4. Hypotheses and Methodology
4.2 Methodology
4.2.2 Tobit Regressions
[Θ ×Θ ]
Θ ×Θ , it measures the technology frontier shift between time period t and t+1. Färe et al. (1992,1994a) point out that a value of FSk greater than one indicates a positive shift or technical progress, a value of FSk less than one indicates a negative shift or technical regress, and value of FSk equal to one indicates no shift in technology frontier.
4.2.2 Tobit Regressions
The Malmquist indices, such as MPI, TEC, and FS, represent the change of productivity of decision make units, and the observations are censored at zero as these variables are ratios of two positive efficiency variables. Thus the Tobit regression models are used to perform the relationships for limited dependent variables, the Malmquist indices of music firms, with independent variables, such as input changes, output changes, exogenous variable changes, and dummy variables, in order to identify the key variables which influence the productivity change of music industry. The Malmquist indices, MPI, TEC, and FS respectively, are set as the dependent variables in (6), and are censored to zero in the Tobit models; the change of attributes, such as piracy rate change, number of internet users change, sales change, number of employees change, and the change of fixed assets, between two years are set to be the independent variables of the Tobit models. Results of these Tobit models will illustrate that how the business
strategies affect the productivity changes of music industry through the internal variables and exogenous variables changes, and fertilize the results of Malmquist indices analysis with two stages evidences.
1 2 3 4 5 j
jt
j
, if Y > 0
Y
= {
0 , if Yα +β PCjt+β IUCjt +β SCjt+β ECjt +β CCjt+εjt ≤0 (6) whereYjt : dependent variable. Dependent variables are set to the TEC, FS, and MPI of the DEA Malmquist Indices in jth firm, respectively.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
The Tobit models, considered dummy variables in addition, are made use of testing the hypotheses one to hypotheses three, listed in chapter 4.1, in our research further. Dummy variables, Di, in these Tobit models are symbolized different business strategies of the music industry; while i=1 represents the events of filing lawsuits; i=2 represents the appearance of Apple’s Online Music Store; and i=3 represents the influence of the MP3 and P2P technology. Results of these Tobit models will test the affection of the business strategies on the productivity changes of music industry, and improve the conclusions making by Malmquist analysis with two stages evidences as well. Model (7) to model (9) represents the relationship between independent variables, dummy variables and the MPI of music firms.
1 2 3 4 5 1 1 j
jt
j
,if Y > 0
MPI
= {
0 , if Yα +β PCjt +β IUCjt+β SCjt+β ECjt+β CCjt+γ D jt+εjt ≤0 (7)Where
MPIjt : dependent variable. MPI of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D1jt : dummy variable in the multi-regression model. D1 represents the events of filing lawsuits, event occurred during year 2000 to year 2002.
1jt
α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
1 2 3 4 5 2 2 j
MPIjt : dependent variable. MPI of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D2jt : dummy variable in the multi-regression model. D2 represents the event period of Apple’s Online Music Store, event occurred during year 2003 to year 2005. 2jt α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
1 2 3 4 5 3 3 j
MPIjt : dependent variable. MPI of the DEA Malmquist Indices in jth firm.
PC : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D3jt : dummy variable in the multi-regression model. D3 represents the event period of Apple’s Online Music Store, event occurred during year 1997 to year 1999. 3jt α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
Model (10) to model (12) represents the relationship between independent variables, dummy variables and the TEC of music firms.
1 2 3 4 5 1 1 j
TECjt : dependent variable. TEC of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D1jt : dummy variable in the multi-regression model. D1 represents the events of filing lawsuits, event occurred during year 2000 to year 2002.
1jt
α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
1 2 3 4 5 2 2 j
jt
j
,if Y > 0
TEC
= {
0 , if Yα +βPCjt+β IUCjt+β SCjt +β ECjt+β CCjt+γ D jt+εjt ≤0 (11)Where
TECjt : dependent variable. MPI of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D2jt : dummy variable in the multi-regression model. D2 represents the event period of Apple’s Online Music Store, event occurred during year 2003 to year 2005. 2jt α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
1 2 3 4 5 3 3 j
TECjt : dependent variable. MPI of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D3jt : dummy variable in the multi-regression model. D3 represents the event period of Apple’s Online Music Store, event occurred during year 1997 to year 1999. 3jt α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
Model (13) to model (15) represents the relationship between independent variables, dummy variables and the FS of music firms.
1 2 3 4 5 1 1 j
FSjt : dependent variable. TEC of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D1jt : dummy variable in the multi-regression model. D1 represents the events of filing lawsuits, event occurred during year 2000 to year 2002.
1jt
α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
1 2 3 4 5 2 2 j
FSjt : dependent variable. MPI of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D2jt : dummy variable in the multi-regression model. D2 represents the event period of Apple’s Online Music Store, event occurred during year 2003 to year 2005. 2jt α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
1 2 3 4 5 3 3 j jt
j
,if Y > 0
FS
= {
0 , if Yα +β PCjt+β IUCjt+β SCjt+β ECjt+βCCjt +γ D jt+εjt ≤0 (15) WhereFSjt : dependent variable. MPI of the DEA Malmquist Indices in jth firm.
PCjt : piracy change in jth firm between period t to t+1.
IUCjt : number of internet user change in jth firm between period t to t+1.
SCjt : sales change in jth firm between period t to t+1.
ECjt : number of employees change in jth firm between period t to t+1.
CCjt : capital change in jth firm between period t to t+1.
D3jt : dummy variable in the multi-regression model. D3 represents the event period of Apple’s Online Music Store, event occurred during year 1997 to year 1999. 3jt
3jt
1 , if t=1997, 1998, 1999 0 , otherwise
{
DD
=
= α : an intercept coefficient.
βk : slope of kth independent variable, for k= 1,…,5.
εjt : the error terms which are assumed to be a normal distribution with mean zero and variance σ2.
t : sampling period, from year 1997 to year 2005.
For the easiness and user friendly functions, the tools of EViews computer program, version 3.0, are applied to compute the empirical results of Tobit models in this research.
4.2.3 Mann-Whitney Test
The Mann-Whitney rank test analysis, which test the hypothesis of launching a new business model do more impact on the productivity than filing a lawsuit against piracy, in this research is done as follows:
1. Ranking order all n DMUs by their Malmquist Productivity Indices in the DEA model. In case of a tie, use the mid-rank for the tied observations.
2. Compute R1 = the sum of rankings of DMUs in the group which the lawsuits were taking place; R2 = the sum of rankings of DMUs in the group which a new business model were performing.
3. Compute the Mann-Whitney rank test statistic:
1 1
1 1 2 1
2 2
2 1 2 2
1 2
( 1) 2 ( 1)
2 ( , )
U n n n n R
n n
U n n R
U Min U U
⋅ +
= ⋅ + −
⋅ +
= ⋅ + −
=
(16)
The Mann-Whitney U statistic displayed in the table is the smaller of these two values.
4. For n1, n2 10 compute the following statistic: ≧
1 2
1 2 1 2
2
( 1)
12 U n n
Z n n n n
− ⋅
= ⋅ + + (17)
5. Z has an approximately standard normal distribution. Therefore the null hypothesis that two strategies have the same distribution of productivity change scores will be rejected at a level of significance α if Z≦-Zα/2 or Z Z≧ α/2, where Zα/2 denotes the upperα/2 percentile of the standard normal distribution.
6. Similarly the single tail null hypothesis, launching a new business model do no more impact on the productivity than filing a lawsuit against piracy, will be rejected while the two tails hypothesis is rejected. The negative Z statistics indicate that the rank sums are lower than their expected values.
For simplifying thetrivial ranking process, SPSS computer program are used to report the normal approximation to the U statistic and the p-values for a two-sided test in our research.