The autoregressive model

Time Series Analysis in R

David S. Matteson

Associate Professor at Cornell University

The autoregressive model - I

The Autoregressive (AR) recursion:

$Today = Constant + Slope * Yesterday + Noise $

Mean centered version:

$(Today - Mean) = $

$ Slope*(Yesterday - Mean) + Noise $

Time Series Analysis in R

The autoregressive model - II

$$(Today - Mean) = $$

$$Slope * (Yesterday - Mean) + Noise$$

More formally: $$ Y_t - \mu = \phi (Y_{t-1} - \mu ) + \epsilon_t$$ where $ \epsilon_t$ is mean zero white noise (WN).

  • The mean $ \mu$
  • The slope $ \phi$
  • The WN variance $ \sigma ^2$
Time Series Analysis in R

AR processes - I

$$Y_t - \mu = \phi(Y_{t-1} - \mu) + \epsilon_t$$

  • If slope $ \phi = 0 $ then: $Y_t = \mu + \epsilon_t $ and

And $Y_t$ is white noise: $(\mu, \sigma _{\epsilon}^2)$

  • If slope $ \phi \neq 0 $ then: $ Y_t$ depends on both $ \epsilon_t$ and $ Y_{t-1} $

And the process {${Y_t}$} is autocorrelated

  • Large values of $ \phi$ lead to greater autocorrelation

  • Negative values of $ \phi$ result in oscillatory time series

Time Series Analysis in R

AR examples

Time Series Analysis in R

Autocorrelations

Time Series Analysis in R

Random walk

If $ \mu = 0$ and slope $ \phi = 1$, then:

$$Y_t = Y_{t-1} + \epsilon_t$$

Which is:

$Today = Yesterday + Noise $

But this is a random walk.

And {$ Y_t $} is not stationary in this case.

Time Series Analysis in R

Let's practice!

Time Series Analysis in R

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