مقاله سال ۲۰۱۶

High-dimensional data sets containing thousands of time series are commonly available and can be accessed at a reasonable cost (Fan, Lv, & Qi, 2011; Stock & Watson, 2002). Recently, there has been a considerable amount of work on exploiting the large amount of information contained in these data sets for forecasting purposes. To handle the dimensionality, various large time series models, containing large numbers of time series relative to the time series length, have been considered. Common approaches include factor models (e.g., Stock & Watson, 2002), Bayesian vector autoregressive (VAR) models (e.g., Banbura, Giannone, & Reichlin, 2010), and reduced-rank VAR models (e.g., Carriero, Kapetanios, & Marcellino, 2011 and Bernardini & Cubadda, 2015), among others. Typically, though, these authors have not accounted for cointegration. Instead, either the time series are transformedin order to achieve stationarity (Bernardini & Cubadda, 2015), or the (non-)stationarity is accounted for in the prior distribution of the autoregressive parameters (Banbura et al., 2010). In cointegration analysis, we estimate long-run equilibrium relationships between several time series, often as implied by economic theory.

This paper develops a cointegration method for highdimensional time series. The vector error correction model (VECM; e.g., Lütkepohl, 2007) is used to estimate and test for the cointegration relationships. Various cointegration tests exist (e.g., Engle & Granger, 1987 and Phillips & Ouliaris, 1990), with the cointegration test of Johansen (1988) being the most popular. However, Johansen’s maximum likelihood approach has various limitations. In a high-dimensional setting, where the number of time series is large compared to the length of the time series, the estimation imprecision will be large. Johansen’s approach is based on the estimation of a VAR model and a canonical correlation analysis. One drawback of the VAR is that the number of parameters that it uses increases quadratically with the number of time series included. As a consequence, the regression parameters will be estimated inaccurately if only limited numbers of time points are available. When the number of time series exceeds the time series length, Johansen’s approach cannot even be applied.

ansen’s approach cannot even be applied. We introduce a penalized maximum likelihood (PML) approach that is designed for estimating the cointegrating vectors in a sparse way, i.e., with some of its components estimated as exactly zero. Sparse estimators have been shown to perform well in various fields, such as economics (e.g., Fan et al., 2011), macroeconomics (e.g., Korobilis, 2013; Liao & Phillips, 2015), finance (e.g., Zhou, Nakajima, & West, 2014), and biostatistics (e.g., Friedman, 2012). Sparse cointegration methods are useful for several reasons. First, sparsity facilitates model interpretation, since only limited numbers of time series, those corresponding to the non-zero coefficients, enter the estimated long-run equilibrium relationships. Second, sparsity improves the forecast performance through a variance reduction. Third, unlike Johansen’s maximum likelihood approach, the sparse approach can still be applied when the number of time series exceeds the time series length.