7.2 Least squares estimation and interpretation of coefficients

Linear regression is a powerful tool for understanding relationships between variables. Least squares estimation finds the best-fitting line by minimizing the sum of squared residuals, providing unbiased estimators of regression coefficients under certain assumptions.

Interpreting regression coefficients is crucial for making sense of the model. The slope coefficient represents the change in the dependent variable for a one-unit increase in the independent variable, while the intercept represents the expected value when all independent variables are zero.

Least squares estimation in linear regression

Minimizing squared residuals

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• Least squares estimation finds the best-fitting line for data points by minimizing the sum of squared residuals
• Calculates vertical distance between each data point and proposed regression line
• Squares these distances and sums them to find total squared error
• Seeks line resulting in smallest possible sum of squared residuals, considered "best fit"
• Provides unbiased estimators of regression coefficients under assumptions of linearity, independence, and homoscedasticity
• Assumes errors (residuals) are normally distributed with mean of zero and constant variance
• Minimizes overall prediction error and maximizes explanatory power of model

Application to linear regression

• Used to determine optimal values for slope and intercept coefficients of regression equation
• Slope coefficient (β1) represents change in dependent variable (Y) for one-unit increase in independent variable (X)
• Intercept coefficient (β0) represents expected value of dependent variable when all independent variables equal zero
• In simple linear regression, finds line equation Y = β0 + β1X that best fits data points
• For multiple regression, extends to find optimal coefficients for multiple independent variables
• Utilizes calculus to find minimum of sum of squared residuals function
• Results in closed-form solution for coefficient estimates in matrix form: β = (X'X)^(-1)X'Y

Computational methods

• Modern statistical software automates least squares calculations
• Iterative algorithms (gradient descent) often used for large datasets or complex models
• Regularization techniques (ridge regression, lasso) modify least squares to prevent overfitting
• Weighted least squares adjusts for heteroscedasticity by giving less weight to observations with higher variance
• Robust regression methods (M-estimation) reduce influence of outliers on coefficient estimates
• Cross-validation techniques assess model performance and generalizability

Interpretation of regression coefficients

Understanding slope coefficients

• Slope coefficient (β1) represents change in dependent variable (Y) for one-unit increase in independent variable (X), holding other variables constant
• Sign of slope coefficient indicates direction of relationship between X and Y (positive or negative)
• Magnitude of slope coefficient indicates strength of relationship between X and Y
• Interpret within range of observed data to avoid extrapolation beyond scope of model
• In multiple regression, each slope coefficient represents partial effect of corresponding independent variable, controlling for effects of other variables
• Standardized coefficients allow comparison of relative importance of predictors measured on different scales
• Interaction terms represent how effect of one variable depends on level of another variable

Interpreting the intercept

• Intercept coefficient (β0) represents expected value of dependent variable when all independent variables equal zero
• May not always have meaningful interpretation, especially if zero values for independent variables are not possible or realistic
• In some cases, centering independent variables (subtracting mean) can make intercept more interpretable
• Useful for making predictions when all independent variables are at their reference levels
• In logistic regression, transformed intercept represents log-odds when all predictors are zero

Contextual considerations

• Interpreting coefficients requires consideration of units of measurement for both dependent and independent variables
• Economic interpretation often involves elasticities or marginal effects
• In time series analysis, coefficients may represent short-term or long-term effects
• Categorical variables require interpretation relative to reference category
• Non-linear transformations (log, polynomial) affect interpretation of coefficients
• Coefficients in generalized linear models (logistic, Poisson) require specific interpretations based on link function

Standard errors of regression coefficients

Calculating standard errors

• Standard error of slope (SE(β1)) calculated using formula: $SE(β1) = s / √(Σ(xi - x̄)²)$, where s is standard error of estimate and xi are individual X values
• Standard error of intercept (SE(β0)) calculated using formula: $SE(β0) = s * √((1/n) + (x̄² / Σ(xi - x̄)²))$, where n is sample size
• For multiple regression, standard errors derived from variance-covariance matrix of coefficient estimates
• Bootstrap methods provide alternative approach to estimating standard errors, especially useful for complex models
• Heteroscedasticity-consistent standard errors (White's standard errors) adjust for non-constant variance

Interpreting standard errors

• Measure precision of estimated coefficients
• Smaller standard errors indicate more precise estimates, larger suggest greater uncertainty
• Used to construct confidence intervals for coefficients
• Typical confidence interval: coefficient ± (critical value * standard error)
• Ratio of coefficient to standard error (t-statistic) tests statistical significance of coefficient
• P-values derived from t-statistics indicate probability of observing coefficient as extreme under null hypothesis
• Standard errors help assess reliability of estimated relationships and overall fit of regression model

Applications in hypothesis testing

• Null hypothesis typically assumes coefficient equals zero (no effect)
• Test statistic (t or z) calculated as coefficient divided by its standard error
• Compare test statistic to critical value from t-distribution (or normal distribution for large samples)
• Confidence intervals that do not include zero indicate statistically significant coefficients
• Multiple testing adjustments (Bonferroni, false discovery rate) control for increased Type I error rate
• Power analysis uses standard errors to determine sample size needed to detect effects of given magnitude