{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T16:41:25Z","timestamp":1770482485586,"version":"3.49.0"},"reference-count":34,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":4577,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1996,8]]},"abstract":"<jats:p>For a first-order autoregressive AR(1) model with zero initial value, x<jats:sub>i<\/jats:sub>= ax<jats:sub>i\u22121<\/jats:sub>,_, + e<jats:sub>i<\/jats:sub>, we provide the bias, mean squared error, skewness, and kurtosis of the maximum likelihood estimator \u00e2. Brownian motion approximations by Phillips (1977, Econometrica 45, 463\u2013485; 1978, Biometrika 65, 91\u201398; 1987, Econometrica 55, 277\u2013301), Phillips and Perron (1988, Biometrika 75, 335\u2013346), and Perron (1991, Econometric Theory 7, 236\u2013252; 1991, Econometrica 59, 211\u2013236), among others, yield an elegant unified theory but do not yield convenient formulas for calibration of skewness and kurtosis. In addition to the usual stationary case |\u03b1| &lt; 1, we include the unstable |\u03b1| = 1 case of the random walk model. For the |\u03b1| &lt; 1 case, we give new exact results for White's (1961, Biometrika 48, 85\u201394) model B, where the initial value x0 is a normal random variable N(0,\u03c32\/(l \u2013 \u03b1<jats:sup>2<\/jats:sup>)). Our expressions are exact for small samples computed by relatively reliable Gaussian quadrature methods, rather than approximate ones in powers of n<jats:sup>\u2212l<\/jats:sup>or \u03b1<jats:sup>2<\/jats:sup>.<\/jats:p>","DOI":"10.1017\/s0266466600006824","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T15:39:09Z","timestamp":1236699549000},"page":"481-499","source":"Crossref","is-referenced-by-count":34,"title":["Exact Moments for Autor1egressive and Random walk Models for a Zero or Stationary Initial 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