Chapter two stationary time series model--ar (p), MA (q), ARMA (P,Q) model and its stationarity

1 White noise process:0 mean, same variance, no autocorrelation (covariance is 0)in the future we encounter the Efshow if not specifically described, is the white noise process. for normal distribution, irrelevant can be introduced independently, so if the white noise is normal distribution, it will also be independent of each other. 2 various and modelsP-Order moving average process:Q-Order autoregressive process:Autoregressive Moving Average model:if the characteristic root of an arma (P,Q) model has at least one value greater than or equal to 1, then {y (t)} is the integration process, at which point the model is called the autoregressive fall Moving Average model (ARIMA)time series Ah, not just to find a formula, and then to find a non-recursive form of expression? (this formula is related to the self-variable t, and then you can get the predicted value of the corresponding y as soon as you know T)3 weakly smooth/covariance stable:The mean and variance are constants (that is, the same variance), and the covariance is only related to the time interval4 self-correlation coefficient:the smoothness of the 5AR (1) model ( first-order difference equation with white noise ): (1) If the initial condition is y0:then the solution is(we judge whether it is smooth by its solution){y (t)} is not stable at this time. · However, if |a1|<1, its t is large enough, {y (t)} is smooth. mean value : Variance: equals Covariance: equals so there is a conclusion:          (2) initial conditions unknown:The General solution is: {y (t)} The condition for smoothing is: 1 |a1|<1 2 and Homogeneous Solution A (A1) ^t is 0: The sequence starts long ago (that is, T is large and combines 1, then 0).or the process is always smooth (a=0)Therefore, the stability of the solution and the smoothness of the sequence is not the same. These two are applicable to all ARMA (P,Q) models. (for arbitrary ARMA (P,Q) models, a homogeneous solution of 0 is a necessary condition for smoothness ) (The homogeneous solution of the ARMA (P,Q) model is or )  6 for the stability of the ARMA (2,1) Model:

The

model expression is: (2.16) (The Intercept term does not affect the smoothness, omitted) set its challenge solution as: (with the undetermined coefficient method)

the coefficients shall satisfy the equation: (2.17)The condition of the convergence of the sequence {alpha i} is the equation (2.16) for the characteristic roots of the homogeneous equation are within the unit circle (because the difference equation in 2.17 is exactly the same for the characteristic equation and equation 2.16 for the characteristic equation)we only consider the special solution because we make homogeneous 0.at this point the Challenge solution/special solution: The mean value is:The variance is: (T is large when the sum of the series)covariance is: equalsso the condition of its smoothness is (t very Large):The characteristic of the characteristic equation of the homogeneous equation corresponding to the 1 model is within the unit circle 2 The homogeneous solution is 0. 7AR (p) Stability of the model: Model:if its characteristic root is within the unit circle, its challenge solution is:{Alpha I} these undetermined coefficients will satisfy the characteristic equation(It is found that the characteristic equations of this characteristic equation and the model correspond to the homogeneous equation are consistent?) )The mean value, variance, and covariance of the solution are:    Therefore, all characteristic roots are within the unit circle, then there is(the necessary conditions for the smoothness of the early higher order difference equations are mentioned), so the mean is a finite length su   so the AR (p) Stationary condition is (t very Large): The characteristic of the characteristic equation of the homogeneous equation corresponding to the 1 model is within the unit circle 2 The homogeneous solution is 0.    8MA (q) The smoothness of the model:the Model is:     The MA (q) is always smooth because its sum of series sums to a finite series sum to Q)     Infinite Ma process The model is:     so as long as the sum of the two series is finite, the MA (Infinity) is stationary.  Smoothness of the 9ARMA (P,Q) Model:
the model is: (2.22)The form of the solution is: (2.23) as long as 2.22 of the characteristic roots are within the unit circle or The characteristic root of inverse characteristic equation is outside the unit circle , theneach item of (2.23) is smooth, so its and also smooth, so the ARMA (P,Q) model is stable. (No more explanations in the book)

from for Notes (Wiz)

Chapter two stationary time series model--ar (p), MA (q), ARMA (P,Q) model and its stationarity