Chapter 17
Time Series Analysis and
Forecasting
©
Time Series
A time series is a set of measurements, ordered
over time, on a particular quantity of interest.
In a time series the sequence of the observations
is important, in contrast to cross section data for
which the sequence of observations is not
important.
Prices and Price Index for Ford Motor
Company Stock Over 12 Weeks
(Figure 17.1)
Week
1
2
3
4
5
6
7
8
9
10
11
12
Price
Price Index
20.250
100.0
19.875
98.1
19.000
93.8
19.750
97.5
20.250
100.0
19.875
98.1
19.375
95.7
19.625
96.9
21.125
104.3
22.375
110.5
25.000
123.5
23.000
113.6
Calculating Price Indices for a
Single Item
Suppose that we have a series of observations over time
on the price of a single item. To form a price index, one
time period is chosen as a base, and the price for every
period is expressed as a percentage of the base period
price. Thus, if p0 denotes the price in the base period
and p1 the price in a second period, the price index for
this second period is
 p1 
100 
 p0 
An Unweighted Price Index
Suppose that we have a series of observations over time on the
prices of a group of K items. As before, one time is chosen as a
base.
The unweighted aggregate index of prices is obtained by
calculating the average price of these items in each time period
and calculating an index for these average prices. That is, the
average price in every period is expressed as a percentage of
the average price in the base period. Let p0i denote the price of
the ith item in the base period and p1i, the price of this item in a
second period. The unweighted aggregate index of prices for
this second period is
K


  p1i 

100 iK1


  p0 i 
 i 1

The Laspeyres Price Index
Suppose that we have a group of K commodities for which
price information is available over a period of time. One period
is selected as the base for an index. The Laspeyres price index
in any period is the total cost of purchasing the quantities
traded in the base period at prices in the period of interest,
expressed as a percentage of the total cost of purchasing these
same quantities in the base period.
Let p0i denote the price and q0i the quantity purchased of the ith
item in the base period. If p1i is the price of the ith item in a
second period, the Laspeyres price index for the period is
 K

  q0i p1i 

100 iK1


  q0i p0i 
 i 1

The Laspeyres Quantity Index
We have quantity data for a set of items collected over a set of
K years. One period is selected as the base period. The
Laspeyres quantity index in any period is then the total cost of
the quantities traded in that period, based on the base period
prices, expressed as a percentage of the total cost of the base
period quantities.
Let p0i and q0i denote the price and quantity of the ith item in
the base period and q1i the quantity of that item in the period of
interest. The Laspeyres quantity index for that period is then
 K

  q1i p0i 

100 iK1


  q0i p0i 
 i 1

The Runs Test
Suppose we have a time series of n observations. Denote
observations above the median with “+” signs and observations
below the median with “-” signs. Use these signs to define the
sequence of observations in the series. Let R denote the number
of runs in the sequence. The null hypothesis is that the series is a
set of random variables. Table 11 of the appendix gives the
smallest significance level against which this null hypothesis can
be rejected against the alternative of positive association between
adjacent observations, as a function of R and n.
If the alternative is the two-sided hypothesis on
nonrandomness, the significance level must be doubled if it is
less than0.5. Alternatively, if the significance level, , read from
the table is bigger than 0.5, the appropriate significance level for
the test against the two-sided alternative is 2(1 - ).
The Runs Test: Large Samples
Given that we have a time series with n observations, n>20,
define the number of runs , R, as the number of sequences
above or below the median. We want to test the null
hypothesis
H0 : The series is random
The following tests have significance level :
(i)
If the alternative hypothesis is positive association between
adjacent observations, the decision rule is:
n
R  1
2
Reject H 0 if Z 
  Z
2
n  2n
4(n  1)
The Runs Test: Large Samples
(continued)
(ii) If the alternative is two-sided, of nonrandomness, the
decision rule is
n
R  1
2
Reject H 0 if Z 
  Z / 2
2
n  2n
4(n  1)
or
n
R  1
2
Z
 Z / 2
2
n  2n
4(n  1)
Time Series Components
Analysis
A time series can be described by models based on the following
components
Tt
Trend Component
St
Seasonal Component
Ct
Cyclical Component
It
Irregular Component
Using these components we can define a time series as the sum
of its components or an additive model
X t  Tt  St  Ct  I t
Alternatively, in other circumstances we might define a time
series as the product of its components or a multiplicative
model – often represented as a logarithmic model
X t  Tt St Ct I t
Simple Centered (2m+1)-Point
Moving Average
Let X1, X2, . . . , Xn be n observations on a time series
of interest. A smoothed series can be obtained by
using a simple centered (2m + 1)-point moving
average
m
1
X t* 
X t j

2m  1 j   m
(t  m  1, m  2,, n  m)
A Simple Moving Average Procedure
for Seasonal Adjustment
Let Xt (t = 1, 2, . . . ,n) be seasonal time series of period s
(s = 4 for quarterly data and s = 12 for monthly
data). A centered s-point moving average series
Xt*, is obtained through the following steps, where it
is assumed that s is even:
(i) Form the s-point moving averages
X
*
t .5

s/2

j   ( s / 2 ) 1
X t j
s s
s
s
(t  ,  1,  2,, n  )
2 2
2
2
(ii) Form the centered s-point moving averages
*
*
X

X
t .5
X t*  t .5
2
s
s
s
(t   1,  2,, n  )
2
2
2
Forecasting Through Simple
Exponential Smoothing
Let X1, X2, . . . , Xn be a set of observations on a nonseasonal
time series with no consistent upward or downward
trend. The simple exponential smoothing method of
forecasting then proceeds as follows
(i) Obtain the smoothed series Xˆ t as
Xˆ 1  X 1
Xˆ  Xˆ
t
t 1
 (1   ) X t
(0    1; t  1,2,, n)
Where  is a smoothing constant whose value is fixed
between 0 and 1.
(ii) Standing at time n, we obtain the forecasts of future
values, X n+h of the series
Xˆ nh  Xˆ n (h  1,2,3)
Forecasting with the Holt-Winters
Method: Nonseasonal Series
Let X1, X2, . . . , Xn be a set of observations on a nonseasonal
time series. The Holt-Winters method of forecasting
proceeds as follows
(i) Obtain estimates of level X̂ tand trend Tt as
Xˆ 1  X 2
T2  X 2  X 1
Xˆ t   ( Xˆ t 1  Tt 1 )  (1   ) X t
(0    1; t  3,4,, n)
Tt  Tt 1  (1   )( Xˆ t  Xˆ t 1 ) (0    1; t  3,4,, n)
Where  and  are smoothing constants whose value are
fixed between 0 and 1.
(ii) Standing at time n, we obtain the forecasts of future
values, X n+h of the series by
Xˆ n h  Xˆ n  hTn
Forecasting with the Holt-Winters
Method: Seasonal Series
Let X1, X2, . . . , Xn be a set of observations on a seasonal time
series of period s (with s = 4 for quarterly data and s = 12
for monthly data). The Holt-Winters method of forecasting
uses a set of recursive estimates from historical series.
These estimates utilize a level factor, , a trend factor, ,
and a multiplicative seasonal factor, . The recursive
estimates are based on the following equations
Xt
ˆ
ˆ
X t   ( X t 1  Tt 1 )  (1   )
Ft  s
(0    1)
Tt  Tt 1  (1   )( Xˆ t  Xˆ t 1 )
(0    1)
Xt
Ft  Ft  s  (1   )
(0    1)
Xˆ
t
Forecasting with the Holt-Winters
Method: Seasonal Series
(continued)
Where X̂ is the smoothed level of series, Tt is the smoothed
t
trend of the series, and Ft is the smoothed seasonal
adjustment for the series. The computational details are left
to a computer.
After the initial procedures generate the level, trend,
and seasonal factors from a historical series we can use the
results to forecast future values at, h, time periods ahead
from the last observation Xn in the historical series. The
forecast equation is
Xˆ nh  ( Xˆ t  hTt ) Ft hs
We note that the seasonal factor, Ft, is the one generated for
the most recent seasonal time period.
Autoregressive Models and Their
Estimation
Let Xt (t = 1, 2, . . ., n) be a time series. A model that can often
be used effectively to represent that series is the
autoregressive model of order p:
X t    1 X t 1  2 X t 2     p X t  p   t
Where , 1 2, . . .,p are fixed parameters and the t are
random variables that have means 0 and constant variance
and are uncorrelated with one another.
The parameters of the autoregressive model are estimated
through a least squares algorithm, as the values of , 1 2, . .
.,p for which the sum of squares
SS 
n
2
(
X




X


X




X
)
 t
1 t 1
2 t 2
p t p
t  p 1
is a minimum.
Forecasting From Estimated
Autoregressive Models
Suppose that we have observations X1, X2, . . .,Xt from a time
series and that an autoregressive model of order p has been
fitted to these data Write the estimated model as:
X t  ˆ  ˆ1 X t 1  ˆ2 X t 2    ˆp X t  p   t
Standing at time n, we obtain forecasts of future values of the
series from
Xˆ t  h  ˆ  ˆ1 Xˆ t  h1  ˆ2 Xˆ t  h2    ˆp Xˆ t  h p (h  1,2,3,)
Where for j > 0,
and for j  0,
is the forecast of X t+j standing at time n
is simply the observed value of X t+j .
Xˆ n  j
Xˆ n  j
Key Words
 ARIMA Models
 Autoregressive Models
and Their Estimation
 Calculating Price Indices
for a Single Item
 Change in Base Period
 Forecasting with the
Holt-Winters Method:
Nonseasonal Series
 Forecasting with the
Holt-Winters Method:
Seasonal Series
 Forecasting From
Estimated
Autoregressive Models
 Forecasting Through
Simple Exponential
Smoothing
 Index Numbers
 Laspeyres Price Index
 Runs Test
 Runs Test: Large
Samples
 Simple Centered (2m+1)Point Moving Averages
 Simple Exponential
Smoothing
 Simple Moving Average
Procedure for Seasonal
Adjustment
Key Words
(continued)
 Spliced Price Index
 Time Series
 Time Series Components
Analysis
 Unweighted Price Index
 Weighted Aggregate
Quantity Index
 Weighted Price Index