I'm estimating a logistic regression model with individuals (n = 800) clustered within countries (k = 6) over several time periods (p = 3). I want to control for time-invariant country-level effects (country fixed effects). I'd like to report predicted unconditional probabilities, and possibly marginal effects to my audience. Because of this, I'm fitting an unconditional likelihood model using 5 country dummies. I know the usual advice in this situation is to use a conditional likelihood model (clogit), because of incidental paramater bias. But I've also read that providing the number of observations in each group (n) is large and the number of clusters (k) small, then unconditional estimates will be unbiased.
For each country, I have a fairly large number of observations (n = 75 to 200). So I compared a model fitted with unconditional likelihood using country dummies to a model fitted with clogit. The estimates and standard errors are identical down to three decimal places, leading me to think that I'm on safe ground using the unconditional model.
However, I'd really like to be able to cite a paper that talks about how large the number of observations need to be within clusters for bias to be negligable. Does anyone have any suggestions?
So far, I've found:
The Stata docs for clogit state that, p 14-15: "Let i = 1,2,...,n denote the groups and let t = 1,2,...,Ti denote the observations for the ith group... If Ti is large for all groups, the bias of the unconditional fixed-effects estimator is not a concern, and we can confidently use logit with an indicator variable for each group"
Which seems to indicate that if the number of observations within clusters (Ti) is large, then estimates are unbiased. But, the docs don't cite a source for this.
However, in Katz (2001) Bias in Conditional and Unconditional Fixed Effects Logit Estimation. Political Analysis 9(4):379-384. On p 379-380: "We observe N units for T time periods and that at each observation we record whether an event occurs... The unconditional maximum-likelihood estimator of the incidental parameters is consistent as T → ∞ for fixed N but inconsistent as N → ∞ for fixed T . The inconsistency arises because the number of incidental parameters increases without bound, while the amount of information about each incidental parameter remains fixed"
The author states that the unconditional model is inconsistent as N goes to infinity, which seems to contradict the Stata doc and my empirical finding.
For each country, I have a fairly large number of observations (n = 75 to 200). So I compared a model fitted with unconditional likelihood using country dummies to a model fitted with clogit. The estimates and standard errors are identical down to three decimal places, leading me to think that I'm on safe ground using the unconditional model.
However, I'd really like to be able to cite a paper that talks about how large the number of observations need to be within clusters for bias to be negligable. Does anyone have any suggestions?
So far, I've found:
The Stata docs for clogit state that, p 14-15: "Let i = 1,2,...,n denote the groups and let t = 1,2,...,Ti denote the observations for the ith group... If Ti is large for all groups, the bias of the unconditional fixed-effects estimator is not a concern, and we can confidently use logit with an indicator variable for each group"
Which seems to indicate that if the number of observations within clusters (Ti) is large, then estimates are unbiased. But, the docs don't cite a source for this.
However, in Katz (2001) Bias in Conditional and Unconditional Fixed Effects Logit Estimation. Political Analysis 9(4):379-384. On p 379-380: "We observe N units for T time periods and that at each observation we record whether an event occurs... The unconditional maximum-likelihood estimator of the incidental parameters is consistent as T → ∞ for fixed N but inconsistent as N → ∞ for fixed T . The inconsistency arises because the number of incidental parameters increases without bound, while the amount of information about each incidental parameter remains fixed"
The author states that the unconditional model is inconsistent as N goes to infinity, which seems to contradict the Stata doc and my empirical finding.