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Consider the following model

$Y_i = f(X_i) + e_i$

from which we observe n iid data points $\left( X_i, Y_i \right)_{i=1}^n$. Suppose that $X_i \in \mathbb{R}^d$ is a $d$ dimensional feature vector. And suppose that a ordinary least squares estimate is fit to data, that is,

$\hat \beta = {\rm arg} \min_{\beta \in \mathbb{R}^d} \sum_i (Y_i - \sum_j X_{ij} \beta_j)^2$

Since a wrong model is estimated, what is the interpretation for the confidence interval around estimated coefficients?

More generally, does it make sense to estimate confidence intervals around parameters in a misspecified model? And what does the confidence interval tell us in such a case?

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    $\begingroup$ You can latex on this site. Please enclose the tex with $ $. See this meta thread: meta.stats.stackexchange.com/questions/218/… $\endgroup$ – user28 Jul 30 '10 at 14:29
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    $\begingroup$ There is no general interpretation for parameters in misspecified models. It depends on the details of the misspecification. A confidence interval maybe useful anyway if the misspecification isn't too bad. $\endgroup$ – Michael Bishop Apr 4 '11 at 18:54
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The confidence interval that you obtain is conditional on the model being correct and the interpretation is also conditional on the model being the correct one. If you know that the model is incorrect then obviously you would not use it to compute the confidence interval.

In reality, you do not know the true model and so you have no way to tell if you have a misspecified model (although you do have ways to assess misspecification, e.g., examine if residuals are normally distributed, diagnostic plots of fitted vs observed values etc). So, to my mind, the real question is if the model is misspecified, to what extent can you rely on confidence intervals as a way to assess where the true parameter is. I suspect that the answer is specific to the degree of misspecification that is coming from f(x) i.e., the degree to which f(x) departs from the assumptions of OLS.

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    $\begingroup$ I'll nitpick your answer I basically agree with. @svadali said: "If you know that the model is incorrect then obviously you would not use it to compute the confidence interval." But to paraphrase George Box: all models are incorrect, sometimes computing confidence intervals is useful anyway. $\endgroup$ – Michael Bishop Apr 4 '11 at 18:49
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model misspecification is irrelevant when you are interpreting co-efficients in a model, but any such interpretation is always conditional on the model. This is because a model is about an association, not about a causal relationship - the OLS coefficients just tells you about the linear association between $X$ and $Y$. It is impossible for this to be "incorrect" per se, and as long as you interpret your results as estimating linear associations you will not be wrong (but possibly irrelevant, answering the wrong questions). There just may be more useful non-linear associations between $X$ and $Y$ that are of interest (i.e. more relevant questions)

And further, if you know your model is misspecified, then surely you must also know something about how it is misspecified, i.e. you must have some information about the model function $f(.)$. So why not use that information and make a better model?

model misspecification is one of those problems which is real, but mostly irrelevant. For if the model is misspecified what can you do about it besides choose a different model which is correctly specified? And if you only have one model, it is either that model or no inference at all (no inference is hardly a good outcome given how much effort and resources it probably took to gather the data).

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  • $\begingroup$ @ probabilityislogic: "model misspecification is irrelevant when you are interpreting co-efficients in a model"--how can it be irrelevant when the addition of new variables such as suppressors could change those coefficients substantially and meaningfully? $\endgroup$ – rolando2 Apr 16 '11 at 17:01
  • $\begingroup$ @rolando2 - new variables means a new model, so my comment still applies. If you are looking to interpret the effect within in a class of models (such as additional variables), then you should take a weighted average of the co-efficient in each model, with the weights being equal to (or proportional to) the plausibility of each model $\endgroup$ – probabilityislogic Apr 16 '11 at 22:52
  • $\begingroup$ @ probabilityislogic I see what you mean, thanks. $\endgroup$ – rolando2 Apr 17 '11 at 2:35
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The answer depends on the degree of misspecification and sample size. In small and moderate samples simplified model will fit (in most cases) better to data then the true model. In moderate and large samples residuals don't have to be normal as due to CLT regression coefficients are normal anyway.

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