131 research outputs found
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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
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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 has a
deficient rank . The formula links the HCIZ integral over to an HCIZ integral over a potentially much smaller unitary group
. We show that the formula generalizes to the integrals over
orthogonal and symplectic groups. In the most general form, it expresses the
hypergeometric function of two matrix
arguments as a repeated contour integral of the hypergeometric function
of two matrix arguments
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Asymptotic analysis of the squared estimation error in misspecified factor models
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 where and 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 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
Model Uncertainty, Ambiguity and the Precautionary Principle: Implications for Biodiversity Management
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
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