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In fitting theoretical models to data, what is the difference between identification, calibration and estimation?

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I agree with 1muflon1, but allow me to add some more nuance.

Identification and calibration can be meant to express a subset of estimation. Any identified coefficient is also an estimate, but not vice-versa.

An identified estimate is any estimate that fulfills certain conditions that make it the true number we want.

For example, any coefficients from (estimating) an OLS regression are estimates. However, a coefficient from an OLS model that fulfills all the OLS assumptions for an unbiased consistent estimate (e.g. no relationship between the error terms and the independent variable) is an identified estimate. Only such a model "achieves identification" or allows authors to state "we identify the effect".

Calibration is relevant when data is used to quantify a theoretical model. There are different ways to do it and hence different potential meanings behind the term.

If you want to produce real-world relevant numbers based on a model (theory) you will need to input certain numbers called "parameters." Inputting those numbers is what it means to say we "calibrate the model by ...". This is what all calibration meanings have in common.

How those numbers are estimated is a different story. Ideally these parameters would also be "identified". Sometimes researchers will calibrate by using parameters from the literature or an identified regression model.

More often calibration is done by trying out different values for the parameters until the model achieves predictions with the least deviation from the data or reproduces some other empirical features. I believe this is the meaning of calibration that is most commonly used in the context of the OP.

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    $\begingroup$ Great addition +1 $\endgroup$ – 1muflon1 May 13 at 20:20
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    $\begingroup$ "Identification and calibration can be meant to express a subset of estimation" is clearly not true. Identification is a statement about a (parametric) model, with respect to the parameter---the injectivity of the mapping from parameter space to distribution of data . Estimation is concerned with mappings from data to parameter space, i.e. estimators. $\endgroup$ – Michael May 14 at 0:10
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    $\begingroup$ @Michael, that is why i say it CAN be meant that. While not strictly true, my personal experience is that a large number of economists use these terms in this fashion. The fact is the statement “we identify x” also implies in practice that the authors “estimate a coefficient for x” (but the reverse is not true). This seems to be what the OP is confused about. Granted this response and question is more about linguistics/jargon than economic/ statistical science. $\endgroup$ – BB King May 14 at 3:38
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Identification and estimation are often used interchangeably (at least that’s my observation from attending conferences and reading papers) but according to econometric literature there is a subtle difference.

For example, in the John Stachurski, a primer in econometric theory the identification is a process of finding out if the parameters are identifiable and identifiability is defined as

“Identifiability means that the parameter vector associated with the unknown distribution can eventually be distinguished from the data.”

I will omit all the other formalism that follows but in essence we can say that parameter is identified if the map from the parameter space to the space of probability distributions of observables can be inverted.

The estimation is actually the process of calculating what the actual parameter $\hat{\beta}$ is.

When it comes to calibration in forecasting it is the comparison of model estimates to actual data with an aim to improve the model fit.

In macroeconomics it is a strategy for finding numerical values for the parameters of artificial economic worlds. A model is calibrated when its parameters are quantified from casual empiricism or unrelated economic studies or are chosen to guarantee that the model mimics some particular feature of the historical data. In other fields it might have different uses as well.

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  • $\begingroup$ I guess if you add something along these lines, I can accept your answer. Calibration is a strategy for finding numerical values for the parameters of artificial economic worlds. A model is calibrated when its parameters are quantified from casual empiricism or unrelated economic studies or are chosen to guarantee that the model mimics some particular feature of the historical data. $\endgroup$ – Beck Batucada May 13 at 17:03
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    $\begingroup$ @BeckBatucada thats pretty much what I meant by the definition used in macro, if you think that this wording improves the answer I can change it. $\endgroup$ – 1muflon1 May 13 at 17:07

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