{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T05:52:32Z","timestamp":1776232352381,"version":"3.50.1"},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T00:00:00Z","timestamp":1287360000000},"content-version":"unspecified","delay-in-days":8113,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econ Theory"],"published-print":{"date-parts":[[1988,8]]},"abstract":"<jats:p>Building on work of McLeish, we present a number of invariance principles for doubly indexed arrays of stochastic processes which may exhibit considerable dependence, heterogeneity, and\/or trending moments. In particular, we consider possibly time-varying functions of infinite histories of heterogeneous mixing processes and obtain general invariance results, with central limit theorems following as corollaries. These results are formulated so as to apply to economic time series, which may exhibit some or all of the features allowed in our theorems. Results are given for the case of both scalar and vector stochastic processes. Using an approach recently pioneered by Phillips, and Phillips and Durlauf, we apply our results to least squares estimation of unit root models.<\/jats:p>","DOI":"10.1017\/s0266466600012032","type":"journal-article","created":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T09:11:46Z","timestamp":1287393106000},"page":"210-230","source":"Crossref","is-referenced-by-count":129,"title":["Some Invariance Principles and Central Limit Theorems for Dependent Heterogeneous Processes"],"prefix":"10.1017","volume":"4","author":[{"given":"Jeffrey M.","family":"Wooldridge","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Halbert","family":"White","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,10,18]]},"reference":[{"key":"S0266466600012032_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF00532611"},{"key":"S0266466600012032_ref009","volume-title":"A unified theory of estimation and inference for nonlinear dynamic models","author":"Gallant","year":"1988"},{"key":"S0266466600012032_ref012","volume-title":"Martingale limit theory and its application","author":"Hall","year":"1980"},{"key":"S0266466600012032_ref028","unstructured":"Wooldridge, J.M. & White, H. . Some invariance and central limit theorems for dependent heterogeneous processes. UCSD Department of Economics Discussion Paper 86\u201318, 1986."},{"key":"S0266466600012032_ref022","unstructured":"Phillips, P.C.B. Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Cowles Foundation Discussion Paper 846, 1987."},{"key":"S0266466600012032_ref024","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177698240"},{"key":"S0266466600012032_ref010","unstructured":"Granger, C.W.J. Models that generate trends. UCSD Department of Economics Discussion Paper, 1985."},{"key":"S0266466600012032_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/3213710"},{"key":"S0266466600012032_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/3213877"},{"key":"S0266466600012032_ref021","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600004126"},{"key":"S0266466600012032_ref005","volume-title":"Convergence of probability measures","author":"Billingsley","year":"1968"},{"key":"S0266466600012032_ref023","doi-asserted-by":"publisher","DOI":"10.2307\/2297602"},{"key":"S0266466600012032_ref027","unstructured":"Wooldridge, J.M. Asymptotic properties of econometric estimators. 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Cowles Foundation Discussion Paper 819, 1987."},{"key":"S0266466600012032_ref011","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.1980.tb00297.x"},{"key":"S0266466600012032_ref018","unstructured":"Phillips, P.C.B. Weak convergence to the matrix stochastic integral \u222b BdB'. Cowles Foundation Discussion Paper 796, 1986."},{"key":"S0266466600012032_ref008","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316719"},{"key":"S0266466600012032_ref020","doi-asserted-by":"publisher","DOI":"10.2307\/1913237"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600012032","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,9]],"date-time":"2021-11-09T05:02:46Z","timestamp":1636434166000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600012032\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,8]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1988,8]]}},"alternative-id":["S0266466600012032"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600012032","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,8]]}}}