\documentstyle[12pt,epsf] {article} %\parskip 10pt \reversemarginpar \topmargin -.4in \oddsidemargin .25in \textheight 8.7in \textwidth 6in \pagestyle{myheadings} \markboth{November 7, 1997}{November 7, 1997} \begin{document} \begin{small} \baselineskip 24pt \parindent 24pt \titlepage \vspace*{12pt} \begin{center} \begin{large} {\bf The Observed Choice Problem in Estimating the Cost of Policies }\\ (PUBLSHED: {\it ECONOMICS LETTERS}, 1998?)\\ \end{large} November 7, 1997 \\ \bigskip Eric Rasmusen\\ {\it Abstract} \end{center} A policy will be used more heavily when its marginal cost is lower. In a regression setting, this can mean that the equation to be estimated is actually $y_i = \beta_i x(\beta_i)$. The analyst who treats times and places as identical will underestimate the policy's average cost. OLS is biased towards small coefficients, and instrumental variables should be used. \begin{small} \noindent \hspace*{20pt} Indiana University, Kelley School of Business,BU 456, 1309 E 10th Street, Bloomington, Indiana, 47405-1701. Office: (812) 855-9219. Fax: 812-855-3354. Email: Erasmuse@indiana.edu. Web: Php.indiana.edu/$\sim$erasmuse. Copies of this paper can be found at Www.bus.indiana.edu/$\sim$erasmuse/@Articles/Unpublished/mchoice.pdf. \end{small} %---------------------------------------------------------------% %---------------- \newpage It is common to estimate policy effects by looking at data from various locations. Suppose $Impact = \beta \cdot Policy$, or \begin{equation} \label{e1} y_i = \beta x_i, \end{equation} and that the impact is undesirable. In this setting, $x_i = x(\beta_i)$ because policies are chosen in recognition of their marginal impacts in particular locations, and $\beta$ varies across locations. This causes a predictable bias in OLS estimation which I call `` the observed choice problem''. This problem has not been directly discussed in the econometrics literature. The closest I have found is Garen (1984). In my own Rasmusen (1996) I develop the problem more fully and apply it to the slightly more complicated case where the policy impact is desirable. The following three-equation model illustrates the bias. \begin{equation} \label{e20} y_i = \beta_i x_i + \epsilon_i \; \end{equation} \begin{equation} \label{e22} \beta_i = \overline{\beta} + v_i \; \end{equation} \begin{equation} \label{e21} x_i= \gamma_1 + \gamma_2 \beta_i + \gamma_3 z_i + u_i\; \end{equation} \noindent Assume that: (i) $\gamma_1 + \gamma_2\overline{\beta} + \frac{\gamma_3 \sum z_i}{N} >0,$ (ii) $\overline{\beta}>0$, (iii) $z$ and $\overline{\beta}$ are nonstochastic, (iv) $\epsilon, u$ and $v$ are independent stochastic disturbances with mean zero and finite variance, (v) $v$ has a symmetric distribution, (vi) $\gamma_2<0$. Assumptions (i) and (ii) are just normalizations, but (vi) represents that $y$ is an undesirable impact of $x$, so $x$ is used less when $\beta_i$ is greater. The OLS estimate of $\overline{\beta}$ is \begin{equation} \label{e7} \widehat{\beta}_{OLS} = \frac{\sum x_i y_i }{\sum x_i^2}, \end{equation} which has the expectation \begin{equation} \label{e9} E \left( \frac{\sum x_i ( \overline{\beta} x_i + v_i x_i + \epsilon_i) }{\sum x_i^2} \right)= E \left( \overline{\beta} \frac{\sum x_i^2}{\sum x_i^2} \right) + E \left( \frac{\sum x_i^2 v_i}{\sum x_i^2} \right) + E \left( \frac{\sum x_i \epsilon_i }{\sum x_i^2} \right)\;. \end{equation} The first and last terms of (\ref{e9}) equal $ \overline{\beta}$ and 0, and the middle term equals 0 if $E (x_i^2 v_i) = 0$. If $x_i$ and $v_i$ are independent, OLS is unbiased. This model, however, violates the OLS assumptions in two ways, each harmless by itself, but bad in combination: random parameters and stochastic regressors. The simpler system of just (\ref{e20}) and (\ref{e22}) has random parameters, and the simpler system of just (\ref{e20}) and (\ref{e21}) (so $\beta_i =\overline{\beta}$) has stochastic regressors, but in each of those two simple systems, OLS would be unbiased. To see that the OLS estimate of $\overline{\beta}$ is biased in the full system, combine equations (\ref{e22}) and (\ref{e21}) to get \begin{equation} \label{e25} x_i= \gamma_1 + \gamma_2 \overline{\beta} + \gamma_2 v_i + \gamma_3 z_i + u_i \; . \end{equation} The critical middle term in equation (\ref{e9}), which for unbiasedness must equal zero, can be written using (\ref{e25}) as \begin{equation} \label{e27} \frac{\sum (\gamma_1 + \gamma_2 \overline{\beta} + \gamma_2 v_i + \gamma_3 z_i + u_i)^2 v_i}{\sum x_i^2}. \end{equation} The summed quantity in the numerator has the expectation \begin{equation} \label{e31} 2\gamma_2[\gamma_1 + \gamma_2 \overline{\beta} + \gamma_3 z_i] \sigma^2_v, \end{equation} since $E (v^3)=0$ by assumption (v), and $u$ and $v$ are independent. Expression (\ref{e31}) has the same sign as $\gamma_2[\gamma_1 + \gamma_2 \overline{\beta} + \gamma_3 z_i]$. Summed across the $n$ observations, this takes the same sign as $\gamma_2$, since the term in square brackets is positive by assumption (i). Since $\gamma_2<0$, $\beta$ is underestimated. This is similar to the folk wisdom that estimation problems lead to coefficients being too small. Instrumental variables can be used to solve the observed-choice problem, as I show in Rasmusen (1996), if the analyst can observe $z$. Figure 1 illustrates the problem. It shows two localities with their own relationships between policy $x$ and impact $y$ depicted as rays through the origin. Localities 1 and 2 have slopes $\beta_1$ and $\beta_2$, an average slope of $\overline{\beta} = \frac{ (\beta_1+\beta_2}{2} $. Policymakers 1 and 2 choose points on their respective rays. If they choose $x$ ignoring local conditions, $x_1$ and $x_2$ have the same expected value, and the expected average of the two observations is on the middle ray. This corresponds to OLS being unbiased. If, however, $y$ is a cost of $x$, and a steeper slope makes a policymaker choose a lower level of $x$, then Locality 1, with a greater marginal cost, chooses a lower $x$ than Locality 2: $x_1< x_2$. If the econometrician draws a line through the origin to lie between the two observations and minimize the squared deviations, that line will have a slope of less than $\overline{\beta}$. OLS underestimates the marginal cost. \epsfysize=6in \epsffile{Choice2.eps} \noindent REFERENCES Garen, John (1984). The returns to schooling: A selectivity bias approach with a continuous choice variable. {\it Econometrica} 52 (September): 1199-1218. Rasmusen, Eric (1996) ``Observed Choice and Optimism in Estimating the Effects of Government Policies,'' forthcoming, {\it Public Choice}. %--------------------------------------------------------------- \end{small} \end{document}