UNIVARIATE TIME SERIES ANALYSIS BRIEFING 1970

Transcription

1 UNIVARIATE TIME SERIES ANALYSIS BRIEFING 1970 Joseph George Caldwell, PhD (Statistics) 1432 N Camino Mateo, Tucson, AZ USA Tel. (001)(520) , jcaldwell9@yahoo.com (File converted to Microsoft Word March 25, 2018, with minor edits.) Copyright 2018 Joseph George Caldwell. All rights reserved. 1

2 ROAD MAP 1. DEFINITIONS / EXAMPLES / FRAMEWORK 2. THE NATURE OF TIME SERIES MODELING 3. TIME SERIES DESCRIPTORS 4. BASIC TIME SERIES MODELS 5. TIME SERIES MODEL BUILDING (ADDITIONAL DETAILS) 6. BOX-JENKINS MODELS 7. PROGRAMS / DATA REQUIREMENTS / REFERENCES 2

3 I. DEFINITIONS / EXAMPLES / FRAMEWORK DEFINITION OF TIME SERIES EXAMPLES OF TIME SERIES TYPES OF TIME SERIES CONSIDERED DEFINITION OF TIME SERIES ANALYSIS 3

4 WHAT IS A TIME SERIES? A SET OF OBSERVATIONS ARRANGED CHRONOLOGICALLY A REALIZATION OF A STOCHASTIC PROCESS STOCHASTIC PROCESS = A FAMILY OF RANDOM VARIABLES: {X(t), tєt} 4

5 EXAMPLES OF TIME SERIES ECONOMIC DAILY COMMODITY OR STOCK PRICES MONTHLY INTEREST RATES SALES AND PRICES CASH BALANCES PHYSICAL DAILY RAINFALL, MEAN DAILY TEMPERATURE INDUSTRIAL PROCESS YIELDS AIRCRAFT VIBRATION ACOUSTIC AND ELECTROMAGNETIC SIGNALS AND NOISE TRACKS OF SHIPS, PLANES, RE-ENTRY VEHICLES BIOLOGICAL DAILY EGG PRODUCTION, POPULATION GROWTH BRAIN WAVES 5

6 TYPES OF TIME SERIES CONSIDERED HERE DISCRETE, EQUIDISTANT POINTS IN TIME STOCHASTIC, NOT DETERMINISTIC E.G., zt = zt-1 + xt-1 + et, NOT zt = a cos(2π bt) EMPIRICAL-THEORETICAL MODELS E.G., NO KALMAN FILTER (MODEL ESSENTIALLY KNOWN) EMPHASIS ON FORECASTING (NOT CONTROL) UNIVARIATE, NOT MULTIVARIATE 6

7 WHAT IS TIME SERIES ANALYSIS? TIME SERIES ANALYSIS IS THE APPLICATION OF MATHEMATICAL TECHNIQUES TO DESCRIBE, PREDICT AND CONTROL TIME SERIES WHICH TECHNIQUE TO USE DEPENDS ON THE TYPE OF ANSWER WANTED TWO CATEGORIES OF TIME SERIES ANALYSIS TECHNIQUES: SPECTRAL ANALYSIS FREQUENCY RESPONSE STUDIES MODEL BUILDING FORECASTING SIMULATION CONTROL 7

8 II. THE NATURE OF TIME SERIES MODELING HOW TIME SERIES MODELS ARE USED PROCEDURE FOR MODEL DEVELOPMENT GENERAL CLASSES OF TIME SERIES MODELS CRITERIA FOR SELECTING A TIME SERIES MODEL CLASS 8

9 HOW TIME SERIES MODELS ARE USED REPRESENT IMPORTANT CHARACTERISTICS OF TIME SERIES BY MATHEMATICAL MODEL DEVELOP OPTIMAL FORECAST OR CONTROL PROCEDURE FOR MODEL APPLY MODEL-OPTIMAL PROCEDURE IN REAL-WORLD SITUATION 9

10 PROCEDURE FOR MODEL DEVELOPMENT ITERATIVE APPROACH TO MODEL BUILDING: 10

11 GENERAL CLASSES OF TIME SERIES MODELS UNIVARIATE SINGLE VARIABLE (E.G., SALES OF A MINOR ITEM UNRELATED TO OTHERS) MULTIPLE VARIABLES, ONE OF WHICH IS DEPENDENT ON THE OTHERS (E.G., CORPORATE QUARTERLY FORECAST; CONTROL PROBLEMS; LEADING INDICATORS) MULTIVARIATE MULTIPLE RELATED VARIABLES FORECAST SALES AND PRICE SIMULTANEOUSLY COORDINATES OF REENTRY VEHICLE:X, Y, Z, X, Y, Z, X, Y, Z 11

12 CRITERIA FOR SELECTING A TIME SERIES MODEL CLASS NATURE OF APPLICATION (E.G., FORECASTING, CONTROL, POLICY ANALYSIS) FORECAST HORIZON (RANGE) PRECISION REQUIREMENTS COST (MODEL DEVELOPMENT AND IMPLEMENTATION, DATA COLLECTION) DATA REQUIREMENTS AND AVAILABILITY ALL OF THE PRECEDING CONSIDERATIONS ARE TAKEN INTO ACCOUNT IN SELECTING AN APPROPRIATE MODEL CLASS 12

13 NATURE OF APPLICATION FORECASTING, CONTROL OR POLICY ANALYSIS UNCONDITIONAL FORECASTING: NO REQUIREMENT FOR EXPLANATORY VARIABLES CONDITIONAL FORECASTING; CONTROL, POLICY ANALYSIS: NEED EXPLANATORY / CONTROL VARIABLES 13

14 FORECAST HORIZON SHORT RANGE: ONE-THREE PERIODS INTO THE FUTURE MEDIUM RANGE: 4-12 PERIODS, 1-2 SEASONS LONG RANGE: BEYOND 14

15 MEASUREMENT OF PRECISION STANDARD ERROR OF THE ESTIMATE: LONG-RANGE VS SHORT-TERM MODELS: 15

16 MODEL COSTS DATA COLLECTION COSTS NUMBER OF VARIABLES, SCOPE HISTORY REQUIRED MODEL DEVELOPMENT COSTS PROFESSIONAL TIME (STATISTICAL ANALYSIS; DATABASE DEVELOPMENT; WEB DEVELOPMENT) COMPUTER SOFTWARE / HARDWARE / SYSTEMS MODEL IMPLEMENTATION COSTS PERSONNEL TIME COMPUTER-RELATED 16

17 III. TIME SERIES DESCRIPTORS BASIC CHARACTERISTICS OF TIME SERIES DESCRIPTORS OF STATIONARY TIME SERIES ESTIMATION OF TIME SERIES DESCRIPTORS 17

18 BASIC CHARACTERISTICS OF TIME SERIES CONTINUOUS VS. DISCRETE EQUISPACED DETERMINISTIC VS. STOCHASTIC COMPONENTS (E.G., MEAN MONTHLY TEMPERATURE VS. SALES) STATIONARY VS. NONSTATIONARY 18

19 STATIONARITY STRONG STATIONARITY (PROBABILITY DISTRIBUTION INDEPENDENT OF SHIFTS): f(x t1,, x tn ) = f(x t1+h,, x tn+h ) for all n, t i, h WEAK STATIONARITY (2 ND MOMENTS INDEPENDENT OF SHIFTS): E(x t 2 ) < for all t; E(x s+t, x s ) = γ t (t th serial covariance) depends only on t, not on s. (NOTE: THIS PRESENTATION USES LOWER-CASE LETTERS BOTH FOR ABSTRACT RANDOM VARIABLES AND THEIR OBSERVED REALIZATIONS.) 19

20 DESCRIPTORS OF STATIONARY TIME SERIES MEAN VARIANCE AUTOCORRELATION FUNCTION (ACF) SPECTRAL DENSITY FUNCTION (SDF) PARTIAL AUTOCORRELATION FUNCTION (PACF OR PAF) 20

21 AUTOCORRELATION FUNCTION (ACF) (CORRELOGRAM) OBSERVATION AT TIME t: z t MEAN: μ = E(z t ) VARIANCE: σ 2 = var(z t ) = E(z t μ) 2 AUTOCOVARIANCE AT LAG k: γ k = cov(z t, z t+k ) = E(z t μ)(z t+k μ) AUTOCORRELATION AT LAG k: ρ k = γ k σ 2 = γ k γ 0 21

22 EXAMPLE MODEL: z t = 10 + a t + a t 1 OR z t = 10 + a t a t 1 WHERE at, at-1, ARE A SEQUENCE OF UNCORRELATED RANDOM NORMAL VARIABLES WITH MEAN 0 AND VARIANCE 1 ( WHITE NOISE ) 22

23 SPECTRAL DENSITY FUNCTION (SDF) THE SPECTRAL DENSITY FUNCTION IS THE FOURIER COSINE TRANSFORM OF THE AUTOCORRELATION FUNCTION: g(f) = 2[1 + 2 ρ k cos (2πfk)], 0 f 1 2. k=1 THE POWER SPECTRUM IS THE FOURIER COSINE TRANSFORM OF THE AUTOCOVARIANCE FUNCTION (CALLED THE PERIODOGRAM FOR SERIES WITH DETERMNISTIC COMPONENTS): p(f) = 2[1 + 2 γ k cos (2πfk)], 0 f 1 k=1 2 = g(f)σ 2. 23

24 ESTIMATION OF THE AUTOCORRELATION FUNCTION (ACF) SAMPLE: z 1,, z n SAMPLE MEAN: z = 1 n n t=1 z t SAMPLE VARIANCE: σ 2 = 1 n (z n t=1 t z ) 2 SAMPLE AUTOCOVARIANCE FUNCTION: n k c k = 1 n (z t z )(z t+k z ), k = 0,1,, k (< n 4 ) t=1 SAMPLE AUTOCORRELATION FUNCTION: r k = c k c 0 24

25 ESTIMATION OF THE SPECTRAL DENSITY FUNCTION (SDF) FOR SERIES WITH DETERMINISTIC COMPONENTS (MIXTURES OF SINE AND COSINE) WAVES AT FIXED FREQUENCIES, BURIED IN NOISE): THE SAMPLE SPECTRUM IS THE ESTIMATE OF THE PERIODOGRAM: n 1 I(f) = 2[c o + 2 c k cos (2πfk)]. k=1 FOR SERIES WITH RANDOM CHANGES OF FREQUENCY, AMPLITUDE AND PHASE: THE SMOOTHED SPECTRUM IS THE ESTIMATE OF THE SPECTRUM: n 1 P (f) = 2[c o + 2 λ k c k cos (2πfk)], WHERE λk ARE SUITABLY CHOSEN WEIGHTS, CALLED A LAG WINDOW. k=1 25

26 REPRESENTATIONAL EQUIVALENCE STATIONARY NORMAL STOCHASTIC PROCESSES ARE CHARACTERIZED BY: THEORETICAL MODEL MEAN, VARIANCE AND AUTOCORRELATION FUNCTION MEAN, VARIANCE AND SPECTRAL DENSITY FUNCTION 26

27 IV. BASIC TIME SERIES MODELS EXAMPLES SUMMARY STATISTICS 27

28 WHITE NOISE (WN) PROCESS MODEL: z t = a t, a t ~N(0, σ 2 ) 28

29 AUTOREGRESSIVE (AR) PROCESS MODEL: z t = φ 1 z t φ p z t p + a t OR (1 φ 1 B φ p B p )z t = a t OR Φ(B)z t = a t WHERE THE BACKWARD SHIFT OPERATOR, B, IS DEFINED BY Bz t = z t 1. 29

30 PARTIAL AUTOCORRELATION FUNCTION (PACF) FOR AUTOREGRESSIVE PROCESSES THE PARTIAL AUTOCORRELATION OF LAG k IS THE SOLUTION φ kk TO ρ j = φ k1 ρ j φ k(k 1) ρ j k+1 + φ kk ρ j k, j = 1,2,, k. FOR AN AUTOREGRESSIVE PROCESS OF ORDER p, φ kk = 0 for k > p. ESTIMATE THE PACF BY THE LEAST-SQUARES ESTIMATE OF φ kk IN THE LINEAR MODEL: z t = φ k1 z t φ kk z t k. 30

31 MOVING AVERAGE (MA) PROCESS MODEL: z t = a t θ 1 a t 1 θ q a t q OR z t = (1 θ 1 B θ q B q )a t OR z t = Θ(B)a t. 31

32 MIXED AUTOREGRESSIVE MOVING AVERAGE (ARMA) PROCESS MODEL: z t = φ 1 z t φ p z t p + a t θ 1 a t 1 θ q a t q OR (1 φ 1 B φ p B p )z t = (1 θ 1 B θ q B q a t OR Φ(B)z t = Θ(B)a t. 32

33 SUMMARY STATISTICS PROCESS ACF PACF SDF WN AR TAILS OFF CUTS OFF VARIES MA CUTS OFF TAILS OFF VARIES ARMA TAILS OFF TAILS OFF VARIES 33

34 V. TIME SERIES MODEL BUILDING (ADDITIONAL DETAILS) ITERATIVE APPROACH REVIEW OF MODEL CLASSES ( FORECASTING METHODS ) COMPARISONS BETWEEN METHODS 34

35 ITERATIVE APPROACH TO MODEL BUILDING THE THEORY-BASED ITERATIVE APPROACH TO DEVELOPMENT OF TIME SERIES MODELS WAS DEVELOPED AND PROMOTED BY G. E. P. BOX AND GWILYM JENKINS, STARTING IN THE 1960s. THE DESCRIPTOR BOX-JENKINS IS APPLIED EITHER TO DESCRIBE THEIR APPROACH TO DEVELOPING TIME-SERIES MODELS, OR TO DESCRIBE THE CLASS OF AUTOREGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) MODELS EMPLOYED IN THEIR APPROACH. 35

36 TYPES OF FORECASTING MODELS General Model type Usual Accuracy Cost to develop Cost to model class forecast horizon Short term (ST) Medium term (MT) Long term (LT) implement Qualitative Delphi LT MED MED MED HI HI Market research MT HI MED MED HI HI Panel consensus LT LO LO LO HI HI Visionary LT LO LO LO HI HI Historical analogy LT LO MED MED HI HI Single variable Moving average ST LO LO LO LO VLO Exponential ST MED LO LO LO VLO smoothing Univariate time ST HI MED LO MED VLO series models ( Box-Jenkins, AR, MA, ARMA, ARIMA) X-11 ST MED MED LO LO LO Trend projections ST MED MED MED LO VLO Neural network ST HI MED LO MED LO Multiple variable univariate Regression model (BJ) MT HI MED LO MED LO Econometric MT MED HI MED HI LO model Anticipation MT MED LO LO HI HI survey Input-output LT LO MED MED VHI MED model Economic inputoutput LT LO HI HI VHI MED Diffusion index MT LO LO LO HI LO Leading indicator ST HI MED LO MED LO (BJ) Life-cycle analysis MT LO LO LO HI HI Neural network ST HI MED LO MED LO Multivariate Econometric simultaneousequation model ( SEM ) Multivariate time series models ( Box Jenkins, VAR, VARMA, VARX, MARMA, MARIMA, Kalman filter) MT HI MED LO VHI HI MT HI MED LO MED LO 36

37 FORECASTS BASED ON UNVALIDATED MODELS MODEL IS FITTED TO DATA LITTLE OR NO DIAGNOSTIC CHECKING (MODEL VALIDATION) MOVING AVERAGE EXPONENTIAL SMOOTHING REGRESSION ON COMPONENTS TREND, SEASONAL PATTERNS, FOURIER COMPONENTS NEURAL NETWORK 37

38 FORECASTS BASED ON VALIDATED MODELS SINGLE VARIABLE o BOX-JENKINS MULTIPLE VARIABLE (THEORETICAL OR EMPIRICAL MODEL) BOX-JENKINS ECONOMETRIC 38

39 SOME FORECASTING ACCURACY COMPARISONS STOCHASTIC VS. INTUITIVE Lead time MSE (Brown) MSE (BJ) Forecast Error Variance ECONOMETRIC VS. STOCHASTIC Theil Coefficient Model Price Quantity Econometric Box-Jenkins Random walk Mean

40 IBM STOCK PRICE SERIES WITH COMPARISON OF LEAD-3 FORECASTS OBTAINED FROM BEST IMA(0,1,1) PROCESS AND BROWN S QUADRATIC FORECAST FOR A PERIOD BEGINNING JULY 11, (REF. BJ p. 168) 40

41 HYBRID MODELS TREND REMOVAL REGRESSION + BOX-JENKINS 41

42 VI. BOX-JENKINS MODELS MODEL IDENTIFICATION PARAMETER ESTIMATION DIAGNOSTIC CHECKING FORECASTING EXAMPLE 42

43 BOX-JENKINS MODEL CLASS AUTOREGRESSIVE INTEGRATED MOVING AVERAGE (ARIMA) MODEL: OR Φ(B) d z t = Θ(B)a t Φ(B)w t = Θ(B)a t WHERE z t = (1 B)z t = z t z t 1 (backward difference operator) AND IS STATIONARY. THAT IS, w t = d z t w t = φ 1 w t φ p + a t θ 1 a t 1 θ q a t q WHERE z t, z t 1, is the observed series w t, w t 1, is a stationary series φ 1,, φ p, θ 1,, θ q are parameters a t, a t 1, is a white noise sequence (mean 0, uncorrelated, variance σ 2 ) 43

44 MODEL IDENTIFICATION APPLY SUCCESSIVE DIFFERENCING (OR OTHER PROCEDURE) TO ORIGINAL DATA UNTIL STATIONARITY IS ACHIEVED (ACF DIES OUT): w t = d z t. EXAMINE THE ACF AND PACF OF wt TO SUGGEST A TENTATIVE MODEL (I.E., IDENTIFY THE DEGREE AND STRUCTURE OF φ(b) AND Θ(B) POLYNOMIALS). 44

45 IDENTIFICATION EXAMPLES BEHAVIOR OF THE AUTOCORRELATION FUNCTIONS FOR THE d-th DIFFERENCE OF AN ARMA PROCESS OF ORDER (p,d,q). REFERENCE: BJ p

46 PARAMETER ESTIMATION IF THE MODEL IS AUTOREGRESSIVE (NO θs), WE HAVEFG A LINEAR STATISTICAL MODEL: z = Zφ + a WHERE z = (zn, zn-1,,zp+1), φ = (φ1,,φp), a = (an, an-1,,ap+1), AND z n 1, z n p Z = [ ], z p,, z 1 SO THE LEAST-SQUARES ESTIMATE OF φ IS φ = (Z Z) 1 Z z. IF θs ARE PRESENT, THE MODEL IS A NONLINEAR STATISTICAL MODEL: I.E., a t = Θ 1 (B)Φ(B)z t a t = a t (φ, θ, z) = a t (β, z) p=q a t β=β0 + (β i β i0 ) a t i=1 β i β=β 0 (Taylor series expansion) IS A LINEAR MODEL WITH PARAMETER δ = β β0. DETERMINE THE PARAMETER ESTIMATES ITERATIVELY. 46

47 TESTS OF MODEL ADEQUACY PROPERTIES OF φ(b) AND Θ(B) POLYNOMIALS (STATIONARITY, INVERTIBILITY, COMMON FACTORS) TESTS OF SIGNIFICANCE OF MODEL PARAMETERS OVERFITTING TEST RESIDUALS FOR WHITENESS INFORMATION CRITERIA (E.G., AKAIKE INFORMATION CRITERION, BAYES INFORMATION CRITERION) DEVELOP MODEL FOR DIFFERENT PERIODS OF DATA 47

48 DIAGNOSTIC CHECKING OF RESIDUALS SIGNIFICANCE OF VARIOUS STATISTICS COMPUTED FROM RESIDUALS: MEAN (t-test) PACF (t-test ON EACH VALUE) a t = Θ 1 (B)Φ (B)z t. ACF (t-test ON EACH VALUE, χ 2 TEST ON ENTIRE FUNCTION) SPECTRUM (GRENANDER-ROSENBLATT TEST, KOLMOGOROV-SMIRNOV TEST) 48

49 MODEL IDENTIFICATION THE PACF AND ACF CAN SUGGEST ADDITION OF φ AND θ COMPONENTS. A PRELIMINARY FITTED MODEL IS: Φ(B)z t = Θ(B)e t. THE et s ARE CORRELATED; A MODEL FOR THE et s IS: THE MODIFIED MODEL IS: Φ e (B)e t = Θ e (B)a t Φ(B)Φ e (B)z t = Θ(B)Θ e (B)a t. 49

50 EXAMPLE OF MODEL IDENTIFICATION SUPPOSE THAT THE CORRECT MODEL IS OF ORDER (0,2,2), BUT THAT THE FITTED MODEL IS z t = (1 +.6B)e t. SUPPOSE THAT THE MODEL SUGGESTED FOR THE RESIDUALS IS THE COMBINED MODEL IS e t = (1.8B)a t. 2 zt = (1.2B.48B 2 )a t. THIS SUGGESTS THAT A MODEL OF ORDER (0,2,2) SHOULD BE EXAMINED. 50

51 MODEL SIMPLIFICATION A MODEL OF THE FORM MIGHT BE REDUCIBLE TO (1 ϕb)(1 B)z t = (1 θb)a t IF θ IS CLOSE TO 1. (1 φb)z t = a t 51

52 CAUTION ESTIMATED AUTOCORRELATIONS MAY BE HIGHLY CORRELATED, AND MAY HAVE LARGE VARIANCES. USE THE ACF ONLY TO SUGGEST MODELS TO FIT (ESTIMATE). RELY ON DIAGNOSTIC CHECKS TO ACCEPT OR REJECT FITTED MODELS. 52

53 OPTIMAL FORECASTER THE OPTIMAL FORECASTER MINIMIZES THE MEAN SQUARED ERROR OF PREDICTION. z t(1) = 1 ahead forecast made from time t WHERE z t(1) = φ 1z t + + φ pz t p θ 1a t 1 θ qa t q a t = z t z t 1(1). 53

54 SEASONALITY A TENTATIVE MODEL IS Φ s (B s )z t = Θ s (B s )e t WHERE et IS CORRELATED WITH et-1, et-2,. SUPPOSE THAT WE CAN REPRESENT THE RESIDUALS BY Φ(B)e t = Θ(B)a t WHERE THE at ARE WHITE. THEN WE MAY COMBINE THESE RESULTS TO OBTAIN Φ s (B s )Φ(B)z t = Θ s (B s )Θ(B)a t. 54

55 EXPONENTIAL SMOOTHING FORECASTER A SPECIAL CASE OF Φ(B)z t = Θ(B)a t, WITH Φ(B) = 1 B, Θ(B) = 1 αb, I,E. THE LEAST-SQUARES FORECASTER IS: z t = z t 1 + a t αa t 1. OR z t(1) = z t αa t z t(1) = (1 α)[z t + αz t 1 + α 2 z t 2 + ]. 55

56 MOVING AVERAGE FORECASTER A SPECIAL CASE OF Φ(B)z t = Θ(B)a t, WITH Φ(B) = 1 1 p B 1 p B2 1 p Bp, Θ(B) = 1. THE LEAST-SQUARES FORECASTER IS z t(1) = 1 p (z t + z t z t p ). 56

57 BOX-JENKINS SAMPLE PROBLEM 57

58 ACF OF SUCCESSIVE DIFFERENCES 58

59 59

60 TENTATIVE MODEL MODEL FORM (ACF SPIKES AT LAGS 1 AND 12) 1 12 z t = (1 θ 1 B)(1 θ 12 )a t. PARAMETER ESTIMATION: θ1 =.40. θ12 =.61; SO FITTED MODEL IS: 1 12 z t = (1.40B)(1.61)a t OR z t = z t 1 + z t 12 z t 13 + a t.40a t 1.61a t a t 13. DIAGNOSTIC CHECKS: 60

61 TOPICS NOT COVERED IN THIS BRIEFING MULTIPLE VARIABLE (TRANSFER FUNCTION) MODELS CROSS CORRELATION FUNCTION (PREWHITENING OF INPUT) CROSS SPECTRAL ANALYSIS DESIGN OF EXPERIMENTS MULTIVARIATE MODELS MULTIPLE CORRELATION COEFFICIENT MULTIPLE COHERENCY SPECTRUM OPTIMAL CONTROL FEED FORWARD CONTROL FEED BACK CONTROL PHYSICAL MODELS KALMAN FILTER NONLINEAR MODELS MODEL EXTENSIONS AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY (ARCH) AND GENERALIZED AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY (GARCH) MODELS 61

62 VII. PROGRAMS / DATA REQUIREMENTS / REFERENCES THE FIRST COMMERCIALLY AVAILABLE GENERAL-PURPOSE BOX-JENKINS FORECASTING PROGRAM WAS TIMES, DEVELOPED BY THE AUTHOR IN UNIVARIATE BOX-JENKINS ANALYSIS PROGRAMS ARE NOW INCLUDED IN ALL MAJOR GENERAL-PURPOSE STATISTICAL SOFTWARE PACKAGES (Stata, SAS, SPSS), AND MULTIVARIATE BOX-JENKINS ANALYSIS PROGRAMS ARE FREELY AVAILABLE IN R. A MINIMUM OF ABOUT 100 DATA POINTS ARE REQUIRED TO DEVELOP A BOX- JENKINS MODEL. THE SEMINAL REFERENCE FOR THE BOX-JENKINS METHODOLOGY IS BOX, G. E. P., AND GWILYM JENKINS, TIME SERIES ANALYSIS, FORECASTING CONTROL, FIRST EDITION HOLDEN-DAY, 1970, LATEST ADDITION IS 5 TH EDITION BY GEORGE E. P. BOX, GWILYM M. JENKINS, GREGORY C. REINSEL AND GRETA M. LJUNG, WILEY, OTHER REFERENCES INCLUDE CRYER, JONATHAN D., AND KUNG-SIK CHAN, TIME SERIES ANALYSIS WITH APPLICATIONS IN R, 2 ND ED., SPRINGER, TSAY, RUEY S., MULTIVARIATE TIME SERIES ANALYSIS WITH R AND FINANCIAL APPLICATIONS, WILEY, 2014 LÜTKEPOHL, HELMUT, NEW INTRODUCTION TO MULTIPLE TIME SERIES ANALYSIS, SPRINGER, 2006 HAMILTON, JAMES D., TIME SERIES ANALYSIS, PRINCETON UNIVERSITY PRESS, 1994 A VERSION OF THE TIMES REFERENCE MANUAL IS POSTED AT INTERNET WEBSITE 62