Extrapolating time series by discounted least squares

An approximating function is fitted to a time series, such as daily observation. The fitting is carried out over all past time by weighted least squares with an exponential weight factor• The approximating function is restricted to be a solution of a certain linear differential equation of the mth order having constant coefficients. The solution which minimizes the least square expression can be continued into the future. In particular tomorrows extrapolated value is defined by this continuation. To obtain an explicit solution of the problem a formula is constructed which gives the extrapolated value as a linear combination of the last m observed values and the last m extrapolated values. The coefficients of this extrapolation formula prove to be simply related to the coefficients of the differential equation. Another extrapolation formula is of vectorial nature. The components of a vector are m independent functionals of the past observations. Then tomorrow's vector is given as a linear function of todays vector and todays scalar observation.