Why is the ordinary least squares method used in a linear regression?
1 Answer
If the Gauss-Markof assumptions hold then OLS provides the lowest standard error of any linear estimator so best linear unbiased estimator
Explanation:
Given these assumptions
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Parameter co-efficents are linear, this just means that
#beta_0 and beta_1# are linear but the#x# variable doesn't have to be linear it can be#x^2# -
The data has been taken from a random sample
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There is no perfect multi-collinearity so two variables are not perfectly correlated.
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#E(u# /#x_j)=0# mean conditional assumption is zero, meaning that the#x_j# variables provide no information about the mean of the unobserved variables. -
The variances are equal for any given level of
#x# i.e.#var(u)=sigma^2#
Then OLS is the best linear estimator in the population of linear estimators or (Best Linear Unbiased Estimator) BLUE.
If you have this additional assumption:
- The variances are normally distributed
Then the OLS estimator becomes the best estimator regardless if it is a linear or non-linear estimator.
What this essentially means is that if assumptions 1-5 hold then OLS provides the lowest standard error of any linear estimator and if 1-6 hold then it provides the lowest standard error of any estimator.