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# # <font color="880000">The Weak Instruments Problem</font>
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# A simple instrumental-variables framework: you fear that your x is correlated with u, so you search for an instrument z:
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# Suppose: $ y = {\beta}x + u $; $ x = {\pi}z + v $; $ u = {\delta}v + w $; with _u, v, and w_ error terms...
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# then:
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# >$ {\beta}_{IV} = \frac{zy}{zx} = \frac{{\beta}xz + uz}{xz} $
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# >$ {\beta}_{IV} = \frac{{\beta}{\pi}zz + {\beta}vz + uz}{{\pi}zz + vz} = \beta + \frac{uz}{{\pi}zz + vz}$
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# >$ {\beta}_{IV} = \beta + \frac{{\delta}vz + wz}{{\pi}zz + vz} = \beta + {\delta}\frac{1}{({\pi}zz/(vz)) + 1} + \frac{wz}{{\pi}zz + vz} $
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# And if $ \pi = 0 $, then: 
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# >$ {\beta}_{IV0} = \beta + \frac{{\delta}vz + wz}{vz} = \beta + \delta + \frac{wz}{vz} $
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# This is the weak instruments problem. As you get more and more data, $ \frac{wz}{vz} $ is not heading for zero, and even if it were your estimated $ {\beta}_{IV0} $ is not headed for $ \beta $ but is rather headed for $ \beta + \delta $
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# Now we would like to apply a file-drawer problem filter: $ \pi $ is zero, but in the sample you have the calculated $ zv $ is large enough that you are happy to run and report the regression. What can we then say?
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# ----
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# ## <font color="880000">The Weak Instruments Problem: Catch Our Breath—Further Notes:</font>
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# <img src="https://tinyurl.com/20190119a-delong" width="300" style="float:right" />
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# **Last Modified 2019-03-06; Created 2019-03-06**
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# * Make three comments
# * Ask three questions
# * Recommend three readings
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# * Weblog Support: <https://github.com/braddelong/WS2019/blob/master/The%20Weak%20Instruments%20Problem.ipynb>
# * nbViewer: <https://nbviewer.jupyter.org/github/braddelong/WS2019/blob/master/The%20Weak%20Instruments%20Problem.ipynb> 
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