10/23/2009

# Differences in differences
before/after (A/B).  For potatoes it's before and after introduction in a country.
control/treatment.  For potatoes it'scountries that are feasible for growing potatoes
  and countries infeasible for growing them.

DD = (y_{at} - y_{bt}) - (y_{ac} - y_{bc})
      change in time   -   change in time
       for treatment        for control
       
So diff in diff assumes that treatment group, without having gotten treatment,
  would have seen same trend as control, and thus subtracts that part out.

Keep in mind that diff in diff only works for level (absolute difference) or
  proportion (percentage difference).  It can't work for both.  For levels, just
  subtract.  For proportions, subtract the logs of the values (which compares
  proportionality)

Doing a randomized trial lets you imply a causal relationship, since any other
  related factors would have applied easily, so it really is causal.

Note: in malaria study introduced last class, we see that the control/treatment
  groups aren't binary.  They are continuous, and so you don't just subtract four
  bins.  Instead, the points you're plotting are slopes of variables in regressions
  that compare the current group born to all other groups born in some way.

# Instrumental Variables
Want to know how X->Y
But may have some unobserved U that affects X and Y.  So need to find
  instrument I that changes Y only by way of X, and no other way.

y(z) = y = y(x(z)) != y(x(z), z)
dy/dz = y'(x) - x'(z)  [there's no dy/dz term on the right hand side!]
since all of our analysis is linear regression, we have
  y = \beta * x + \epsilon [ ]
  x = \pi * z + \psi    [ reduced form ]
    so
    y = \beta * \pi * z + \epsilon + \beta * \psi
    
  Effect of x on y is [Reduced]/...