The Derivative of the Natural Logarithm

The derivative of the natural logarithm is defined the following way:

f(x) = ln (x)

f'(x)=\frac{1}{x}

The formal proof of the derivative is provided at the bottom of this post.

The following example further explains the derivative of the natural logarithm. Remember that Continue reading The Derivative of the Natural Logarithm

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Robust Standard Errors in STATA

”Robust” standard errors is a technique to obtain unbiased standard errors of OLS coefficients under heteroscedasticity. In contrary to other statistical software, such as R for instance, it is rather simple to calculate robust standard errors in STATA. All you need to is add the option robust to you regression command. That is:

Continue reading Robust Standard Errors in STATA

Robust Standard Errors in R

One can calculate robust standard errors in R in various ways. However, one can easily reach its limit when calculating robust standard errors in R, especially when you are new in R. It always bordered me that you can calculate robust standard errors so easily in STATA, but you needed ten lines of code to compute robust standard errors in R. I decided to solve the problem myself and modified the summary() function in R so that it replicates the simple way of STATA. I added the parameter robust to the summary() function that calculates robust standard errors if one sets the parameter to true. With the new summary() function you can get robust standard errors in your usual summary() output. All you need to do is to set the robust parameter to true:

summary(lm.object, robust=T) Continue reading Robust Standard Errors in R

Robust Standard Errors in R – Function

One can calculate robust standard errors easily in STATA. However, one can easily reach its limit when calculating robust standard errors in R. Although there exist several possibilities to calculate heteroscedasticity consistent standard errors most of them are not easy to implement, especially for beginners. I modified the summary() function in R so that it replicates the simple way of STATA. You can find the new summary() function below. Furthermore, I uploaded the function to a github.com repository. This makes it easy to load the function into your R session. In order to see how you can import the new summary() function into your R session and how you can use it see this post here. Continue reading Robust Standard Errors in R – Function

Violation of CLRM – Assumption 4.2: Consequences of Heteroscedasticity

Violating assumption 4.2, i.e. \sigma_{i}^{2} \neq \sigma_{j}^{2} \text{ for } i \neq j leads to heteroscedasticity. Recall, under heteroscedasticity the OLS estimator still delivers unbiased and consistent coefficient estimates, but the estimator will be biased for standard errors. Increasing the number of observations will not solve the problem in this case.

Fortunately, several ways exist to deal with heteroscedasticity: Continue reading Violation of CLRM – Assumption 4.2: Consequences of Heteroscedasticity