10.4 Least squares estimation

Least squares estimation is a powerful statistical method for fitting linear regression models. It finds the best-fitting line by minimizing the sum of squared differences between observed and predicted values. This approach provides unbiased, efficient, and consistent estimates of model parameters.

The method relies on key assumptions like linearity, independence, and normality of errors. It allows for hypothesis testing, confidence intervals, and predictions. While sensitive to outliers and multicollinearity, alternatives like robust regression and regularization techniques can address these limitations.

Definition of least squares estimation

• Least squares estimation is a statistical method used to estimate the parameters of a linear regression model
• Aims to find the values of the parameters that minimize the sum of the squared differences between the observed values and the predicted values
• Commonly used in regression analysis to fit a line or curve to a set of data points

Principles of least squares estimation

• The goal is to find the best-fitting line or curve that minimizes the discrepancies between the observed data and the model predictions
• Assumes that the errors or residuals (differences between observed and predicted values) are normally distributed with a mean of zero and constant variance
• Provides a closed-form solution for estimating the parameters of a linear regression model

Minimizing sum of squared residuals

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• The objective is to minimize the sum of the squared residuals, where residuals are the differences between the observed values and the predicted values from the model
• Squaring the residuals ensures that positive and negative residuals do not cancel each other out and gives more weight to larger residuals
• The least squares estimates are obtained by finding the values of the parameters that minimize this sum of squared residuals

Derivation of least squares estimators

• The least squares estimators are derived by solving a set of normal equations that result from setting the partial derivatives of the sum of squared residuals with respect to each parameter equal to zero
• The solution to these normal equations provides the least squares estimates of the parameters

For simple linear regression

• In simple linear regression, there is only one independent variable (predictor) and the model is represented by the equation $y = \beta_0 + \beta_1x + \epsilon$
• The least squares estimators for the intercept ($\beta_0$) and slope ($\beta_1$) can be derived using the formulas:
  • $\hat{\beta}_1 = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}$
  • $\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}$

For multiple linear regression

• In multiple linear regression, there are multiple independent variables (predictors) and the model is represented by the equation $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_px_p + \epsilon$
• The least squares estimators for the parameters can be obtained using matrix algebra by solving the normal equations $(\mathbf{X}^T\mathbf{X})\hat{\boldsymbol{\beta}} = \mathbf{X}^T\mathbf{y}$, where $\mathbf{X}$ is the design matrix containing the predictor variables and $\mathbf{y}$ is the vector of observed response values

Properties of least squares estimators

• The least squares estimators possess several desirable statistical properties when the assumptions of the linear regression model are satisfied

Unbiasedness

• The least squares estimators are unbiased, meaning that their expected values are equal to the true values of the parameters
• On average, the least squares estimates will be centered around the true parameter values

Efficiency

• The least squares estimators are the most efficient among all unbiased linear estimators
• They have the smallest variance among all unbiased estimators, making them the best linear unbiased estimators (BLUE)

Consistency

• As the sample size increases, the least squares estimators converge in probability to the true parameter values
• With a large enough sample, the least squares estimates will be close to the true values of the parameters

Assumptions of least squares estimation

• The validity and optimality of the least squares estimators rely on several assumptions about the linear regression model

Linearity

• The relationship between the dependent variable and the independent variables is assumed to be linear
• The model can be represented by a linear equation with additive error terms

Independence

• The observations or errors are assumed to be independently distributed
• There should be no correlation or dependence between the residuals

Homoscedasticity

• The variance of the errors is assumed to be constant across all levels of the independent variables
• The spread of the residuals should be consistent throughout the range of the predictors

Normality

• The errors or residuals are assumed to follow a normal distribution with a mean of zero
• This assumption is necessary for valid hypothesis testing and confidence interval estimation

Interpretation of least squares estimates

• The least squares estimates provide information about the relationship between the dependent variable and the independent variables

Slope coefficients

• The slope coefficients ($\beta_1, \beta_2, ..., \beta_p$) represent the change in the dependent variable for a one-unit increase in the corresponding independent variable, holding all other variables constant
• They indicate the magnitude and direction of the effect of each predictor on the response variable

Intercept term

• The intercept term ($\beta_0$) represents the expected value of the dependent variable when all independent variables are zero
• It is the point where the regression line intersects the y-axis

Assessing goodness of fit

• The goodness of fit measures how well the least squares model fits the observed data

Coefficient of determination (R-squared)

• R-squared is the proportion of the variance in the dependent variable that is explained by the independent variables in the model
• It ranges from 0 to 1, with higher values indicating a better fit
• Calculated as the ratio of the explained sum of squares (ESS) to the total sum of squares (TSS): $R^2 = \frac{ESS}{TSS}$

Adjusted R-squared

• Adjusted R-squared is a modified version of R-squared that takes into account the number of predictors in the model
• It penalizes the addition of unnecessary predictors and provides a more conservative measure of model fit
• Useful for comparing models with different numbers of predictors

Hypothesis testing with least squares estimates

• Hypothesis testing allows us to assess the statistical significance of the least squares estimates and the overall model

T-tests for individual coefficients

• T-tests are used to test the significance of individual regression coefficients
• The null hypothesis is that the coefficient is equal to zero ($H_0: \beta_j = 0$), indicating no significant relationship between the predictor and the response
• The test statistic is calculated as $t = \frac{\hat{\beta}_j - 0}{SE(\hat{\beta}_j)}$, where $SE(\hat{\beta}_j)$ is the standard error of the coefficient estimate

F-test for overall model significance

• The F-test is used to assess the overall significance of the regression model
• The null hypothesis is that all regression coefficients (except the intercept) are simultaneously equal to zero ($H_0: \beta_1 = \beta_2 = ... = \beta_p = 0$)
• The test statistic is calculated as $F = \frac{MSR}{MSE}$, where $MSR$ is the mean square regression and $MSE$ is the mean square error

Confidence intervals for least squares estimates

• Confidence intervals provide a range of plausible values for the true regression coefficients
• They are constructed using the least squares estimates and their standard errors
• A 95% confidence interval for a coefficient $\beta_j$ is given by $\hat{\beta}_j \pm t_{\alpha/2, n-p-1} \cdot SE(\hat{\beta}_j)$, where $t_{\alpha/2, n-p-1}$ is the critical value from the t-distribution with $n-p-1$ degrees of freedom

Prediction using least squares models

• Least squares models can be used to make predictions for new observations based on the estimated regression equation

Confidence intervals for predictions

• Confidence intervals for predictions provide a range of plausible values for the mean response at a given set of predictor values
• They take into account the uncertainty in the estimated regression coefficients and the variability of the data

Prediction intervals

• Prediction intervals provide a range of plausible values for an individual future observation at a given set of predictor values
• They are wider than confidence intervals because they account for both the uncertainty in the estimated coefficients and the inherent variability of individual observations

Limitations of least squares estimation

• Least squares estimation has some limitations and potential issues that should be considered

Sensitivity to outliers

• Least squares estimates can be heavily influenced by outliers or extreme observations
• Outliers can pull the regression line towards them, distorting the estimates and reducing the model's robustness

Multicollinearity

• Multicollinearity occurs when there is a high correlation among the independent variables in the model
• It can lead to unstable and unreliable estimates of the regression coefficients
• Multicollinearity can make it difficult to interpret the individual effects of the predictors on the response variable

Alternatives to least squares estimation

• There are alternative estimation methods that can be used when the assumptions of least squares estimation are violated or when dealing with specific challenges

Robust regression methods

• Robust regression methods, such as M-estimation and least absolute deviation (LAD) regression, are less sensitive to outliers compared to least squares estimation
• They minimize different loss functions that are less affected by extreme observations

Ridge regression

• Ridge regression is a regularization technique used when multicollinearity is present
• It adds a penalty term to the least squares objective function, shrinking the coefficient estimates towards zero
• The penalty term is controlled by a tuning parameter, which balances the trade-off between fitting the data and reducing the complexity of the model

Lasso regression

• Lasso (Least Absolute Shrinkage and Selection Operator) regression is another regularization technique that can handle multicollinearity and perform variable selection
• It adds an L1 penalty term to the least squares objective function, which can shrink some coefficient estimates exactly to zero
• Lasso regression can effectively identify and exclude irrelevant predictors from the model