17 Endogeneity Bias - Working's Example

17 Endogeneity Bias - Working's Example - Economics 241B...

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Economics 241B Endogeneity Bias - The Example of Working The classic illustration of the biases created by endogeneity dates to Working in 1927. Consider the simple model of demand and supply (for co/ee, say) Q d t = & 0 + 1 P t + U t Q s t = ± 0 + ± 1 P t + V t Q d t = Q s t : The error term U t demand, other than price, such as the public±s mood for co/ee. Depending on the value of U t , the demand curve in the price-quantity plane shifts up or down. Similarly, V t represents supply factors other than price. As the intercepts capture the mean of the error terms, we assume E ( U t ) = 0 and E ( V t ) = 0 . To avoid inessential complications, we also assume Cov ( U t ;V t ) = 0 . If we de²ne the market-clearing quantity to be Q t = Q d t = Q s t , then the system reduces to two equations Q t = & 0 + 1 P t + U t (demand) Q t = ± 0 + ± 1 P t + V t : (supply) A regressor is endogenous if it is not orthogonal to the error term, that is, if it does not satisfy the orthogonality condition. With an intercept in the equation, endogeneity arises if and only if the regressor is correlated with the error. In the present example, the regressor P t is necessarily endogenous in both equations. To see why, treat the demand and supply equations as a system of two simultaneous equations and solve for ( P t ;Q t ) as P t = ± 0 & 0 1 ± 1 + V t U t 1 ± 1 Q t = 1 ± 0 & 0 ± 1 1 ± 1 + 1 V t ± 1 U t 1 ± 1 : Because price is a function of the two error terms, price is clearly correlated with each error.
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We can take the argument one step further and calculate the covariance of price with each error. The covariance of price with the demand shifter U t and the supply shifter V t is Cov ( P t ;U t ) = V ar ( U t ) 1 ± 1 ; Cov ( P t ;V t ) = V ar ( V t ) 1 ± 1 : If the demand curve is downward sloping ( 1 < 0 ) and the supply curve is upward
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17 Endogeneity Bias - Working's Example - Economics 241B...

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