TS_Fall_2011_Lecture_5

# TS_Fall_2011_Lecture_5 - FIN 271 FINANCIAL MODELING AND...

This preview shows pages 1–9. Sign up to view the full content.

FIN 271 FINANCIAL MODELING AND ECONOMETRICS LECTURE SET 5 TIME SERIES MODELING REFIK SOYER THE GEORGE WASHINGTON UNIVERSITY

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 AUTOREGRESSIVE INTEGRATED MOVING AVERAGE PROCESSES. • ARIMA(p,d,q) Processes Nonstationary Processes Ê Difference stationary TS (or unit root series): Differenced series is stationary. If is not stationary, but if the first differences of ]] tt W( 1 B ) ttt 1 t œ] ] œ ] is stationary then we may represent W by a stationary ARMA process. t Example: The Dow Jones data (Random Walk) ]] œG 1 t % ] t is not stationary, but the first differenced series is white noise. Ê] can be represented by an ARIMA(0,1,0) model. t 1
3 In general, if is an ARIMA(p,d,q) process, then ] t is nonstationary, but the d th differences of the series ] ] tt W( 1 B ) d œ ] can be represented by an stationary ARMA(p,q) process. What do we mean by d th differences ? d1 W ( 1B ) œÊ œ ] ] œ] ttt 1 t d 2 (1 B)W W W ( ) ( ) (1 B) œ œ ] ] ] œ ] t 1 t t 1t 1 t 2 t 2  If is described by an ARIMA(p,d,q) process then we write ] t 9) % (B) (1 B) (B) . ] œ d Equivalently, W , the d th difference of the series can be represented by a stationary and invertible ARMA process % (B)W (B) œ where W (1 . d ] 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 If the differenced series has nonzero mean, the we can write 9) % (B) W (B) . tt œ where W W . œ . Examples : • ARIMA (0, 1, 1): ]] œ   1 t1 t 1  .% ) % • ARIMA (1, 1, 0): ( ) ( ) œG ] ] t 1 t1 t2 t 9% (1 B) (1 B) Ê] œ G ] t 1 t EX. Price of the Crude oil. • ARIMA (1, 1, 1): ( ) ( )   t t 1t1 ) % (1 B) (1 (1 B) Ê ] œ G % 1t 1 t • How about higher order differencing ? Ex. d 2 (SAS Example Analysis of PIRC data). œ 3
Analysis of PIRC Data OPTIONS LS=80; *QUARTERLY TIME SERIES; DATA IRCALL; INFILE 'C:\TSDATA\IRC.TXT'; INPUT IRC PIRC YD RCP RAAA; PROC ARIMA; IDENTIFY VAR=PIRC; IDENTIFY VAR=PIRC(1); IDENTIFY VAR=PIRC(1,1); RUN; Pl ot 60 80 100 120 140 160 180 200 0 10 20 30 40 50 60 70 80 90 110 PI RC 1 4

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Name of variable = PIRC. Standard deviation = 32.27038 Number of observations = 101 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0 1041.378 1.00000 | |********************| 1 987.826 0.94858 | . |******************* | 2 933.834 0.89673 | . |****************** | 3 881.977 0.84693 | . |***************** | 4 833.674 0.80055 | . |**************** | 5 789.514 0.75814 | . |*************** | 6 745.101 0.71549 | . |************** | 7 703.467 0.67552 | . |************** | 8 662.211 0.63590 | . |************* | 9 623.851 0.59906 | . |************. | 10 588.171 0.56480 | . |*********** . | 11 554.106 0.53209 | . |*********** . | 12 521.008 0.50031 | . |********** . | 13 489.406 0.46996 | . |********* . | 14 457.373 0.43920 | . |********* . | 15 425.996 0.40907 | . |******** . | 16 393.545 0.37791 | . |******** . | 17 360.578 0.34625 | . |******* . | 18 328.828 0.31576 | . |****** . | 19 297.590 0.28577 | . |****** . | 20 268.327 0.25767 | . |***** . | 2 5
Name of variable = PIRC. Period(s) of Differencing = 1. Standard deviation = 1.6697 Number of observations = 100 NOTE: The first observation was eliminated by differencing. Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0 2.787899 1.00000 | |********************| 1 2.140614 0.76782 | . |*************** | 2 2.070001 0.74249 | . |*************** | 3 1.800982 0.64600 | . |************* | 4 1.558508 0.55903 | . |*********** | 5 1.604676 0.57559 | . |************ | 6 1.525965 0.54735 | . |*********** | 7 1.411033 0.50613 | . |********** | 8 1.323173 0.47461 | . |*********. | 9 1.139464 0.40872 | . |******** . | 10 1.101324 0.39504 | . |******** . | 11 0.980445 0.35168 | . |******* . | 12 0.920560 0.33020 | . |******* . | 13 1.058342 0.37962 | . |******** . | 14 0.990563 0.35531 | . |******* . | 15 1.114757 0.39986 | . |******** . | 16 1.079192 0.38710 | . |******** . | 17 1.009858 0.36223 | . |******* . | 18 0.924783 0.33171 | . |******* . | 19 0.828745 0.29727 | . |****** . | 3 6

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Name of variable = PIRC. Period(s) of Differencing = 1,1. Mean of working series = 0.029293 Standard deviation = 1.12216 Number of observations = 99 NOTE: The first 2 observations were eliminated bydifferencing.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 39

TS_Fall_2011_Lecture_5 - FIN 271 FINANCIAL MODELING AND...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online