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Model Building For ARIMA time series Consists of three steps 1.Identification 2.Estimation 3.Diagnostic checking

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ARIMA Model building Identification Determination of p, d and q

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To identify an ARIMA(p,d,q) we use extensively the autocorrelation function { h : - < h < } and the partial autocorrelation function, { kk : 0 k < }.

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The definition of the sample covariance function {C x (h) : - < h < } and the sample autocorrelation function {r h : - < h < } are given below: The divisor is T, some statisticians use T – h (If T is large, both give approximately the same results.)

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It can be shown that: Thus Assuming k = 0 for k > q

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The sample partial autocorrelation function is defined by:

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It can be shown that:

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Identification of an Arima process Determining the values of p,d,q

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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.

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To determine the value of d Note: the autocorrelation function for a stationary ARMA time series satisfies the following difference equation The solution to this equation has general form where r 1, r 2, r 1, … r p, are the roots of the polynomial

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For a stationary ARMA time series Therefore The roots r 1, r 2, r 1, … r p, have absolute value greater than 1. If the ARMA time series is non-stationary some of the roots r 1, r 2, r 1, … r p, have absolute value equal to 1, and

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stationary non-stationary

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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.

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Identification Determination of the values of p and q.

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To determine the value of p and q we use the graphical properties of the autocorrelation function and the partial autocorrelation function. Again recall the following:

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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

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Patterns of the ACF and PACF of MA(2) Time Series In the shaded region the roots of the MA operator are complex

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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.

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Summary: To determine p and q. Use the following table. MA(q)AR(p)ARMA(p,q) ACFCuts after qTails off PACFTails offCuts after pTails off 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.)

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Examples

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The data

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Possible Identifications 1.d = 0, p = 1, q= 1 2.d = 1, p = 0, q= 1

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ACF and PACF for x t, x t and 2 x t (Sunspot Data)

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Possible Identification 1.d = 0, p = 2, q= 0

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ACF and PACF for x t, x t and 2 x t (IBM Stock Price Data)

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Possible Identification 1.d = 1, p =0, q= 0

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Estimation of ARIMA parameters

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Preliminary Estimation Using the Method of moments Equate sample statistics to population paramaters

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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 1, 2, …, q we solve the system of equations:

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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

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For q = 2 the equations become:

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Estimation of parameters of an ARMA(p,q) series We use a similar technique. Namely: Obtain an expression for h in terms 1, 2,..., p ; 1, 1,..., q of and set up q + p equations for the estimates of 1, 2,..., p ; 1, 2,..., q by replacing h by r h.

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Estimation of parameters of an ARMA(p,q) series Example: The ARMA(1,1) process The expression for 1 and 2 in terms of 1 and 1 are: Further

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Thus the expression for the estimates of 1, 1, and 2 are : and

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Hence or This is a quadratic equation which can be solved

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Example (ChemicalConcentration Data) the time series was identified as either an ARIMA(1,0,1) time series or an ARIMA(0,1,1) series. If we use the first identification then series x t is an ARMA(1,1) series.

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Identifying the series x t is an ARMA(1,1) series. The autocorrelation at lag 1 is r 1 = and the autocorrelation at lag 2 is r 2 = Thus the estimate of 1 is 0.495/0.570 = Also the quadratic equation becomes which has the two solutions and Again we select as our estimate of 1 to be the solution -0.48, resulting in an invertible estimated series.

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Since = (1 - 1 ) the estimate of can be computed as follows: Thus the identified model in this case is x t = 0.87 x t-1 + u t u t

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If we use the second identification then series x t = x t – x t-1 is an MA(1) series. Thus the estimate of 1 is: The value of r 1 = Thus the estimate of 1 is: The estimate of 1 = -0.53, corresponds to an invertible time series. This is the solution that we will choose

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The estimate of the parameter is the sample mean. Thus the identified model in this case is: x t = u t u t or x t = x t-1 + u t u t This compares with the other identification: x t = 0.87 x t-1 + u t u t (An ARIMA(1,0,1) model) (An ARIMA(0,1,1) model)

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Preliminary Estimation of the Parameters of an AR(p) Process

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and The regression coefficients 1, 2, …., p and the auto correlation function h satisfy the Yule- Walker equations:

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and The Yule-Walker equations can be used to estimate the regression coefficients 1, 2, …., p using the sample auto correlation function r h by replacing h with r h.

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Example Considering the data in example 1 (Sunspot Data) the time series was identified as an AR(2) time series. The autocorrelation at lag 1 is r 1 = and the autocorrelation at lag 2 is r 2 = The equations for the estimators of the parameters of this series are which has solution Since = ( 1 - 1 - 2 ) then it can be estimated as follows:

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Thus the identified model in this case is x t = x t x t-2 + u t +14.9

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Maximum Likelihood Estimation of the parameters of an ARMA(p,q) Series

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The method of Maximum Likelihood Estimation selects as estimators of a set of parameters 1, 2,..., k, the values that maximize L( 1, 2,..., k ) = f(x 1,x 2,..., x N ; 1, 2,..., k ) where f(x 1,x 2,..., x N ; 1, 2,..., k ) is the joint density function of the observations x 1,x 2,..., x N. L( 1, 2,..., k ) is called the Likelihood function.

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It is important to note that: finding the values - 1, 2,..., k - to maximize L( 1, 2,..., k ) is equivalent to finding the values to maximize l( 1, 2,..., k ) = ln L( 1, 2,..., k ). l( 1, 2,..., k ) is called the log-Likelihood function.

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Again let {u t : t T} be identically distributed and uncorrelated with mean zero. In addition assume that each is normally distributed. Consider the time series {x t : t T} defined by the equation: (*)x t = 1 x t-1 + 2 x t p x t-p + + u t + 1 u t-1 + 2 u t q u t-q

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Assume that x 1, x 2,...,x N are observations on the time series up to time t = N. To estimate the p + q + 2 parameters 1, 2,..., p ; 1, 2,..., q ; , 2 by the method of Maximum Likelihood estimation we need to find the joint density function of x 1, x 2,...,x N f(x 1, x 2,..., x N | 1, 2,..., p ; 1, 2,..., q, , 2 ) = f(x| , , , 2 ).

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We know that u 1, u 2,...,u N are independent normal with mean zero and variance 2. Thus the joint density function of u 1, u 2,...,u N is g(u 1, u 2,...,u N ; 2 ) = g(u ; 2 ) is given by.

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It is difficult to determine the exact density function of x 1,x 2,..., x N from this information however if we assume that p starting values on the x-process x* = (x 1-p,x 2-p,..., x o ) and q starting values on the u-process u* = (u 1-q,u 2-q,..., u o ) have been observed then the conditional distribution of x = (x 1,x 2,..., x N ) given x* = (x 1- p,x 2-p,..., x o ) and u* = (u 1-q,u 2-q,..., u o ) can easily be determined.

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The system of equations : x 1 = 1 x 0 + 2 x p x 1-p + + u 1 + 1 u 0 + 2 u q u 1-q x 2 = 1 x 1 + 2 x p x 2-p + + u 2 + 1 u 1 + 2 u q u 2-q... x N = 1 x N-1 + 2 x N p x N-p + + u N + 1 u N-1 + 2 u N q u N-q

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can be solved for: u 1 = u 1 (x, x*, u*; , , ) u 2 = u 2 (x, x*, u*; , , )... u N = u N (x, x*, u*; , , ) (The jacobian of the transformation is 1)

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Then the joint density of x given x* and u* is given by:

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Let: = “conditional likelihood function”

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“conditional log likelihood function” =

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The values that maximize are the values that minimize with

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Comment: Requires a iterative numerical minimization procedure to find: The minimization of: Steepest descent Simulated annealing etc

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Comment: for specific values of The computation of: can be achieved by using the forecast equations

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Comment: assumes we know the value of starting values of the time series {x t | t T} and {u t | t T} The minimization of : Namely x* and u*.

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Approaches: 1.Use estimated values: 2.Use forecasting and backcasting equations to estimate the values:

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Backcasting: If the time series {x t |t T} satisfies the equation: It can also be shown to satisfy the equation: Both equations result in a time series with the same mean, variance and autocorrelation function: In the same way that the first equation can be used to forecast into the future the second equation can be used to backcast into the past:

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Approaches to handling starting values of the series {x t |t T} and {u t |t T} 1.Initially start with the values: 2.Estimate the parameters of the model using Maximum Likelihood estimation and the conditional Likelihood function. 3.Use the estimated parameters to backcast the components of x*. The backcasted components of u* will still be zero.

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4.Repeat steps 2 and 3 until the estimates stablize. This algorithm is an application of the E-M algorithm This general algorithm is frequently used when there are missing values. The E stands for Expectation (using a model to estimate the missing values) The M stands for Maximum Likelihood Estimation, the process used to estimate the parameters of the model.

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Some Examples using: Minitab Statistica S-Plus SAS

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