{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T10:45:43Z","timestamp":1774521943114,"version":"3.50.1"},"reference-count":69,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2017,1,9]],"date-time":"2017-01-09T00:00:00Z","timestamp":1483920000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["www.cambridge.org"],"crossmark-restriction":true},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2018,2]]},"abstract":"<jats:p>This paper examines the ordinary least squares (OLS) estimator of the structural parameters in a simple macroeconomic model in which agents are boundedly rational and use an adaptive learning rule to form expectations of the endogenous variable. The popularity of learning models has recently increased amongst applied economists and policy makers who seek to estimate them empirically. Yet the econometrics of learning models is largely uncharted territory. We consider two prominent learning algorithms, namely constant gain and decreasing gain learning. For each of the two learning rules, our analysis proceeds in two stages. First, the paper derives the asymptotic properties of agents\u2019 expectations. At the second stage, the paper derives the asymptotics of OLS in the structural model, taking the first stage learning dynamics as given. In the case of constant gain learning, the structural model effectively amounts to a stationary, cointegrating, or co-explosiveness regression. With decreasing gain learning, the regressors are asymptotically collinear such that OLS does not satisfy, in general, the Grenander conditions for consistent estimability. Nevertheless, this paper shows that the OLS estimator remains consistent in all models considered. It also shows, however, that its asymptotic distribution, and hence any inference based upon it, may be nonstandard.<\/jats:p>","DOI":"10.1017\/s0266466616000529","type":"journal-article","created":{"date-parts":[[2017,1,9]],"date-time":"2017-01-09T08:13:24Z","timestamp":1483949604000},"page":"68-111","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":12,"title":["ESTIMATING STRUCTURAL PARAMETERS IN REGRESSION MODELS WITH ADAPTIVE LEARNING"],"prefix":"10.1017","volume":"34","author":[{"given":"Norbert","family":"Christopeit","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Massmann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,1,9]]},"reference":[{"key":"S0266466616000529_ref48","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1016\/j.jmoneco.2006.11.007","article-title":"Expectations, learning and macroeconomic persistence","volume":"54","author":"Milani","year":"2007","journal-title":"Journal of Monetary Economics"},{"key":"S0266466616000529_ref68","doi-asserted-by":"crossref","first-page":"1397","DOI":"10.2307\/1913104","article-title":"Asymptotic normality, when regressors have a unit root","volume":"56","author":"West","year":"1988","journal-title":"Econometrica"},{"key":"S0266466616000529_ref7","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.econlet.2005.09.005","article-title":"A simple recursive forecasting model","volume":"91","author":"Branch","year":"2006","journal-title":"Economics Letters"},{"key":"S0266466616000529_ref38","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/0047-259X(82)90071-9","article-title":"Asymptotic properties of projections with applications to stochastic regression problems","volume":"12","author":"Lai","year":"1982b","journal-title":"Journal of Multivariate Analysis"},{"key":"S0266466616000529_ref24","doi-asserted-by":"crossref","DOI":"10.1515\/9781400824267","volume-title":"Learning and Expectations in Macroeconomics","author":"Evans","year":"2001"},{"key":"S0266466616000529_ref32","doi-asserted-by":"crossref","first-page":"1104","DOI":"10.1080\/01621459.1968.11009327","article-title":"Series expansions of the doubly noncentral t-distribution","volume":"63","author":"Krishnan","year":"1968","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466616000529_ref61","first-page":"877","volume-title":"The New Palgrave Dictionary of Economics","author":"Sargent","year":"2008"},{"key":"S0266466616000529_ref17","doi-asserted-by":"crossref","unstructured":"Christopeit, N. & Massmann, M. 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