Linear Regression Models with Logarithmic Transformations

Linear Regression Models with Logarithmic TransformationsKenneth Benoit Methodology InstituteLondon School of 17, 20111 Logarithmic Transformations of variablesConsidering the simple bivariate Linear modelYi= + Xi+ i,1there are four possible com-binations of Transformations involving logarithms: the Linear case with no Transformations , thelinear-log model , the log- Linear model2, and the log-log Yi= + Xi Yi= + logXilogYlog-linearlog-loglog Yi= + Xilog Yi= + logXiTable 1: Four varieties of Logarithmic transformationsRemember that we are usingnaturallogarithms, where the base ise Logarithms mayhave other bases, for instance the decimal logarithm of base 10. (The base 10 logarithm is used inthe definition of the Richter scale, for instance, measuring the intensity of earthquakes as Richter=log(intensity).)

24 68 0 20 40 60 80 100 Log(Expenses) 3 Interpreting coefficients in logarithmically models with logarithmic transformations 3.1 Linear model: Yi = + Xi + i Recall that in the linear regression model, logYi = + Xi + i, the coefficient gives us directly the change in Y for a one-unit change in X.No additional interpretation is required beyond the

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