Unconditional cross-nested 4-level model

[Marco Tacchi]

Dear all,

I would like to replicate a paper (https://onlinelibrary.wiley.com/doi/10.1002/smj.3057) that uses an unconditional HLM model to decompose variation in firm performance. Firm performance is observed over time. Particularly, the authors write:

"We used a 4-level nested HLM model of years (level 1), nested within board chairs and CEOs (level 2), nested within firms (level 3), which is finally nested within industries (level 4). It should be noted that both the board chair effect and CEO effect are at the same level of analysis (level 2), as there is not a nested relationship between these two effects. Following the terminology used by Bryk and Raudenbush (1992), we used level 1 to denote the lowest level, and level 4 to denote the highest level. In the HLM analysis, we followed Singer’s (1998) guidelines and selected the “unstructured” covariance structure."

If there is only CEO (level 2) or board chair (level 2), then the model seems intuitive to estimate using the -mixed- command, i.e.,

mixed performance || industry: || firm: || ceo:

However, I am not sure how to enter both board chair and CEO that are both on the same level and calculate the proportion of each respective variance.

Would be grateful for your help.

[Felix Bittmann]

I am not familiar with the underlying data or corporate structures so this is a guess. What I read from the description is that CEOs and boards are at the same level. All you need is a unique identifier for each combination. For example,

Company A - Board T - CEO Y: Code 01 (Original state)
Company A - Board T - CEO Z: Code 02 (CEO changes)
Company A - Board L - CEO Z: Code 03 (Board changes)

So whenever the Board or the CEO changes (a new unique combination on the firm level), you have a new ID for this identifier level.

[Marco Tacchi]

I am not familiar with the underlying data or corporate structures so this is a guess. What I read from the description is that CEOs and boards are at the same level. All you need is a unique identifier for each combination. For example,

Company A - Board T - CEO Y: Code 01 (Original state)
Company A - Board T - CEO Z: Code 02 (CEO changes)
Company A - Board L - CEO Z: Code 03 (Board changes)

So whenever the Board or the CEO changes (a new unique combination on the firm level), you have a new ID for this identifier level.

Dear Felix,

Thank you for the reply. I have a unique identifier for CEO and board chair. I was wondering what Stata command involving -mixed- could do the estimation. Would this work, for example?

mixed performance || industry: || firm: || _all: R.ceo || _all: R.board

As I understand, using the -_all- prefix is one way to address a cross-nested structure in the data.

[Felix Bittmann]

I am not sure about the data structure (from a theoretical point of view). Is it possible that a CEO changes from company A to company B? Same goes for boards. My guess is that a cross-nested structure only applies if this is the case.

[Marco Tacchi]

I am not sure about the data structure (from a theoretical point of view). Is it possible that a CEO changes from company A to company B? Same goes for boards. My guess is that a cross-nested structure only applies if this is the case.

Sorry for a delayed reply. In their paper, the authors write the following: "As the CEO and board chair effects are indistinguishable from firm effects if the CEO and board chair do not change at any point in the study window, we excluded from the sample firms that experience no change in the board chair or CEO position during the period under study. In order to differentiate the board chair effect from the CEO effect, we also excluded from the sample firms in which the CEO and board chair positions were held by the same person." So, I suppose the answer to your question is "Yes".

My broader question with the -mixed- command is how to apply it to model the situations when there is no clear hierarchical order as in my case (i.e. CEOs and board chairs are at the same level). In the Stata manual, it is supposedly done utilizing the -_all:- syntax. Do you think my syntax above is correct?

mixed performance || industry: || firm: || _all: R.ceo || _all: R.board