as blue noise. This similarly acts as a high-pass filter, everything except for blue light will be filtered. AR(2) AR(2) processes can be split into three groups depending on the characteristics of their roots: When , the process has a pair of complex-conjugate roots, creating a mid-frequency peak at: Otherwise the process has real roots, and: When it acts as a low-pass filter on the white noise with a spectral peak at When it acts as a high-pass filter on the white noise with a spectral peak at . The process is non-stationary when the roots are outside the unit circle. The process is stable when the roots are within the unit circle, or equivalently when the coefficients are in the triangle . The full PSD function can be expressed in real form as: Implementations in statistics packages R, the stats package includes an ar function. MATLAB's Econometrics Toolbox and System Identification Toolbox includes autoregressive models Matlab and Octave: the TSA toolbox contains several estimation functions for uni-variate, multivariate and adaptive autoregressive models. PyMC3: the Bayesian statistics and probabilistic programming framework supports autoregressive modes with p lags. bayesloop supports parameter inference and model selection for the AR-1 process with time-varying parameters. Python: implementation in statsmodels. Impulse response The impulse response of a system is the change in an evolving variable in response to a change in the value of a shock term k periods earlier, as a function of k. Since the AR model is a special case of the vector autoregressive model, the computation of the impulse response in vector autoregression#impulse response applies here. n-step-ahead forecasting Once the parameters of the autoregression have been estimated, the autoregression can be used to forecast an arbitrary number of periods into the future. First use t to refer to the first period for which data is not yet available; substitute the known preceding values Xt-i for i=1, ..., p into the autoregressive equation while setting the error term equal to zero (because we forecast Xt to equal its expected value, and the expected value of the unobserved error term is zero). The output of the autoregressive equation is the forecast for the first unobserved period. Next, use t to refer to the next period for which data is not yet available; again the autoregressive equation is used to make the forecast, with one difference: the value of X one period prior to