November 6, 2000
Mistakes and Extensions for:
Eric Rasmusen, ``Observed Choice, Estimation, and Optimism About
Policy Changes, '' Public Choice, (October 1998) 97: 65-91.
(1) There is a mistake on page 73. The paper says,
"If the econometrician draws a line through the
origin to go through the two observations and minimize
the squared deviations, that line will have a
*negative* slope. OLS underestimates the marginal
benefit, and in fact give an impossible result"
This is obviously wrong-- a line through the origin
through the middle of positive datapoints cannot have
a negative slope. I was confusing the cases where the
intercept is or is not constrained to equal zero. I
should have said,
"If the econometrician draws a line through the origin
to go through the two observations and minimize the
sum of squared deviations, that line will have a
slope gentler than the true value. OLS underestimates
the marginal benefit."
I could have added:
"If the econometrician does not constrain the line to
go through the origin, minimizing the sum of squared
deviations will yield a regression line with
*negative* slope-- the impossible result that more of
the policy leads to less benefit."
(2) The paper uses a model with zero intercept and one
regressor, so
y_i = beta_i x_i + epsilon_i
The observed choice bias still exists in the same
way if the intercept is allowed to vary among
observations, so the model is
y_i = alpha_i + beta_i x_i + epsilon_i
The slope coefficient betabarhat will be biased as in
the simple model with the zero intercept. It doesn't
matter if the intercept is assumed to take the same
nonzero value for all observations or allowed to vary.
The intercept alphabarhat will also be biased, but
with the opposite bias of the slope coefficient. This
is because it takes the value
alphabarhat = xbar - betabarhat* ybar.
We must, however, make one additional assumption
for the presence of an intercept to make no
difference: that every observation does use a
nonzero value of the policy X. Also, I ignore the
information that the intercept must be non-negative,
which prior information might perhaps make some
technique fancier than OLS optimal. One way to
interpret this assumption is that we assume that even
if the policy is chosen to be X=0, there is still some
benefit or cost Y-- a true fixed cost or benefit, as
opposed to a cost or benefit that jumps from 0 to some
higher level if any policy X>0 is used. This is not a
big assumption, and it is, of course, easily checked
in the observed data on X.
(3) Typo: On page 67, the first line after equation
(3) should have an "N" so it reads "N(0," rather than
"(0,". This is an obvious typo that will not mislead
anyone.
(4) Missing information, p. 90: The variable
"Welfare Income Equivalent" is taken from Tanner,
Moore, and Hartman (1995).
(5) A related article is Summers & Pritchett (1993).
They note that countries do not choose policies
randomly. They are worried about things such as the
tendency for a country that deliberately picks a
policy to be in bad shape anyway and to deteriorate
despite the policy. See Summers, Lawrence and Lant
Pritchett (1993) ``The structural-adjustment debate,''
The American Economic Review, May 1993; 83: 383-
390.
(6) An idea. This paper involves the bother of whether
to use expected values or plims, unbiasedness or
consistency, in evaluating estimators. I switch to
plims and consistency when setting out the
instrumental variables estimator because with infinite
supports of the normal distribution for regressors,
expected values have existence problems. I perhaps
should have done that for the OLS part too, since I
have stochastic regressors there.
Another way to tackle that problem, I speculate, is
to use truncated normal distributions for the error
terms instead of normal distributions. That creates
problems for small-sample hypothesis testing, perhaps,
but not for expected values or consistency