Theoretical description of an hierarchical model (xtmixed)

[Thomas Lindner]

Hello everyone,

I have a panel with unbalanced data (using Stata 12.0 for Windows) In particular, I have the performance of firms for different years, and those firms work in an industries and have certain CEOs. Therefore, I have a nested structure with CEOs being nested in firms, and firms being nested in industries. To get a better feel for the hierarchical structure see the data example:

list industry firm ceo year roa in 40/60, clean

       industry   firm   ceo   year    performance  
 40.          3      5     8   2011    3.2960024  
 41.          2      6     9   2002    4.2481346  
 42.          2      6     9   2003     2.708844  
 43.          2      6     9   2004    4.4785706  
 44.          2      6     9   2005    3.2659824  
 45.          2      6     9   2006    2.6625132  
 46.          2      6     9   2007    2.5259193  
 47.          2      6     9   2008    4.6811333  
 48.          2      6     9   2009    2.5424542  
 49.          2      6    10   2010   -3.6346642  
 50.          2      6    10   2011    1.5950693  
 51.          3      8    11   2002    8.8132951  
 52.          3      8    11   2003    9.0472671  
 53.          3      8    11   2004    3.6794818  
 54.          3      8    11   2005   -4.5508126  
 55.          3      8    11   2006   -7.1551291  
 56.          3      8    11   2007    2.3615752  
 57.          3      8    12   2008    4.7706465  
 58.          3      8    13   2009    2.7570734  
 59.          3      8    13   2010    3.7838164  
 60.          3      8    13   2011    6.9582259

Furthermore, I have the following Stata command to analyze the data (assume it’s given):

xtmixed  performance  i.year || industry: || firm: || ceo:, var  

I’m now looking for the “formal/theoretical” description of the model. Is the following correct?

Where i denotes industry, j denotes firm, k denotes CEO. Gamma₀₀₀ is supposed to be the grand mean. Or am I missing the “year” level and it should be:

Where t denotes time, i denotes industry, j denotes firm, k denotes CEO. Gamma₀₀₀₀ is again the grand mean.

Thanks in advance for any advice!

[wbuchanan]

Do firms have more than one CEO over the duration? What research question are you trying to answer? Given the granularity of the time measurement it might be worth considering cross-classified models to account for the outcome dependence in cases where a CEO moves from one firm to another within a given year. That said, if the CEO is stable across time, how would you distinguish their effect from that of the firm?

[Thomas Lindner]

Thanks for your answer!

I'm trying to estimate the CEO-effect, i.e. how much variance in the firm-performance can be explained by CEOs. As you can see, CEOs can change the company within the entire timespan (2002-2011), but not during the year. As I said, the Stata model is given (although one can obviously discuss about its accuracy), and I'm now interested in how the "theoretical" model description would look like. Any idea if the first or second suggestion is good?