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Model Building For ARIMA time series

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## Model Building For ARIMA time series

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**Model Building For ARIMA time series**Consists of three steps Identification Estimation Diagnostic checking**ARIMA Model building**Identification Determination of p, d and q**To identify an ARIMA(p,d,q) we use extensively**the autocorrelation function {rh : - < h < } and the partial autocorrelation function, {Fkk: 0 k < }.**The divisor is T, some statisticians use T – h (If T is**large, both give approximately the same results.) The definition of the sample covariance function {Cx(h) : - < h < } and the sample autocorrelation function {rh: - < h < } are given below:**It can be shown that:**Thus Assuming rk = 0 for k > q**Identification of an Arima process**Determining the values of p,d,q**Recall that if a process is stationary one of the roots of**the autoregressive operator is equal to one. • This will cause the limiting value of the autocorrelation function to be non-zero. • Thus a nonstationary process is identified by an autocorrelation function that does not tail away to zero quickly or cut-off after a finite number of steps.**Note: the autocorrelation function for a stationary ARMA**time series satisfies the following difference equation To determine the value of d The solution to this equation has general form where r1, r2, r1, … rp, are the roots of the polynomial**For a stationary ARMA time series**The roots r1, r2, r1, … rp, have absolute value greater than 1. Therefore If the ARMA time series is non-stationary some of the roots r1, r2, r1, … rp, have absolute value equal to 1, and**stationary**non-stationary**If the process is non-stationary then first differences of**the series are computed to determine if that operation results in a stationary series. • The process is continued until a stationary time series is found. • This then determines the value of d.**Identification**Determination of the values of p and q.**To determine the value of pand q we use the graphical**properties of the autocorrelation function and the partial autocorrelation function. Again recall the following:**More specically some typical patterns of the autocorrelation**function and the partial autocorrelation function for some important ARMA series are as follows: Patterns of the ACF and PACF of AR(2) Time Series In the shaded region the roots of the AR operator are complex**Patterns of the ACF and PACF of MA(2) Time Series**In the shaded region the roots of the MA operator are complex**Patterns of the ACF and PACF of ARMA(1.1) Time Series**Note: The patterns exhibited by the ACF and the PACF give important and useful information relating to the values of the parameters of the time series.**Summary: To determine p and q.**Use the following table. Note: Usually p + q ≤ 4. There is no harm in over identifying the time series. (allowing more parameters in the model than necessary. We can always test to determine if the extra parameters are zero.)**Possible Identifications**• d = 0, p = 1, q= 1 • d = 1, p = 0, q= 1**Possible Identification**• d = 0, p = 2, q= 0**Possible Identification**• d = 1, p =0, q= 0**Estimation**of ARIMA parameters**Preliminary Estimation**Using the Method of moments Equate sample statistics to population paramaters**Estimation of parameters of an MA(q) series**The theoretical autocorrelation function in terms the parameters of an MA(q) process is given by. To estimate a1, a2, … , aq we solve the system of equations:**This set of equations is non-linear and generally very**difficult to solve For q = 1 the equation becomes: Thus or This equation has the two solutions One solution will result in the MA(1) time series being invertible**Estimation of parameters of anARMA(p,q) series**We use a similar technique. Namely: Obtain an expression for rh in terms b1, b2 , ... , bp ; a1, a1, ... , aq of and set up q + p equations for the estimates of b1, b2 , ... , bp ; a1, a2, ... , aqby replacing rh by rh.**Estimation of parameters of an ARMA(p,q) series**Example: The ARMA(1,1) process The expression for r1 and r2 in terms of b1 and a1 are: Further**Thus the expression for the estimates of b1, a1,**and s2 are : and**Hence**or This is a quadratic equation which can be solved