Supplemental A. Identification.
We have engaged in the following exercises in order to address identification:
1. Recovering the parameters from the full and restricted versions of our model
considering a simple 1-country 1-technology framework. We run simulations for a
full grid of parameter values (we test for 3x3x3x3=81 initial starting values). See
exercise 1 below.
2. We run a total of 10 simulations for the model using the parameter values estimated
in the paper under a variety of error term assumptions, various shocks for both
product markets and for all 19 countries. For every simulation we use the created
time series to re-estimate the model and recover the original estimated values of the
53 parameters. See exercise 2 below.
We ran simulations with different parameter combinations illustrating how stronger
network effects (through a higher ) delays the adoption levels in the early stages relative
to the diffusion peak and that these patterns are uniquely identified. In short, the Bass-type
structure is a specific case of ours (with  = 0). Our proposed model specification is an
extension of the simple 1-country 1-technology network structure.
Exercise 1:
To illustrate, for the simplest framework with one country and one technology (and just 1
gamma):
  N ( t ) 
M̂ ( t )

 1  ˆ exp ˆ 
S( t )
  S ( t ) 

(1)


N( t  1 ) 
n̂( t )  ˆ  ˆ
 M̂ ( t  1 )  N ( t  1 )   t ,
M̂
(
t

1
)


Here, there are 4 important parameters that may vary:
1) Coefficient of external influence, 
2) Coefficient of internal influence, 
3) Early-adopter coefficient, 
4) Network parameter, 
(2)
Consequently, we ran the “initial identification exercise” for 3 alternative values for each
parameter, yielding 3x3x3x3=81 exercises. Thus, in order to test for potential identification
issues, for the endogenous market-potential structure (equation 1), we ran 3x3x3x3
combinations of each parameter spanning a reasonable set of parameter values. For the
model calibration, the parameter grid was chosen in line with the value ranges obtained in
the estimations of the full model: =0.1, 0.2, 0.3; = 0, 0.2, 0.4; =0.33, 0.66, 0.99; and =0.5,
1, and 2. For each and every combination of parameters, a 25-period time series of i.i.d.
random shocks normally distributed was generated. The potential market was normalized
to M ( t )  1, t  1,...,25. For period 1, an initial adoption level of 0.1% (N(1)=0.001) was
assumed. For every combination of parameters, the adoption time series were created by
using the difference equation system (1)-(2). With the simulated data, both the full and the
restricted model (=0) were estimated. The results of the simulation exercise were the
following:
-
For the estimation of the original model, in all the 81 exercises, the four original
parameter values: , , , and , lied within the 95% confidence interval of the
estimated ones.
-
-
-
For the estimation of the model with no network effects (=0) the estimated
parameters for , , and , are consistently biased, as none of the original values lie
within the 95% confidence interval of the estimated ones.
In all cases, for both estimations – the full and the restricted model – the level of fit
was very high, with R2 over 95%. However, in all cases the log-likelihood ratio test
rejects the null hypothesis, =0 at a 0.05% level.
Despite the high level of fit attained for the model with no network effects, a simple
analysis of the differences between the estimated diffusion curve and the simulated
dataset reveals the important limitation of the Bass model to resemble the dynamics
generated by the model with a growing potential market. Assuming a constant
potential market, the Bass model consistently overestimates the adoption levels
during the first periods, compensating with lover adoption rates after the diffusion
peak.
The following example illustrates the results:
Time series of 50 periods, random shocks normally distributed, error variance 0.001, the
parameters have been calibrated according to the values obtained for the diffusion of the
Internet: =0.25 and = 0.45. The other parameters were set to =0.99, and =2. For
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period 1, an adoption level of 0.1% (N(1)=0.001) was assumed. The estimated parameters
for both specifications with and without network effect are reported below
Parameter estimates
External influence
()
Internal influence
()
Early adoption Coefficient
()
Without NE
With NE
0.0059 ***
(0.0006)
0.2326 ***
(0.0243)
0.4076 ***
(0.0035)
0.4228 ***.
(0.0325)
0.2006 ***
(0.0073)
0.9876 ***.
(0.0024)
Network Effects
( )
1.9964 ***
(0.0040)
Adjusted R2
Log Likelihood
0.9974
0.9985
262.7971
280.7234
Note: * p<0.05, ** p<0.01, *** p<0.001 ; Standard Errors in parentheses
As with the previous 81 cases, for the model with NE, the original parameter values: , , ,
and , lies within the 95% confidence interval of the estimated ones. Despite the high level
of fit presented by the restricted version with no NE, the LL ratio test rejects the null
hypothesis at a 0.1% level. The following figures illustrate the simulated dataset and the
estimated diffusion curves with both models. The model with a constant potential market
produces a biased diffusion curve, with a higher number of new adopters before the
diffusion peak and a lower one later. This result is clearly illustrated in the analysis of the
differences between the data and the estimated adoption paths, plotted in the last figure.
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Estimated diffusion curves
Differences between data and the
estimated paths
Exercise 2:
An identification test has also been run for the full model, with local, cross country, and
indirect effects, containing a total of 53 parameters. In every simulation a total of 38 time
series of fifteen periods were generated, corresponding to the adoption levels for both
technologies across the 19 countries. A total of 10 simulations were run, obtaining 530
estimated parameters. For every simulation, two 15x19 matrices of normally distributed
shocks were generated, with zero mean and standard deviations equal to the ones
estimated for each market. Departing from the real adoption levels in the first year (1991),
the dynamic equations of the full model were used to create the 15-period simulated
diffusion paths for both technologies and all 19 countries:

N ( t  1) 
N xi ( t )  N xi ( t  1 )  ˆ xi  ˆ xi xi
M xi ( t  1 )  N xi ( t  1 )   xit , i  [ 1 : 19 ],
M xi ( t  1 ) 


N yi ( t  1 ) 
N yi ( t )  N yi ( t  1 )  ˆ yi  ˆ yi
 M yi ( t  1 )  N yi ( t  1 )   yit , i  [ 1 : 19 ],
M yi ( t  1 ) 



t  [ 1991 : 2005 ]
t  [ 1991 : 2005 ]
Where the potential markets evolve according to the equations:
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

  N xj ( t ) 
 N (t ) ~
M xi ( t )
  ˆˆ  N yi ( t ) 
  ˆ x  j i
 1  ˆ x exp ˆ x  xi
xy 

  S xj ( t ) 
S xi ( t )

 S xi ( t ) 
 S yi ( t ) 
 j i




  N yj ( t ) 
 N yi ( t )  ~
  ˆˆ  N xi ( t ) 
  ˆ y  j i
 1  ˆ y exp ˆ y 
yx 

 S (t ) 
  S yj ( t ) 
S yi ( t )

yi
 S xi ( t ) 


j

i



M yi ( t )
For every simulation the parameters were re-estimated, testing if the original parameter
values lied within the 95% confidence interval of the estimated ones. In 95.85% of the
comparisons the original parameters lied within the 95% confidence interval (508 out of
530 estimated parameters).
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