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Linear regression theory
- 2. What &Why 1 What is Regression? Formulation of a functional relationship between a set of Independent or Explanatory variables (X’s) with a Dependent or Response variable (Y). Y = f(X) Why Regression? Knowledge of Y is crucial for decision making. • Will he/she buy or not? • Shall I offer him/her the loan or not? • ……… X is available at the time of decision making and is related to Y, thus making it possible to have a prediction of Y.
- 3. 2 Types of Regression Y Continuous E.g., SalesVolume, Claim Amount, % of sales growth etc. Binary (0/1) E.g., Buy/No-Buy, Survive/Not- Survive,Win/Loss etc Ordinary Least Square (OLS) Regression Logistic Regression
- 4. • Regression analysis is used to: • Predict the value of a dependent variable based on the value of at least one independent variable • Explain the impact of changes in an independent variable on the dependent variable • Dependent variable: the variable we wish to explain, usually denoted by Y. • Independent variable: the variable used to explain the dependent variable. Usually denoted by X. 3 Intro to RegressionAnalysis
- 5. 4 Regression Example Predict the fitness of a person based on one or more parameters.
- 7. • Only one independent variable, x • Relationship between x and y is described by a linear function • Changes in y are assumed to be caused by changes in x 6 Simple Linear Regression Model
- 8. 7 Assumptions for Simple Linear Regression E(ε) = 0
- 9. 8 Assumptions for Multiple Regression
- 10. 9 Assumptions for Multiple Regression ݅ 2 ߝ ଶ
- 11. 10 i]j0,)ε[E(ε ji ≠= Assumptions for Multiple Regression
- 13. 12 Simple Linear Regression Model
- 14. 13 Beta Zero
- 18. 17 The Simple Linear Regression Model
- 19. 18 The Multiple Linear Regression Model
- 20. 19 Model for Multiple Regression
- 21. 20 Positive Linear Relationship Negative Linear Relationship No Relationship Relationship NOT Linear Types of Regression Relationships
- 22. 21 Unknown Relationship Population Random Sample Y Xi i i= + +β β ε0 1 ☺ ☺ ☺ ☺ ☺ ☺ ☺ Population & Sample Regression Models
- 23. 22 PredictedValue ofY for Xi Intercept = β0 Random Error for this x value Y X uXββY 10 ++= xi Slope = β1 ui Individual person's marks Population Linear Regression
- 24. 23 Linear component Population y intercept Population Slope Coefficient Random Error term, or residual Dependent Variable Independent Variable Random Error component uXββY 10 ++= But can we actually get this equation? If yes what all information we will need? Population Regression Function
- 25. 24 PredictedValue ofY for Xi Intercept = β0 Random Error for this x value Y Xxi Slope = β1 exbby 10 ++= ei ObservedValue of y for xi Sample Regression Function
- 26. 25 exbby 10i ++= Estimate of the regression intercept Estimate of the regression slope Independent variable Error term Notice the similarity with the Population Regression Function Can we do something of the error term? Sample Regression Function
- 27. • Represents the influence of all the variable which we have not accounted for in the equation • It represents the difference between the actual y values as compared the predicted y values from the Sample Regression Line • Wouldn't it be good if we were able to reduce this error term? • By the way - what are we trying to achieve by Sample Regression? 26 The ErrorTerm (Residual)
- 28. 27 HowWell A Model Fits the Data
- 29. 28 Comparing the Regression Model to a Baseline Model
- 30. 29 Comparing the Regression Model to a Baseline Model
- 31. • The sum of the residuals from the least squares regression line is zero. • The sum of the squared residuals is a minimum. Minimize( ) • The simple regression line always passes through the mean of the y variable and the mean of the x variable • The least squares coefficients are unbiased estimates of β0 and β1 30 0)ˆ( =−∑ yy 2 )ˆ( yy∑ − OLS Regression Properties
- 32. • Parameter Instability - This happens in situations where correlations change over a period of time.This is very common in financial markets where economic, tax, regulatory, and political factors change frequently. • Public knowledge of a specific regression relation may cause a large number of people to react in a similar fashion towards the variables, negating its future usefulness. • If any of the regression assumptions are violated, predicted dependent variables and hypothesis tests will not hold valid. 31 Limitations of RegressionAnalysis
- 33. • In simple linear regression, the dependent variable was assumed to be dependent on only one variable (independent variable) • In General Multiple Linear Regression model, the dependent variable derives its value from two or more than two variable. • General Multiple Linear Regression model take the following form: where: Yi = ith observation of dependent variableY Xki = ith observation of kth independent variable X b0 = intercept term bk = slope coefficient of kth independent variable εi = error term of ith observation n = number of observations k = total number of independent variables 32 ikikiii XbXbXbbY ε+++++= .........22110 General Multiple Linear Regression Model
- 34. • As we calculated the intercept and the slope coefficient in case of simple linear regression by minimizing the sum of squared errors, similarly we estimate the intercept and slope coefficient in multiple linear regression. • Sum of Squared Errors is minimized and the slope coefficient is estimated. • The resultant estimated equation becomes: • Now the error in the ith observation can be written as: 33 ∑= n i i 1 2 ε kikiii XbXbXbbY ∧∧∧∧∧ ++++= .........22110 ++++−=−= ∧∧∧∧∧ kikiiiiii XbXbXbbYYY .........22110ε Estimated Regression Equation
- 35. 34 Assumptions of Multiple Regression Model • There exists a linear relationship between the dependent and independent variables. • The expected value of the error term, conditional on the independent variables is zero. • The error terms are homoskedastic, i.e. the variance of the error terms is constant for all the observations. • The expected value of the product of error terms is always zero, which implies that the error terms are uncorrelated with each other. • The error term is normally distributed. • The independent variables doesn't have any linear relationships between each other.
- 36. Thank you!