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Suppose the model is

$$ \ln(\frac{P_{ij}}{1-P_{ij}}) = X_{ij}\beta_0 + A_i + A_j + \epsilon_{ij}, $$

The unit of observation is at $ij$ level, and $A_i$ and $A_j$ are coefficients on the dummy for each $i$ and $j$ to account for unobserved effect of $i$ and $j$ on the probability $P_{ij}$.

Suppose I got the estimates of $\hat{\beta}\rightarrow\beta_0$ after conditioning out the incidental parameters, $A_i$ and $A_j$. Since I do not have the estimates of $\hat{A}_i$ and $\hat{A}_j$, I cannot "predict" based on the available estimates.

Can I partial out the effect of $X_{ij}\hat{\beta}$ in order to get $\hat{A}_{i}$ and $\hat{A}_j$? I just need the point estimates of A, not the standard errors.

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    $\begingroup$ A known term in a glm linear model is called an "offset". Offsets are a standard feature of glms and any glm or logistic regression software will tell you how to include them in your model fit. So handling the first term is straightforward. On the other hand, I have no idea what you mean by a "high dimensional fixed effect term" or why $A$ has no coefficient in your model. $\endgroup$ – Gordon Smyth Jun 16 at 4:36
  • $\begingroup$ Please clarify your post as in the comments by @Gordon Smyth $\endgroup$ – kjetil b halvorsen Jun 16 at 17:15
  • $\begingroup$ @kjetilbhalvorsen I have clarified the post! $\endgroup$ – user325721 Jun 28 at 5:05
  • $\begingroup$ This is still some what unclear, are the $A$'s the dummys, and you only need estimates for the coefficients of the dummys? Please clarify further, but I reopen now. $\endgroup$ – kjetil b halvorsen Jun 28 at 5:46
  • $\begingroup$ @kjetilbhalvorsen Yes that's correct! I edited the post. Thank you so much for your patience! $\endgroup$ – user325721 Jun 28 at 10:09
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If I understand this correctly, you can use an offset term in your model specification. That is a term used in R, if other systems use different terminology I do not know, but they should have an option. An offset is a term used in a linear predictor with a known coefficient of 1. In a traditional linear model, this concept is unnecessary, since you could just subtract that term from $Y$, but in logistic regression and other glm's (generalized linear models), estimation is iterative, and the linear predictor does not predict the outcome directly, but through the link function, so this is not possible.

So:

  • make a new variable xx by multiplying x with $\hat{\beta}_0$
  • A is a factor variable, in R A <- factor(A)
  • In R you can use
 glm(Y ~ A + offset(xx), family=binomial, data=your_df)
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