Autoregression

An autoregression is a time series model that typically uses the previous values of the same series as an explanatory factor for the regression in order to predict the next value. Let's say that we have measured and kept track of a metric over time, called y_t, which is measured at time t when this value is regressed on previous values from that same time series. For example, y_t on ${\mathrm{yt-1}}^{}$:

As shown in the preceding equation, the previous value y_t-1 has become the predictor here and y_t is the response value that is to be predicted. Also, ε_t is normally distributed with a mean of zero and variance of 1. The order of the autoregression model is defined by the number of previous values that are being used by the model to determine the next value. Therefore, the preceding equation is a first-order autoregression, or AR(1). If we have to generalize it, a order autoregression, written as AR(k), is a multiple linear regression in which the value of the series at any time (t) is a (linear) function of the values at times $t-1,t-2,\dots ,$

The following list shows what the following values means for an AR(1) model:

• When β₁ = 0, ${\mathrm{yt,\; it}}^{}$is equivalent to white noise
• When β₁ = 1 and β₀= 0, y_t, it is equivalent to a random walk
• When β₁ = 1 and β₀ ≠ 0, y_t, it is equivalent to a random walk with drift
• When β₁ < 1, y_t, it tends to oscillate between positive and negative values