· Annual yield of a crop in a country for a number of years,
· Annual profit before tax of a firm,
· Daily temperature of a city,
· Annual rainfall of a country,
· Total monthly sales receipts in a departmental store,
· Daily closing prices of a share on the stock exchange,
· Annual balance of trade of a country,
· Weekly consumer price index number, etc.
(a) to examine the patterns of change over time, and
(b) to use these patterns to forecast and predict future values.
Graph of a Time Series –
Histogram:
A time series trend is graphically presented by plotting observed values against corresponding time points and joining these points by straight lines. This graph is commonly known as ‘Histogram’.
Example 1:
Plot the following observed values on a graph:
Table I – Quarterly Sales (In million Rs.)
Time 
Sale 

2003 
I 
219 

II 
357 

III 
645 

IV 
513 
2004 
I 
549 

II 
640 

III 
701 

IV 
590 
2005 
I 
657 

II 
394 

III 
543 

IV 
600 
Solution:
Signal and Noise:
Y = f(t) + U
Where Y : time series variable,
f(t) : systematic sequence signal, and
U : random sequence noise.
The change or variations in the observations of a time series are due to one or more of the four factors called ‘components of the time series’ or ‘characteristic movements of a time series’. These components are:
(a) Secular trend (T)
(b) Seasonal variations (S)
(c) Cyclical variations (C)
(d) Random variations (R)
(a) Secular Trend (T): The word ‘secular’ is used to mean ‘longterm’ or ‘relating to long periods of time’. Thus, the secular trend refers to the movement of a time series in one direction over a fairly long period of time. The movement is smooth, steady and regular in nature. Such a movement characterises the general pattern of increase or decrease in an economic or social phenomenon.
(b) Seasonal Variations (S): Such movements refer to shortterm variations which a time series usually follows during corresponding months or seasons of successive years. It is refer to any variation of repeating nature, within a period of one year, caused by recurring events. For example, increased demand for woollen clothes during winter, increased sales at a departmental store before Eid, increased sale of candies before Christmas, etc.
(c) Cyclical Variations (C): Such movements refer to longterm oscillations or swings about the trend line or curve. Since the movements take the form of upward and downward swings, they are also called ‘cycles’. The movements are considered cyclical only if they recur after a period of more than one year. The term ‘cycle’ is used for ‘business cycle’, which is consist of four phases, i.e., prosperity recession, depression and revival.
(d) Random Variations (R): These movements refer to fluctuations of irregular nature caused by chance events such as war, flood, storm, earthquake, accidents, strikes, etc. They are also known as ‘irregular’, ‘accidental’ or ‘erratic’ movements.
The first three components, i.e., secular trend, seasonal variations and cyclical variations, follow regular patterns of variation, therefore, fall under signal. While the random variations follow irregular patterns of movements, therefore, it falls under noise.
Analysis of Time Series / Time
Series Model:
1. An approach to represent time series data is to multiply the four components . this is called ‘Multiplicative Time Series Model’:
Y = T × C × S × I
Where Y : observed value of a time series at a particular time point
T : Trend (secular)
C : Cyclical variations
S : Seasonal variations
R or I : Irregular or random variations
2. The second approach is based on additive law, known as ‘Additive Model’:
Y = T + C + S + R
3. This analysis is often called the ‘decomposition’ of a time series into basic component movements.
Measurement of Secular Trend:
(a) Freehand curve method,
(b) Semiaverages method,
(c) Movingaverages method, and
(d) Leastsquares method.
Freehand, semiaverages and moving averages methods are used to study the pattern of change over a long period of time. They remove the shortterm changes or smooth out the series. Least squares method is used to forecast the future values.
(a) Freehand Curve Method: In this method the data are plotted on a graph measuring the time units (years, months, etc.) along the xaxis and the values of the time series variable along the yaxis. A trend line or curve is drawn through the graph in such a way that it shows the general tendency of the values.
Example 2:
Take the data from Example 1, and plot the observed on a graph and draw a trend line using ‘freehand curve method’.
Solution:
(b) SemiAverages Method: In this method, the data are divided into two parts. If the number of values is odd, either the middle value is left out or the series is divided unevenly. The averages for each part are computed and placed against the centre of each part. The averages are plotted and joined by line. The line is extended to cover the whole data.
Example 3:
For the following data, calculate the trend values and plot them on a graph using ‘semiaverages method’:
Table II
Pakistan’s
Per Capita Income (In Rs.)
Year

1996 
1997 
1998 
1999 
2000 
2001 
2002 
2003 
2004 
2005 
PCI 
15522 
17393 
18901 
20377 
25244 
27227 
28769 
31572 
35196 
41008 
Solution:
Year 
PCI 
Semi Total 
Semi Average 
Trend Value 
1996 
15522 


14180.6 
1997 
17393 


16834.0 
1998 
18901 
97437 
19487.4 
19487.4 
1999 
20377 


22140.8 
2000 
25244 


24794.2 
2001 
27227 


27447.6 
2002 
28769 


30101.0 
2003 
31572 
163772 
32754..4 
32754.4 
2004 
35196 


35407.8 
2005 
41008 


38061.2 
Note:
Difference between semiaverages: 32754.4 – 19487.4 = 13267
Average increase in income per year: 13267 ÷ 5 = 2653.4
Trend values for 1997: 19487.4 – 2653.4 = 16834
Trend values for 1999: 19487.4 + 2653.4 = 22140.8
(c) MovingAverages Method: Moving averages method is appropriate only when the trend is linear. It is also used to eliminate seasonal, cyclical and irregular fluctuations in the data. In this method, we find the simple average successively taking a specific number of values at a time. For e.g., if we want to find 3year moving average, we shall find the average of the first three values, then drop the first value and include the fourth value. The process will be continued till all the values in the series are exhausted.
Example 4:
For the following data, calculate 3year and 5year moving averages and plot them on a graph using ‘movingaverages method’:
Annual
GDP Growth Rate (in percent)
Year 
1996 
1997 
1998 
1999 
2000 
2001 
2002 
2003 
2004 
2005 
% 
6.6 
1.7 
3.5 
4.2 
3.9 
1.8 
3.1 
4.8 
6.4 
8.4 
Solution:
Year 
Values 
3
year moving total 
3
year moving average 
5
year moving total 
5
year moving average 
1996 
6.6 




1997 
1.7 
11.8 
3.93 


1998 
3.5 
9.4 
3.13 
19.9 
3.98 
1999 
4.2 
11.6 
3.87 
15.1 
3.02 
2000 
3.9 
9.9 
3.3 
16.5 
3.3 
2001 
1.8 
8.8 
2.93 
17.8 
3.56 
2002 
3.1 
9.7 
3.23 
20 
4 
2003 
4.8 
14.3 
4.77 
24.5 
4.9 
2004 
6.4 
19.6 
6.53 


2005 
8.4 




(d) Least Squares Method: The method of least squares states that of all the curves which can possibly be drawn to approximate the given data, the best fitting curve is the one for which the sum of squares of deviations is the least.
Method of least squares is used to fit a linear trend given by the straight line and nonlinear trend given by a second and higher degree curves. In this method, an algebraic equation is fitted to the observed data. This equation may be linear, quadratic or exponental depending upon the pattern of time series graph.
Let be the fitted equation, where gives the estimated value of y. The difference is called ‘error’. Least square method finds the constants of the equation such that:
· sum of errors is zero, that is .
· sum of squared errors is minimum, that is
(i) Fitting of Linear Equation: If the graph of a time series show a linear trend, a linear equation is fitted:
Where = the trend value for any time point,
a & b = constants of the equation.
The normal equations are:
Σy = na + bΣx
and
Σxy = aΣx + bΣx^{2}
To reduce the computations involved in the solution of there normal equations, timepoints are coded according to the following scheme, denoted by x:
For odd numbers of timepoints, codes are:
…………….. –4, –3, –2, –1, 0, 1, 2, 3, 4, ………………….
For even numbers of timepoints, codes are:
…………….. –7, –5, –3, –1, 1, 3, 5, 7, …………………….
In both cases, the sum of codes is zero.
Since the sum of codes (i.e., Σx) is zero, therefore the Σx^{3} is also equal to zero, and the formulae of a and b are as follows:
Example 5:
For the following data, calculate the trend values and plot them on a graph using ‘least squares method’ by fitting linear equation:
Table
III
Pakistan’s
GDP (In Trillion Rs.)
Year 
1996 
1997 
1998 
1999 
2000 
2001 
2002 
2003 
2004 
2005 
GDP 
3.10 
3.15 
3.26 
3.40 
3.53 
3.59 
3.71 
3.89 
4.14 
4.48 
Solution:
Year 
GDP(Y) 
X 
XY 
X^{2} 
Trend Y
= 3.625 + 0.07118ּX 
1996 
3.10 
–9 
–27.9 
81 
2.984 
1997 
3.15 
–7 
–22.05 
49 
3.127 
1998 
3.26 
–5 
–16.3 
25 
3.269 
1999 
3.40 
–3 
–10.2 
9 
3.411 
2000 
3.53 
–1 
–3.53 
1 
3.554 
2001 
3.59 
1 
3.59 
1 
3.696 
2002 
3.71 
3 
11.13 
9 
3.839 
2003 
3.89 
5 
19.45 
25 
3.981 
2004 
4.14 
7 
28.98 
49 
4.123 
2005 
4.48 
9 
40.32 
81 
4.266 
Total 
36.25 
0 
23.49 
330 

The equation of the straight line is: Y
= a + bX
The equation would be: Y = 3.625 +
0.07118ּX
(ii) Fitting of Quadratic Equation: The data may often be better expressed by a quadratic trend of the type:
which is a second degree curve called ‘parabola’. Values of a, b and c are obtained by solving the normal equations by the principle of least squares:
Σy
= na + bΣx + cΣx^{2} = na + cΣx^{2}
Σxy
= aΣx + bΣx^{2} + cΣx^{3} = bΣx^{2}
Σx^{2}y
= aΣx^{2} + bΣx^{3} + cΣx^{4} = aΣx^{2}
+ cΣx^{4}
Example 6:
For the following data, calculate the trend values and plot them on a graph using ‘least squares method’ by fitting quadratic equation:
Table
III
Pakistan’s
GDP (In Trillion Rs.)
Year 
1996 
1997 
1998 
1999 
2000 
2001 
2002 
2003 
2004 
2005 
GDP 
3.10 
3.15 
3.26 
3.40 
3.53 
3.59 
3.71 
3.89 
4.14 
4.48 
Solution:
Year 
GDP(Y) 
X 
XY 
X^{2} 
X^{3 } 
X^{4 } 
X^{2}Y 
Trend 
1996 
3.10 
–9 
–27.9 
81 
–729 
6561 
251.1 
3.126 
1997 
3.15 
–7 
–22.05 
49 
–343 
2401 
154.35 
3.174 
1998 
3.26 
–5 
–16.3 
25 
–125 
625 
81.5 
3.246 
1999 
3.40 
–3 
–10.2 
9 
–27 
81 
30.6 
3.341 
2000 
3.53 
–1 
–3.53 
1 
–1 
1 
3.53 
3.460 
2001 
3.59 
1 
3.59 
1 
1 
1 
3.59 
3.602 
2002 
3.71 
3 
11.13 
9 
27 
81 
33.39 
3.768 
2003 
3.89 
5 
19.45 
25 
125 
625 
97.25 
3.957 
2004 
4.14 
7 
28.98 
49 
343 
2401 
202.86 
4.170 
2005 
4.48 
9 
40.32 
81 
729 
6561 
362.88 
4.407 
Total 
36.25 
0 
23.49 
330 
0 
19338 
1221.05 

Σy = na + cΣx^{2}
Σxy = bΣx^{2}
Σx^{2}y = aΣx^{2}
+ cΣx^{4}
Substituting the values:
36.25
=
10a
+ 330c
23.49
=
330b
1221.05
=
330a
+ 19338c
Solving the above equations, we get a = 3.528, b = 0.07118 and c = 0.00294
The equation of the fitted parabola is:
Thus, the equation is:
(iii) Fitting of Exponental Equation: Let the exponental equation be fitted is:
Taking log on both sides of the exponental equation:
or
Where
A and B are computed using the two normal equations:
and converted to a and b by the relations:
a
= 10^{A} and b = 10^{B}
Example 7:
Fit an exponental curve to the following data, using method of least squares:
Pakistan
Cotton Production (in thousand tonnes)
Year 
1997 
1998 
1999 
2000 
2001 
2002 
2003 
2004 
2005 
Values 
9374 
9184 
8790 
11240 
10732 
10613 
10211 
10048 
14265 
Solution:
Year 
x 
y 
log
y 
xּlog
y 
x^{2 } 
Trend 
1997 
–4 
9374 
3.9719 
–15.8876 
16 
8980 
1998 
–3 
9184 
3.9630 
–11.889 
9 
9315 
1999 
–2 
8790 
3.9440 
–7.888 
4 
9663 
2000 
–1 
11240 
4.0508 
–4.0508 
1 
10023 
2001 
0 
10732 
4.0307 
0 
0 
10397 
2002 
1 
10613 
4.0258 
4.0258 
1 
10785 
2003 
2 
10211 
4.0091 
8.0182 
4 
11187 
2004 
3 
10048 
4.0021 
12.0063 
9 
11604 
2005 
4 
14265 
4.1543 
16.6172 
16 
12037 
Total 
0 

36.1517 
0.9521 
60 

Normal equations are:
Substituting the values:
36.1517 = 9A → A = 4.0169
0.9521 = 60B → B = 0.0159
Now
a = 10^{A} = 10^{4.0169} = 10397
b = 10^{B} = 10^{0.0159} = 1.0373
The exponental curve is:
Now, substituting the values: