for , one can show that: , and . Choosing the maximum lag The partial autocorrelation of an AR(p) process equals zero at lags larger than p, so the appropriate maximum lag p is the one after which the partial autocorrelations are all zero. Calculation of the AR parameters There are many ways to estimate the coefficients, such as the ordinary least squares procedure or method of moments (through Yule–Walker equations). The AR(p) model is given by the equation It is based on parameters where i = 1, ..., p. There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the Yule–Walker equations. Yule–Walker equations The Yule–Walker equations, named for Udny Yule and Gilbert Walker, are the following set of equations. where , yielding equations. Here is the autocovariance function of Xt, is the standard deviation of the input noise process, and is the Kronecker delta function. Because the last part of an individual equation is non-zero only if , the set of equations can be solved by representing the equations for in matrix form, thus getting the equation which can be solved for all The remaining equation for m = 0 is which, once are known, can be solved for An alternative formulation is in terms of the autocorrelation function. The AR parameters are determined by the first p+1 elements of the autocorrelation function. The full autocorrelation function can then be derived by recursively calculating Examples for some Low-order AR(p) processes p=1 Hence p=2 The Yule–Walker equations for an AR(2) process are Remember that Using the first equation yields Using the recursion formula yields Estimation of AR parameters The above equations (the Yule–Walker equations) provide several routes to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values. Some of these variants can be described as follows: Estimation of autocovariances or autocorrelations. Here each of these terms is estimated separately, using conventional estimates. There are different ways of doing this and the choice between these affects the properties of the estimation scheme. For example, negative estimates of the variance can be produced by some choices. Formulation as a least squares regression problem in which an ordinary least squares prediction problem is constructed, basing prediction