AR(1) process

Summary

• A stochastic process \(\{ Y_t \}\) is called an autoregressive process of order 1 or an AR(1) process if

\[Y_t=δ+θY_{t-1}+ε_t\]

• where \(δ\) and \(θ\) are parameters and \(\{ ε_t \}\) is white noise.
• The AR(1) is started at \(t=0\) if \(Y_t=δ+θY_{t-1}+ε_t\) for \(t≥1\) . An assumption must then be made about the starting value \(Y_0\) ( \(Y_0\) may be a constant or a random variable)
• An AR(1) process satisfies the stability condition if \(\left| θ \right|<1\) .
• An AR(1) process is stationary if and only if it satisfies the stability condition.
• The stability condition is a necessary but not sufficient condition for stationarity for a started AR(1) process
• An AR(1) process which does not satisfies the stability condition must be a started process.
• Result: The expected value of a stationary AR(1) process \(Y_t=δ+θY_{t-1}+ε_t\) :

\[E\left( Y_t \right)= \frac{δ}{1-θ}\]

• If \(Y_t\) follows an AR(1) process, \(Y_t=δ+θY_{t-1}+ε_t\) , then \(y_t=Y_t-E\left( Y_t \right)\) will follow an AR(1) process with intercept 0

\[y_t=θy_{t-1}+ε_t\]

• This is notationally much more convenient. We have

\[E\left( y_t \right)=0\]

• Result: The variance of a stationary AR(1) process \(y_t=θy_{t-1}+ε_t\) :

\[Var\left( y_t \right)= \frac{σ^2}{1-θ^2}\]

• Result: The covariances of a stationary AR(1) process \(y_t=θy_{t-1}+ε_t\) :

\[Cov\left( y_t,y_{t-k} \right)=ρ^k \frac{σ^2}{1-θ^2}\]

• Result: The correlations (or autocorrelations) of a stationary AR(1) process \(y_t=θy_{t-1}+ε_t\) :

\[corr\left( y_t,y_{t-k} \right)=θ^k\]

• Autocorrelations viewed as a function of \(k\) is called the autocorrelation function (ACF).