# Statistics for Environmental Engineers

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If we accept this model, the intervention estimation problem becomes how to determine the weighting factor в. Fortunately, an alternate formulation of the model makes this reasonably simple. The white noise-random walk model is equivalent to an ARIMA (0,1,1) model (Box et al., 1994):

yt = yt_ 1 + a, _ ва,_!

where 0 < в < 1, and a, = independent random noise distributed as N(0,o<2).

The white noise and the random walk error components are related to o2 as follows:

o2 = во2

o2 = (1 _ в)2о2

The equivalence of the two forms of the model and the derivations are given in Pallesen et al. (1985).

A recursive iteration is done separately for each section of data before and after the intervention to estimate в. The method is to:

1. Choose a starting value for в and use at= yt— yt-1+ ва,-1 to recursively calculate the residuals at t = 2, 3,…, T. Assume at=0= 0 to start the calculations.

2.    Calculate the residual sum of squares, RSS(e) = Y*a] for each section. Add these to get the total RSS for the entire series. If a gap has been used to account for a transition period, data in the gap are omitted from these calculations.

3.    Search over a range of в to minimize RSS(e).

4.    Use the minimum RSS to estimate al = RSS(e), where n is the total number of residuals

n