Ordinary Least Squares Regression
HO #22
Projecting Future Sales Activity
You know from AGEC 317 that the Ordinary Least Squares (OLS) regression model can be specified as follows:
(1) Y = B0 + B1X1 + B2X2 + … + e
where:
Y dependent variable
Xi ith independent variable
B0 intercept
Bi coefficient for ith variable
e error term
You also know from Handout #6 (Enhanced Investment Expenditures) earlier this semester that theory of the firm suggests that a firm’s desired capital stock is given by:
(2) K*t = ([Pet Yet]/cet
where:
K*t Desired capital stock at end of year t
( Partial elasticity of production for capital in year t
Pet Expected product price in year t
Yet Expected output in year t
cet Expected rental price of capital in year t
Handout #6 also indicated that net investment expenditures for depreciable inputs like farm machinery would be given by:
(3) NIt = K*t - Kt-1 = It - Dt
If we want to project the demand for say farm machinery, therefore we can estimate the following equation using ordinary least squares regression:
(4) NIt = B0 + B1(RATIOt) + B2(Kt-1)
where:
RATIOt E(Pt*Yt)/E(ct)
Kt-1 Existing capital stock
Bi Coefficients to be estimated
Once we have estimated equation (4), we can project the level of net investment in farm machinery by inserting our expectations for farm revenue (E(Pt*Yt)) and the cost of capital E(ct) into the right-hand side and solving for net investment expenditures (NIt).
Gross investment expenditures – or total sales in the industry – is finally found by:
(5) It = NIt + Dt
Forecasting Future Sales Activity
Given the superior econometric results from the full model based upon economic theory, the next step is to solve this equation beyond the sample period to forecast future sales activity.
Point forecast based on a single set of expectations:
Given:
RATIOt 4.19 (historical mean)
NIt-1 9.50 (lagged sales)
(6) NIt = -0.455 + 0.351(RATIOt) + 0.888(NIt-1)
= -0.455 + 0.351(4.19) + 0.888(9.5)
= $9.45
or $9.45 billion in expected machinery sales next year. Given a standard deviation of 1.49, we see that there is a significant probability that sales could be as low as $8.93 billion or as high as $9.97 billion.
(7) NIt = -0.455 + 0.351(RATIOt) + 0.888(NIt-1)
= -0.455 + 0.351(4.19 – 1.49) + 0.888(9.5) = $8.93
= -0.455 + 0.351(4.19 + 1.49) + 0.888(9.5) = $9.97
High, Low and Expected Sales Next Year.
[pic]
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$8.93 $9.45 $9.97
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