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Identification of a model is the simple property that when two likelihoods (for the same data), or Joint CDF (for the same random variables), if equal, then they must imply that the parameters must also be equal, i.e. we cannot have two different set of parameter values which fit data. (This is probably imprecise, sorry).

I'm wondering just how many types of identification there are. I've heard of weak, semi-strong, and strong... And what do they mean?

If possible, I would like the answer to also link to a bibliographical reference, where the authors would explain in some detail most of these concepts in a general setting.

Thanks in advance.

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  • $\begingroup$ I am bit confused by the question, identification in econometrics often refers to identification of parameters. That’s why often empirical sections of papers are called identification strategy. Weak, or strong etc are not as much types of identification but refer to how good the model is in identifying the parameters for example in context of IV or GMM weak identification would mean that even in large samples your parameter estimates have non normal distributions etc. in that sense any identification can be weak, strong etc depending on how it’s applied $\endgroup$ – 1muflon1 Apr 16 at 12:04
  • $\begingroup$ @1muflon1 hum... could you elaborate more in an answer? maybe I'm mixing both. Thanks for the comment though. ;) $\endgroup$ – An old man in the sea. Apr 16 at 12:08
  • $\begingroup$ I can try i just don’t know how to respond because I am not sure if I understand the question. Maybe it would help if you would show the references to the things you say in first paragraph $\endgroup$ – 1muflon1 Apr 16 at 12:10
  • $\begingroup$ @1muflon1 I think it's just the usual definition, like you state in your first comment. en.wikipedia.org/wiki/Identifiability $\endgroup$ – An old man in the sea. Apr 16 at 12:12
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I think to answer this it is best to first go over definition of identification.

Following Stachurski (2016) identification or identifiability (I omit formal description since its also in that wikipedia article you provided in your comment):

means that the parameter vector associated with unknown distribution can eventually be distinguised from the data.

Hence identification is more or less what people usually call estimation. For example, in OLS ($y= X\beta+e$) where the $\beta$ coefficient is:

$$\hat{\beta} = (X' X)^{-1}X'y$$

it can be proven that $\hat{\beta}$ can only be identified when the $X'X$ matrix is invertible otherwise $(X'X)^{-1}$ is not defined and you simply wont be able to calculate the $\beta$ or R or Python or Stata would give you error message, like for example where you have perfect multicolinearity.

Every model you can think of has some identification conditions - hence its not really appropriate to talk about types of identification, identification means the model can estimate the parameters and every model has its own conditions for identification of the parameters.

Consequently the process of estimating any coefficients is usually called identification strategy. If by asking about different types of identification you mean different models/ identificaiton strategies, a good-all-purpose textbook is Pesaran (2015). Verbeek (2008) is good intermediate introduction textbook, if you look only for time series analysis Hamilton (1994) is classic although bit dated now. For treatment evaluation I recommend Angrist & Pischke (2008). I wont go over all possible types of models as nowadays you have such a variety its impossible to make exhaustive list in answer on SE.

Next when we talk about weak or strong identification we mean by weak identification that the estimators and test statistics are not well approximated by their standard asymptotic limits because of limited information in the data. Basically, the point is that identification in itself does not guarantee the coefficients are consistent and unbiased only that they can be estimated and usually the term weak identification is applied when the aforementioned are not guaranteed and strong identification when they are. Also I mostly seen these terms being applied to the IV, GMM or other instrument using estimator where weak identification is often used as a synonym for the fact that the first stage or instruments are weak. But I dont see people denoting weak or strong as a type so I dont know if this is what you mean by that. To see when the coefficient estimates are consistent and unbiased you can again for each separate model see in the literature I recommended above.

References:

Angrist, J. D., & Pischke, J. S. (2008). Mostly harmless econometrics: An empiricist's companion. Princeton university press.

Hamilton, J. D. (1994). Time series analysis (Vol. 2, pp. 690-696). New Jersey: Princeton.

Pesaran, M. H. (2015). Time series and panel data econometrics. Oxford University Press.

Stachurski, J. (2016). A primer in econometric theory. Mit Press.

Verbeek, M. (2008). A guide to modern econometrics. John Wiley & Sons.

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    $\begingroup$ Hum... I'm not sure I agree with your intuition of identification. Usually it's more in a sense related to injectivity of parameters, with $P_{\phi}=P_{\theta}\Rightarrow \phi=\theta$. I may not necessarily mean we're able to estimate them. However, if we are able to correctly estimate it, then to be identified, there cannot be more than one value. en.wikipedia.org/wiki/Identifiability#Example_1 $\endgroup$ – An old man in the sea. Apr 16 at 14:50
  • $\begingroup$ About the last paragraph helped, though. so thanks ;) I'll just wait a bit and see if nothing else shows up. $\endgroup$ – An old man in the sea. Apr 16 at 14:53
  • $\begingroup$ I don't know the topic well enough to give an answer but weak, just, over usually applies in a simultaneous equations context and refers to whether to the reduced form can be transformed back to the structural form ( or even estimated ). Not certain but I bet Peter Kennedy's text discusses this topic. And, if it does, it will be discussed clearly. Check it for a discussion. He died probably close to 10 years ago and left that book as a valuable gift to all of us. It's not the most detailed mathematically like a greene or a wooldridge but there are nuggets of gold in the various discussions. $\endgroup$ – mark leeds Apr 16 at 15:21
  • $\begingroup$ Identification and estimation are different things, and it is unfortunate they get conflated... $\endgroup$ – Richard Hardy Sep 24 at 7:37

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