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 Statistical Considerations for Variables Which are the Cumulative Sums of Random Disturbances

Variables which have a stochastic structure of the form

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

where the U(t)'s are random variables require different methods of analysis from those variables which have a stochastic structure of the form

#### Y(t) = f(t) + U(t)

where f(t) is a known function of time and the U(t)'s are random variables. This is a deterministic structure with random disturbances.

The name that has been applied to the first model shown above is random walk. Such data will appear to have trends even if the expected value of U[t] is zero for all t. This is illustrated below: (Click on REFRESH to get a new sample and new time series.)

Although almost every sample appears to show trends there is no long term trend in the variable.

For a contrast the diagram below shows what the time path looks like for a variable which has random disturbances about a linear trend.

There is of course the very important case which combines both of these stochastic structures; i.e.,

#### T(t) = T(t-1) + f(t) + U(t)

This is sometimes called a random walk with drift.

The diagram below illustrates the behavior of a variable which follows a random walk with drift.

There is also another important stochastic structure.

#### T[t] = T[t-1] + U[t] where U[t] = λU[t-1] + V[t]

and where V[t] is a random variable uncorrelated with V[s] for s≠t. Such data will appear to have trends even if the expected value of U[t] is zero for all t. This is illustrated below: (Click on REFRESH to get a new sample and new time series.)

This model can be called a random walk with autocorrelated disturbances. Again, although almost every sample appears to show trends there is no long term trend in the variable.

The following webpages explore the special statistical problems for variables which are cumulative sums of random disturbances. To repeat, one of the important aspects of such variables is that they display the appearance of short term trends even when there is no long term trend. This leads people to believe they are seeing a long term trend even when none exists.