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# st: RE: Cragg-Donald statistic test for weak instruments

 From "Schaffer, Mark E" To Subject st: RE: Cragg-Donald statistic test for weak instruments Date Sun, 4 Dec 2011 14:33:52 -0000

```Elizabeth,

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of
> Lim, Elizabeth
> Sent: 25 November 2011 17:46
> To: 'statalist@hsphsun2.harvard.edu'
> Subject: st: Cragg-Donald statistic test for weak instruments
>
> Hello,
>
> I've read the postings in the archives on the subject of
> Cragg-Donald statistic for multiple endogenous regressors but
> would like to seek further clarifications.  According to the
> Baum, Schaffer, and Stillman (2007) paper on page 23, the
> Cragg and Donald (1993) statistic is [(N-L)/L2)(r2k1/1-r2k1].
>  I currently have the following info:-
>
> Number of observations N=542
> Number of regressors K=30
> Number of instruments L=36
> Number of excluded instruments L1=8
> Number of endogenous regressors K1 =2
>
> I'm interested in finding out more about the following issues:
>
> (1) How to compute Cragg-Donald statistic manually?  Looking
> at the formula provided by Baum et al (2007), I know I have a
> few pieces of the info (N and L) required to plug into the
> formula, but how and where do I get the rest of the info to
> successfully compute the C-D?  Or is there an alternative way
> to get the C-D statistic based on the existing pieces of info
> I've provided above?

For your case (2 endogenous regressors), the C-D stat is closely related
to tests of the rank of a matrix.  You can do it "by hand" in Stata's
matrix language(s), or you can do it "by hand" by deriving via
Anderson's canonical correlations tests.  There are references in BSS
and also in the help files for ivreg2 and ranktest.  But you can't do it
"by hand" using just the information you've provided above.

> (2) When to reject the null hypothesis of weak instruments
> using the Cragg-Donald statistic?  I know I should compare
> the Cragg-Donald statistic with the Stock and Yogo's (2005)
> range of critical values for maximal IV relative bias (5%,
> 10%, 20%, 30%) and maximal IV size (10%, 15%, 20%, 25%), but
> at what point do I know I have weak instruments? Other
> statistics use a single p-value so it's easy to know that the
> null hypothesis will  be rejected when p<0.05, but with a
> range of critical values, what is the threshold for rejection
> of the null (within these range of critical values)?  For
> example, if my C-D statistic falls between a maximal IV
> relative bias of 20% and 30% or if I have a maximal IV size
> of >25% maximal IV size, etc, does it mean that I've got weak
> instruments?

The question you're asking - can I reject the null of weak IVs? - is not
can I reject the null that the maximal relative bias (due to IV
"weakness") is 10%?

> (3) Since there are two sets of S-Y critical values, which
> one should I use to reject my null?

Up to you!  Are you more worried about relative bias or size
distortions?  If you don't have strong views and are just worried in
general, then the natural thing do to is map the results onto your own
Worry Index.  "The results suggest weak identification is only a
moderate concern...."  "The results suggest the IV coefficients and SEs
are unreliable because of weak identification...."

> Will I always get the
> same conclusion (i.e., reject null or not) regardless of
> whether I use the critical values for "maximal IV relative
> bias" or "maximal IV size"?

Hard to see how you could get the same conclusion with two different
sets of critical values.  The test statistic is the same but you are
applying two different sets of cutoffs.

> Sorry if my questions sounded fundamental, but this is a new
> topic for me, and I'm still trying to grasp the logic behind
> the statistics.

No problem, hope this helps.

--Mark

>
> Best,
> Elizabeth
>
>
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