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Appendix 1 
ARIMA and SARIMA models 
An extensive review of ARIMA and SARIMA models 
can be found in, e.g., Box et al. (2008) and Brockwell 
and Davis (2002). A mean-centered time series x t can 
be modeled as an ARIMA(p,d,q), where p, d, q are non- 
negative integers, if it can be adequately fitted with the 
process equation 
(\>(B)(\-B) d X t = Q(B)Z t , 
where for a time interval T, (X t ) teT is a sequence of 
random variables, B is a backshift differencing opera- 
tor B h X t =X t _ h (h nonnegative integer), (l-B) d X t - ^ d x X t is 
stationary, (j)(B) and 0(B) are linear filters defined as 
<j>(B)= 1- ^ B - <p 2 B 2 -...- (p p BP and 0(B)=1+ 0 X B+ 0 2 B 2 + ... 
+ 6 q BP and ( Z t ) teT is a sequence of uncorrelated random 
variables with zero mean and variance d 2 (termed white 
noise). In ARIMA models the orders p , q , and d define 
the structure of the model, by specifying the autoregres- 
sive (AR) and moving average (MA) components of an 
autoregressive-moving average process (ARMA[p,q]). 
d is the degree of differencing (g?> 1) required for X t to 
become stationary. This differencing involves the loss 
of d observations in the series. 
The SARIMA (p,d,q)x(P,D,Q) s models, where P, D, Q , 
and S are nonnegative integers, extend the modeling ca- 
pabilities of ARIMA(p,d,q) models to seasonal processes. 
The SARIMA process equation is given by 
<t>(B)&(B s )(l-B) d a-B s ) D X t =O(B)0(B s )Z t , 
where X t , Z t , <p(B) and 0(B) are defined as above, (1 -B) d 
(l-B s ) D X t = V d VgX t is stationary, and &(B S ) and 0(B S ) 
are seasonal linear filters defined as &(B S ) = 1-& 1 B S - 
0 2 B 2S - ... -0 P B PS and 0(B S ) = 1 + 0 1 B S + 0 2 B 2S + ... 
+ 0qB® s . In SARIMA, P defines the seasonal autore- 
gressive component of the model (SAR) and Q the sea- 
sonal moving average component of the model (SMA). S 
represents the seasonal period (e.g., 12 months) and D 
is the degree of seasonal differencing. Together, S and 
D account for seasonal nonstationarity in X t through a 
data transformation that involves the loss of DS obser- 
vations in the series. 
Appendix 2 
Selection of ARIMA and SARIMA models 
Box-Jenkins approach ARIMA and SARIMA models 
are usually fitted by using a sequence of three gen- 
eral steps collectively known as the Box-Jenkins (BJ) 
method: 1) identification of the model; 2) estimation 
of the model; and 3) a diagnostic check of the model 
(Box et al., 2008). In the identification stage, a model 
structure (p,d,q)x(P,D,Q) s is selected by comparisons 
of sample ACF and PACF with theoretical ACF/PACF 
profiles of AR, MA and ARMA processes. In the esti- 
mation stage, the model structure is fitted to the data 
and its parameters are estimated, generally by using 
conditional sum of squares or maximum likelihood 
methods. In the diagnostic check stage, the goodness- 
of-fit and assumptions for the model are evaluated 
and, if necessary, the BJ procedure is repeated until 
a suitable model is found. This model is then used 
to forecast future values (Box et al., 2008). In-depth 
theoretical coverage of the BJ method is given in Box 
et al. (2008) and extensive practical applications are 
provided in Pankratz (1983) and Brockwell and Davis 
( 2002 ). 
The model identification stage of the BJ method is 
widely considered its most subjective step because it 
relies primarily on graphical interpretations of ACF/ 
PACF estimates obtained from a single sample. This 
interpretation requires substantial analytical expertise 
and knowledge of the time series (both of which are 
problematic in data-poor scenarios) and is trouble- 
some when complex ARMA processes have generated 
the data (Harvey, 1989; Shumway and Stoffer, 2006). 
Furthermore, it can also be confounded by existing 
correlations among ACF/PACF estimates (Box et al., 
2008). The minimum sample size generally advised for 
SARIMA model fitting is 50 observations (Pankratz, 
1983; Chatfield, 1996b), but see Hyndman and Kosten- 
ko (2007) for an absolute lower limit. When sample 
size is large (e.g., n >100), ACF/PACF estimates have 
lower variability and are more likely to approximate 
the theoretical ACF/PACF estimates of the underly- 
ing process. In such cases, less subjectivity exists in 
identification of the model. However, when sample size 
is small, the interpretation of ACF/PACF patterns be- 
comes increasingly confounded by the large variance 
of the sample estimates, particularly at larger lags 
(>n/4) (Box et al., 2008). This variability substantially 
increases the subjectivity of the model identification 
