1. Chapter 9 Model Building
NOTE: Some slides have blank sections. They are based on a teaching style in which the corresponding blank (derivations, theorem proofs, examples, …) are worked out in class on the board or overhead projector.

2. Check for Model Appropriateness
Residuals should be white noise
or, based on parameter estimates
- these are calculated after estimates
(tswge uses backcasting)

3. Testing Residuals for White Noise
Check sample autocorrelations of residuals vs 95% limit lines.
Ljung-Box test (Portmanteau test)
portmanteau [port man tō]
1. a large travelling case made of stiff leather, esp one hinged at the back so as to open out into two compartments
2. embodying several uses or qualities

4. Testing Residuals for White Noise
Check sample autocorrelations of residuals vs 95% limit lines.
Ljung-Box test (Portmanteau)
- Test the hypotheses:
- Test Statistic:

5. 3. Other tests for randomness
- runs test
- etc.

6. Examining Residuals for Some Examples
Demo: Generate realization from ARMA(2,1):
Example Simulated seasonal data (fig8.8)
Example Airline data (airlog)

7. tswge demo d1=gen.arma.wge(n=200,phi=c(1.2,-.8),theta=.9)
plotts.sample.wge(d1)
# overfit AR models
est.ar.wge(d1,p=10,type='burg') #
est.ar.wge(d1,p=12,type='burg')
aic.wge(d1,p=0:6,q=0:2)
d1.est=est.arma.wge(d1,p=
# check residuals
plotts.sample.wge(d1.est$res,arlimits=TRUE)
ljung.wge(d1.est$res,p=
# final model
mean(d1)

8. Example 8.7 Simulated Seasonal Data
Sample Autocorrelations
Residuals
Residual Sample Autocorrelations

9. tswge demo dd1=gen.aruma.wge(n=200,s=12,phi=c(1.25,-.9)) dd1=dd1+50
plotts.sample.wge(dd1)
# overfit AR models
ov14=est.ar.wge(dd1,p=14, type='burg') #
ov16=est.ar.wge(dd1,p=16, type='burg')
dd1.12=artrans.wge(dd1,phi.tr=c(0,0,0,0,0,0,0,0,0,0,0,1))
plotts.sample.wge(dd1.12)
aic.wge(dd1.12,p=0:6,q=0:2)
# estimate model parameters
dd1.12.est=est.arma.wge(dd1.12,p=
# check residuals
plotts.sample.wge(dd1.12.est$res,arlimits=TRUE)
ljung.wge(dd1.12.est$res,p=
# final model
mean(dd1)

10. Example 8.8 Airline Data Data Sample Autocorrelations Residuals
Res. Sample Autocorrelations

11. twge demo log airline data data(airlog) plotts.sample.wge(airlog)
# overfit AR models
ov14=est.ar.wge(airlog,p=14, type='burg') #
ov16=est.ar.wge(airlog,p=16, type='burg')
# transform data
la.12=artrans.wge(airlog,phi.tr=c(0,0,0,0,0,0,0,0,0,0,0,1))
plotts.sample.wge(la.12)
aic.wge(la.12,p=0:13,q=0:2)
# estimate parameters of stationary part
la.12.est=est.arma.wge(la.12,p=
# check residuals
plotts.sample.wge(la.12.est$res,arlimits=TRUE)
ljung.wge(la.12.est$res,p=
# final model

12. Another Important Check for Model Appropriateness
Does the model make sense?
stationarity vs. nonstationarity
seasonal vs. non-seasonal
correlation based vs signal-plus-noise model
are characteristics of fitted model consistent with those of the data
forecasts and spectral estimates make sense?
do realizations and their characteristics behave like the data

13. Stationarity vs Nonstationarity?
A decision that must be consciously made by the investigator
Tools:
overfitting
unit roots tests (Dickey-Fuller test, etc.)
- designed to test the null hypothesis
- i.e. decision to include a unit root in the model is based on a decision not to reject a null hypothesis
- in reality the models (1 - B)Xt = at and ( B)Xt = at will both create realizations for which the null hypothesis is usually not rejected using the Dickey-Fuller test
an understanding of the physical problem and properties of the model selected

14. Modeling Global Temperature Data
(a) Stationary model:
data(hadley)
mean(hadley)
plotts.sample.wge(hadley)
aic.wge(hadley,p=0:6,q=0:1)
# estimate stationary model
had.est=est.arma.wge(hadley,p=
# check residuals
plotts.sample.wge(had.est$res,arlimits=TRUE)
ljung.wge(had.est$res,p=
# other realizations from this model
demo=gen.arma.wge(n=160,phi=had.est$phi,theta=had.est$theta)
plotts.sample.wge(demo)

15. (b) Nonstationary model:
data(hadley)
plotts.sample.wge(hadley)
# Overfit AR models
d8=est.ar.wge(hadley,p=8, type='burg') #
d12=est.ar.wge(hadley,p=12, type='burg')
# Difference the data
h.dif=artrans.wge(hadley,phi.tr=1)
plotts.sample.wge(h.dif)
aic.wge(h.dif,p=0:6,q=0:1)
h.dif.est=est.arma.wge(h.dif,p=
# Examine residuals
plotts.sample.wge(h.dif.est$res,arlimits=TRUE)
ljung.wge(h.dif.est$res,p=
# other realizations from this model
demo=gen.aruma.wge(n=160,d=1,phi=h.dif.est$phi,theta=h.dif.est$theta)
plotts.sample.wge(demo)

16. Forecasts using Stationary Model
data(hadley)
fore.arma.wge(hadley,phi=c(1.27,-.47,.19), theta=.63,n.ahead=25,limits=FALSE)
Forecasts using Nonstationary Model
data(hadley)
fore.aruma.wge(hadley,d=1,phi=c(.33,-.18), theta=.7,n.ahead=25,limits=FALSE)

17. Notes:
the two models are quite similar but produce very different forecasts
it is important to understand the properties of the selected model
- the selection of a stationary model will automatically produce forecasts that eventually tend toward the mean of the observed data
(i.e. it was the decision to use the stationary model that produced these results)
beware of results by investigators who choose a model in order to produce desired results

18. Deterministic Signal-plus-Noise Models
Example Signals:
, C constant

19. Recall -- sometimes it’s not easy to tell whether a deterministic signal is present in the data
Is there a deterministic signal?

20. Realizations
- is there a deterministic signal?
Recall - Sometimes it’s not easy to tell whether a deterministic signal is present in the data.
Global Temperature Data

21. Realizations from the stationary model fit to temperature data

22. Another Possible Model
Question: Should observed increasing temperature trend be predicted to continue?
- based on standard ARMA/ARUMA fitting the answer is “No”
Another Possible Model
“deterministic signal + noise model”
nonstationary due to non-constant mean

23. Common Strategy for Assessing which Model is Appropriate
• assume Z t  AR(p) with zero mean
note that this is different from usual regression since the noise is correlated
in the presence of noise with positive autocorrelations, using the incorrect procedure of testing H0: b = 0 using usual regression methods, results in inflated observed significance levels
testing H0: b = 0 is a difficult problem

24. Important Points:
- realizations from AR (ARMA/ARUMA) models have random trends

25. Cochrane-Orcutt Method

26. Question: Technique: Test:
Is there a deterministic trend in the data (That should be predicted to continue)?
Technique:
Test:
IF we conclude b=0, fit AR model & trend not predicted to continue
IF we conclude b 0, trend is predicted to continue

27. Cochrane-Orcutt Test Note: Test H0: b = 0 using the test statistic
Woodward and Gray (1993) Journal of Climate showed:
- when using the Cochrane Orcutt method to remove the correlated errors, the resulting test still has inflated observed significance levels.
- same is true with ML methods

28. Simulation Results Observed Significance Levels for Tests H 0 : b = 0
-- b = 0 (i.e. null hypothesis of zero slope is true)
replicates generated from each model
-- hypothesis of zero slope tested at a = .05 for each realization using Cochrane-Orcutt
Observed Significance Levels for Tests H 0 : b = 0
(nominal level = 5%)

29. Bootstrap Method
Woodward, Bottone, and Gray (1997) -- Journal of Agricultural, Biological, and Environmental Statistics (JABES),
Given a time series realization that may have a “deterministic trend”: To test H 0 : b = 0
1. Based on the observed data, calculate a test statistic Q that is a test for trend (e.g. Cochrane-Orcutt statistic) for which large Q suggests a trend
Fit stationary AR model (with constant mean) to observed time series data and generate K realizations from this model
For each realization in (2) calculate the test statistic in (1)
Reject H 0 : b = 0 if Q in (1) exceeds the 95th percentile of the bootstrap-based test statistics in (3)

30. Simulation Results Observed Significance Levels for Tests H 0 : b = 0
-- b = 0 (i.e. null hypothesis of zero slope is true)
replicates generated from each model
Observed Significance Levels for Tests H 0 : b = 0
5.9
6.4
3.8
4.1
8.1
5.3
5.1
3.8
10.0
7.6
6.4
6.1
black - Cochrane-Orcutt
red - Bootstrap

31. Comments:
1. If a significant slope is detected, then forecasts from the model Xt = a + bt + Zt will forecast an existing trend to continue
- such a model should be used with caution if more than short-term forecasts are needed
2. Jon Sanders (2009) developed bootstrap-based procedures to test for significant
- monotonic trend
- nonparametric trend

32. Checking Realization Characteristics
Do realizations generated from the model have characteristics consistent with those of the actual data?
- similar realizations?
- similar sample autocorrelations?
- similar spectral densities (nonparametric)?
- ...

33. Sunspot Data: (ss08)

34. Note: This data set ss08 contains sunspot numbers for the years not included in the sunspot data sunspot.classic
We consider 2 models:
- AR(2)
- AR(9)

35. Comparing Realizations
Sunspot Data
Realizations from the AR(2) Model
Realizations from the AR(9) Model

36. Comparing Sample Autocorrelations
Sunspot Data
Figure 9.10
AR(2) Model
AR(9) Model

37. Comparing Nonparametric Spectral Densities
Sunspot Data
Figure 9.11
AR(2) Model
AR(9) Model

38. Time Series Model Checking using Parametric Bootstraps
Tsay - Applied Statistics (1992)
generates multiple realizations from the fitted model
checks to see whether actual data are consistent with the realizations from the fitted model
Woodward and Gray – Journal of Climate (1995)
Given a time series realization
- generate bootstrap realizations from each of two candidate models
- use discriminant analysis to ascertain which model generates realizations that best match characteristics of the observed realization.

39. Comprehensive Analysis of Time Series Data
Involves:
Examination
-- of data
-- of sample autocorrelations
Obtaining a model
-- stationary or nonstationary
-- correlation-based or signal-plus-noise
-- identifying p and q
-- estimating coefficients
III. Checking for model appropriateness
IV. Obtaining forecasts, spectral estimates, etc as dictated by the situation

40. Important Note: Previous discussion only focused on deciding among
- ARMA(p,q)
- ARUMA(p,d,q)
- signal-plus-noise
There are MANY other models and tools available to the time series analyst, among which are:
- long memory models (Ch. 11)
- multivariate and state-space models (Ch. 10)
- ARCH/GARCH models (Ch. 4)
- wavelet analysis (Ch. 12)
- models for data with time varying frequencies (TVF) (Ch. 13)