<p>I have commonly seen references to a Kleibergen and Paap F-statistic. <a href="https://pdf.sciencedirectassets.com/271689/1-s2.0-S0304407606X02750/1-s2.0-S0304407605000850/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEIz%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIQCLP8C9wtvA9ctIaK5tpD3u6MU54FJuBLBPkyn9ONTrEwIgKbckj2e8P%2FIKFuQVAdwvyCkyr3Oql7qNhIkl1wBzFyEqgwQIpf%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARAEGgwwNTkwMDM1NDY4NjUiDL1f3ijZkhUASrbkEyrXAzeOfxkrCR1G59T1DM4TDV4FOVbN1uJ%2FJE3MYbM4dhKefiOG6TUD2mmMs9K9E5gxKU9bYOBHAORw%2BVkSHyDc3FoM4zbwWbgD6GhCzfEMw6d75O%2B2SjDzL3FfcxOg4zer3pxpwacMADO0hL7QdlV2vs4z91S7AcP%2B9rcHXowTkjpRts3snY%2BAjCz6DXB%2FPs20KbX5QxFSfoBmImMr0tdkqIYvPsS8j4RBrLhZuOoEzq1dx1IMTxvTVRwR8KXEc0w2wIg3qf19CEhle2KVOO8erG5osc4DUiv6Q54B24UOQ7gDY0OT%2FLt4zn6tsl4Q4Ft9rQ0Xec0cRbx6qkeA6pRWGNdHHF1%2BnettrCxHO45tKsmzQPVD58TIiF9E7OTDvXJWzEe%2FNUb0%2FrmvKIyOs2zScgux9I9Wu477pqjCqjSauBE2qpzFobRe2bDlZSOoppYz8C0jUefgcilPv%2FROOCegeZzPi2RlWed%2BGehXAdXtlX0cfwA8EQPBB52y4QJg07mHhdeUpryrag7rBlB4bTFXEUi1XmD4Xpvs82IS32x8wPzq9z6TzUk6GgzTSU6I%2FUzzdL9l%2BuufSxYk2o5KDKbp%2BR8E0HDqw9yld91pZSrUYZ2ikJz5JUNo3TDxp8SIBjqlAecfPVz2zMgl0J5LhXIIzJsZIwB0y29dbjIV41cQcF5TbruFqSk59IDHY2PsSrYI9YGQn6q8Suj9hp9skA%2FsShQYJGzbguDAbtb82fRRG9H162d6d50EcIEtitmpn08wG%2Bfz20NZSpiB6jd7d0LGdLST5UAM6gBPOL4VSL%2FjpyTGieRaQNOzRS841Zpnmz2DOyFTiloXHtE%2FJUDwMH4zCBd48xiW5Q%3D%3D&amp;X-Amz-Algorithm=AWS4-HMAC-SHA256&amp;X-Amz-Date=20210809T123032Z&amp;X-Amz-SignedHeaders=host&amp;X-Amz-Expires=300&amp;X-Amz-Credential=ASIAQ3PHCVTY7OLW6XAA%2F20210809%2Fus-east-1%2Fs3%2Faws4_request&amp;X-Amz-Signature=b566ca7980583888a06ec71f39f87a528c29a7fff41fe5f10474f0088f148c95&amp;hash=4c771f7119a770aa364f0b0c6e375f672dbcd964dfe47a2065c6fd2851991272&amp;host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&amp;pii=S0304407605000850&amp;tid=spdf-066bb1a0-6bfc-454c-9b59-e59a99c1e4bb&amp;sid=73f0c2bf858de1472d5af7f0a18a417dc810gxrqa&amp;type=client" rel="nofollow noreferrer">In their paper, I see a derivation of a</a> chi-squared statistic, but don't see any references to the F-distribution or F-stat. <a href="http://www.repec.org/bocode/i/ivreg2.html" rel="nofollow noreferrer">In other sources</a>, I have seen that with one endogenous variable, the Kleibergen-Paap F-stat reduces to an F-test that the excluded instruments are jointly significant in the first stage, but I haven't actually been able to find a formula for this F-statistic. Can someone point me in the right direction?</p>

CC BY-SA 4.0
2021-08-10T14:03:35.300Z
When there is only one endogenous regressor, the singular value decomposition is trivial and the test seems to reduce to F or chi-square. It's just their Assumption 1 or (17) with $q=0$, and there is nothing special. Apply $m=1$ and $q=0$ to (1) so $\Lambda_q$ is $k\times 1$, where $k$ is the number of extra instruments.
chan1142
CC BY-SA 4.0
2021-08-12T12:17:41.517Z
@chan1142  Thank you for the reply. It is helpful, but I am still struggling. My understanding is that an F distribution is the ratio of Chi-squareds (each divided by their degrees of freedom). At no point in the paper do I observe a ratio of chi squareds referenced. Can you provide more clarification?
Michael Gmeiner
CC BY-SA 4.0
2021-08-12T12:22:59.450Z
$F_{m,n}=(\chi_m^2/m)/(\chi_n^2/n)$. When $n\to\infty$, the LLN implies $\chi_n^2 / n \to_p 1$. Thus, for large samples ($n$ is the residual degrees of freedom), $F_{m,n} \approx \chi_m^2/m$. So, once the denominator chi-square has been established, the numerator does not really matter much asymptotically. That's what I think about F and chi-square.
chan1142
CC BY-SA 4.0
2021-08-12T12:30:33.113Z
Regarding KP, KP is an extension of the first-stage F test (or asymptotically equivalently chi-square) to multiple endogenous regressors. What KP derive is chi-square, but it does not really matter to distinguish chi-square and F (except the fact that the numerator of F divides chi-square by dof) here. If you want to derive F in the KP setting, I think you can do it in the traditional way because then everything is plain. When $m=1$, there is only one null hypothesis (that $q=0$).
chan1142
CC BY-SA 4.0
2021-08-12T12:36:11.770Z
If you want to get a ratio, you need to assume that the error term is iid normal and estimate $\sigma_v^2$ by SSR/df, like we usually do for OLS, e.g., https://stats.stackexchange.com/questions/258461/proof-that-f-statistic-follows-f-distribution
chan1142
CC BY-SA 4.0
2021-08-13T02:15:34.730Z
Oops I found numerator and denominator are messed up up there
chan1142
<econometrics><instrumental-variable>
Kleibergen Paap F-statistic
Michael Gmeiner
# Kleibergen Paap F-statistic

(Asked by: username_0 on 2021-08-09)

I have commonly seen references to a Kleibergen and Paap F-statistic. [In their paper, I see a derivation of a](https://pdf.sciencedirectassets.com/271689/1-s2.0-S0304407606X02750/1-s2.0-S0304407605000850/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEIz%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIQCLP8C9wtvA9ctIaK5tpD3u6MU54FJuBLBPkyn9ONTrEwIgKbckj2e8P%2FIKFuQVAdwvyCkyr3Oql7qNhIkl1wBzFyEqgwQIpf%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARAEGgwwNTkwMDM1NDY4NjUiDL1f3ijZkhUASrbkEyrXAzeOfxkrCR1G59T1DM4TDV4FOVbN1uJ%2FJE3MYbM4dhKefiOG6TUD2mmMs9K9E5gxKU9bYOBHAORw%2BVkSHyDc3FoM4zbwWbgD6GhCzfEMw6d75O%2B2SjDzL3FfcxOg4zer3pxpwacMADO0hL7QdlV2vs4z91S7AcP%2B9rcHXowTkjpRts3snY%2BAjCz6DXB%2FPs20KbX5QxFSfoBmImMr0tdkqIYvPsS8j4RBrLhZuOoEzq1dx1IMTxvTVRwR8KXEc0w2wIg3qf19CEhle2KVOO8erG5osc4DUiv6Q54B24UOQ7gDY0OT%2FLt4zn6tsl4Q4Ft9rQ0Xec0cRbx6qkeA6pRWGNdHHF1%2BnettrCxHO45tKsmzQPVD58TIiF9E7OTDvXJWzEe%2FNUb0%2FrmvKIyOs2zScgux9I9Wu477pqjCqjSauBE2qpzFobRe2bDlZSOoppYz8C0jUefgcilPv%2FROOCegeZzPi2RlWed%2BGehXAdXtlX0cfwA8EQPBB52y4QJg07mHhdeUpryrag7rBlB4bTFXEUi1XmD4Xpvs82IS32x8wPzq9z6TzUk6GgzTSU6I%2FUzzdL9l%2BuufSxYk2o5KDKbp%2BR8E0HDqw9yld91pZSrUYZ2ikJz5JUNo3TDxp8SIBjqlAecfPVz2zMgl0J5LhXIIzJsZIwB0y29dbjIV41cQcF5TbruFqSk59IDHY2PsSrYI9YGQn6q8Suj9hp9skA%2FsShQYJGzbguDAbtb82fRRG9H162d6d50EcIEtitmpn08wG%2Bfz20NZSpiB6jd7d0LGdLST5UAM6gBPOL4VSL%2FjpyTGieRaQNOzRS841Zpnmz2DOyFTiloXHtE%2FJUDwMH4zCBd48xiW5Q%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20210809T123032Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTY7OLW6XAA%2F20210809%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=b566ca7980583888a06ec71f39f87a528c29a7fff41fe5f10474f0088f148c95&hash=4c771f7119a770aa364f0b0c6e375f672dbcd964dfe47a2065c6fd2851991272&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0304407605000850&tid=spdf-066bb1a0-6bfc-454c-9b59-e59a99c1e4bb&sid=73f0c2bf858de1472d5af7f0a18a417dc810gxrqa&type=client) chi-squared statistic, but don't see any references to the F-distribution or F-stat. [In other sources](http://www.repec.org/bocode/i/ivreg2.html), I have seen that with one endogenous variable, the Kleibergen-Paap F-stat reduces to an F-test that the excluded instruments are jointly significant in the first stage, but I haven't actually been able to find a formula for this F-statistic. Can someone point me in the right direction?

**Tags:** econometrics, instrumental-variable
> **Comments:**
> **username_1**: When there is only one endogenous regressor, the singular value decomposition is trivial and the test seems to reduce to F or chi-square. It's just their Assumption 1 or (17) with $q=0$, and there is nothing special. Apply $m=1$ and $q=0$ to (1) so $\Lambda_q$ is $k\times 1$, where $k$ is the number of extra instruments.  
> **username_0**: @chan1142  Thank you for the reply. It is helpful, but I am still struggling. My understanding is that an F distribution is the ratio of Chi-squareds (each divided by their degrees of freedom). At no point in the paper do I observe a ratio of chi squareds referenced. Can you provide more clarification?  
> **username_1**: $F_{m,n}=(\chi_m^2/m)/(\chi_n^2/n)$. When $n\to\infty$, the LLN implies $\chi_n^2 / n \to_p 1$. Thus, for large samples ($n$ is the residual degrees of freedom), $F_{m,n} \approx \chi_m^2/m$. So, once the denominator chi-square has been established, the numerator does not really matter much asymptotically. That's what I think about F and chi-square.  
> **username_1**: Regarding KP, KP is an extension of the first-stage F test (or asymptotically equivalently chi-square) to multiple endogenous regressors. What KP derive is chi-square, but it does not really matter to distinguish chi-square and F (except the fact that the numerator of F divides chi-square by dof) here. If you want to derive F in the KP setting, I think you can do it in the traditional way because then everything is plain. When $m=1$, there is only one null hypothesis (that $q=0$).  
> **username_1**: If you want to get a ratio, you need to assume that the error term is iid normal and estimate $\sigma_v^2$ by SSR/df, like we usually do for OLS, e.g., https://stats.stackexchange.com/questions/258461/proof-that-f-statistic-follows-f-distribution  
> **username_1**: Oops I found numerator and denominator are messed up up there  

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