Xiaoheng 'Kevin' Zhang wrote:
> I have a serier of index for 10 years. It is a Herfindahl index of
> concentration and I would like to test if the change of this index over
> time is significant.
> I am not sure how to translate this real problem into a statistics
> problem. Since it looks like a decreasing trend, I used linear regression
> of index on year and found the slope is statistically different from 0.
> But I am worrying about sample size......
>
> The indices are
> year index
> 1993 0.149552855
> 1994 0.146646187
> 1995 0.143958559
> 1996 0.145009261
> 1997 0.147389484
> 1998 0.145309026
> 1999 0.144218297
> 2000 0.142834716
> 2001 0.140957544
> 2002 0.140444707
There are two things about the Herfindahl-Hirschman index of market
concentration (to give it its full title), and its use as a response
variable in OLS that you need to be aware of:
(1) Since the index (H) is a fixed 0-1 scale, where 0 = perfect competition
and 1 = a monopoly, the use of -reg- is invalid under the Gauss-Markov
assumptions underpinning OLS;
and
(2) calculating the logit transformation of H gives you a new index (H*)
whose scale stretches from -infinity to +infinity. This makes it a much
more useful - and valid - index for OLS model fitting. Unlike H's scale,
H*'s scale is also _linear_.
Inputting your data and generating H*
. clear
. input year index
year index
1. 1993 0.149552855
2.
. 1994 0.146646187
3.
. 1995 0.143958559
4.
. 1996 0.145009261
5.
. 1997 0.147389484
6.
. 1998 0.145309026
7.
. 1999 0.144218297
8.
. 2000 0.142834716
9.
. 2001 0.140957544
10.
. 2002 0.140444707
11. end
. g logindex=ln(index/(1-index))
and then looking at the relationship graphically via
. twoway line logindex year
shows that H* decreased by nearly -0.08 over 10 years, indicating that
competition within whatever market you're measuring _increased_. But was
that decrease in H* statistically significant over this period?
. reg logindex year, eform(OR)
Source | SS df MS Number of obs = 10
-------------+------------------------------ F( 1, 8) = 22.75
Model | .003446335 1 .003446335 Prob > F = 0.0014
Residual | .001211688 8 .000151461 R-squared = 0.7399
-------------+------------------------------ Adj R-squared = 0.7074
Total | .004658023 9 .000517558 Root MSE = .01231
----------------------------------------------------------------------------
logindex | OR Std. Err. t P>|t| [95% Conf. Interval]
-----------+----------------------------------------------------------------
year | .9935576 .0013462 -4.77 0.001 .990458 .9966668
----------------------------------------------------------------------------
Yes: H* significantly decreased by six-thousandths of 1 percent every year
in the period (notice the use of the -eform()- option to obtain this).
Whether this is important enough to care about is, of course, your call.
Although there doesn't appear to be any real improvement in model fit over
the standard OLS model I suspect you fitted (R^2 for I = .7056), you are
at least fitting a much more valid model. The model fit itself is pretty
impressive.
But then there's the pesky problem of your small N. The only way to
improve this is by having more data (you don't say where this data comes
from). Do you have it? Also, other variables need to be used if they're
available: e.g., if this is market data, then information on, say, whether
any new laws tightening or relaxing market competition would be very
useful to have.
I hope this helps.
CLIVE NICHOLAS |t: 0(044)7903 397793
Politics |e: [email protected]
Newcastle University |http://www.ncl.ac.uk/geps
Whereever you go and whatever you do, just remember this. No matter how
many like you, admire you, love you or adore you, the number of people
turning up to your funeral will be largely determined by local weather
conditions.
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