gvlma function may be worth a look.

1 Gauss Markov Assumptions

The Gauss-Markov assumptions, also known as the Gauss-Markov conditions or classical linear regression assumptions, are a set of key assumptions that form the basis for Ordinary Least Squares (OLS) regression. When these assumptions are met, OLS estimators are unbiased, efficient, and possess the minimum variance among the class of linear unbiased estimators.

Here are the key Gauss-Markov assumptions:

  1. Linearity: The relationship between the dependent variable and the independent variables is linear. The model should be correctly specified in terms of its functional form.

    • Mathematically, \(Y_i​ = \beta_0 + \beta_​1 ​X_{1i​} + \beta_​2 ​X_{2i} ​+… + \beta_​k ​X_{ki} ​+ \epsilon_i​\)

  2. Independence: The independence assumption in linear regression pertains to the independence of errors across observations. It means that the error for one observation is not correlated with the error for another observation.

    • Mathematically, it is expressed as \(Cov(\epsilon_i​,\epsilon_j​)=0\) for all \(i \neq j\).

  3. Homoscedasticity: The variance of the errors (residuals) should be constant across all levels of the independent variables. This implies that the spread of residuals is the same for all predicted values.

    • Mathematically, it is expressed as $Var(\epsilon_i​)=\sigma $ for all \(i\).

  4. No Perfect Multicollinearity: The independent variables should not be perfectly correlated with each other. This means that there should not be a perfect linear relationship among any subset of the independent variables.

    • This is often expressed using the correlation matrix or the variance inflation factor (VIF) to check for high correlation among the independent variables.

  5. Zero Conditional Mean (or Expected Value of Residuals): The expected value of the residuals is zero for all values of the independent variables. In mathematical terms, \(E(\epsilon_i​∣X)=0\), where \(\epsilon_i\) is the error term for the ith observation and X\ represents the matrix of independent variables.

    • Mathematically, \(E(\epsilon_i​∣X)=0\) implies \(Cov(X_i,\epsilon_ji)=0\).

    • AKA exogenity/No endogeneity: The independent variables are assumed to be exogenous, meaning they are not influenced by the error term.

  6. Normality of Errors (Optional): While OLS does not require the normality of the errors for unbiasedness and efficiency, it is often assumed for hypothesis testing and confidence interval construction. This assumption is less critical for large sample sizes due to the Central Limit Theorem.

Meeting these assumptions enhances the reliability of OLS estimates and statistical inferences drawn from regression analysis. If the assumptions are violated, it can lead to biased estimates and inefficient standard errors. Researchers often perform diagnostic tests and checks to assess the validity of these assumptions before relying on regression results.

2 What does OLS is BLUE mean?

OLS stands for Ordinary Least Squares, and BLUE stands for Best Linear Unbiased Estimator. The phrase “OLS is BLUE” is a statement in the context of linear regression analysis.

The Ordinary Least Squares (OLS) method is a technique used to estimate the parameters of a linear regression model by minimizing the sum of the squared differences between the observed and predicted values. When certain assumptions, known as the Gauss-Markov assumptions, are satisfied, OLS provides the Best Linear Unbiased Estimators (BLUE) i.e. the OLS estimators are not only unbiased but also have the minimum variance among the class of linear unbiased estimators.

The Gauss-Markov assumptions are the conditions under which OLS estimators become BLUE. When these assumptions are satisfied, OLS produces unbiased estimates with the minimum variance, making it the best among the class of linear unbiased estimators. However, it’s crucial for researchers to be aware of the assumptions and to check for their validity in practice to ensure the reliability of OLS results. If the assumptions are violated, alternative estimation methods or corrections may be necessary.

When these assumptions hold, OLS provides efficient and unbiased estimates of the parameters, making it the Best Linear Unbiased Estimator. The phrase “OLS is BLUE” is a way of expressing this property succinctly.

2.1 EXTRA - The relationship between OLS being BLUE and the Gauss-Markov assumptions:

  1. Unbiasedness (Gauss-Markov Assumption): One of the Gauss-Markov assumptions is that the expected value of the residuals (errors) is zero for all levels of the independent variables. When this assumption is met, OLS estimators are unbiased, meaning that, on average, they accurately estimate the true population parameters.

  2. Efficiency (Gauss-Markov Assumption): The Gauss-Markov assumptions also include the assumption that the variances of the errors are constant (homoscedasticity) and that there is no perfect multicollinearity. In the presence of these conditions, OLS estimators are efficient, meaning they have the minimum variance among the class of linear unbiased estimators.

  3. Best (Gauss-Markov Assumption): The term “Best” in BLUE stands for Best Linear Unbiased Estimators. This means that among all linear unbiased estimators, OLS provides the estimates with the smallest possible variance when the Gauss-Markov assumptions hold.

3 In max three paragraphs, why should we take the log of a variable in your linear regression? There are many reasons, including easy interpretation or better fit, but please stick with the a few only.

Taking the logarithm of a variable in linear regression can offer several advantages. One primary reason is to handle non-linear relationships and heteroscedasticity. In cases where the relationship between the dependent and independent variables is not strictly linear, taking the log of one or more variables can transform the data, making the relationship more linear and thus improving the model’s fit. Additionally, by addressing heteroscedasticity (unequal variance of errors), logging can help achieve constant variance across the range of predictors, meeting one of the assumptions of linear regression.

Another crucial reason is to interpret coefficients in percentage terms. When you take the log of a variable, the coefficients in the regression equation represent percentage changes rather than absolute changes. This can enhance the interpretability of the results, especially when dealing with variables that exhibit exponential growth or decay. For example, if the dependent variable is logged, a coefficient of 0.02 can be interpreted as a 2% change in the dependent variable associated with a one-unit change in the independent variable.

Lastly, logging variables can be useful when dealing with skewed or highly skewed data. Applying the logarithm can help normalize the distribution, making it closer to a normal distribution and improving the robustness of statistical inference in linear regression.

https://library.virginia.edu/data/articles/interpreting-log-transformations-in-a-linear-model