Adventures in Forecasting David A. Dickey NC State University

Transcription

1 (Previously presented at SAS Global Forum, Dallas, 2015) Adventures in Forecasting David A. Dickey NC State University Learning objectives: Understand ARIMA models. Interpret ARIMA output from PROC ARIMA Forecast with forecast intervals Understand when to difference data Understand advantages/disadvantages of deterministic vs. stochastic inputs Compare forecasts from deterministic versus stochastic input models. Incorporate Trends Incorporate Seasonality (optional) Introduce Cointegration Note: Examples here are run in SAS TM TM SAS and its products is the registered trademark of SAS Institute, Cary NC, USA 1

2 Pg. Topic Brief index 3 Overview 4 Autoregressive Models 8 Model checking 10 Prediction Intervals 11 Moving Average Models 14 Model Selection - AIC 19 Stationarity Unit Roots 26 Determining Lag Differences for Unit Root Tests 33 Models with Inputs (PROC AUTOREG) 47 Detecting Outliers 52 Seasonal Models 63 (optional) Nonlinear Trends 67 (optional) St. Petersburg Visitor Example 72 (optional) Seasonal Unit Roots 77 (optional) Cointegrated Series Real data examples Silver pg. 5, 24, 36 Iron &Steel 12 Brewers 14, 23 Corn Yields 24 Harley 33 NCSU Energy 40 Deer Crash 52 Ice Classic 63 St. Pete Visits 67 T-bill rates 78 2

3 Overview of Time Series and Forecasting: Data taken over time (usually equally spaced) Yt = data at time t = mean (constant over time) Simplest model: Yt = + et where et ~ N(0, 2 ) independent. Forecast: Y S Example 0: Yt = + Zt, corr(zt,zt-1)=0.8 Model (Yt (Yt 1 et, et ~ N(0, 2 ) independent 3

4 ARIMA Models: AR(p): Autoregressive ( Yt ) 1( Yt 1 ) 2( Yt 2 ) ( Y ) e p t p t et independent, constant variance: White Noise How to find p? Regress Y on lags. PACF Partial Autocorrelation Function (1) Regress Yt on Yt-1 then Yt on Yt-1 and Yt-2 then Yt on Yt-1,Yt-2, Yt-3 etc. (2) Plot last lag coefficients versus lags. Red series example: Partial Autocorrelations 4

5 Example 1: Supplies of Silver in NY commodities exchange: Getting PACF (and other identifying plots). SAS code: PROC ARIMA data=silver plots(unpack) = all; IDENTIFY var=silver; run; 5

6 PACF Spikes outside 2 standard error bands are statistically significant Two spikes p=2 ( Y ) ( Y ) ( Y ) e t 1 t 1 2 t 2 t How to estimate and s? PROC ARIMA s ESTIMATE statement. Use maximum likelihood (ml option) PROC ARIMA data=silver plots(unpack) = all; identify var=silver; ESTIMATE p=2 ml; 6

7 Maximum Likelihood Estimation Parameter Estimate Standard Error t Value Approx Pr > t Lag MU < AR1, < AR1, < ( Y ) ( Y ) ( Y ) e t 1 t 1 2 t 2 t ( Y 668) 1.57( Y 668) 0.67( Y 668) e t t 1 t 2 t ( Y 668) 1.57( Y 668) 0.67( Y 668) e t t 1 t 2 t 7

8 Backshift notation: B(Yt)=Yt-1, B 2 (Yt)=B(B(Yt))=Yt-2 2 (1 1.57B 0.67 B )( Y 668) e SAS output: (uses backshift) t t Autoregressive Factors Factor 1: B**(1) B**(2) Checks: (1) Overfit (try AR(3) ) ESTIMATE p=3 ml; Parameter Maximum Likelihood Estimation Estimate Standard Error t Value Approx Pr > t Lag MU < AR1, < AR1, AR1, (2) Residual autocorrelations Residual rt Residual autocorrelation at lag j: Corr(rt, rt-j) = (j) 8

9 Box-Pierce Q statistic: Estimate, square, and sum k of these. Multiply by sample size n. PROC ARIMA: k in sets of 6. Limit distribution Chi-square if errors independent. Later modification: Box-Ljung statistic for H0:residuals uncorrelated n k j 1 n n 2 j 2 j SAS output: To Lag Chi- Square Autocorrelation Check of Residuals DF Pr > ChiSq Autocorrelations Residuals uncorrelated Residuals are White Noise Residuals are unpredictable 9

10 SAS computes Box-Ljung on original data too. To Lag Chi- Square Autocorrelation Check for White Noise DF Pr > ChiSq Autocorrelations < < Data autocorrelated predictable! Note: All p-values are based on an assumption called stationarity discussed later. How to predict? ( Y ) ( Y ) ( Y ) e t 1 t 1 2 t 2 t One step prediction Y ( Y ) ( Y ), future error e t 1 1 t 2 t 1 t 1 Two step prediction Yt ( Yt 1 ) 2( Yt ), error et 1et Prediction error variance ( 2 = variance(et) ), (1 ),

11 From prediction error variances, get 95% prediction intervals. Can estimate variance of et from past data. SAS PROC ARIMA does it all for you! Moving Average, MA(q), and ARMA(p,q) models MA(1) Yt = + et et-1 Variance (1+ 2 ) 2 Yt-1 = + et-1 et-2 (1)=- /(1+ 2 ) Yt-2 = + et-2 et-3 (2)=0/(1+ 2 )=0 11

12 Autocorrelation function ACF ( (j)) is 0 after lag q for MA(q). PACF is useless for identifying q in MA(q). PACF drops to 0 after lag 3 AR(3) p=3 ACF drops to 0 after lag 2 MA(2) q=2 Neither drops ARMA(p,q) p= q= ( Y ) ( Y )... ( Y ) e e e t 1 t 1 p t p t 1 t 1 q t q Example 2: Iron and Steel Exports. PROC ARIMA plots(unpack)=all; IDENTIFY var=export; 12

13 ACF: could be MA(1) PACF: could be AR(1) Spike at lags 0, 1 (No spike displayed at lag 0) ESTIMATE P=1 ML; ESTIMATE Q=2 ML; ESTIMATE Q=1 ML; Maximum Likelihood Estimation Approx Parameter Estimate t Value Pr> t Lag MU < AR1, MU < MA1, MA1, MU < MA1, How to choose? AIC - smaller is better 13

14 AIC: -2 ln(lmax)+2(# parameters) Lmax = max of likelihood function AIC (MA(1)) AIC (AR(1)) AIC (MA(2)) FORECAST lead=5 out=out1 id=date interval=year; Example 3: Brewers Proportion Won Mean of Working Series Standard Deviation

15 Autocorrelations Lag Correlation Std Error ******************** ********** **** ** ** *** *** * * **** **** ***** "." marks two standard errors Could be MA(1) Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq Autocorrelations NOT White Noise! SAS Code: PROC ARIMA data=brewers; IDENTIFY var=win_pct nlag=12; run; ESTIMATE q=1 ml; 15

16 Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > t Lag MU < MA1, AIC Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations Estimated Mean Moving Average Factors Factor 1: B**(1) Partial Autocorrelations Lag Correlation ********** ** ** * *** *** **** * ****. OR could be AR(1) 16

17 ESTIMATE p=1 ml; Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > t Lag MU < AR1, < AIC (vs ) Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations Model for variable Win_pct Estimated Mean Autoregressive Factors Factor 1: B**(1) Conclusions for Brewers: Both models have statistically significant parameters. Both models are sufficient (no lack of fit) 17

18 Predictions from MA(1): First one uses correlations The rest are on the mean. Predictions for AR(1): Converge exponentially fast toward mean Not much difference but AIC prefers AR(1) 18

19 Stationarity (1) Mean constant (no trends) (2) Variance constant (3) Covariance (j) and correlation (j) = (j)/ (0) between Yt and Yt-j depend only on j ARMA(p,q) model: ( Y ) ( Y )... ( Y ) e e e t 1 t 1 p t p t 1 t 1 q t q Stationarity guaranteed whenever solutions of equation (roots of characteristic polynomial ) X p 1X p-1 2X p-2 p =0 are all <1 in magnitude. 19

20 Examples (1) Yt =.8(Yt-1 ) + et X-.8=0 X=.8 stationary (2) Yt = 1.00(Yt-1 ) + et nonstationary Note: Yt= Yt-1 + et Random walk (3) Yt = 1.6(Yt-1 ) 0.6(Yt-2 )+ et characteristic polynomial X 2 1.6X+0.6=0 X=1 or X=0.6 nonstationary (unit root X=1) (Yt ) (Yt-1 ) =0.6[(Yt-1 ) (Yt-2 )]+ et (Yt Yt-1) =0.6(Yt-1 Yt-2) + et First differences form stationary AR(1) process! No mean no mean reversion no gravity pulling toward the mean. 20

21 (4) Yt = 1.60(Yt-1 ) 0.63(Yt-1 )+ et X X+0.63=0 X=0.9 or X=0.7 roots <1 stationary (Yt ) (Yt-1 ) = 0.03(Yt-1 ) [(Yt-1 ) (Yt-2 )]+ et Yt Yt-1 = 0.03(Yt-1 ) (Yt-1 Yt-2)+ et * Unit Root testing (H0: Series has a unit root) Regress Yt Yt-1 on Yt-1 and (Yt-1 Yt-2) Look at t test for Yt-1. If it is significantly negative then stationary. *Note: If X=1 then (X X+0.63) = 0.3 (always equals lag Y coefficient so 0 unit root) 21

22 Problem: Distribution of t statistic on Yt-1 is not t distribution under unit root hypothesis. Distribution looks like this histogram: (1 million random walks of length n=100) Overlays: N(sample mean & variance) N(0,1) Correct distribution: Dickey-Fuller test in PROC ARIMA is the correct (left) 5 th %ile 46% of t s are less than (the normal 5 th percentile) 22

23 Example 3 revisited: Brewers PROC ARIMA data=brewers; IDENTIFY var=wins nlag=12 stationarity=(adf=0); run; Dickey-Fuller Unit Root Tests Type Lags Rho Pr < Rho Tau Pr < Tau Zero Mean Single Mean Trend Why single mean? Series has nonzero mean and no trend. Conclusion reject H0:unit roots so Brewers series is stationary (mean reverting). 0 lags do not need lagged differences in model (just regress Yt-Yt-1 on Yt-1) 23

24 Example 1 revisited: Stocks of silver Needed AR(2) (2 lags) so regress Yt-Yt-1 (D_Silver) on Yt-1 (L_Silver) and Yt-1-Yt-2 (D_Silver_1) PROC REG: Parameter Variable DF Estimate t Value Pr> t Intercept L_Silver wrong distn. D_Silver_ <.0001 OK 24

25 PROC ARIMA: Augmented Dickey-Fuller Unit Root Tests Type Lags Rho Pr<Rho Tau Pr<Tau Zero Mean Single Mean OK Trend Same t statistic, corrected p-value! Conclusion: Unit root difference the series. 1 lag need 1 lagged difference in model (regress Yt-Yt-1 on Yt-1 and Yt-1-Yt-2 ) PROC ARIMA data=silver; IDENTIFY var=silver(1) stationarity=(adf=(0)); ESTIMATE p=1 ml; FORECAST lead=24 out=outn ID=date Interval=month; 25

26 Unit root forecast & forecast interval HOW MANY LAGGED DIFFERENCES? Regression: (Yt Yt 1) = b0 + b1yt 1 + b2(yt 1 Yt 2) + + bp(yt p 1 Yt p) not. standard distributions for these standard Dickey & Fuller (1979) 26

27 Lagged difference coefficients b2 bp have standard (asymptotically normal) distributions. Trust their t test p-values in PROC REG. b0 and b1 have t statistics with same nonstandard limit distributions as in the AR(1) model. Implication: Just use PROC REG to determine appropriate number of lagged differences. o Too few => invalid tests o Too many => loss of power Said & Dickey (1984, 1985) prove that methods work even if moving average terms are present. Chang and Dickey (1993) show that the Inverse Autocorrelation Function (IACF) can be used to check for overdifferencing. Y t = (Y t 1 ) + e t e t 1 ( <1) Autoregressive Moving Average Dual model: Y t = (Y t 1 ) + e t e t 1 Definition: Inverse Autocorrelation Function is Autocorrelation Function of dual model. 27

28 IACF estimation: (a) Fit long autoregression, (b) move coefficients to moving average (MA) side, (c) calculate ACF as if estimated MA is true. Chang (1993) Moving average unit root (e.g. =1) slow decay in IACF (Inverse AutoCorrelation Function) Differencing whenever you see a trend is NOT appropriate: Y Y t t 1 Y t e t ( t e t t 1) e e t 1 e t t N(0, ) IACF from generated linear trend plus white noise: ~ independent ( Non invertible moving average ) 28

29 Example 4: Corn yields in the U.S. (bushels per acre and ) Analysis of post 1942 yields. Levels data: PROC ARIMA; IDENTIFY var=yield stationarity=(adf=0); 29

30 Dickey-Fuller Unit Root Tests Type Lags Tau Pr<Tau Zero Mean Single Mean Trend <.0001 Autocorrelation Check of Residuals (from linear trend plus white noise) To Chi- Pr > Lag Square DF ChiSq --Autocorrelations Suppose we difference anyway: IACF 30

31 Example 2 revisited again: Silver Series DATA CHECK; SET SILVER; Lag_silver = LAG(silver); Diff = silver-lag_silver; OUTPUT; RETAIN; Diff5=Diff4; Diff4=Diff3; Diff3=Diff2; Diff2=Diff1; Diff1=Diff; PROC REG; MODEL Diff = Lag_silver Diff1-Diff5; REMOVE_2andup: TEST Diff2=0, Diff3=0, Diff4=0, Diff5=0; run Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept X Lag_silver X Diff OK ** Diff OK Diff OK Diff OK Diff OK Test REMOVE_2andup Results for Dependent Variable Diff 31

32 Mean Source DF Square F Value Pr > F Numerator OK Denominator What actually happened next in Silver series? (1) Fit stationary (AR(2)) and nonstationary models (differences~ AR(1)) to the data. (2) Compute forecasts, stationary and nonstationary 32

33 PROC AUTOREG Fits a regression model (least squares) Fits stationary autoregressive model to error terms Refits accounting for autoregressive errors. Example 5-A: AUTOREG Harley-Davidson closing stock prices 2009-present. PROC AUTOREG data=harley; MODEL close=date/ nlag=15 backstep; run; 33

34 One by one, AUTOREG eliminates insignificant lags then: Estimates of Autoregressive Parameters Lag Coefficient Standard Error t Value Final model: Parameter Estimates Variable DF Estimate Standard Error t Value Approx Pr > t Intercept <.0001 Date <.0001 In PROC AUTOREG model is Zt+ Zt-1=et rather than Zt Zt-1=et so with = , error term Zt satisfies Zt 0.97Zt-1=et. 34

35 ARIMA Harley-Davidson closing stock prices 01/01/2009through 05/13/ (vs. AUTOREG) Apparent upward movement: Linear trend or nonstationary? Regress Yt Yt-1 on 1, t, Yt-1 (& lagged differences) H0: Yt= + Yt-1 + et random walk with drift H1: Yt= t + Zt with Zt stationary AR(p) * New distribution for t-test on Yt-1 35

36 With trend * Yt= t + Zt general model so Yt Yt-1 = t-(t-1) + Zt Zt-1 and if Zt=Zt-1+et (unit root) then Yt Yt-1 = + et or Yt Yt-1 + et Random walk with drift 36

37 Without trend With trend 1 million simulations - runs in 7 seconds! 37

38 SAS code for Harley stock closing price PROC ARIMA data=harley; IDENTIFY var=close stationarity=(adf) crosscor=(date) noprint; ESTIMATE input=(date) p=1 ml; FORECASE lead=120 id=date interval=weekday out=out1; run; Stationarity test (0,1,2 lagged differences): Augmented Dickey-Fuller Unit Root Tests Type Lags Rho Pr < Rho Tau Pr < Tau Zero Mean Single Mean Trend Conclusion: stationary around a linear trend. 38

39 Estimates: trend + AR(1) Parameter Maximum Likelihood Estimation Estimate Standard Error t Value Approx Pr > t Lag Variable Shift MU < Close 0 AR1, < Close 0 NUM < Date 0 39

40 Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq Autocorrelations What actually happened? 40

41 Example 6 (with inputs): NCSU Energy Demand Type of day Class Days Work Days (no classes) Holidays & weekends. Temperature Season of Year Step 1: Make some plots of energy demand vs. temperature and season. Use type of day as color. Seasons: S = A sin(2 t/365), C=B cos(2 t/365) Temperature Season of Year 41

42 Step 2: PROC AUTOREG with all inputs: PROC AUTOREG data=energy; MODEL demand = temp tempsq class work s c /nlag=15 backstep dwprob; output out=out3 predicted = p predictedm=pm residual=r residualm=rm; run; Estimates of Autoregressive Parameters Lag Coefficient Standard Error t Value Autocorrelation at 1, 7, 14, and others. 42

43 After autocorrelation adjustments, trust t tests etc. Parameter Estimates Variable DF Estimate Standard Error t Value Approx Pr > t Intercept <.0001 TEMP <.0001 TEMPSQ <.0001 CLASS <.0001 WORK <.0001 S <.0001 C Need better model? Big negative residual on Jan. 2 rm JUL79 01AUG79 01SEP79 01OCT79 01NOV79 01DEC79 01JAN80 01FEB80 01MAR80 01APR80 01MAY80 01JUN80 01JUL80 DATE WC non work work class Residuals from regression part. Large residual on workday January 2. Add dummy variable. 43

44 Same idea: PROC ARIMA Step 1: Graphs Step 2: Regress on inputs, diagnose residual autocorrelation: Not white noise (bottom right) Activity (bars) at lag 1, 7, 14 44

45 Step 3: Estimate resulting model from diagnostics plus trial and error: e input = (temp tempsq class work s c) p=1 q=(1,7,14) ml; Parameter Maximum Likelihood Estimation Estimate Standard Error t Value Approx Pr > t Lag Variable Shift MU < DEMAND 0 MA1, DEMAND 0 MA1, DEMAND 0 MA1, DEMAND 0 AR1, < DEMAND 0 NUM < TEMP 0 NUM < TEMPSQ 0 NUM < CLASS 0 NUM < WORK 0 NUM < S 0 NUM C 0 Note: class days get class effect 913 plus work effect Note 2: Lags are sensible. 45

46 Step 4: Check model fit (stats look OK): Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq Autocorrelations

47 Looking for outliers that can be explained PROC ARIMA, OUTLIER statement Available types (1) Additive (single outlier) (2) Level shift (sudden change in mean) (3) Temporary change (level shift for k contiguous time points you specify k) NCSU energy: tested every point 365 tests. Adjust for multiple testing Require p < 0.05/365 = (Bonferroni) OUTLIER type=additive alpha= id=date; FORMAT date weekdate.; run; /***************************************** January 2, 1980 Wednesday: Hangover Day :-) March 3,1980 Monday: On the afternoon and evening of March 2, 1980, North Carolina experienced a major winter storm with heavy snow across the entire state and near blizzard conditions in the eastern part of the state. Widespread snowfall totals of 12 to 18 47

48 inches were observed over Eastern North Carolina, with localized amounts ranging up to 22 inches at Morehead City and 25 inches at Elizabeth City, with unofficial reports of up to 30 inches at Emerald Isle and Cherry Point (Figure 1). This was one of the great snowstorms in Eastern North Carolina history. What made this storm so remarkable was the combination of snow, high winds, and very cold temperatures. May 10,1980 Saturday. Graduation! ****************************************/; Outlier Details Obs Time ID Type Estimate Chi-Square Approx Prob>ChiSq 186 Wednesday Additive < Saturday Additive < Monday Additive <.0001 Outlier Details Obs Time ID Type Estimate Chi-Square Approx Prob>ChiSq JAN-1980 Additive < MAY-1980 Additive < MAR-1980 Additive <

49 Outliers: Jan 2 (hangover day!), March 3 (snowstorm), May 10 (graduation day). AR(1) produces 3 rebound outlying next day residuals ( ). Add dummy variables for explainable outliers data next; merge outarima energy; by date; hangover = (date="02jan1980"d); storm = (date="03mar1980"d); graduation = (date="10may1980"d); 49

50 PROC ARIMA data=next; IDENTIFY var=demand crosscor=(temp tempsq class work s c hangover graduation storm) noprint; ESTIMATE input = (temp tempsq class work s c hangover graduation storm) p=1 q=(7,14) ml; FORECAST lead=0 out=outarima2 id=date interval=day; run; Parameter Maximum Likelihood Estimation Estimate Standard Error t Value Approx Pr > t Lag Variable Shift MU < DEMAND 0 MA1, < DEMAND 0 MA1, DEMAND 0 AR1, < DEMAND 0 NUM < TEMP 0 NUM < TEMPSQ 0 NUM < CLASS 0 NUM < WORK 0 NUM < S 0 NUM C 0 NUM < hangover 0 NUM < graduation 0 NUM < storm 0 50

51 Model looks fine. Comparison: Workday = Non workday Jan 2 = Workday 3473 = Non workday Jan 2 is like a non workday Class day = Non workday AUTOREG - regression with AR(p) errors versus ARIMA regression with differencing, ARMA(p,q) errors. 51

52 SEASONALITY Many economic and environmental series show seasonality. (1) Very regular ( deterministic ) or (2) Slowly changing ( stochastic ) Example 7: NC accident reports involving deer. Method 1: regression PROC REG data=deer; MODEL deer=date X11; run; (X11: 1 in Nov, 0 otherwise) Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 X <

53 Looks like December and October need dummies too! PROC REG data=deer; MODEL deer=date X10 X11 X12; run; Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 X <.0001 X <.0001 X <.0001 Average of Jan through Sept. is 929 crashes per month. Add 1391 in October, 2830 in November, 1377 in December. 53

54 Try dummies for all but one month (need average of rest so must leave out at least one) PROC REG data=deer; MODEL deer=x1-x11; OUTPUT out=out1 predicted=p residual=r; run; Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X X <

55 Average of rest is just December mean Subtract 886 in January, add 1452 in November. October (X10) is not significantly different than December. Residuals for Deer Crash data: Looks like a trend add trend (date): PROC REG data=deer; MODEL deer=date X1-X11; OUTPUT out=out1 predicted=p residual=r; run; 55

56 Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X X <.0001 date <.0001 Trend is 0.22 more accidents per day (1 per 5 days) and is significantly different from 0. What about autocorrelation? Method 2: PROC AUTOREG PROC AUTOREG data=deer; MODEL deer=date X1-X11/nlag=13 backstep; run; 56

57 Backward Elimination of Autoregressive Terms Lag Estimate t Value Pr > t Preliminary MSE Estimates of Autoregressive Parameters Standard Lag Coefficient Error t Value

58 Parameter Estimates Standard Approx Variable DF Estimate Error t Value Pr > t Intercept date <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X <.0001 X X <.0001 Method 3: PROC ARIMA PROC ARIMA plots=(forecast(forecast)); IDENTIFY var=deer crosscor= (date X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11); ESTIMATE p=1 ML input= (date X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11); FORECAST lead=12 id=date interval=month; run; 58

59 Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr> t Lag Variable MU deer AR1, deer NUM < date NUM < X1 NUM < X2 NUM < X3 NUM < X4 NUM < X5 NUM < X6 NUM < X7 NUM < X8 NUM < X9 NUM X10 NUM < X11 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations Autoregressive Factors Factor 1: B**(1) 59

60 Method 4: Differencing Compute and model Dt = Yt-Yt-12 Removes seasonality Removes linear trend e t Y t Use (at least) q=(12) et- et-12 (A) if near 1, you ve overdifferenced (B) if 0< <1 this is seasonal exponential smoothing model. Y t Y e e t 12 t t 12 Y t Y ( e ) Y ( ) t Y Y Y e t t t 12 t 12 t 24 t 24 ( 1 )[ Y Y 2Y 3Y ] e t 12 t 24 t 36 t 48 t Forecast is a weighted (exponentially smoothed) average of past values: Yˆ t (1 )[ Y t 12 Y t 24 2Y t 36 3Y t 48 ] 60

61 IDENTIFY var=deer(12) nlag=25; ESTIMATE P=1 Q=(12) ml; run; Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr> t Lag MU < MA1, AR1, < Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations Lag 12 MA somewhat close to 1 with large standard error, model OK but not best. Variance estimate 15,122 (vs. 13,431 for dummy variable model). 61

62 Forecasts are similar 2 years out. 62

63 OPTIONAL (time permitting) Trend Breaks Accounting for changes in trend Example 7: Nenana Ice Classic data (trend break) Exact time (day and time) of thaw of the Tanana river in Nenana Alaska: 1917 Apr 30 11:30 a.m May 11 9:33 a.m May 3 2:33 p.m. (more data) 2010 Apr 29 6:22 p.m May 04 4:24 p.m Apr 23 7:39 p.m. When the tripod moves downstream, that is the unofficial start of spring. 63

64 Get ramp with PROC NLIN / X= PROC NLIN data=all; PARMS point=1960 int=126 slope=-.2; X = (year-point)*(year>point); MODEL break = int + slope*x; OUTPUT out=out2 predicted=p residual=r; Approx Approximate 95% Confidence Parameter Estimate Std Error Limits point int slope

65 PROC SGPLOT data=out2; SERIES Y=break X=year; SERIES Y=p X=year/ lineattrs = (color=red thickness=2); REFLINE / axis=x; run;quit; What about autocorrelation? Final ramp: Xt = (year )*(year>1965); PROC ARIMA; IDENTIFY var=break crosscor=(ramp) noprint; ESTIMATE input=(ramp); FORECAST lead=5 id=year out=out1; run; 65

66 PROC ARIMA generates diagnostic plots: Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

67 Optional: more on seasonal, time permitting: Example 9: Visitors to St. Petersburg/Clearwater Model 1: Seasonal dummy variables + trend + AR(p) (REG - R 2 >97%) PROC REG data=aaem.stpete; MODEL visitors=t m1-m11; OUTPUT out=out4 predicted = P residual=r UCL=u95 LCL=l95; run; 67

68 PROC SGPLOT data=out3; BAND lower=l95 upper=u95 X=date; SERIES Y=P X=date; SCATTER Y=visitors X=date / datalabel=month datalabelattrs=(color=red size=0.3 cm); SERIES Y=U95 X=date/ lineattrs=(color=red thickness=0.8); SERIES Y=L95 X=date/ lineattrs=(color=red thickness=0.8); REFLINE "01apr2013"d / axis=x; where 2011<year(date)<2015; run; 68

69 PROC SGPLOT data=out4; NEEDLE Y=r X=date; run; Definitely autocorrelated Slowly changing mean? Try seasonal span difference model 69

70 PROC ARIMA data=stpete plots=forecast(forecast); IDENTIFY var=visitors(12); Typical ACF for ARMA(1,1) has initial dropoff followed by exponential decay try ARMA(1,1) on span 12 differences. ESTIMATE P=1 Q=1 ml; FORECAST lead=44 id=date interval = month out=outarima; run; 70

71 Standard Approx Parameter Estimate Error t Value Pr > t Lag MU MA1, AR1, < Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq Autocorrelations

72 PROC SGPLOT data=outarima; ; SERIES X=date Y=residual; run; We have seen deterministic (dummy variables) and dynamic (seasonal differences) models for St Pete visitors. How do decide between them? Seasonal unit root test! 72

73 Optional: Seasonal Multiplicative Model. Accommodates slowly changing seasonal patterns. Contrast - Indicator (dummy) variables assume same effect for every January (etc.) no matter what year. Simplest model seasonal random walk. Yt Yt-s+et Yt Yt-s=et Seasonal AR: (1 s )Yt=et Yt Yt-s +et Yt Yt-s=( -1)(Yt-s et Idea: Regress Yt Yt-s on (Yt-s to estimate and get a t test. Distribution of t is far from standard normal. Modification: replace Yt with deviations yt from seasonal means. 73

74 Visitors data: Estimate of : = Parameter Standard Variable DF Estimate Error t Value Pr> t Intercept <.0001 Y_ <.0001 XX Is t significant? What is the (null) distribution? Empirical (left) and N(0,1) (right) densities with means. Area to left of leftmost line ( =-4.50, see next model) is Critical value for 5% is -5.84, median is (Dickey, Hasza & Fuller, 1984, table 7). DHF 74

75 Conclusion: Insufficient evidence against seasonal dynamic model (seasonal unit roots). What if it s seasonal with more lags? Multiplicative seasonal model is popular. (1 B s )(1 1B 2B 2 pb p )(Yt- s)=et s=seasonal mean Step 1: Regress Dt=Yt Yt-s on Dt-1, Dt-2, Dt-p to estimate s. Residuals are et. Using these estimates, estimate filtered lags Ft-s= (1 1B 2B 2 pb p )(Yt-s- s) (note t-s subscript). Step 2: Regress Dt on Ft-s, Dt-1, Dt-2,, Dt-p to get test statistic and improvements for q estimates. For visitor data with p=2: 75

76 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept Filter <.0001 D D From normal curve graph above, p-value is about , not <.0001 Optional (time permitting) Cointegration Two unit root processes Xt and Yt (nonstationary) are said to be cointegrated if there is a linear combination of them St=aXt+bYt such that St is stationary. Usually a and b are unknown but sometimes the situation suggests values and in that case all that is needed to show Cointegration is a unit root test on St. 76

77 Cointegration analogy: Drunk man walking a dog Man > unit root process Dog > unit root process Distance from man to dog stationary! Red path = man Blue path = dog Example 10: T-bill rates for two maturities Data: 10 year t bill yields 30 year t bill yields (in logarithms) 77

78 Unit root tests show (1) Both series are nonstationary (can t reject unit roots using trend tests) (2) Neither requires more than 1 difference Let St = log(30 year) log(10 year) = log(30 year rate / 10 year rate) 78

79 Graph of St Unit root test on St Augmented Dickey-Fuller Unit Root Tests Type Lags Rho Pr < Rho Tau Pr < Tau Zero Mean Single Mean Trend

80 Final model: log(30 yr. rate/10 yr. rate) is stationary with estimated mean and autoregressive order 1 structure with =0.92 Summary: Use ACF, PACF to identify p=# autoregressive lags and q= # moving average lags. Stationarity mean reverting models versus unit roots (random walk type models). Graphics and DF test (and others) available. Diagnostics errors should be white noise Ljung Box test to check. Regression with autoregressive or ARMA errors Nonlinear regression to estimate slope changes (least squares). Seasonal models dynamic or deterministic. 80

81 References for unit roots: Chang, M. C. and D. A. Dickey, (1993) "Recognizing Overdifferenced Time Series," Journal of Time Series Analysis, 15, 1-8. Dickey, D. A. and W. A. Fuller (1979). Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association, 74, p Dickey, D. A. and W. A. Fuller (1981). Likelihood Ratio Statistics for Autoregressive Time Series with a unit Root. Econometrica 49, Dickey, D. A., D. P. Hasza, and W. A. Fuller (1984). Testing for Unit Roots in Seasonal Time Series, Journal of the American Statistical Association, 79, Said, S. E. and D. A. Dickey (1984). Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order, Biometrika, 71, Said, S. E. and D. A. Dickey (1985). Hypothesis Testing in ARIMA (p, 1, q) Models, Journal of the American Statistical Association, 80, Dickey, D. A., W.R. Bell and R. B. Miller (1986). Unit Roots in Time Series Models: Tests and Implications, American Statistician 40,