1 the disturbance term in logarithmic models thus far, nothing has been said about the disturbance...

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THE DISTURBANCE TERM IN LOGARITHMIC MODELS

Thus far, nothing has been said about the disturbance term in nonlinear regression models.

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For the regression results in a linearized model to have the desired properties, the disturbance term in the transformed model should be additive and it should satisfy the regression model conditions.


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To be able to perform the usual tests, it should be normally distributed in the transformed model.


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In the case of the first example of a nonlinear model, there was no problem. If the disturbance term had the required properties in the original model, it would have them in the regression model. It has not been affected by the transformation.

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In the discussion of the logarithmic model, the disturbance term was omitted altogether.


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However, implicitly it was being assumed that there was an additive disturbance term in the transformed model.

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For this to be possible, the random component in the original model must be a multiplicative term, eu.

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We will denote this multiplicative term v.

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When u is equal to 0, not modifying the value of log Y, v is equal to 1, likewise not modifying the value of Y.

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Positive values of u correspond to values of v greater than 1, the random factor having a positive effect on Y and log Y. Likewise negative values of u correspond to values of v between 0 and 1, the random factor having a negative effect on Y and log Y.

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Besides satisfying the regression model conditions, we need u to be normally distributed if we are to perform t tests and F tests.


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This will be the case if v has a lognormal distribution, shown above.

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The mode of the distribution is located at v = 1, where u = 0.

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The same multiplicative disturbance term is needed in the semilogarithmic model.

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Note that, with this distribution, one should expect a small proportion of observations to be subject to large positive random effects.

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Here is the scatter diagram for earnings and schooling using Data Set 21. You can see that there are several outliers, with the four most extreme highlighted.


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Here is the scatter diagram for the semilogarithmic model, with its regression line. The same four observations remain outliers, but they do not appear to be so extreme.


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The histogram above compares the distributions of the residuals from the linear and semi-logarithmic regressions. The distributions have been standardized, that is, scaled so that they have standard deviation equal to 1, to make them comparable.

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It can be shown that if the disturbance term in a regression model has a normal distribution, so will the residuals.

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It is obvious that the residuals from the semilogarithmic regression are approximately normal, but those from the linear regression are not. This is evidence that the semi-logarithmic model is the better specification.

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What would happen if the disturbance term in the logarithmic or semilogarithmic model were additive, rather than multiplicative?


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If this were the case, we would not be able to linearize the model by taking logarithms. There is no way of simplifying log(b1Xb + u). We should have to use some nonlinear regression technique.

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Copyright Christopher Dougherty 2012.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2012.11.03

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1 the disturbance term in logarithmic models thus far, nothing has been said about the disturbance...

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