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514The Linear Model and Its Parameters
515What Makes a 'Best Fit' Line?
516Residuals: The Distance from Prediction
517The Least Squares Criterion
518Deriving the Least Squares Estimators
519Computing β₁: The Slope Estimate
520Computing β₀: The Intercept Estimate
521Properties of Least Squares Estimators
522Implementing Least Squares from Scratch
523The Meaning of the Intercept
524The Meaning of the Slope
525Units and Scale in Interpretation
526When the Intercept Has No Meaning
527Positive, Negative, and Zero Slopes
528Standardized vs Unstandardized Coefficients
529Practical vs Statistical Significance
530Communicating Results to Non-Technical Audiences
531What is R-Squared?
532Calculating R-Squared
533Interpreting R-Squared Values
534R-Squared vs Correlation Squared
535Limitations of R-Squared
536Residual Standard Error (RSE)
537When R-Squared is Not Enough
538What Are Fitted Values?
539What Are Residuals?
540The Residual Formula
541Properties of Residuals
542Computing Fitted Values and Residuals
543Residuals as Unexplained Variation
544The Role of Residuals in Diagnostics
545Extracting Residuals and Fitted Values in Python
546The Five Core Assumptions of Linear Regression
547Linearity: The Relationship Must Be Linear
548Independence of Observations
549Homoscedasticity: Constant Variance of Residuals
550Normality of Residuals
551No Perfect Multicollinearity in Simple Regression
552Zero Conditional Mean of Errors
553Exogeneity: X Must Be Independent of Errors
554Consequences of Violating Assumptions
555When Regression Is and Isn't Appropriate

514The Linear Model and Its Parameters
515What Makes a 'Best Fit' Line?
516Residuals: The Distance from Prediction
517The Least Squares Criterion
518Deriving the Least Squares Estimators
519Computing β₁: The Slope Estimate
520Computing β₀: The Intercept Estimate
521Properties of Least Squares Estimators
522Implementing Least Squares from Scratch
523The Meaning of the Intercept
524The Meaning of the Slope
525Units and Scale in Interpretation
526When the Intercept Has No Meaning
527Positive, Negative, and Zero Slopes
528Standardized vs Unstandardized Coefficients
529Practical vs Statistical Significance
530Communicating Results to Non-Technical Audiences
531What is R-Squared?
532Calculating R-Squared
533Interpreting R-Squared Values
534R-Squared vs Correlation Squared
535Limitations of R-Squared
536Residual Standard Error (RSE)
537When R-Squared is Not Enough
538What Are Fitted Values?
539What Are Residuals?
540The Residual Formula
541Properties of Residuals
542Computing Fitted Values and Residuals
543Residuals as Unexplained Variation
544The Role of Residuals in Diagnostics
545Extracting Residuals and Fitted Values in Python
546The Five Core Assumptions of Linear Regression
547Linearity: The Relationship Must Be Linear
548Independence of Observations
549Homoscedasticity: Constant Variance of Residuals
550Normality of Residuals
551No Perfect Multicollinearity in Simple Regression
552Zero Conditional Mean of Errors
553Exogeneity: X Must Be Independent of Errors
554Consequences of Violating Assumptions
555When Regression Is and Isn't Appropriate

Lesson 518 of 2,145·14. Simple Linear RegressionPro lesson

Deriving the Least Squares Estimators

Walk through the mathematical derivation of the slope and intercept formulas using calculus and the least squares criterion.

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