intervals of the initial partition of the support $[\alpha ,\beta ]$ of the series of Spain and Germany after removal of exponential trend and weekly seasonality.

Table 5 Elements of the initial partition of the support of the exponentially detrended and deseasonalized series of electricity prices of Spain and Germany