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7.2 Variants for the Computation of Malmquist Indices
To neutralize the effect of technological change, the comparison to the frontier of
excellence can be repeated, taking as reference, successively, the production
frontier of the two consecutive years between which the progress of the production
unit is being evaluated. This means realizing two computations:
first, taking as
reference the frontier of the
first year and dividing the distance to this frontier
(measured in terms of DEA efficiency score or of probability of being the best) of
the vector formed with the production
figures of the subsequent year by that dis-
tance of the vector formed with the production
figures of the initial year; then,
calculating the ratio between the distances of the same two vectors to the frontier of
the second year. The Malmquist index most used is formed by the geometric mean
between these two ratios.
Thus, to calculate the index, for each evaluated alternative, the computation of
four measures of distance to the frontier is required. Using the notation d(a, t, u) to
denote the distance of the vector of observations of the a-th alternative at instant t to
the frontier formed using the observations of the various alternatives at instant u,
these measures are:
(1)
the distance to the production frontier of time t of the vector observed in the
a-th alternative at time t, denoted by d(a, t, t);
(2)
the distance to the production frontier of time t of the vector observed in the
a-th alternative at time t + 1, denoted by d(a, t + 1, t);
(3)
the distance to the production frontier of time t + 1 of the vector observed in
the a-th alternative at time t, denoted by d(a, t, t + 1);
(4)
the distance to the production frontier of time t + 1 of the vector observed in
the a-th alternative at time t + 1, denoted by d(a, t + 1, t + 1);
By taking as reference only the frontier of time t, the index of productivity
evolution is de
ned by d(a, t +1, t)/d(a, t, t). By sticking to the frontier of time t + 1,
the index will be de
ned by d(a, t + 1, t + 1)/d(a, t, t + 1). To take into account the
evolution of the frontier, the geometric mean of these two indices in the geometric
mean Malmquist index is
finally used:
1
=
2
½
ð
da
ð
;
t
þ
1
;
t
Þ=
da
ð
;
t
;
t
Þ
Þ
ð
da
ð
;
t
þ
1
;
t
þ
1
Þ=
da
ð
;
t
;
t
þ
1
Þ
Þ
:
Here, the distance to the frontier is given in probabilistic terms. Thus, denoting
by P(a, t, u) the probabilistic evaluation of the a-th alternative when the values
observed at time t are introduced into the analysis, together with all the values
observed at time u, the Malmquist probabilistic rate of improvement is given by the
square root of
the product of
the ratios P a
ð
;
t
þ
1
;
t
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Pa
ð
;
t
;
t
Þ
and P a
ð
;
t
þ
1
;
t
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1
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Pa
;
t
;
t
þ
1
ð
Þ
:
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