CHAPTER 1: An introduction to time series and forecasting

CHAPTER 1: An introduction to time series
and forecasting
Basic Questions:
1. What is a time series?
2. What are the purposes of time series analysis?
3. what are the difference between classical (Independent Identically Distributed (IID)) statistical
analysis (e.g. inference and modelling) and time series analysis?
1 Time Series
A time series is a sequence of observations over time.
Example: Records of a person’s height:
age : 1 2 3 4 5 6 7
height(m) : 0.4 0.5 0.8 1.0 1.1 1.2 1.4
(notation : y 1 y 2 y 3 y 4 y 5 y 6 y 7 )
For this example, we have n = 7 observations. We call n the number of observations or
the length of a times series. We denote the observation at time t by y t (or x t etc.)
The time series can be then denoted as
{0.4, 0.5, 0.8, 1.0, 1.1, 1.2, 1.4}
or
{y t : t = 1, 2, ..., n}
Plot of a time series
1

1.5
height
1
0.5
0
1 2 3 4 5 6 7 8 9 10
age
1.5
height
1
0.5
0
1 2 3 4 5 6 7 8 9 10
age
Figure 1:
Note that the above observations are taken over equally time intervals. We can also observe
the variable with unequally time intervals
age : 0.5 1 1.5 2 3 5 7
height(m) : 0.35 0.4 0.45 0.5 0.8 1.1 1.4
Theoretically, we can observe the data continuously and get a “continuous-time” time series.
Remarks: We are mainly interested in discrete-time time series with equally fixed time
intervals. e.g. observations made monthly, daily, weekly, etc.
2

Example (Unemployment Rate (%) in Singapore)
year rate year rate year rate
1973 4.4 1984 2.7 1995 2.7
1974 3.9 1985 4.1 1996 3.0
1975 4.5 1986 6.5 1997 2.4
1976 4.4 1987 4.7 1998 3.2
1977 3.9 1988 3.3 1999 4.6
1978 3.6 1989 2.2 2000 4.4
1979 3.3 1990 1.7 2001 3.4
1980 3.5 1991 1.9 2002 5.2
1981 2.9 1992 2.7 2003 5.4
1982 2.6 1993 2.7 2004 5.3
1983 3.2 1994 2.6
7
6
5
4
3
2
1
1970 1975 1980 1985 1990 1995 2000 2005
Figure 2:
More time series [what can you observe in the time series?]
7000
Canadian Lynx captured (1828-1934)
6000
number of lynx
5000
4000
3000
2000
1000
0
0 20 40 60 80 100 120
time (year)
Figure 3:
3

35
Temperature in Hong Kong (1994-1997)
daily temperature in HK
30
25
20
15
10
5
0 200 400 600 800 1000 1200
Number of patients with respiratory problems in Hong Kong (1994)
300
no. of patients
250
200
150
100
0 50 100 150 200 250 300 350 400
time (daily)
10000
Measles cases in London (1944-1978)
cases of measles
8000
6000
4000
2000
0
0 100 200 300 400 500 600 700 800 900 1000
time (week)
DOW stock index (1992-2005)
15000
DOW index
10000
5000
0
1992 1994 1996 1998 2000 2002 2004
time
return: z t
= log(y t
) −log(y t−1
)
0.1
0.05
0
−0.05
−0.1
1992 1994 1996 1998 2000 2002 2004
time
4

2 Forecasting
A major objective of time series analysis is forecasting of future values of the series
e.g. what will be the unemployment rate next year?
Is there a trend in global temperature?
what is the seasonal effect?
what is the relationship between GDP and interest rate?
Forecasting methods:
1. Qualitative forecasting methods: use the opinions of experts to predict future events
subjectively.
2. Quantitative forecasting methods: Based the historical data, use statistical methods to
predict future values of a variable.
3 The difference between the time series and IID statistics
Time series data are dependent
1. there is an order for the observation of time series.
2. time series data are dependent. e.g. this month’s unemployment rate will be correlated
with the last month’s.
The problem with dependence:
Consider the IID case
X 1 , X 2 , · · · , X n random sample with mean µ and variance σ 2 . Then we estimate µ by
ˆµ = (X 1 + X 2 + · · · + X n )/n
5

The variance of ˆµ is
we have V ar(ˆµ) → 0 as n → ∞.
V ar(ˆµ) = 1 n 2 (V ar(X 1) + V ar(X 2 ) + ...V ar(X n )) = σ2
n
Imagine the situation where all the X i ’s are “perfectly” correlated, i.e.
Cov(X i , X j ) = σ 2
Corr(X i , X j ) = 1
We still estimate µ by
ˆµ = (X 1 + X 2 + · · · + X n )/n
the variance is then
V ar(ˆµ) = 1 n 2 V ar(X 1 + X 2 + · · · + X n )
= 1 n∑
n { V ar(X 2 i ) + 2 ∑ Cov(X i , X j )}
i=1
i