Question

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

Answer 1

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 = β_0cap + β_1capX, …………………….(2)

where β_1cap = Sxy/Sxx = r.√(Syy/Sxx) = r.(SDy/SDx) and β_0cap = Ybar – β_1cap.Xbar..…………………………….…..(3)

Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i ………………………………………….……….….(4)

Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i ………………………………………….……….….(5)

Sxx = Σ(i = 1 to n)(x_i – Xbar)² ………………………………………………..…………....(6)

Syy = Σ(i = 1 to n)(y_i – Ybar)² ……………………………………………..………………(7)

Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – 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.