De-Biased Graphical Lasso for High-Frequency Data

Abstract

This paper develops a new statistical inference theory for the precision matrix of high-frequency data in a high-dimensional setting. The focus is not only on point estimation but also on interval estimation and hypothesis testing for entries of the precision matrix. To accomplish this purpose, we establish an abstract asymptotic theory for the weighted graphical Lasso and its de-biased version without specifying the form of the initial covariance estimator. We also extend the scope of the theory to the case that a known factor structure is present in the data. The developed theory is applied to the concrete situation where we can use the realized covariance matrix as the initial covariance estimator, and we obtain a feasible asymptotic distribution theory to construct (simultaneous) confidence intervals and (multiple) testing procedures for entries of the precision matrix.[19] and subsequently built up by, among others, [2,17,20]. Other common methods used in i.i.d. settings have also been investigated in the literature of high-frequency financial econometrics. Hautsch et al. [25] and Morimoto & Nagata [41] formally apply eigenvalue regularization methods based on random matrix theory to high-frequency data. Lam et al. [37] accommodate the non-linear shrinkage estimator of [39] to a high-frequency data setting with the help of the spectral distribution theory for the realized covariance matrix developed in [58]. Brownlees et al. [8] employ the ℓ 1 -penalized Gaussian MLE, which is known as the graphical Lasso, to estimate the precision matrix (the inverse of the covariance matrix) of high-frequency data. The latter approach is closely related to the methodology we will focus on.Despite the recent advances in this topic as above, most studies in this area focus only on point estimation of covariance and precision matrices, and there are little work about interval estimation and hypothesis testing for these objects. A few exceptions are [36,44] and [34]. The first two articles are concerned with continuous-time factor models: Kong & Liu [36] propose a test for the constancy of the factor loading matrix, while Pelger [44] assumes constant loadings and develops an asymptotic distribution theory to make inference for the factors and loadings. Meanwhile, Koike [34] establishes a high-dimensional central limit theorem for the realized covariance matrix which allows us to construct simultaneous confidence regions or carry out multiple testing for entries of the high-dimensional covariance matrix of high-frequency data. The aim of this study is to develop such a statistical inference theory for the precision matrix of high-frequency data. This is naturally motivated by the fact that the precision matrix of asset returns plays an important role in mean-variance analysis of portfolio allocation (see e.g. [16, Chapter 5]). We accomplish this purpose by imposing a sparsity assumption on the precision matrix. Such an assumption has a clear interpretation in connection with Gaussian graphical models: For a Gaussian random v...

Inferential Theory for Granular Instrumental Variables in High Dimensions

“…However, these approaches rule out the presence of an approximate factor structure, as they assume unconditional sparsity of the composite error term v it . It is clear that sparsity of v it fails given our (pervasive) factor structure, as pointed out by Barigozzi et al (2018), Brownlees et al (2018) and Koike (2020).…”

Learning from Forecast Errors: A New Approach to Forecast Combinations

“…Since our interest is in constructing weights for the forecast combination, our goal is to estimate a precision matrix of the forecast errors. However, as pointed out by Koike (2020), when common factors are present across the forecast errors, the precision matrix cannot be sparse because all pairs of the forecast errors are partially correlated given other forecast errors through the common factors.…”

“…Therefore, we impose a sparsity assumption on the precision matrix of the idiosyncratic errors, Θ ε , which is obtained using the estimated residuals after removing the co-movements induced by the factors (see Barigozzi et al (2018); Brownlees et al (2018); Koike (2020)).…”

“…They show that when the dimension of the system is large, the largest eigenvalues of the precision converge to a positive affine transfor-mation. Koike (2020) uses the Weighted Graphical Lasso to estimate a sparse covariance matrix of the idiosyncratic component for a factor model with observable factors for high-frequency data.…”