Asymptotics of the principal components estimator of large factor models with weak factors and i.i.d. Gaussian noise.

We consider large factor models where factors' explanatory power does not strongly dominate the explanatory power of the idiosyncratic terms asymptotically. We find the first and second order asymptotics of the principal components estimator of such a weak factors as the dimensionality of the data and the number of observations tend to infinity proportionally. The principal components estimator is inconsistent but asymptotically normal

Detection of weak signals in high-dimensional complex-valued data

This paper considers the problem of detecting a few signals in high-dimensional complex-valued Gaussian data satisfying Johnstone's (2001) \textit{spiked covariance model}. We focus on the difficult case where signals are weak in the sense that the sizes of the corresponding covariance spikes are below the \textit{phase transition threshold} studied in Baik et al (2005). We derive a simple analytical expression for the maximal possible asymptotic probability of correct detection holding the asymptotic probability of false detection fixed. To accomplish this derivation, we establish what we believe to be a new formula for the \textit{% Harish-Chandra/Itzykson-Zuber (HCIZ) integral} \int_{\mathcal{U}(p)}e^{\tr(AGBG^{-1})}dG , where $A$ has a deficient rank $r<p$. The formula links the HCIZ integral over $\mathcal{U}(p)$ to an HCIZ integral over a potentially much smaller unitary group $\mathcal{U}(r)$. We show that the formula generalizes to the integrals over orthogonal and symplectic groups. In the most general form, it expresses the hypergeometric function $_{0}F_{0}^{(\alpha)}$of two $p\times p$ matrix arguments as a repeated contour integral of the hypergeometric function $_{0}F_{0}^{(\alpha)}$of two $r\times r$ matrix arguments

Testing in high-dimensional spiked models

We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are proportional to high dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the non-centrality or the covariance parameter of $H$ has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-specific phase transition threshold, none of the sample eigenvalues separate from the bulk, making the testing problem challenging. Using a unified strategy for the six cases, we show that the log likelihood ratio processes parameterized by the value of the sub-critical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike

Testing in High-Dimensional Spiked Models

We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of {code} where H and E are proportional to high dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the non-centrality or the covariance parameter of H has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-specific phase transition threshold, none of the sample eigenvalues separate from the bulk, making the testing problem challenging. Using a unified strategy for the six cases, we show that the log likelihood ratio processes parameterized by the value of the sub-critical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike

Signal detection in high dimension: The multispiked case

This paper deals with the local asymptotic structure, in the sense of Le Cam's asymptotic theory of statistical experiments, of the signal detection problem in high dimension. More precisely, we consider the problem of testing the null hypothesis of sphericity of a high-dimensional covariance matrix against an alternative of (unspecified) multiple symmetry-breaking directions (\textit{multispiked} alternatives). Simple analytical expressions for the asymptotic power envelope and the asymptotic powers of previously proposed tests are derived. These asymptotic powers are shown to lie very substantially below the envelope, at least for relatively small values of the number of symmetry-breaking directions under the alternative. In contrast, the asymptotic power of the likelihood ratio test based on the eigenvalues of the sample covariance matrix is shown to be close to that envelope. These results extend to the case of multispiked alternatives the findings of an earlier study (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximations considered in the single-spiked context do not extend to the multispiked case

Alternative Asymptotics for Cointegration Tests in Large VARs

Johansen's (1988, 1991) likelihood ratio test for cointegration rank of a Gaussian VAR depends only on the squared sample canonical correlations between current changes and past levels of a simple transformation of the data. We study the asymptotic behavior of the empirical distribution of those squared canonical correlations when the number of observations and the dimensionality of the VAR diverge to infinity simultaneously and proportionally. We find that the distribution almost surely weakly converges to the so-called Wachter distribution. This finding provides a theoretical explanation for the observed tendency of Johansen's test to find "spurious cointegration". It also sheds light on the workings and limitations of the Bartlett correction approach to the over-rejection problem. We propose a simple graphical device, similar to the scree plot, for a preliminary assessment of cointegration in high-dimensional VARs

Factor demand linkages, technology shocks, and the business cycle

This paper argues that factor demand linkages can be important for the transmission of both sectoral and aggregate shocks. We show this using a panel of highly disaggregated manufacturing sectors together with sectoral structural VARs. When sectoral interactions are explicitly accounted for, a contemporaneous technology shock to all manufacturing sectors implies a positive response in both output and hours at the aggregate level. Otherwise there is a negative correlation, as in much of the existing literature. Furthermore, we find that technology shocks are important drivers of the business cycle

Instabilities and robust control in natural resource management

Most renewable natural resources exhibit marked demographic and environmental stochasticities, which are exarcebated in management decisions by the uncertainty regarding the choice of an appropriate model to describe system dynamics. Moreover, demand and supply analysis often indicates the presence of instabilities and multiple equilibria, which may lead to management problems that are intensified by uncertainty on the evolution of the resource stock. In this paper the fishery management problem is used as an example to explore the potential of robust optimal control, where the objective is to choose a harvesting rule that will work under a range of admissible specifications for the stock-recruitment equation. The paper derives robust harvesting rules leading to a unique equilibrium, which could be helpful in the design of policy instruments such as robust quota systems.info:eu-repo/semantics/publishedVersio