Time series Analysis:
Introduction:
The analysis of experimental data
that have been observed at different points in time leads to new and unique
problems in statistical modeling and inference. The obvious correlation
introduced by the sampling of adjacent points in time can severely restrict the
applicability of the many conventional statistical methods traditionally
dependent on the assumption that these adjacent observations are independent
and identically distributed. The systematic approach by which one goes about
answering the mathematical and statistical questions posed by these time
correlations is commonly referred to as time series analysis. (Robert H. Shumway
and David S. Sto‑ er.,2016)
Time series is defined as set of
quantitative data taken at equal successive interval of time period. Time
series analysis is defined as the process of decomposing (break down) time
series data for estimation of growth, decay, prediction and random fluctuation.
Some examples of time series data:
1.
Price over time
2.
Quantity produced of a crop over time
3.
Migration of Nepalese labor to foreign over time
4.
Increase in income of people in an area over
time
5.
Increase
in temperature over time (Global warming)
Time series looks upon following
components during its procedure:
1.
Seasonal variation
2.
Annual variation
3.
Trend variation
4.
Cyclical variation
5.
Random/ Irregular variation
1.
Seasonal variation:
Seasonal variation is the
tendency of change in time series data over different seasons in repeated
fashion for numbers of year.
2.
Annual variation:
Annual variation is the change in
value of time series indicator year after year.
3.
Random variation:
Random variation is the irregular
movement in time series data resulted from unpredictable factors.E.g Political
reasons, natural calamities etc.
4.
Cyclical variation:
Cyclical variation is the
tendency of time series data to converge or diverge across time period. For
this type of study, we need data of long time. For example, Business cycle and
Cob-Web price model (i.e. 10-60 years).
5.
Trend variation:
Trend variation is pattern of
change in value of time series data over longer duration of time estimated
generally through regression function
After having these concepts on
components of time series, next step is to estimate their presence in the data
matrix.
I)
Estimate yearly average
II)
Estimate seasonal average
III)
Estimate overall average
IV)
Estimate seasonal index
Seasonal index= seasonal average/overall average = y/z
(i.e. y1/z, y2/z, y3/z, y4/z)
V)
Multiply the different values of seasons by
respective seasonal index. It gives deseasonalized value (i.e. seasonalization
is omitted)
(Note: Do not do deseasonalization when seasonal
effects are to be measured. Refer the dummy regression technique to measure the
effects of seasons on time series data.)
VI)
Restructure data matrix in to single column
|
Year
|
Season
|
Value (descending
value)
|
|
2011
|
q1
|
X11
|
|
|
q2
|
X12
|
|
|
q3
|
X13
|
|
|
q4
|
X14
|
|
2012
|
q1
|
X21
|
|
|
q2
|
X22
|
|
|
q3
|
X23
|
|
|
q4
|
X24
|
VII)
Carryout moving average to remove irregular
variation
E. g:
80= x11
82=x12
83=x13
85=x14
90=x15
93=x16
(We lose first 2 and last 2 data if the period is 4,
period 4 – avg. of 4 data taken)
In even period averaging technique, we will lose k no
of observation where k is no. of period.
VIII)
Trend: Estimate trend for the obtained data
(deseasonalized and smoothened) using the following methods:
A)
Semi average technique
B)
Layman technique
C)
Regression technique
A) Semi
average technique:
Take the average
of the first half data and the last half data separately and plot them in the
graph.
B) Scattered
diagram:
Plot the
information in scattered diagram.
C) Regression
technique:
Employ
regression technique and estimate trend using normal equation.
∑y=na+b∑x
∑xy=a∑x+b∑x2
(Normal
equations)
Price-Y
Time-x
Estimate the
values of ‘a’ and ‘b’ in above equation and place in trend function:
Y=a+bx
i.e. P=a+bT
ꝺP/ꝺT=b
Comments
Post a Comment