Forecasting (prediction) limits
Example Linear deterministic trend estimated by least-squares
Yt   0   t  t  et
Yˆt l  E Yt l Y1 ,  , Yt   b0  b1  t  l   b0 Y1 ,  , Yt   b1 Y1 ,  , Yt   t  l 
eˆt l  Yt l  Yˆt l
 
Var eˆt l   Yt l and Yˆt l independent  Var Yt l   Var Yˆt l 
2


t

1



t  l 
 
2  
1
 From regression analysis t heory   e2   e2    

2 
t
t 
t 1 

s


 


s 1
2

 

2

t

1

 

t  l 

1
2


 
  e2  1  
2 
t
t 
t

1


s 
 


s 1
2  


Note! The average of the numbers 1, 2, … , t is
1
1 t  t  1 t  1
t
 s 1 s  

t
t
2
2
Hence, calculated prediction limits for Yt+l become
t 1

t  l 

1
2

Yˆt l  c  ˆ e  1   
2
t
t 
t 1
s 1  s  2 
where c is a quantile of a proper sampling distribution emerging from the use of
ˆ e2 as an estimatorof  e2 and the requested coverage of the limits.
For t large it suffices to use the standard normal distribution and a good
approximation is also obtained even if the term
t 1

t  l 

1
2 


2
t
t 
t 1
s1  s  2 
is omitted under the square root
 Yˆt l  z 2  ˆ e
PrN 0,1  z 2    2
ARIMA-models
Yˆt l  z 2  ˆ e 

l 1
2
ˆ

j
j 0
where ˆ 0 ,,ˆ l 1 are functions of the
parameter estimatesˆ ,, ˆ and ˆ ,,ˆ
1
p
1
q
Using R
ts=arima(x,…)
for fitting models
plot.Arima(ts,…)
for plotting fitted models with 95%
prediction limits
See documentation for plot.Arima . However, the generic command plot
can be used.
forecast.Arima
Install and load package “forecast”. Gives
more flexibility with respect to prediction
limits.
Seasonal ARIMA models
Example “beersales” data
A clear seasonal pattern and also a trend, possibly a quadratic trend
Residuals from detrended data
beerq<-lm(beersales~time(beersales)+I(time(beersales)^2))
plot(y=rstudent(beerq),x=as.vector(time(beersales)),type="b",
pch=as.vector(season(beersales)),xlab="Time")
Seasonal pattern, but possibly no long-term trend left
SAC and SPAC of the residuals:
SAC
Spikes at or close
to seasonal lags
(or half-seasonal
lags)
SPAC
Modelling the autocorrelation at seasonal lags
Pure seasonal variation:
Yt    1  Yt 12  et
Seasonal AR(1)12 - model
Stationary if 1  1  Roots to the characteristic equation 1  12
1 0
outside the unit circle
1k 12
ρk  
 0
k  0,12,24,36,...
otherwise
Yt    et  1  et 12
Seasonal MA(1)12 - model
Invertible if 1  1
 Root s to the characteristic equation 1  12
1 0
outside the unit circle
 1
  
1
ρk  
2
1


1

 0
k 0
k  12
otherwise
Non-seasonal and seasonal variation:
AR(p, P)s or ARMA(p,0)(P,0)s
Yt    1  Yt 1    p  Yt  p  1  Yt s   P  Yt Ps  et
However, we cannot discard that the non-seasonal and seasonal variation
“interact”  Better to use multiplicative Seasonal AR Models
1 B  B 1  B
p
1
p
1
s

 P BPs Yt    et
Example:
1  0.3  B 1  0.2  B12 Yt  et


 1  0.3  B  0.2  B12  0.3  0.2  B13 Yt  et
 Yt  0.3  Yt 1  0.2  Yt 12  0.05 Yt 13  et
Multiplicative MA(q, Q)s or ARMA(0,q)(0,Q)s



Yt    11B q Bq 1  1Bs  Q BQs et
Mixed models:
1   B     B 1   B     B Y   
 1   B     B 1   B     B e
p
1
p
1
P
q
1
P s
s
q
t
Q s
s
1
Q
t
Many terms! Condensed expression:
 B B s Yt     B B s et
 B   1  i 1i  B i ; B s   1  i 1  i  B s 
p
i
P
 B s   1   j 1 j  B j ; B   1   j 1  j  B
q
Q

s j
ARMA p, q P, Qs
Non-stationary Seasonal ARIMA models
Non-stationary at non-seasonal level:
Model dth order regular differences:
dYt  Yt   1  B Yt
d
Non-stationary at seasonal level:
Seasonal non-stationarity is harder to detect from a plotted times-series. The
seasonal variation is not stable.
Model Dth order seasonal differences:
Example First-order monthly differences:
12Yt  1  Bs Yt  Yt  Yt 12
can follow a stable seasonal pattern


 sDYt   s  s   sYt   1  B s Yt
D
The general Seasonal ARIMA model
 B  B s 1  B d 1  B s  Yt     B B s et
D
It does not matter whether regular or seasonal differences are taken first
ARIMA p, d , q P, D, Qs
Model specification, fitting and diagnostic checking
Example “beersales” data
Clearly nonstationary at nonseasonal level, i.e.
there is a longterm trend
Investigate SAC and SPAC of original data
Many substantial spikes
both at non-seasonal and at
seasonal levelCalls for differentiation at
both levels.
Try first-order seasonal differences first. Here: monthly data


Wt  1  B12 Yt  Yt  Yt 12
beer_sdiff1 <- diff(beersales,lag=12)
Look at SAC and SPAC again
Better, but now we need to try regular differences
Take first order differences in seasonally differenced data


Ut  1  B 1  B12 Yt  1  BWt  Wt Wt 1  Yt  Yt 12  Yt 1  Yt 13 
beer_sdiff1rdiff1 <- diff(beer_sdiff1,lag=1)
Look at SAC and SPAC again
SAC starts to look “good”, but SPAC not
Take second order differences in seasonally differenced data
Since we suspected a non-linear long-term trend


Vt  1  B  1  B12 Yt  1  B U t  U t  U t 1 
2
 Wt  Wt 1  Wt 1  Wt  2   Wt  2Wt 1  Wt  2 
Yt  Yt 12  2Yt 1  Yt 13   Yt  2  Yt 14
beer_sdiff1rdiff2 <- diff(diff(beer_sdiff1,lag=1),lag=1)
Non-seasonal part
Seasonal part
Could be an ARMA(2,0)(0,1)12 or an ARMA(1,1) (0,1)12
These models for original data becomes
ARIMA(2,2,0) (0,1,1)12 and ARIMA(1,2,1) (0,1,1)12
model1 <-arima(beersales,order=c(2,2,0),
seasonal=list(order=c(0,1,1),period=12))
Series: beersales
ARIMA(2,2,0)(0,1,1)[12]
Coefficients:
ar1
-1.0257
s.e.
0.0596
ar2
-0.6200
0.0599
sma1
-0.7092
0.0755
sigma^2 estimated as 0.6095: log likelihood=-216.34
AIC=438.69
AICc=438.92
BIC=451.42
Diagnostic checking can be used in a condensed way by function tsdiag. The
Ljung-Box test can specifically be obtained from function Box.test
tsdiag(model1)
standardized residuals
SPAC(standardized residuals)
P-values of Ljung-Box test with K = 24
Box.test(residuals(model1), lag = 12, type = "Ljung-Box",
fitdf = 3)
p+q+P+Q
(how many degrees of freedom withdrawn from K)
K
(how many lags
included)
Box-Ljung test
data: residuals(model1)
X-squared = 30.1752, df = 9, p-value = 0.0004096
For seasonal data with season length s the L-B test is usually calculated for
K = s, 2s, 3s and 4s
Box.test(residuals(model1), lag = 24, type = "Ljung-Box",
fitdf = 3)
Box-Ljung test
data: residuals(model1)
X-squared = 57.9673, df = 21, p-value = 2.581e-05
Box.test(residuals(model1), lag = 36, type = "Ljung-Box",
fitdf = 3)
Box-Ljung test
data: residuals(model1)
X-squared = 76.7444, df = 33, p-value = 2.431e-05
Box.test(residuals(model1), lag = 48, type = "Ljung-Box",
fitdf = 3)
Box-Ljung test
data: residuals(model1)
X-squared = 92.9916, df = 45, p-value = 3.436e-05
Hence, the residuals from the first model are not satisfactory
model2 <-arima(beersales,order=c(1,2,1),
seasonal=list(order=c(0,1,1),period=12))
print(model2)
Series: beersales
ARIMA(1,2,1)(0,1,1)[12]
Coefficients:
ar1
-0.4470
s.e.
0.0678
ma1
-0.9998
0.0176
sma1
-0.6352
0.0930
sigma^2 estimated as 0.4575: log likelihood=-192.86
AIC=391.72
AICc=391.96
BIC=404.45
Better fit ! But is it good?
tsdiag(model2)
Not good! We should maybe try second-order seasonal differentiation too.
Time series regression models
The classical set-up uses deterministic trend functions and seasonal indices
Yt  mt   S t   et
Examples:
12
Yt   0  1  t    s , j  x j t   et linear trend in monthly data
j 2
1 if t is in month j
where x j t   
otherwise
0
Yt   0  1  t   2  t 2
quatadic trend, no seasonal variation
The classical set-up can be extended by allowing for autocorrelated error
terms (instead of white noise). Usually it is sufficient with and AR(1) or
AR(2). However, the trend and seasonal terms are still assumed deterministic.
Dynamic time series regression models
To extend the classical set-up with explanatory variables comprising other time
series we need another way of modelling.
Note that a stationary ARMA-model
Yt   0  1  Yt 1     p  Yt  p  et  1  et 1     q  et  q

1    B    
1



p
q

B
Y



1



B





B
et
p
t
0
1
1

 B Yt   0   B et
can also be written
 B 
Yt   0 
et
 B 
 0


  0 
  B 

The general dynamic regression model for a response time series Yt with one
covariate time series Xt can be written
C   B  b
 B 
Yt   0 
B Xt 
et
 B 
 B 
Special case 1:
Xt relates to some event that has occurred at a certain time points (e.g. 9/11)
It can the either be a step function
1 t  T
X t T   
0 t  T
St T 
or a pulse function
1 t  T
X t T   
0 t  T
Pt T 
Step functions would imply a permanent change in the level of Yt . Such a change
can further be constant or gradually increasing (depending on (B) and (B) ). It
can also be delayed (depending on b )
Pulse functions would imply a temporary change in the level of Yt . Such a
change may be just at the specific time point gradually decreasing (depending
on (B) and (B) ).
Strep and pulse functions are used to model the effects of a particular event, as
so-called intervention.
 Intervention models
For Xt being a “regular” times series (i.e. varying with time) the models are called
transfer function models