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"# The Weak Instruments Problem\n",
"\n",
"A simple instrumental-variables framework: you fear that your x is correlated with u, so you search for an instrument z:\n",
"\n",
"Suppose: $ y = {\\beta}x + u $; $ x = {\\pi}z + v $; $ u = {\\delta}v + w $; with _u, v, and w_ error terms...\n",
"\n",
"then:\n",
"\n",
">$ {\\beta}_{IV} = \\frac{zy}{zx} = \\frac{{\\beta}xz + uz}{xz} $\n",
"\n",
">$ {\\beta}_{IV} = \\frac{{\\beta}{\\pi}zz + {\\beta}vz + uz}{{\\pi}zz + vz} = \\beta + \\frac{uz}{{\\pi}zz + vz}$\n",
"\n",
">$ {\\beta}_{IV} = \\beta + \\frac{{\\delta}vz + wz}{{\\pi}zz + vz} = \\beta + {\\delta}\\frac{1}{({\\pi}zz/(vz)) + 1} + \\frac{wz}{{\\pi}zz + vz} $\n",
"\n",
"And if $ \\pi = 0 $, then: \n",
"\n",
">$ {\\beta}_{IV0} = \\beta + \\frac{{\\delta}vz + wz}{vz} = \\beta + \\delta + \\frac{wz}{vz} $\n",
"\n",
"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 $\n",
"\n",
"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?\n",
"\n",
"----\n",
"\n"
]
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"## The Weak Instruments Problem: Catch Our Breathâ€”Further Notes:\n",
"\n",
"\n",
"\n",
"**Last Modified 2019-03-06; Created 2019-03-06**\n",
"\n",
"* Make three comments\n",
"* Ask three questions\n",
"* Recommend three readings\n",
"\n",
" \n",
"\n",
"----\n",
"\n",
"* Weblog Support: \n",
"* nbViewer: \n",
"\n",
" \n",
"\n",
"----"
]
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