OLS Estimator: Derivation in Regression Analysis

Study Notes

OLS Estimator: Derivation

Ordinary Least Squares (OLS) Overview:
- A method used in regression analysis to estimate the coefficients of a linear relationship.
- Minimizes the sum of the squared differences between observed and predicted values.
Model Specification:
- The linear regression model is represented as:
  - ( Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_kX_k + \epsilon )
- Where:
  - ( Y ) = dependent variable
  - ( X_i ) = independent variables
  - ( \beta_i ) = coefficients
  - ( \epsilon ) = error term
Objective Function:
- The goal is to minimize the residual sum of squares (RSS):
  - ( RSS = \sum (Y_i - \hat{Y}_i)^2 )
  - Where ( \hat{Y}_i ) is the predicted value from the model.
Taking Derivatives:
- To find the OLS estimates, take the partial derivative of the RSS with respect to each coefficient ( \beta_j ) and set it to zero:
  - ( \frac{\partial (RSS)}{\partial \beta_j} = 0 )
Normal Equations:
- For each coefficient ( \beta_j ), the resulting equations after differentiation lead to the normal equations:
  - ( \sum (Y_i - \hat{Y}i) X{ij} = 0 )
Matrix Representation:
- In matrix form, the model can be expressed as:
  - ( Y = X\beta + \epsilon )
- Where:
  - ( Y ) = vector of observations
  - ( X ) = design matrix of predictors (including a column of ones for the intercept)
  - ( \beta ) = vector of coefficients
Solution Using Linear Algebra:
- The OLS estimator can be derived as:
  - ( \hat{\beta} = (X'X)^{-1}X'Y )
- Where:
  - ( X' ) = transpose of matrix ( X )
  - ( (X'X)^{-1} ) = inverse of the product of ( X' ) and ( X )
Conditions for OLS:
- Requires assumptions:
  - Linearity
  - Independence of errors
  - Homoscedasticity (constant variance of errors)
  - No multicollinearity among predictors
Interpretation:
- The OLS estimator provides the best linear unbiased estimates (BLUE) under the Gauss-Markov theorem when the above assumptions are met.

Ordinary Least Squares (OLS)

OLS regression is a common method to find the best fitting line for a set of data points.
OLS minimizes the difference between the predicted and actual values by using the sum of squared residuals (RSS).
The model forOLS regression is:
- ( Y = \beta_0 + \beta_1X_1 + \beta_2X_2 +...+ \beta_kX_k + \epsilon )
- Where ( Y ) is the dependent variable, ( X_i ) are independent variables, ( \beta_i ) are the coefficients, and ( \epsilon ) represents the error term.

Minimizing the Residual Sum of Squares (RSS)

The RSS formula represents the sum of the squared differences between the observed values (Y) and predicted values ( \hat{Y} )
- ( RSS = \sum (Y_i - \hat{Y}_i)^2 )
To find the minimum RSS, OLS takes partial derivatives of the RSS with respect to each coefficient ( \beta_j ) and sets them to zero.
- ( \frac{\partial (RSS)}{\partial \beta_j} = 0 )

OLS Equations

This differentiation process ultimately leads to the normal equations:
- ( \sum (Y_i - \hat{Y}i) X{ij} = 0 )
The normal equations are used to solve for the coefficients ( \beta_j ) that result in the best fit.

Matrix Representation of OLS

The OLS model can be expressed in matrix form as follows:
- ( Y = X\beta + \epsilon )
Where ( Y ) is the vector of observations, ( X ) is the design matrix of predictors (including a column of ones for the intercept), and ( \beta ) is the vector of coefficients.

The OLS Formula

Using linear algebra, the OLS estimator can be derived as the product of the inverse of the product of ( X' ) and ( X ) with the product of ( X' ) and ( Y )
- ( \hat{\beta} = (X'X)^{-1}X'Y )

OLS Assumptions

OLS requires certain assumptions to be met for the estimators to be statistically accurate. These assumptions include:
- Linearity: The relationship between dependent and independent variables must be linear.
- Independence of errors: The error terms in the model must be independent of each other.
- Homoscedasticity: The variance of the error terms must be constant across all values of the independent variables.
- No Multicollinearity: The independent variables should not be perfectly correlated with each other, which could lead to instability in the estimates.

Interpreting OLS

The OLS estimator provides the best linear unbiased estimates (BLUE) under the Gauss-Markov assumptions.
This means that the OLS estimates are the most efficient estimates available, given that the assumptions are met.

Description

This quiz covers the derivation of the Ordinary Least Squares (OLS) estimator in the context of regression analysis. It explores model specification, the objective function, and taking derivatives to find estimates. Test your understanding of these key concepts in linear regression.