Supplementary Notes -- Trend Forecasting -- Constant Percent Rate of Growth

 

Given this data:

 

Sales    Time Period (year)

6500                15

6500                14

5750                13

5250                12

4750                11

3750                10

3500                9

3000                8

3000                7

2750                6

2500                5

2250                4

2000                3

1750                2

1750                1

 

 

 

 

 

A linear trend model assumes that there is a constant dollar amount of increase in sales per time period   The implementation of the simple linear trend regression model is easy: the dependent variable data points are the sales figures and the independent variable data points are the years.  Thus the relationship is postulated to be :  Sales = a + bT, where T is time (year).  Performing this regression using the above data yields the equation:   Sales = 759 + 363T  (roughly)

 

Given the seemingly curved pattern of the data above, when plotted, maybe a different assumption makes better sense -- that there is a constant percent inccrease in sales per time period.  That is, a constant rate of growth.  This approach is often better in time-series work, such as trend projection.

                                    a.         but to explore this idea, you can not use "regular" regression, because the function would not be linear in this case.

b.         the general functional relationship for constant percent growth is:

                                                Q_t  =  Q₀(1+g)^t  ,where Q₀ is base period sales 

                                 where t is time period

                                 where g is the growth rate

                                    c.         the answer is to use a logarithmic form:

                                              (using the rules of logs)

log Q_t    =  log Q₀  +  t log(1+g)

 

here, the coeffic. ( b ) which comes out of the regression will be the log of (1+g), where g is the growth rate (% growth).

 

                                                The data to be input will be the logs of the Sales figs. at each time t (dep. vlb.) , and the t's (1,2,3,...) will be the independent variable.

The reported (by program) constant term will be the log of sales in the period before t=1.  ( Qo )

                                                (logs to base 10 used here)

d.         The result of running the regression is:

approximately:  Q_t  =  3.169  + .044 t 

                                                                                         (245.8)   (31.1)

                                                                   R square = .99, ts in paren.

                            

e.         but, to interpret the equation, convert back from logs:

Q_t  = antilog(3.169) x [antilog(.044)]^t       

Log₁₀                                        which is:

                                                                        Q_t  = $1475.7 (1.107)^t             

 

                       for t=23 :

                                                                        Q  = $1475.7 (1.107)²³   =$15,300  

 

            3.         Compare this with the projection for yr=23 using the simple linear trend model regression (const. annual $ increase )

a. the figure, using orig. regression equation

                           (for yr =23),  is 9117 mil. [759.52 + (363.39x23)]

 

                        b. using the first approach results in a drastic underestimate when projecting a distant  time period.