In this notebook I describe evaluating the rank of a model matrix X by forming a pivoted Cholesky factorization of X'X. I use this form because the X'X matrix will be evaluated whether or not the rank is needed. I would appreciate if those who know the numerical linear algebra involved could tell …

Your link isn’t rendered correctly (use nbviewer), but realistically if you want a response you should post the code directly

You have: As explained in the documentation for dpstrf, this value [tol] should be negated to be used as a relative tolerance. Otherwise it is used as an absolute tolerance which usually is not what you want. This is not quite right. If tol is less than zero, dpstrf uses the recommended default v…

Usually one cares more about identifying linear dependent variables in the model matrix rather than just the rank. This way, if there are linearly dependent variables, these can be dropped such that one obtains a model matrix that is full rank. If the approach to solving the problem is through inver…

[image] Nosferican: This way, if there are linearly dependent variables, these can be dropped such that one obtains a model matrix that is full rank. This approach can be somewhat brittle (the estimates change discontinuously as a function of the data). In practice, using a vague but proper p…

I use it mostly for instrumental variable estimators such as 2SLS or after transformations such as fixed effects. In those cases, it makes sense for the user to specify a model which for some estimations the model matrix has to be trimmed to a subset which is full rank.

@Ralph_Smith Thank you for the correction on my reading of the documentation. The reason that I chose the square root of eps() is because the positive semidefinite matrix is constructed as X'X and it is rank deficiency of X that I want to detect. This also related to the next comment by @Nosferica…

@Tamas_Papp I agree that the use of hierarchical/multilevel models or, more generally, mixed-effects models is more robust. This calculation is being performed within the MixedModels package whose purpose is to fit exactly those models. I have been contacted by users who have some factors they ar…

Tolerance specification for pivoted Cholesky

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Tolerance specification for pivoted Cholesky

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