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