Demand Foresting/ Business Forecasting
(Least Square Method)
1. Method of Least Squares- Fitting Linear Trend (For Odd Years)
Given the following data, forecast the estimated value of sales for the year 2017 using the least
square method.
Year Sales (Y)
(‘000 units)
2010 125
2011 128
2012 133
2013 135
2014 140
2015 141
2016 143
Find estimated sales for the year 2017?
N=7 (No. of observations/ observed values)
Answer:
Year Sales (Y) X XY 𝑋 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑌
(‘000 units) (𝑌)
2010 125 -3 -375 9 125.67
2011 128 -2 -256 4 128.78
2012 133 -1 -133 1 131.89
2013 135 0 0 0 135
2014 140 1 140 1 138.11
2015 141 2 282 4 141.22
2016 143 3 429 9 144.33
∑ 𝑌 =945 𝑋=0 𝑋𝑌 = 87 𝑋 = 28
2017 4 147.44
2018 5 150.55
2019 6 153.66
2020 7 156.77
, Here we need to convert “Year” into “X” variable and “X” is considered as the independent
variable for our analysis.
𝑌𝑒𝑎𝑟 − 𝑂𝑟𝑖𝑔𝑖𝑛
𝑋=
𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙
Where, origin is the mid-year i.e. 2013
Interval is the year gap i.e. 1.
So for the year 2010, X will be
𝑌𝑒𝑎𝑟 − 𝑂𝑟𝑖𝑔𝑖𝑛 2010 − 2013
𝑋= = = −3
𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 1
Similarly, for the years 2011, 2012, 2013, 2014, 2015, and 2016, the X values will be -2, -1, 0, 1,
2, and 3, respectively.
Here estimated Y i.e.
Actual Y i.e.
In order to get the estimated Y values, first we have to find out the values of “a” and “b”. For this
we need to solve two normal equations as follows:
∑ 𝑌 = 𝑁𝑎 + 𝑏 ∑ 𝑋 (1)
∑ 𝑋𝑌 = 𝑎 ∑ 𝑋 + 𝑏 ∑ 𝑋 (2)
Solving the above two normal equations with the help of above table, we can find out the values
of “a” and “b” as follows:
From equation (1), we can get,
(Least Square Method)
1. Method of Least Squares- Fitting Linear Trend (For Odd Years)
Given the following data, forecast the estimated value of sales for the year 2017 using the least
square method.
Year Sales (Y)
(‘000 units)
2010 125
2011 128
2012 133
2013 135
2014 140
2015 141
2016 143
Find estimated sales for the year 2017?
N=7 (No. of observations/ observed values)
Answer:
Year Sales (Y) X XY 𝑋 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑌
(‘000 units) (𝑌)
2010 125 -3 -375 9 125.67
2011 128 -2 -256 4 128.78
2012 133 -1 -133 1 131.89
2013 135 0 0 0 135
2014 140 1 140 1 138.11
2015 141 2 282 4 141.22
2016 143 3 429 9 144.33
∑ 𝑌 =945 𝑋=0 𝑋𝑌 = 87 𝑋 = 28
2017 4 147.44
2018 5 150.55
2019 6 153.66
2020 7 156.77
, Here we need to convert “Year” into “X” variable and “X” is considered as the independent
variable for our analysis.
𝑌𝑒𝑎𝑟 − 𝑂𝑟𝑖𝑔𝑖𝑛
𝑋=
𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙
Where, origin is the mid-year i.e. 2013
Interval is the year gap i.e. 1.
So for the year 2010, X will be
𝑌𝑒𝑎𝑟 − 𝑂𝑟𝑖𝑔𝑖𝑛 2010 − 2013
𝑋= = = −3
𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 1
Similarly, for the years 2011, 2012, 2013, 2014, 2015, and 2016, the X values will be -2, -1, 0, 1,
2, and 3, respectively.
Here estimated Y i.e.
Actual Y i.e.
In order to get the estimated Y values, first we have to find out the values of “a” and “b”. For this
we need to solve two normal equations as follows:
∑ 𝑌 = 𝑁𝑎 + 𝑏 ∑ 𝑋 (1)
∑ 𝑋𝑌 = 𝑎 ∑ 𝑋 + 𝑏 ∑ 𝑋 (2)
Solving the above two normal equations with the help of above table, we can find out the values
of “a” and “b” as follows:
From equation (1), we can get,
