In: Statistics and Probability
The annual revenues collected in each of the past ten years for the Orange County Solid Waste Division are provided below. If the revenue total for 2014 is $42,843,901, which revenue projection method (SMA, TMA, or regression) is the most accurate in this case? Use APE to justify your answer. Use the last three years of revenues for moving averages and all years for regression
Year |
Revenue |
2005 |
33,120,989 |
2006 |
36,979,392 |
2007 |
36,390,302 |
2008 |
35,678,632 |
2009 |
37,986,901 |
2010 |
39,697,702 |
2011 |
37,639,287 |
2012 |
39,479,675 |
2013 |
40,099,709 |
SOLUTION:
[Moving average of the given last 3 years = (37739287 + 39479674 + 40199709)/3 = 39139556
Estimated regression equation is: y2014 = 33934691.44 + (710496.73 x 10) = 41039659.
Clearly, 41039659 is closer to 2014 actual of 43844901]
Back-up Theory
The linear regression model Y = β0 + β1X + ε, ………………………………………..(1)
where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ2.
Estimated Regression of Y on X is given by: Y = β0cap + β1capX, …………………….(2)
where β1cap = Sxy/Sxx = r.√(Syy/Sxx) = r.(SDy/SDx) and β0cap = Ybar – β1cap.Xbar..…………………………….…..(3)
Mean X = Xbar = (1/n) Σ(i = 1 to n)xi ………………………………………….……….….(4)
Mean Y = Ybar = (1/n) Σ(i = 1 to n)yi ………………………………………….……….….(5)
Sxx = Σ(i = 1 to n)(xi – Xbar)2 ………………………………………………..…………....(6)
Syy = Σ(i = 1 to n)(yi – Ybar)2 ……………………………………………..………………(7)
Sxy = Σ(i = 1 to n){(xi – Xbar)(yi – Ybar)} ……………………………………………….(8)
Now to work out the solution,
Take the given first year as x = 1, next year as x = 2, and so on, last year as x = 9 and revenue as y.
n |
9 |
Xbar |
5.00 |
ybar |
37487175.1111 |
Sxx |
60 |
Syy |
4.07794E+13 |
Sxy |
42629804 |
β1cap |
710496.7333 |
β0cap |
33934691.44 |
Estimated regression equation is: ycap = 33934691.44 + 710496.73x.