Asymptotics of Cointegration Tests for High-Dimensional VAR(kk)

The paper studies non-stationary high-dimensional vector autoregressions of order $k$, VAR($k$). Additional deterministic terms such as trend or seasonality are allowed. The number of time periods, $T$, and number of coordinates, $N$, are assumed to be large and of the same order. Under such regime the first-order asymptotics of the Johansen likelihood ratio (LR), Pillai-Barlett, and Hotelling-Lawley tests for cointegration is derived: Test statistics converge to non-random integrals. For more refined analysis, the paper proposes and analyzes a modification of the Johansen test. The new test for the absence of cointegration converges to the partial sum of the Airy$_1$ point process. Supporting Monte Carlo simulations indicate that the same behavior persists universally in many situations beyond our theorems. The paper presents an empirical implementation of the approach to the analysis of stocks in S$\&$P$100$ and of cryptocurrencies. The latter example has strong presence of multiple cointegrating relationships, while the former is consistent with the null of no cointegration.Comment: v3: 45 pages, 11 figures (new simulations added

Boundary Limit Theory for Functional Local to Unity Regression

This paper studies functional local unit root models (FLURs) in which the autoregressive coefficient may vary with time in the vicinity of unity. We extend conventional local to unity (LUR) models by allowing the localizing coefficient to be a function which characterizes departures from unity that may occur within the sample in both stationary and explosive directions. Such models enhance the flexibility of the LUR framework by including break point, trending, and multi-directional departures from unit autoregressive coefficients. We study the behavior of this model as the localizing function diverges, thereby determining the impact on the time series and on inference from the time series as the limits of the domain of definition of the autoregressive coefficient are approached. This boundary limit theory enables us to characterize the asymptotic form of power functions for associated unit root tests against functional alternatives. Both sequential and simultaneous limits (as the sample size and localizing coefficient diverge) are developed. We find that asymptotics for the process, the autoregressive estimate, and its $t$ statistic have boundary limit behavior that differs from standard limit theory in both explosive and stationary cases. Some novel features of the boundary limit theory are the presence of a segmented limit process for the time series in the stationary direction and a degenerate process in the explosive direction. These features have material implications for autoregressive estimation and inference which are examined in the paper

Point Optimal Testing with Roots That Are Functionally Local to Unity

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases. It is shown that against FLUR\ alternatives, conventional constant point optimal tests can have extremely low power, particularly when the departure from unity occurs early in the sample period. Simulation results are reported and some implications for empirical practice are examined

Point optimal testing with roots that are functionally local to unity

Limit theory for regressions involving local to unit roots (LURs) is now used extensively in time series econometric work, establishing power properties for unit root and cointegration tests, assisting the construction of uniform confidence intervals for autoregressive coefficients, and enabling the development of methods robust to departures from unit roots. The present paper shows how to generalize LUR asymptotics to cases where the localized departure from unity is a time varying function rather than a constant. Such a functional local unit root (FLUR) model has much greater generality and encompasses many cases of additional interest that appear in practical work, including structural break formulations that admit subperiods of unit root, local stationary and local explosive behavior within a given sample. Point optimal FLUR tests are constructed in the paper to accommodate such cases and demonstrate how the power envelope changes in situations of practical interest. Against FLUR alternatives, conventional constant point optimal tests can be asymptotically infinitely deficient in power, with poor finite sample power perfor- mance particularly when the departure from unity occurs early in the sample period. New analytic explanation for this phenomenon is provided. Simulation results are reported and some implications for empirical practice are examine

Boundary limit theory for functional local to unity regression

This paper studies functional local unit root models (FLURs) in which the autoregressive coefficient may vary with time in the vicinity of unity. We extend conventional local to unity (LUR) models by allowing the localizing coefficient to be a function which characterizes departures from unity that may occur within the sample in both stationary and explosive directions. Such models enhance the flexibility of the LUR framework by including break point, trending, and multi-directional departures from unit autoregressive coefficients. We study the behavior of this model as the localizing function diverges, thereby determining the impact on the time series and on inference from the time series as the limits of the domain of definition of the autoregressive coefficient are approached. This boundary limit theory enables us to characterize the asymptotic form of power functions for associated unit root tests against functional alternatives. Both sequential and simultaneous limits (as the sample size and localizing coefficient diverge) are developed. We find that asymptotics for the process, the autoregressive estimate, and its $t$ statistic have boundary limit behavior that differs from standard limit theory in both explosive and stationary cases. Some novel features of the boundary limit theory are the presence of a segmented limit process for the time series in the stationary direction and a degenerate process in the explosive direction. These features have material implications for autoregressive estimation and inference which are examined in the paper

Spatial and temporal variation in macroparasite communities of three-spined stickleback

Patterns in parasite community structure are often observed in natural systems and an important question in parasite ecology is whether such patterns are repeatable across time and space. Field studies commonly look at spatial or temporal repeatability of patterns, but they are rarely investigated in conjunction. We use a large dataset on the macroparasites of the three-spined stickleback, Gasterosteus aculeatus L., collected from 14 locations on North Uist, Scotland over an 8-year period to investigate: (1) repeatability of patterns in parasite communities among populations and whether variation is consistent across years, (2) whether variation between years can be explained by climatic variation and progression of the season and (3) whether variation in habitat characteristics explain population differences. Differences in relative abundance and prevalence across populations were observed in a number of parasites investigated indicating a lack of consistency across years in numerous parasite community measures; however, differences between populations in the prevalence and abundance of some parasites were consistent throughout the study. Average temperature did not affect parasite community, and progression of the season was only significant for two of 13 community measures. Two of the six habitat characteristics investigated (pH and calcium concentration) significantly affected parasite presence

MRPS18CP2 alleles and DEFA3 absence as putative chromosome 8p23.1 modifiers of hearing loss due to mtDNA mutation A1555G in the 12S rRNA gene

<p>Abstract</p> <p>Background</p> <p>Mitochondrial DNA (mtDNA) mutations account for at least 5% of cases of postlingual, nonsyndromic hearing impairment. Among them, mutation A1555G is frequently found associated with aminoglycoside-induced and/or nonsyndromic hearing loss in families presenting with extremely variable clinical phenotypes. Biochemical and genetic data have suggested that nuclear background is the main factor involved in modulating the phenotypic expression of mutation A1555G. However, although a major nuclear modifying locus was located on chromosome 8p23.1 and regardless intensive screening of the region, the gene involved has not been identified.</p> <p>Methods</p> <p>With the aim to gain insights into the factors that determine the phenotypic expression of A1555G mutation, we have analysed in detail different genetic and genomic elements on 8p23.1 region (<it>DEFA3 </it>gene absence, <it>CLDN23 </it>gene and <it>MRPS18CP2 </it>pseudogene) in a group of 213 A1555G carriers.</p> <p>Results</p> <p>Family based association studies identified a positive association for a polymorphism on <it>MRPS18CP2 </it>and an overrepresentation of <it>DEFA3 </it>gene absence in the deaf group of A1555G carriers.</p> <p>Conclusion</p> <p>Although none of the factors analysed seem to have a major contribution to the phenotype, our findings provide further evidences of the involvement of 8p23.1 region as a modifying locus for A1555G 12S rRNA gene mutation.</p

Author Correction: Cross-ancestry genome-wide association analysis of corneal thickness strengthens link between complex and Mendelian eye diseases.

Emmanuelle Souzeau, who contributed to analysis of data, was inadvertently omitted from the author list in the originally published version of this Article. This has now been corrected in both the PDF and HTML versions of the Article