Sparse Vector Autoregressive Modeling

Abstract

The vector autoregressive (VAR) model has been widely used for modeling temporal dependence in a multivariate time series. For large (and even moderate) dimensions, the number of AR coefficients can be prohibitively large, resulting in noisy estimates, unstable predictions and difficult-to-interpret temporal dependence. To overcome such drawbacks, we propose a 2-stage approach for fitting sparse VAR (sVAR) models in which many of the AR coefficients are zero. The first stage selects non-zero AR coefficients based on an estimate of the partial spectral coherence (PSC) together with the use of BIC. The PSC is useful for quantifying the conditional relationship between marginal series in a multivariate process. A refinement second stage is then applied to further reduce the number of parameters. The performance of this 2-stage approach is illustrated with simulation results. The 2-stage approach is also applied to two real data examples: the first is the Google Flu Trends data and the second is a time series of concentration levels of air pollutants.which satisfies the recursions,see Brockwell and Davis (1991) and Reinsel (1997), which implies that Z t is independent of Y s for s < t. Without loss of generality, we also assume that the vector process {Y t } has mean 0, i.e., µ = 0 in (2.1). Sparse vector autoregressive models (sVAR)The temporal dependence structure of the VAR model (2.1) is characterized by the AR coefficient matrices A 1 , . . . , A p . Based on T observations Y 1 , . . . , Y T from the VAR model, we want to estimate these AR matrices. However, a VAR(p) model, when fully-parametrized, has K 2 p AR parameters that need to be estimated. For large (and even moderate) dimension K, the number of parameters can be prohibitively large, resulting in noisy estimates, unstable predictions and difficult-to-interpret descriptions of the temporal dependence. It is also generally believed that, for most applications, the true model of the series is sparse, i.e., the number of non-zero coefficients is small. Therefore it is preferable to fit a sparse VAR (sVAR) model in which many of its AR parameters are zero. In this paper we develop a 2-stage approach of fitting sVAR models. The first stage selects non-zero AR coefficients by screening pairs of distinct marginal series that are conditionally correlated. To compute direction-free conditional correlation between components in the time series, we use tools from the frequency-domain, specifically the partial spectral coherence (PSC). Below we introduce the basic properties related to PSC. Let {Y t,i } and {Y t,j } (i = j) denote two distinct marginal series of the process {Y t }, and {Y t,−ij } denote the remaining (K − 2)-dimensional process. To compute the conditional correlation between two time series {Y t,i } and {Y t,j }, we need to adjust for the linear effect from the remaining marginal series {Y t,−ij }. The removal of the linear effect of {Y t,−ij } from each of {Y t,i } and {Y t,j } can be achieved by using results of linear filters, e.g., see Brillinge...

Very-Short-Term Probabilistic Wind Power Forecasts by Sparse Vector Autoregression

“…At each stage the set of coefficients selected is that which minimises the Bayesian information criterion (BIC). This approach is detailed in the remainder of this section, for further discussion see Davis et al [20].…”

“…A 2-stage procedure for fitting a sparse vector autoregressive model has been proposed by Davis et al in [20]. The first stage selects symmetric pairs of coefficients to be included in the sparse model based on the corresponding pair of time series' conditional dependence.…”

“…The periodogram smoothed by a modified Daniell kernel is used here, as in [20], though alternative spectral density estimates could be employed.…”

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