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What is BLUE (Best Linear Unbiased Estimator), and how is it derived?

What is BLUE (Best Linear Unbiased Estimator), and how is it derived?


What is BLUE (Best Linear Unbiased Estimator), and how is it derived?


Answer: "BLUE stands for Best Linear Unbiased Estimator, which is a key concept in statistics, particularly in the context of linear regression models. Let’s break it down:

1. Best: It means that the estimator has the smallest variance among all the unbiased estimators. In other words, it's the most efficient estimator.

2. Linear: The estimator is a linear function of the observed data (i.e., it is expressed as a linear combination of the data).

3. Unbiased: An estimator is called unbiased if the expected value of the estimator is equal to the true parameter being estimated. This ensures that, on average, the estimator gives the correct results.

4. Estimator: It's a formula or rule that gives an estimate of the unknown parameter.

In simpler terms, BLUE is the best possible estimator for the coefficients in a linear regression model that is linear in nature, unbiased, and has the least variance among all unbiased estimators.

Derivation of BLUE (Gauss-Markov Theorem):

The Gauss-Markov theorem states that, under certain assumptions, the Ordinary Least Squares (OLS) estimator in a linear regression model is BLUE. These assumptions are:

  1. Linearity: The relationship between the independent variables and the dependent variable is linear.

  2. No Endogeneity: The independent variables are not correlated with the errors (no omitted variable bias).

  3. Homoscedasticity: The error terms have constant variance (no heteroscedasticity).

  4. Independence: The error terms are uncorrelated with each other (no autocorrelation).

  5. No Perfect Multicollinearity: The independent variables are not perfectly correlated with each other.

Derivation Process:

  • In linear regression, we have the model:

y=X β+ϵ 

where y is the vector of the dependent variable, X is the matrix of independent variables, β is the vector of coefficients, and ϵ is the error term.

The Ordinary Least Squares (OLS) estimator minimizes the sum of squared errors and is given by:

β^​OLS​=((X'X))^(−1) X'y

  • The Gauss-Markov theorem shows that, under the assumptions mentioned, this OLS estimator, β^OLS, has the smallest variance among all unbiased linear estimators, meaning it is BLUE.

Conclusion:

In summary, BLUE is the Best Linear Unbiased Estimator, which is typically achieved through the OLS estimator in linear regression, provided the key assumptions of the Gauss-Markov theorem are met. This makes OLS highly desirable in regression analysis because it guarantees the most efficient and unbiased estimates for the regression coefficients."


 

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