ANNEX D
(to Recommendation E.506)
Example of a top down modelling method
The model for forecasting telephone traffic from Norway to the European
countries is divided into two separate parts. The first step is an econometric
model for the total traffic from Norway to Europe. Thereafter, we apply a model
for the breakdown of the total traffic on each country.
D.1 Econometric model of the total traffic from Norway to Europe
With an econometric model we try to explain the development in telephone
traffic, measured in charged minutes, as a function of the main explanatory
variables. Because of the lack of data for some variables, such as tourism, these
variables have had to be omitted in the model.
The general model may be written:
Xt = eK . eq GNP \s(a,t) . eq P \s(b,t) . eq A \s(c,t) . eut (t =
1, 2, . . ., N) (D-1)
where:
Xt is the demand for telephone traffic from Norway to Europe at time t
(charged minutes).
GNPt is the gross national product in Norway at time t (real prices).
Pt is the index of charges for traffic from Norway to Europe at time t
(real prices).
At is the percentage direct-dialled telephone traffic from Norway to
Europe (to take account of the effect of automation). For
statistical reasons (i.e. impossibility of taking logarithm of
zero) At goes from 1 to 2 instead of from 0 to 1.
K is the constant.
a is the elasticity with respect to GNP.
b is the price elasticity.
c is the elasticity with respect to automation.
ut is the stochastic variable, summarizing the impact of those
variables that are not explicitly introduced in the model and whose
effects tend to compensate each other (expectation of ut = 0 and
var ut = s2).
By applying regression analysis (OLSQ) we have arrived at the coefficients
(elasticities) in the forecasting model for telephone traffic from Norway to
Europe given in Table D-1/E.506 (in our calculations we have used data for the
period 1951-1980).
The t statistics should be compared with the Student's Distribution with N
- d degrees of freedom, where N is the number of observations and d is the number
of estimated parameters. In this example, N = 30 and d = 4.
The model "explains" 99.7% of the variation in the demand for telephone
traffic from Norway to Europe in the period 1951-1980.
Fascicle II.3 - Rec. E.506 PAGE1
From this logarithmic model it can be seen that:
- an increase in GNP of 1% causes an increase in the telephone traffic of
2.80%,
- an increase of 1% in the charges, measured in real prices, causes a
decrease in the telephone traffic of 0.26%, and
- an increase of 1% in At causes an increase in the traffic of 0.29%.
We now use the expected future development in charges to Europe, in GNP,
and in the future automation of traffic to Europe to forecast the development in
telephone traffic from Norway to Europe from the equation:
Xt = et-16.095 . GNPt2.80 . Ptu-0.26 . At0.29 (D-2)
TABLE D-1/E.506
Coefficients Estimated values t statistics
K -16.095 -4.2
a 2.799 8.2
b - 0.264 -1.0
c 0.290 2.1
D.2 Model for breakdown of the total traffic from Norway to Europe
The method of breakdown is first to apply the trend to forecast the
traffic to each country. However, we let the trend become less important the
further into the period of forecast we are, i.e. we let the trend for each
country converge to the increase in the total traffic to Europe. Secondly, the
traffic to each country is adjusted up or down, by a percentage that is equal to
all countries, so that the sum of the traffic to each country equals the
forecasted total traffic to Europe from equation (D-2).
Mathematically, the breakdown model can be expressed as follows:
Calculation of the trend for country i:
Rit = bi + ai . t, i = 1, . . ., 34 t = 1, . . ., N (D-3)
where
Rit = eq \f( Xit,Xt), i.e country i's share of the total traffic to Europe.
Xit is the traffic to country i at time t
Xt is the traffic to Europe at time t
t is the trend variable
ai and bi are two coefficients specific to country i; i.e. ai is country i's
trend. The coefficients are estimated by using regression analysis, and we have
based calculations on observed traffic for the period 1966-1980.
The forecasted shares for country i is then calculated by
Rit = RiN + ai . (t - N) . e-eq \f(t-5,40) (D-4)
where N is the last year of observation, and e is the exponential function.
The factor e-eq \f(t-5,40) is a correcting factor which ensures that the
growth in the telephone traffic to each country will converge towards the growth
of total traffic to Europe after the adjustment made in Equation (D-6).
To have the sum of the countries' shares equal one, it is necessary that
eq \i\su(i, , ) Rit = 1 (D-5)
This we obtain by setting the adjusted share, eq \x\to(R)it, equal to
eq \x\to(R)it = Rit eq \f(1,\i\su(i, , )Rit) (D-6)
Each country's forecast traffic is then calculated by multiplying the
total traffic to Europe, Xt, by each country's share of the total traffic:
Xit = eq \x\to(R)it x Xt (D-7)
PAGE4 Fascicle II.3 - Rec. E.506
D.3 Econometric model for telephone traffic from Norway to Central and South
America, Africa, Asia, and Oceania.
For telephone traffic from Norway to these continents we have used the
same explanatory variables and estimated coefficients. Instead of gross national
product, our analysis has shown that for the traffic to these continents the
number of telephone stations within each continent are a better and more
significant explanatory variable.
After using cross-section/time-series simultaneous estimation we have
arrived at the coefficients in Table D-2/E.506 for the forecasting model for
telephone traffic from Norway to these continents (for each continent we have
based our calculations on data for the period 1961-1980):
TABLE D-2/E.506
Coefficients Estimated values t statistics
Charges -1.930 -5.5
Telephone stations 2.009 4.2
Automation 0.5 -
We then have R2 = 0.96. The model may be written:
Xkt = eK . (TSkt)2.009 . (Pkt)1.930 . (Akt)0.5 (D-8)
where
Xkt is the telephone traffic to continent k (k = Central America, .
., Oceania) at time t,
eK is the constant specific to each continent. For telephone traffic
from Norway to:
Central America: K1 = -11.025
South America: K2 = -12.62
Africa: K3 = -11.395
Asia: K4 = -15.02
Oceania: K5 = -13.194
TSkt is the number of telephone stations within continent k at time t,
Pkt is the index of charges, measured in real prices, to continent k at
time t, and
Akt is the percentage direct-dialled telephone traffic to continent k.
Equation (D-8) is now used - together with the expected future development
in charges to each continent, future development in telephone stations on each
continent and future development in automation of telephone traffic from Norway
to the continent - to forecast the future development in telephone traffic from
Norway to the continent.
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Fascicle II.3 - Rec. E.506 PAGE1
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PAGE4 Fascicle II.3 - Rec. E.506