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Log-level and Log-log transformations inLinear Regression ModelsA. Joseph GuseWashington and Lee UniversityFall 2012, Econ 398 Public Finance SeminarLevel-LevelA Level-level Regression Specification.y = 0 + 1x1 + This is called a level-level specification because rawvalues (levels) of y are being regressed on raw values of x .Level-LevelA Level-level Regression Specification.y = 0 + 1x1 + This is called a level-level specification because rawvalues (levels) of y are being regressed on raw values of x .How do we interpret 1?Level-LevelA Level-level Regression Specification.y = 0 + 1x1 + This is called a level-level specification because rawvalues (levels) of y are being regressed on raw values of x .How do we interpret 1?Differentiate w.r.t. x1 to find the marginal effect of x on y . Inthis case, IS the marginal effect.dydx= Log-LevelA Log-level Regression Specification.log(y) = 0 + 1x1 + This is called a log-level specification because the naturallog transformed values of y are being regressed on rawvalues of x .Log-LevelA Log-level Regression Specification.log(y) = 0 + 1x1 + This is called a log-level specification because the naturallog transformed values of y are being regressed on rawvalues of x .You might want to run this specification if you think thatincreases in x lead to a constant percentage increase in y .(wage on education? forest lumber volume on years?)Log-Level ContinuedHow do we interpret 1?Log-Level ContinuedHow do we interpret 1?First solve for y .log(y) = 0 + 1x1 + y = e0+1x1+Log-Level ContinuedHow do we interpret 1?First solve for y .log(y) = 0 + 1x1 + y = e0+1x1+Then differentiate to get the marginal effect:dydx1= e0+1x1+ = 1ySo the marginal effect depends on the value of y , while itself represents the growth rate.1 =dydx11yFor example, if we estimated that 1 is .04, we would saythat another year increases the volume of lumber by 4%.Log-LogA Log-Log Regression Specification.log(y) = 0 + 1 log(x1) + You might want to run this specification if you think thatpercentage increases in x lead to constant percentagechanges in y . (e.g. constant demand elasiticity).Log-LogA Log-Log Regression Specification.log(y) = 0 + 1 log(x1) + You might want to run this specification if you think thatpercentage increases in x lead to constant percentagechanges in y . (e.g. constant demand elasiticity).To caculate marginal effects. Solve for y ...log(y) = 0 + 1 log(x1) + y = e0+1 log(x1)+... and differentiate w.r.t. xdydx1=1x1e0+1 log(x1)+ = 1yx1Log-Log Continued.Log-Log Regression Specification Continued...log(y) = 0 + 1 log(x1) + From previous slide the marginal effect isdydx1= 1yx1Log-Log Continued.Log-Log Regression Specification Continued...log(y) = 0 + 1 log(x1) + From previous slide the marginal effect isdydx1= 1yx1Solving for 1 we get1 =dydx1x1yHence 1 is an elasticity. If x1 is price and y is demand andwe estimate 1 = .6, it means that a 10% increase in theprice of the good would lead to a 6% decrease in demand.