Least-Squares Regression

Definition

Least-squares regression is a statistical method used to determine the best-fitting linear equation that describes the relationship between a dependent variable and one or more independent variables. It aims to minimize the sum of the squared differences between the observed values and the predicted values from the regression model.

5 Must Know Facts For Your Next Test

1. Least-squares regression is the most common method used to fit a linear regression model, as it provides the best linear unbiased estimator (BLUE) of the regression coefficients.
2. The goal of least-squares regression is to find the line of best fit that minimizes the sum of the squared vertical distances between the observed data points and the predicted values from the regression line.
3. Least-squares regression assumes that the errors (residuals) are normally distributed, have constant variance, and are independent of each other.
4. The regression coefficients estimated by the least-squares method represent the average change in the dependent variable associated with a one-unit change in the independent variable, holding all other variables constant.
5. Least-squares regression can be used for both simple linear regression (one independent variable) and multiple linear regression (multiple independent variables).

Review Questions

• Explain the purpose of least-squares regression and how it differs from other regression methods.
  • The purpose of least-squares regression is to find the linear equation that best fits the relationship between a dependent variable and one or more independent variables. It does this by minimizing the sum of the squared differences between the observed values and the predicted values from the regression model. This differs from other regression methods, such as ordinary least squares (OLS) or maximum likelihood estimation, which may have different objectives or assumptions. Least-squares regression is the most commonly used method because it provides the best linear unbiased estimator (BLUE) of the regression coefficients, meaning the estimates have the lowest variance among all unbiased linear estimators.
• Describe the assumptions required for least-squares regression and the implications if these assumptions are violated.
  • Least-squares regression assumes that the errors (residuals) are normally distributed, have constant variance, and are independent of each other. If these assumptions are violated, the regression coefficients may be biased or inefficient, and the statistical inferences (such as confidence intervals and hypothesis tests) may not be valid. For example, if the errors have non-constant variance (heteroscedasticity), the standard errors of the regression coefficients will be biased, leading to incorrect conclusions about the statistical significance of the independent variables. Violations of the assumptions can be detected through diagnostic tests and may require the use of alternative regression methods, such as weighted least squares or robust regression, to address the issues.
• Explain how the least-squares regression line can be used to make predictions about the dependent variable, and discuss the concept of prediction intervals.
  • The least-squares regression line can be used to make predictions about the dependent variable by plugging in values for the independent variable(s) and calculating the corresponding predicted value. However, these predictions are subject to uncertainty, as the actual observed value may differ from the predicted value due to the presence of residuals. Prediction intervals provide a range of values within which the true (unobserved) value of the dependent variable is expected to fall, with a specified level of confidence. The width of the prediction interval depends on the variability of the residuals and the distance of the new observation from the mean of the independent variable(s). Prediction intervals are wider than confidence intervals for the mean response, as they account for both the uncertainty in the regression coefficients and the inherent variability in the data.

Related terms

Linear Regression:

A statistical technique used to model the linear relationship between a dependent variable and one or more independent variables.

Regression Coefficients: The estimated parameters in a regression model that represent the change in the dependent variable associated with a one-unit change in the independent variable, holding all other variables constant.

Residuals:

The differences between the observed values and the predicted values from a regression model, which represent the unexplained variation in the dependent variable.