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         {\bf  The Observed Choice    Problem in Estimating the Cost of Policies     }\\ 

 (PUBLSHED:  {\it ECONOMICS LETTERS}, 1998?)\\
 
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November 7, 1997  \\   
        \bigskip   
 Eric Rasmusen\\   
     

        {\it Abstract}   
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 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. 
   
     
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          \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       Www.bus.indiana.edu/$\sim$erasmuse/@Articles/Unpublished/mchoice.pdf. 
      

   	    
 
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    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.   
 
 
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\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}.  
 
      
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