Apparent Trends in Variables Which are the Cumulative Sum of Random Disturbances

San José State University Department of Economics

applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

Apparent Trends in Variables Which
are the Cumulative Sum of
Random Disturbances

Consider a variable, such as the temperature of a body, which is the cumulative sum of another variable, such as the net heat input; i.e.,

T(t+1) = T(t) + U(t)

where T is temperature, t is time, and U(t) is the net heat input over the interval t to t+1. The following chart shows the simulation of such a variable:

Typically such variables display periods of apparent trends even though there may be no long term trend involved for the variable. The topic addressed in this material is the analytic quantification of these periods of apparent trends; i.e. what is their average length and what is their trend rate.
The effect of operations such as summing or averaging on random disturbances can be handled satisfactory by means of the spectrum of a series. For more see Spectra of Various Transformations of White Noise. The characteristics of cycles such as cycle period and amplitude do not provide information on trend length.
The matter of trend length can be addressed more directly in terms of runs or quasi-runs of rates of increase. The expected lengths of runs based upon this analysis are far shorter than what one perceives in displays such as the one above. For more on expected run lengths see Runs.
An unsuccessful attempt to construct an algorithm for determining the trend lengths leads to the conjecture that the trends are perceived rather than being inherent in the data. The observer of the plotted data seems to visually carryout a smoothing operation. The diagram below shows the series and a simply three-period moving average of the series.

Having smoothed the series the observer visually perceives the slopes of the smoothed series. The slopes are then visually smoothed and the trend is assessed from the smoothed rates of increase.

The operation of smoothing, differencing and then smoothing tends to creat cycles where none exist in the original data. A variable following a sinusoidal pattern spends a good deal of its time near the maximu and the minimum. Thus the trend level and the time spent at that level are determined by the period and amplitude of dominant cyles. Such cycles are shown by the spectrum for the smoothed rate of growth.
Below is shown the spectrum for the smoothed growth rates from a series generated by 2000 random values. Each time the screen is refreshed the spectrum for a new sample is computed.
The peaks in the sample spectra are determined by the smoothing and differencing operations and not due to any cyclicity in the random variable. These peaks correspond to spurious cyclicities introduced by the smoothing and differencing operations. For more on this topic see Spectra of Various Transformations of White Noise.
(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins