Property 5: The Yule-Walker equations also hold where k = 0 provided we add a σ 2 term to the sum. These are known as the Yule-Walker equations. Here we assume that γ h = γ -h and ρ h = ρ -h if h < 0, and ρ 0 = 1. Similarly the autocorrelation at lag k > 0 can be calculated as
It turns out that such a process is stationary when |φ 1| 0 can be calculated as Similar to the ordinary linear regression model, we assume that the error terms are independently distributed based on a normal distribution with zero mean and a constant variance σ 2 and that the error terms are independent of the y values. Thinking of the subscripts i as representing time, we see that the value of y at time i+1 is a linear function of y at time i plus a fixed constant and a random error term. In a simple linear regression model, the predicted dependent variable is modeled as a linear function of the independent variable plus a random error term.Ī first-order autoregressive process, denoted AR(1), takes the form
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