5.3 Least Squares Estimation Using Matrices

Least squares estimation using matrices is a powerful method for analyzing linear relationships between variables. It involves solving normal equations to find parameter estimates that minimize the sum of squared residuals, providing efficient and interpretable results.

The matrix approach offers computational advantages and yields estimators with desirable properties like unbiasedness and efficiency. Understanding these concepts is crucial for applying linear regression effectively and interpreting results accurately in various fields of study.

Least squares estimates

Solving the normal equations

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• The normal equations are a set of linear equations that can be solved to obtain the least squares estimates of the regression parameters in a linear regression model
• Derived by minimizing the sum of squared residuals, which measures the difference between the observed values and the predicted values from the regression model
• In matrix notation, the normal equations are expressed as $(X'X)β = X'y$, where:
  • $X$ is the design matrix
  • $X'$ is the transpose of $X$
  • $y$ is the vector of observed values
  • $β$ is the vector of regression parameters to be estimated
• The solution to the normal equations is given by $β̂ = (X'X)⁻¹X'y$, where:
  • $β̂$ represents the least squares estimates of the regression parameters
  • $(X'X)⁻¹$ is the inverse of the matrix product $X'X$
  • The existence of $(X'X)⁻¹$ requires that the columns of $X$ are linearly independent, ensuring a unique solution to the normal equations
• The matrix approach to solving the normal equations offers computational efficiency, particularly when dealing with a large number of predictor variables or observations

Interpreting the estimates

• The least squares estimates obtained from the matrix approach represent the values of the regression parameters that minimize the sum of squared residuals
• Each element in the $β̂$ vector corresponds to a specific regression parameter, such as the intercept or the coefficient associated with a predictor variable
• The intercept estimate represents the expected value of the response variable when all predictor variables are zero, assuming it is meaningful within the context of the problem
• The coefficient estimates represent the change in the expected value of the response variable for a one-unit increase in the corresponding predictor variable, holding all other predictors constant
• The signs of the coefficient estimates indicate the direction of the relationship between the predictor variables and the response variable:
  • Positive values indicate a positive relationship
  • Negative values indicate a negative relationship
• The magnitude of the coefficient estimates provides information about the strength of the relationship between the predictor variables and the response variable, with larger absolute values indicating a stronger relationship
• Interpreting the least squares estimates should be done within the context of the problem and considering the units of measurement for the variables involved

Fitted values and residuals

Calculating fitted values and residuals

• The fitted values, denoted as $ŷ$, are the predicted values of the response variable obtained from the regression model using the least squares estimates of the parameters
• In matrix notation, the fitted values can be calculated as $ŷ = Xβ̂$, where:
  • $X$ is the design matrix
  • $β̂$ is the vector of least squares estimates
• The residuals, denoted as $e$, are the differences between the observed values of the response variable and the fitted values obtained from the regression model
• In matrix notation, the residuals can be calculated as $e = y - ŷ$, where:
  • $y$ is the vector of observed values
  • $ŷ$ is the vector of fitted values
• The sum of squared residuals, denoted as $SSR$, measures the discrepancy between the observed values and the fitted values and can be calculated as $SSR = e'e$, where $e'$ is the transpose of the residual vector

Computational efficiency

• The matrix approach to calculating the fitted values and residuals offers computational efficiency
• It allows for easy manipulation of the results for further analysis, such as calculating goodness-of-fit measures or conducting diagnostic tests
• Example: Calculating the coefficient of determination ($R²$) using the fitted values and residuals
  • $R² = 1 - \frac{SSR}{SST}$, where $SST$ is the total sum of squares

Properties of estimators

Desirable properties

• The least squares estimators obtained from the matrix approach have several desirable properties, making them widely used in linear regression analysis
• Unbiasedness: The least squares estimators are unbiased, meaning that the expected value of the estimators is equal to the true value of the parameters being estimated, $E(β̂) = β$
• Consistency: As the sample size increases, the least squares estimators converge in probability to the true value of the parameters, $plim(β̂) = β$
• Efficiency: Among all linear unbiased estimators, the least squares estimators have the smallest variance, making them the best linear unbiased estimators (BLUE) under certain assumptions
  • The Gauss-Markov theorem states that the least squares estimators are BLUE when the errors are uncorrelated, have constant variance (homoscedasticity), and have a mean of zero

Normality and assumptions

• If the errors are normally distributed, the least squares estimators are also normally distributed, enabling the construction of confidence intervals and hypothesis tests based on the normal distribution
• The properties of the least squares estimators in the matrix context are derived using the assumptions of the classical linear regression model, such as:
  • Linearity
  • Independence
  • Homoscedasticity
  • Normality of errors
• Violations of these assumptions can affect the properties of the least squares estimators and may require the use of alternative estimation methods or remedial measures to address the issues
  • Example: Using robust standard errors or transforming variables to address heteroscedasticity