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**Model Building For ARIMA time series**

Consists of three steps Identification Estimation Diagnostic checking

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**Determination of p, d and q**

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 {rh : - < h < } and the partial autocorrelation function, {Fkk: 0 k < }.

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**The definition of the sample covariance function **

{Cx(h) : - < h < } and the sample autocorrelation function {rh: - < 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 rk = 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 r1, r2, r1, … rp, are the roots of the polynomial

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**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

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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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**Determination of the values of p and q.**

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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**Patterns of the ACF and PACF of AR(2) Time Series **

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) ACF Cuts after q Tails off PACF Cuts after p 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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The data

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**Possible Identifications**

d = 0, p = 1, q= 1 d = 1, p = 0, q= 1

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**ACF and PACF for xt ,Dxt and D2xt (Sunspot Data)**

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**Possible Identification**

d = 0, p = 2, q= 0

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**ACF and PACF for xt ,Dxt and D2xt (IBM Stock Price Data)**

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**Possible Identification**

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 a1, a2, … , aq 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 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, ... , aq by replacing rh by rh.

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**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

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**Thus the expression for the estimates of b1, a1, and s2 are :**

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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 xt is an ARMA(1,1) series.

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**Identifying the series xt is an ARMA(1,1) series.**

The autocorrelation at lag 1 is r1 = and the autocorrelation at lag 2 is r2 = Thus the estimate of b1 is 0.495/0.570 = 0.87. Also the quadratic equation becomes which has the two solutions and Again we select as our estimate of a1 to be the solution -0.48, resulting in an invertible estimated series.

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**Since d = m(1 - b1) the estimate of d can be computed as follows:**

Thus the identified model in this case is xt = 0.87 xt-1 + ut ut

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**If we use the second identification then series**

Dxt = xt – xt-1 is an MA(1) series. Thus the estimate of a1 is: The value of r1 = Thus the estimate of a1 is: The estimate of a1 = -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 m is the sample mean. **

Thus the identified model in this case is: Dxt = ut ut or xt = xt-1 + ut ut This compares with the other identification: (An ARIMA(0,1,1) model) xt = 0.87 xt-1 + ut ut (An ARIMA(1,0,1) model)

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**Preliminary Estimation**

of the Parameters of an AR(p) Process

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**The regression coefficients b1, b2, …**

The regression coefficients b1, b2, …., bp and the auto correlation function rh satisfy the Yule-Walker equations: and

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The Yule-Walker equations can be used to estimate the regression coefficients b1, b2, …., bp using the sample auto correlation function rh by replacing rh with rh. and

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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 r1 = and the autocorrelation at lag 2 is r2 = The equations for the estimators of the parameters of this series are which has solution Since d = m( 1 -b1 - b2) then it can be estimated as follows:

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**Thus the identified model in this case is**

xt = xt xt-2 + ut +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 q1,q2, ... , qk , the values that maximize L(q1,q2, ... , qk) = f(x1,x2, ... , xN;q1,q2, ... , qk) where f(x1,x2, ... , xN;q1,q2, ... , qk) is the joint density function of the observations x1,x2, ... , xN. L(q1,q2, ... , qk) is called the Likelihood function.

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**It is important to note that:**

finding the values -q1,q2, ... , qk- to maximize L(q1,q2, ... , qk) is equivalent to finding the values to maximize l(q1,q2, ... , qk) = ln L(q1,q2, ... , qk). l(q1,q2, ... , qk) is called the log-Likelihood function.

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Again let {ut : t ÎT} be identically distributed and uncorrelated with mean zero. In addition assume that each is normally distributed . Consider the time series {xt : t ÎT} defined by the equation: (*) xt = b1xt-1 + b2xt bpxt-p + d + ut +a1ut-1 + a2ut aqut-q

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**Assume that x1, x2, ...,xN are observations on the time series up to time t = N.**

To estimate the p + q + 2 parameters b1, b2, ... ,bp ; a1, a2, ... ,aq ; d , s2 by the method of Maximum Likelihood estimation we need to find the joint density function of x1, x2, ...,xN f(x1, x2, ..., xN |b1, b2, ... ,bp ; a1, a2, ... ,aq , d, s2) = f(x| b, a, d ,s2).

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**We know that u1, u2, ...,uN are independent normal with mean zero and variance s2.**

Thus the joint density function of u1, u2, ...,uN is g(u1, u2, ...,uN ; s2) = g(u ; s2) is given by.

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**It is difficult to determine the exact density function of x1,x2,**

It is difficult to determine the exact density function of x1,x2, ... , xN from this information however if we assume that p starting values on the x-process x* = (x1-p,x2-p, ... , xo) and q starting values on the u-process u* = (u1-q,u2-q, ... , uo) have been observed then the conditional distribution of x = (x1,x2, ... , xN) given x* = (x1-p,x2-p, ... , xo) and u* = (u1-q,u2-q, ... , uo) can easily be determined.

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**The system of equations :**

x1 = b1x0 + b2x bpx1-p + d + u1 +a1u0 + a2u aqu1-q x2 = b1x1 + b2x bpx2-p + d + u2 +a1u1 + a2u aqu2-q ... xN= b1xN-1 + b2xN bpxN-p + d + uN +a1uN-1 + a2uN aquN-q

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can be solved for: u1 = u1 (x, x*, u*; b, a, d) u2 = u2 (x, x*, u*; b, a, d) ... uN = uN (x, x*, u*; b, a, d) (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: The minimization of: Requires a iterative numerical minimization procedure to find: Steepest descent Simulated annealing etc

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

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

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

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**Backcasting: If the time series {xt|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 {xt|t T} and {ut|t T}**

Initially start with the values: Estimate the parameters of the model using Maximum Likelihood estimation and the conditional Likelihood function. Use the estimated parameters to backcast the components of x*. The backcasted components of u* will still be zero.

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**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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