Limit theory involving stochastic integrals is now widespread in time series econometrics and relies on a few key results on function space weak convergence. In establishing weak convergence of sample covariances to stochastic integrals, the literature commonly uses martingale and semimartingale structures. While these structures have wide relevance, many applications in econometrics involve a cointegration framework where endogeneity and nonlinearity play a major role and lead to complications in the limit theory. This paper explores weak convergence limit theory to stochastic integral functionals in such settings. We use a novel decomposition of sample covariances of functions of I (1) and I (0) time series that simplifies the asymptotic development and we provide limit results for such covariances when linear process, long memory, and mixing variates are involved in the innovations. The limit results extend earlier findings in the literature, are relevant in many econometric applications, and involve simple conditions that facilitate implementation in practice. A nonlinear extension of FM regression is used to illustrate practical application of the methods.
Good understanding of the genes controlling root development is required to engineer root systems better adapted to different soil types. In wheat (Triticum aestivum L.), the 1RS.1BL wheat–rye (Secale cereale L.) translocation has been associated with improved drought tolerance and a large root system. However, an isogenic line carrying an interstitial segment from wheat chromosome arm 1BS in the distal region of the 1RS arm (1RSRW) showed reduced grain yield and shorter roots both in the field and in hydroponic cultures relative to isogenic lines with the complete 1RS arm. In this study, we used exome capture to characterize 1RSRW and its parental lines T‐9 and 1B+40. We show that 1RSRW has a 7.0 Mb duplicated 1RS region and a 4.8 Mb 1BS insertion colinear with the 1RS duplication, resulting in triplicated genes. Lines homozygous for 1RSRW have short seminal roots, while lines heterozygous for this chromosome have roots of intermediate length. By contrast, near‐isogenic lines carrying only the 1BS distal region or the 1RS‐1BS duplication have long seminal roots similar to 1RS, suggesting a limited effect of the 1BS genes. These results suggest that the dosage of duplicated 1RS genes is critical for seminal root length. An induced deletion encompassing 38 orthologous wheat and rye duplicated genes restored root length and confirmed the importance of gene dosage in the short‐root phenotype. We explored the expression profiles and functional annotation of these genes and discuss their potential as candidate genes for the regulation of seminal root length in wheat.
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