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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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To identify an ARIMA(p,d,q) we use extensively

the autocorrelation function

{rh: -< h< }

andthe partial autocorrelation function,

{Fkk: 0 k< }.

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It can be shown that:

t

kttkhhTrrCov rr1

,

Thus

q

tt

tth rTTrVar 1

22

21

11

r

Assumingrk= 0 for k> q

q

t

tr rT

sh

1

2211

Let

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The sample partial autocorrelation function is defined

by:

1

1

1

1

1

21

21

11

21

21

11

F

kk

k

k

kkk

kk

rr

rr

rr

rrr

rr

rr

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It can be shown that:

T

Var kk1 F

Ts

kk

1Let

F

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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 theroots of the autoregressive operator is equal toone.

This will cause the limiting value of the

autocorrelation function to be non-zero.

Thus a nonstationary process is identified byan autocorrelation function that does not tail

away to zero quickly or cut-off after a finitenumber 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 equation1 1 2 2h h h p h pr r r r

The solution to this equation has general form

1 2

1 2

1 1 1h ph h h

p

c c cr r r

r

where r1, r2, r1, rp, are the roots of the polynomial

21 21 p

px x x x

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For a stationary ARMA time series

Therefore

1 2

1 2

1 1 10 ash ph h h

p

c c c hr r r

r

The roots r1, r2, r1, rp, have absolute value greaterthan 1.

If the ARMA time series is non-stationary

some of the roots r1, r2, r1, rp, have absolute value

equal to 1, and

1 21 2

1 1 10 as

h ph h hp

c c c a hr r r

r

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0

0.5

1

0 3 6 9 12 15 18 21 24 27 30

stationary

0

0.5

1

0 3 6 9 12 15 18 21 24 27 30

non-stationary

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If the process is non-stationary then firstdifferences 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 ofp and q.

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To determine the value ofpand qwe use the

graphical properties of the autocorrelation

function and the partial autocorrelation function.

Again recall the following:

Auto-correlationfunction

PartialAutocorrelationfunction

Cuts off

Cuts off

Infinite. Tails off.

Damped Exponentials

and/or Cosine waves

Infinite. Tails off.

Infinite. Tails off.Infinite. Tails off.

Dominated by damped

Exponentials & Cosine

waves.

Dominated by damped

Exponentials & Cosine waves

Damped Exponentials

and/or Cosine wavesafter q-p.

after p-q.

Process MA(q) AR(p) ARMA(p,q)

Properties of the ACF and PACF of MA, AR and ARMA Series

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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 determinep and q.

Use the following table.

MA(q) AR(p) ARMA(p,q)

ACF Cuts after q Tails off Tails offPACF Tails off Cuts afterp Tails off

Note: Usuallyp + q 4. There is no harm in overidentifying 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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2001000

16

17

18

Ex ample A: "Uncontrolled" Concentration, Two-Hourly Readings:

Chemical Process

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The data

1 17.0 41 17.6 81 16.8 121 16.9 161 17.1

2 16.6 42 17.5 82 16.7 122 17.1 162 17.13 16.3 43 16.5 83 16.4 123 16.8 163 17.1

4 16.1 44 17.8 84 16.5 124 17.0 164 17.45 17.1 45 17.3 85 16.4 125 17.2 165 17.2

6 16.9 46 17.3 86 16.6 126 17.3 166 16.97 16.8 47 17.1 87 16.5 127 17.2 167 16.9

8 17.4 48 17.4 88 16.7 128 17.3 168 17.0

9 17.1 49 16.9 89 16.4 129 17.2 169 16.710 17.0 50 17.3 90 16.4 130 17.2 170 16.911 16.7 51 17.6 91 16.2 131 17.5 171 17.3

12 17.4 52 16.9 92 16.4 132 16.9 172 17.813 17.2 53 16.7 93 16.3 133 16.9 173 17.8

14 17.4 54 16.8 94 16.4 134 16.9 174 17.615 17.4 55 16.8 95 17.0 135 17.0 175 17.5

16 17.0 56 17.2 96 16.9 136 16.5 176 17.017 17.3 57 16.8 97 17.1 137 16.7 177 16.9

18 17.2 58 17.6 98 17.1 138 16.8 178 17.1

19 17.4 59 17.2 99 16.7 139 16.7 179 17.220 16.8 60 16.6 100 16.9 140 16.7 180 17.421 17.1 61 17.1 101 16.5 141 16.6 181 17.5

22 17.4 62 16.9 102 17.2 142 16.5 182 17.923 17.4 63 16.6 103 16.4 143 17.0 183 17.0

24 17.5 64 18.0 104 17.0 144 16.7 184 17.025 17.4 65 17.2 105 17.0 145 16.7 185 17.0

26 17.6 66 17.3 106 16.7 146 16.9 186 17.227 17.4 67 17.0 107 16.2 147 17.4 187 17.3

28 17.3 68 16.9 108 16.6 148 17.1 188 17.4

29 17.0 69 17.3 109 16.9 149 17.0 189 17.430 17.8 70 16.8 110 16.5 150 16.8 190 17.031 17.5 71 17.3 111 16.6 151 17.2 191 18.0

32 18.1 72 17.4 112 16.6 152 17.2 192 18.233 17.5 73 17.7 113 17.0 153 17.4 193 17.6

34 17.4 74 16.8 114 17.1 154 17.2 194 17.835 17.4 75 16.9 115 17.1 155 16.9 195 17.7

36 17.1 76 17.0 116 16.7 156 16.8 196 17.237 17.6 77 16.9 117 16.8 157 17.0 197 17.4

38 17.7 78 17.0 118 16.3 158 17.4

39 17.4 79 16.6 119 16.6 159 17.240 17.8 80 16.7 120 16.8 160 17.2

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18901860183018001770

0

100

200

Example B: Annual Sunspot Numbers

(1790-1869)

Example B: Sunspot Numbers: Yearly

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The data

Example B: Sunspot Numbers: Yearly

1770 101 1795 21 1820 16 1845 40

1771 82 1796 16 1821 7 1846 64

1772 66 1797 6 1822 4 1847 98

1773 35 1798 4 1823 2 1848 124

1774 31 1799 7 1824 8 1849 961775 7 1800 14 1825 17 1850 66

1776 20 1801 34 1826 36 1851 64

1777 92 1802 45 1827 50 1852 54

1778 154 1803 43 1828 62 1853 39

1779 125 1804 48 1829 67 1854 21

1780 85 1805 42 1830 71 1855 7

1781 68 1806 28 1831 48 1856 41782 38 1807 10 1832 28 1857 23

1783 23 1808 8 1833 8 1858 55

1784 10 1809 2 1834 13 1859 94

1785 24 1810 0 1835 57 1860 96

1786 83 1811 1 1836 122 1861 77

1787 132 1812 5 1837 138 1862 591788 131 1813 12 1838 103 1863 44

1789 118 1814 14 1839 86 1864 47

1790 90 1815 35 1840 63 1865 30

1791 67 1816 46 1841 37 1866 16

1792 60 1817 41 1842 24 1867 7

1793 47 1818 30 1843 11 1868 37

1794 41 1819 24 1844 15 1869 74

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3002001000

300

400

500

600

700Daily IBM Common Stock Closing Prices

May 17 1961-November 2 1962

Day

Price($)

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Example C: IBM Common Stock Closing Prices: Daily (May 17 1961- Nov 2 1962)

460 471 527 580 551 523 333 394 330457 467 540 579 551 516 330 393 340452 473 542 584 552 511 336 409 339459 481 538 581 553 518 328 411 331462 488 541 581 557 517 316 409 345459 490 541 577 557 520 320 408 352463 489 547 577 548 519 332 393 346479 489 553 578 547 519 320 391 352493 485 559 580 545 519 333 388 357490 491 557 586 545 518 344 396492 492 557 583 539 513 339 387498 494 560 581 539 499 350 383499 499 571 576 535 485 351 388497 498 571 571 537 454 350 382496 500 569 575 535 462 345 384490 497 575 575 536 473 350 382489 494 580 573 537 482 359 383478 495 584 577 543 486 375 383487 500 585 582 548 475 379 388491 504 590 584 546 459 376 395

487 513 599 579 547 451 382 392482 511 603 572 548 453 370 386487 514 599 577 549 446 365 383482 510 596 571 553 455 367 377479 509 585 560 553 452 372 364478 515 587 549 552 457 373 369479 519 585 556 551 449 363 355477 523 581 557 550 450 371 350479 519 583 563 553 435 369 353475 523 592 564 554 415 376 340479 531 592 567 551 398 387 350476 547 596 561 551 399 387 349

478 551 596 559 545 361 376 358479 547 595 553 547 383 385 360477 541 598 553 547 393 385 360476 545 598 553 537 385 380 366475 549 595 547 539 360 373 359473 545 595 550 538 364 382 356474 549 592 544 533 365 377 355474 547 588 541 525 370 376 367474 543 582 532 513 374 379 357465 540 576 525 510 359 386 361466 539 578 542 521 335 387 355467 532 589 555 521 323 386 348

471 517 585 558 521 306 389 343Read downwards

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Chemical Concentration data:

2001000

16

17

18

Example A: "Uncontr olled" Concentration, Two-Hourly Readings:

Chemical Pr ocess

par Summary Statistics

d N Mean Std. Dev.

0 197 17.062 0.398

1 196 0.002 0.369

2 195 0.003 0.622

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ACF and PACF for xt ,xt and 2xt (Chemical Concentration Data)

h10 20 30 40

hr

-1.0

0.0

1.0x t

-1

0

1

10 20 30 40

Fkk

^

x t

k

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-1.0

0.0

1.0hr

10 20 30 40h

x t

-1.0

0.0

1.0

Fkk

^x t

10 20 30 40

k

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-1.0

0.0

1.0

10 20 30 40h

x t2

hr

-1.0

0.0

1.0

10 20 30 40

Fkk

k

x t2

^

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Possible Identifications

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

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

Sunspot Data:

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Sunspot Data:

18901860183018001770

0

100

200

Example B: Annual Sunspot Numbers

(1790-1869)

Summary Statistics for the Sunspot Data

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ACF and PACF for xt,xtand 2xt (SunspotData)

-1.0

0.0

1.0

rh

h

10 20 30 40

x t

-1.0

0.0

1.0

x tF

kk

10 20 30 40k

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-1.0

0.0

1.0

x t

10 20 30 40h

rh

-1.0

0.0

1.0

x t

10 20 30 40

Fkk

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-1.0

0.0

1.0

x t2

rh

10 20 30 40h

-1.0

0.0

1.0

10 20 30 40

Fkk

k

x t2

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Possible Identification

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

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IBM stock data:

3002001000

300

400

500

600

700 Daily IBM Common StockClosing Prices

May 17 1961-Nove mber 2 1962

Day

Price($)

Summary Statistics

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ACF and PACF for xt,xtand 2xt (IBM Stock PriceData)

-1.0

0.0

1.0

x t

rh

10 20 30 40h

-1.0

0.0

1.0

10 20 30 40k

Fkkx t

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-1.0

0.0

1.0

rh x t

10 20 30 40h

-1.0

0.0

1.0

x t

Fkk

10 20 30 40k

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-1.0

0.0

1.0rh2 x t

10 20 30 40h

-1.0

0.0

1.0

2 x t

Fkk

10 20 30 40k

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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 populationparamaters

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Estimation of parameters of an MA(q) series

The theoretical autocorrelation function in terms theparameters of an MA(q) process is given by.

qh

qhq

qhqhh

h

0

11 22221

11

r

To estimate 1, 2, , qwe solve the system of

equations:

qhrq

qhqhh

h

1

1

22

2

2

1

11

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This set of equations is non-linear and generally verydifficult to solve

For q = 1 the equation becomes:

Thus

2

1

1

11

r

01 1121 r

or 0 112

11 rr

This equation has the two solutions

14

1

2

1

2

11

1 rr

One solution will result in the MA(1) time series being

invertible

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For q = 2 the equations become:

2

2

2

1

2111

1

r

2

2

2

1

22

1

r

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Estimation of parameters of an

ARMA(p,q) series

We use a similar technique.

Namely: Obtain an expression for rhin terms 1,

2, ... , p; 1, 1, ... , qof and set up q +p

equations for the estimates of 1, 2, ... , p; 1,

2, ... , qby replacingrhby rh.

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Estimation of parameters of an ARMA(p,q) series

112

11

2

1

11111

211

rr

r

Example: The ARMA(1,1) process

The expression for r1and r2in terms of 1and

1are:

Further

021

1

11

2

1

2

12

xtuVar s

s

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112

11

2

1

1111

1

21

1

rr

r

Thus the expression for the estimates of 1, 1,

and s2are :

and

021

1

11

2

1

2

12

xC

s

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1111112111

2

1

121

and

r

r

r

Hence

or

1

21

1

21

1

21

2

11 121

r

r

r

r

r

rr

This is a quadratic equation which can be solved

0

12

1

2112

1

2

22

2

1

1

21

r

rrr

rrr

rr

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Example (ChemicalConcentration Data)

the time series was identified as either anARIMA(1,0,1) time series or an ARIMA(0,1,1)

series.

If we use the first identification then seriesxtis an

ARMA(1,1) series.

Id tif i th i i ARMA(1 1) i

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Identifying the seriesxtis an ARMA(1,1) series.

The autocorrelation at lag 1 is r1= 0.570 and the

autocorrelation at lag 2 is r2= 0.495 .

Thus the estimate of 1is 0.495/0.570 = 0.87.

Also the quadratic equation

becomes

0121

2112

1

2

22

21

1

21

rrr

rrr

rrr

02984.0

7642.0

2984.0 12

1

which has the two solutions -0.48 and -2.08. Again we

select as our estimate of 1to be the solution -0.48,

resulting in an invertibleestimated series.

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Since d= m(1 - 1) the estimate of dcan be computed as

follows:

Thus the identified model in this case is

xt= 0.87xt-1+ ut- 0.48 ut-1+ 2.25

25.2)87.01(062.171 1 d x

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If we use the second identification then series

xt = xtxt-1 is an MA(1) series.

Thus the estimate of 1is:

14

1

2

1

2

11

1 rr

The value of r1 = -0.413.Thus the estimate of 1is:

53.0

89.11

413.04

1

413.02

1

21

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 mis the sample mean.

Thus the identified model in this case is:

xt= ut- 0.53 ut-1+ 0.002 or

xt=xt-1 + ut- 0.53 ut-1+ 0.002

This compares with the other identification:

xt= 0.87xt-1+ ut- 0.48 ut-1+ 2.25(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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pprr

ss

11

2

10

111 1 pprr

2112 pprrr

and

111 ppp rr

The regression coefficients 1, 2, ., pand theauto correlation function rhsatisfy the Yule-Walker equations

:

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ppx rrC s 10 112

1111 pprr

2112

pprrr

and

1 11 ppp rr

The Yule-Walker equations can be used to estimatethe regression coefficients 1, 2, ., pusing thesample auto correlation function rhby replacing rhwith rh.

Example

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Example

Considering the data in example 1 (Sunspot Data) the time serieswas identified as an AR(2) time series .

The autocorrelation at lag 1 is r1= 0.807 and the autocorrelationat lag 2 is r2= 0.429 .

The equations for the estimators of the parameters of this seriesare

429000018070

807080700001

21

21

...

...

which has solution

6370321.1

2

1

.

Since d= m( 1 -1- 2) then it can be estimated as follows:

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Thus the identified model in this case is

xt= 1.321xt-1-0.637xt-2+ ut+14.9

9.14637.0321.11590.46

1

21 x d

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Maximum Likelihood

Estimationof the parameters of an ARMA(p,q)

Series

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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 T} be identically distributed

and uncorrelated with mean zero. In addition

assume that each is normally distributed .

Consider the time series {xt: t T} defined by

the equation:

(*) xt= 1xt-1+ 2xt-2 +... +pxt-p+ d+ ut

+1ut-1+ 2ut-2 +... +qut-q

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Assume thatx1,x2, ...,xN are observations on the

time series up to time t=N.

To estimate thep+ q+ 2 parameters 1, 2, ...

,p; 1, 2, ... ,q; d, s2by the method of

Maximum Likelihood estimation we need to find

the joint density function ofx1,x2, ...,xN

f(x1,x2, ...,xN|1, 2, ... ,p; 1, 2, ... ,q, d, s2)

= f(x| , , d,s2).

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We know that u1, u2, ...,uNare independent

normal with mean zero and variance s2.

Thus the joint density function of u1, u2, ...,uNis

g(u1, u2, ...,uN; s2) = g(u; s2) is given by.

N

t

t

n

N uguug1

2

2

22

12

1exp21;;,

ssss u

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The system of equations :

x1= 1x0+ 2x-1 +... +px1-p+ d+ u1+1u0

+ 2u-1 +... + qu1-q

x2= 1x1+ 2x0 +... +px2-p+ d+ u2+1u1

+ 2u0 +... +qu2-q...

xN= 1xN-1+ 2xN-2 +... +pxN-p+ d+ uN

+1uN-1+ 2uN-2 +... + quN-q

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can be solved for:u1= u1(x, x*, u*; , , d)u2= u2(x, x*, u*; , , d)...

uN= uN(x, x*, u*; , , 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:

2,,,*,*, sduxxf

N

t

t

n

u

1

2

2,,*,*,

2

1exp

2

1d

ss

ux

d

ss,,*

2

1exp

2

12

S

n

N

t

tuS1

2 ,,*,*,,,*where dd ux

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Let:

2

**, ,,, sduxxL

N

t

t

n

u1

2

2,,*,*,

2

1exp

2

1d

ssux

dss

,,*2

1exp

2

12

Sn

N

t

tuS1

2 ,,*,*,,,*again dd ux

= conditional likelihood function

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2

**,

2

**, ,,,ln,,, sdsd uxxuxx Ll

N

t

tunn

1

2

2

2,,*,*,

2

1ln

22d

ss ux

ds

s ,,*2

12ln

22ln

2 22

Snn

conditional log likelihood function =

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2

**,

2

**, ,,,and,,, sdsd uxxuxx Ll

N

t

tuS1

2,,*,*,,,* dd ux

The values that maximize

are the values

that minimize

d,,

dds ,,*1,,*,*,11

22ux S

nu

n

N

t

t

with

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N

t

tuS1

2,,*,*,,,* dd ux

Comment:

Requires a iterative numerical minimization

procedure to find:

The minimization of:

d,,

Steepest descent

Simulated annealing

etc

C

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N

t

tuS1

2,,*,*,,,* dd ux

Comment:

for specific values of

The computation of:

can be achieved by using the forecast

equations

d,,

1 1 ttt xxu

C

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N

t

tuS1

2,,*,*,,,* dd ux

Comment:

assumes we know the value of starting values

of the time series {xt| tT} and {ut| tT}

The minimization of :

Namely x* and u*.

A h

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*ofcomponentsfor the0

*ofcomponentsfor the

u

xx

Approaches:

1. Use estimated values:

2. Use forecasting and backcasting equations toestimate the values:

Backcasting:

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Backcasting:

If the time series {xt|tT} satisfies the equation:

2211 qtqttt uuuu

2211

d ptpttt

xxxx

It can also be shown to satisfy the equation:

2211 qtqttt uuuu 2211 d

ptpttt xxxx

Both equations result in a time series with the samemean, variance and autocorrelation function:

In the same way that the first equation can be used toforecast into the future the second equation can beused to backcast into the past:

A h t h dli t ti l f

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*ofcomponentsfor the0

*ofcomponentsfor the

u

xx

Approaches to handling starting values of

the series {xt|t T} and {ut|t T}

1. Initially start with the values:

2. Estimate the parameters of the model usingMaximum Likelihood estimation and the

conditional Likelihood function.

3. Use the estimated parameters to backcast thecomponents 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.

TheE stands for Expectation (using a model to estimate

the missing values)

TheM stands for Maximum Likelihood Estimation, theprocess used to estimate the parameters of the model.

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Some Examples using:

Minitab

Statistica

S-Plus

SAS