# Detrending using moving average

In Terrence C. Mills book Im currently reading The Foundation of Modern Time Series Analysis on while discussing Person's detrending methods, he mentions the following result.

... Concider the deviation of $x_t$ from a three year moving average centered on $$x_t:x_t-(x_{t+1}+x_t+x_{t-1})/3$$. This is equivalent to: $$-\frac{1}{3}(x_{t+1}-2x_t+x_{t-1})=-\frac{1}{3}\Delta^2x_{t+1}$$

so the correlation between the two series will be identical to the correlation between the deviations of the three-year moving averages...

pages:41-42

Im not sure how the above equality is derived. How does this work mathematically?

• I just want to make doubly sure: the thing you want demonstrated is that this $-\frac{1}{3}(x_{t+1}-2x_t+x_{t-1})$ equals this $-\frac{1}{3}\Delta^2x_{t+1}$? (I presume you know the definition of $\Delta$ being used there?) – Glen_b Nov 12 '17 at 5:27
• @Glen_b yeah I want to see how this equality exists. I know $\Delta$ refers to the number of times the data was differenced. – EconJohn Nov 12 '17 at 16:01