# Why taking the first difference is the same as an AR(1) filter?

Today, my professor said that for highly correlated time series, taking the first differentiation is like applying an AR(1) filter.. Unfortunately I a was not able to ask him after the lecture.

I am confused...

Formula for a differenced time series is

$Y'_{t} = Y_{t}-Y_{t-1}$.

For a stationary time series $Y'_{t}$ should be random noise so $\epsilon_{t}$. So the differenced time series becomes

$\epsilon_{t}$ =$Y_{t}-Y_{t-1}$.

Rearranging this we get:

$Y_{t}=Y_{t-1}+$ $\epsilon_{t}$

AR(1) is defined as

$Y_{t}=\theta_{1}$$Y_{t-1}$+$\epsilon_{t}$

So as $\theta_{1}$ approaches 1 (highly correlated) applying an AR(1) becomes like a differenced time series.