LANGUAGEFortran 66 DESCRIPTION AND PURPOSE The algorithm presented here enables the exact likelihood function of a stationary autoregressive-moving average (ARM A) process to be calculated by means ofthe Kalman filter; see Harvey andPhillips (1976, 1979). Two subroutines are basic to the algorithm. The first, subroutine STARMA, casts the ARMA model into the "state space" form necessary for Kalman filtering, and computes the covariance matrix associated with the initial value ofthe state vector. The second subroutine, KARMA, carries out the recursions and produces a set of standardized prediction errors, together with the determinant of the covariance matrix of the observations. These two quantities together yield the exact likelihood, and this may be maximized by an iterative procedure based on a numerical optimization algorithm which does not require analytic derivatives.Subroutine KARMA contains a device whereby the likelihood may be approximated to a level of accuracy which is under the control of the user. This enables a considerable amount of computing time to be saved, with very little attendant loss in precision.Finally, another subroutine, KALFO R, may be used to compute predictions offuture values of the series, together with the associated conditional mean square errors. THEORY An autoregressive-moving average process is defined by (1) where the sr's are normally and independently distributed with mean zero and variance (J2, and WI is observable. Such a process will be referred to as an ARMA(p, q) process and the set of parameters (¢1>""¢P, O!, ...,8 q ) will be denoted by (¢,O).An ARMA(p, q) process may be put in "state space" form by defining an r x 1 vector, lXI' which obeys the "transition equation"where r = max(p,q+ 1) and 312 T= APPLIED STATISTICS , and R = (3) Note that, unless p = q + 1, some of the

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARYThe regression model with autoregressive-moving average disturbances may be cast in a form suitable for the application of Kalman filtering techniques. This enables the generalized least squares estimator to be calculated without evaluating and inverting the covariance matrix of the disturbances. The problem of forecasting future values of the dependent variable is also effectively solved when the Kalman filter technique is applied. Furthermore, the properties of the residuals produced by the filter suggest that they may be useful for diagnostic checking of the model. The Kalman filter algorithm also forins the basis of a method for the exact maximum likelihood estimation of the model. This may well have computational, as well as theoretical, advantages over other methods.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Wiley,

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The small sample bias of the least-squares coefficient estimator is examined in the dynamic multiple linear regression model with normally distributed whitenoise disturbances and an arbitrary number of regressors which are all exogenous except for the one-period lagged-dependent variable. We employ large sample (T → ∞) and small disturbance (σ → 0) asymptotic theory and derive and compare expressions to O(T−1) and to O(σ2), respectively, for the bias in the least-squares coefficient vector. In some simulations and for an empirical example, we examine the mean (squared) error of these expressions and of corrected estimation procedures that yield estimates that are unbiased to O(T−l) and to O(σ2), respectively. The large sample approach proves to be superior, easily applicable, and capable of generating more efficient and less biased estimators.