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st: Compare coefficients across groups in panel data


From   Simon Pickert <simon.pickert@t-online.de>
To   statalist@hsphsun2.harvard.edu
Subject   st: Compare coefficients across groups in panel data
Date   Tue, 17 Dec 2013 13:14:21 +0100

Dear Statlisters,

[This is not a speficic Stata question, but rather a statistics question, for which I’d appreciate your help very much!]

I want to compare regression coefficients across two groups. I have a panel data set and I use a fixed-effects regression, so things are a little bit more tricky than with a standard regression in which I could easily use interaction terms to test across for differences in coefficients across groups. In the final (panel) data set, I can’t identify my groups anymore, because I’m aggregating over them (in order to end up at a daily level).

Please, have a look at this example:

Research question: Is the relation between cost of ice-cream and ice-cream rating stronger for adults than for children?

Imagine 2 groups of people: Adults and children. Over 1 month, you record how much every person spend on ice-cream (4 sorts), how they rated the ice-cream and what size of ice-cream they got. Your dependent variable is rating, your independent variables are cost and size.

   +—————————————————————+
    | day | ice-cream   | rating | cost  | size  | group |
    +—————————————————————+
 1.  |  1  |  vanilla      |  2     |      3   |    4    |     1      |
 2.  |  1  |  choco      |  1     |      4    |    3    |    0      |
 3.  |  1  |  vanilla     |  2     |      2    |    1    |    1       |
 4.  |  1  |  hazelnut  |  9     |      2    |    1    |    0       |
 5.  |  2  |  hazelnut  |  2     |      1    |    1    |    0       |
 6.  |  2  |  hazelnut  |  5     |      1    |    1    |    0       |
 7.  |  2  |  vanilla     |  6     |      5    |    2    |    1       |
 8.  |  3  |  berry       |  1     |      6    |    3    |    0       |
 9.  |  3  |  berry       |  6     |      4    |    2    |    0       |
10. |  3  |  berry       |  7     |      3    |    4    |    1       |
11. |  3  |  berry       |  5     |      2    |    1    |    1       |
12. |  3  |  vanilla     |  4     |      1    |   2     |    0       |
13. |  3  |  vanilla     |  4     |      5    |   2     |    0       |
etc |  .   |  ......         |  .      |        .   |  …     |       .     |   
    +—————————————————————+

Not caring about groups at all, I could just aggregate the data to a day-icecream level (because I want to consider fixed effects at the ice-cream level!). My panel dataset would look like this:

    +—————————————————+
    | day | ice-cream | rating | cost  |    size  | 
    +—————————————————+
 1. |  1  |  vanilla      |  2       |  2.5  |  2.5       | 
 2. |  1  |  choco      |  1       |  4      |  3         | 
 3. |  1  |  hazelnut  |  9       |  2      |  1         | 
 4. |  2  |  hazelnut  |  3.5    |  1      |  1         | 
 5. |  2  |  vanilla     |  6       |  5      |  2         | 
 6. |  3  |  berry       |  4.75  |  3.7   |  2.5      | 
 7. |  3  |  vanilla     |  4       |  3      |  2         | 
etc |  .  |  ......         |  .        |  .       |  ...        | 
    +—————————————————+

However, this doesn’t allow me to calculate a regression model for each group.

So, I followed a different path: I split the raw data set according to groups (one set for each group) and aggregated to the daily level afterwards. The two panel datasets looks like this:

Group 1 (Adults)

    +———————————-————————+
    | day | ice-cream | rating | cost  | size  | group |
    +———————————-————————+
 1. |  1  |  vanilla       |  1.5   |  2.5  |  2.5  |     1      |
 2. |  2  |  vanilla       |  6     |  5     |  2      |     1      |
 3. |  3  |  berry         |  7     |  3     |  4      |     1      |
 4. |  3  |  vanilla       |  6     |  2.5  |  2.5   |     1     |
etc |  .  |  ......           |  .      |  .      |  …     |       .    |   
    +———————————————————+

Group 2 (Children)

    +———————————-————————+
    | day | ice-cream | rating | cost  | size  | group|
    +———————————-————————+
 1. |  1  |  choco      |      1     |  4    |    3    |    0    |
 2. |  1  |  hazelnut |      9      |  2    |    1    |    0    |
 3. |  2  |  hazelnut |     3.5    |  1    |    1    |    0    |
 4. |  3  |  berry      |     3.5    |  5    |   2.5  |    0    |
 5. |  3  |  vanilla    |     4       |  3    |    2    |    0    |
etc |  .  |  ......        |  .           |  .     |  …     |    .    |   
    +———————————-————————+

For each dataset, I run a fixed-effects regression. This leaves me with two model outputs which I want to compare and see whether the coefficients are different.

Any ideas how to test coefficients across models in this case?

Thank you
Simon


P.S. I asked this question at cross-validated as well, in case the tables above are difficult to read: http://stats.stackexchange.com/questions/78431/compare-coefficients-across-groups-in-panel-data-set
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