Predicting the Turning Points of a Time Series

Abstract

Students of economic time series recognize the notion of a -a point in time when a series which had been increasing reverses and, for a time, decreases.' An example of a turning point is given in figure 1 where the time series of quarterly, seasonally adjusted real GNP is shown. The time points 1957:3 and 1973:4 can be considered turning points since they each mark the beginning of a downturn. For economists, turning points have long been an object of study and comment (Zarnowitz 1972). Turning points have received relatively less attention from statisticians. Thus, the term is not found in the indexes of Wiener (1949), Yaglom (1962), Whittle (1963), Box and Jenkins (1970), Hannan (1970), Anderson (1971), Brillinger (1975), or Fuller (1976). More to the point (though more subjectively), the concept of a turning point is absent as well. In Section II of this paper I discuss the relationship between the theory of minimum mean square error linear prediction and the turning point prediction problem. I show why the linear prediction theory does not give a solution to the Linear least-squares prediction methods are not directly applicable to the prediction of time series points. In this paper the linear least-squares technique is extended to allow computation of the distribution of the turning points of a time series, conditional on past observations. The method is illustrated using quarterly seasonally adjusted GNP.

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