1 / 19

Linear Regression

Linear Regression. A method of calculating a linear equation for the relationship between two or more variables using multiple data points. Linear Regression. General form: Y = a + bX + e “Y” is the dependent variable “X” is the explanatory variable “a” is the intercept parameter

binta
Download Presentation

Linear Regression

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Linear Regression • A method of calculating a linear equation for the relationship between two or more variables using multiple data points.

  2. Linear Regression • General form: Y = a + bX + e • “Y” is the dependent variable • “X” is the explanatory variable • “a” is the intercept parameter • “b” is the slope parameter • “e” is an error term or residual

  3. Regression Results • Y and X come from data • A computer program calculates estimates of a and b • e is the difference between a + bX and the actual value of Y corresponding to X • OLS estimates of a and b minimize the sum of the squared residuals ∑e2 • “OLS” is Ordinary Least Squares

  4. The Regression Line and the Residual

  5. Electricity Demand Example • Data for residential customers in U. S. states • Y is millions of kilo-watt-hours sold • X is population • Other data include per capita income, price of electricity (cents/kwh) and price of natural gas

  6. Variable Means 2004

  7. Variable Means 2012

  8. Excel Regression: Milkwh

  9. Actual vs. Predicted Milkwh

  10. Useful Statistics and Tests • t-statistic: is estimated coefficient significantly different from zero? • Coefficient of determination or R-square: % variation explained • F-statistic: statistical significance of the entire regression equation; OR is the R-square significantly different from zero? • Find these on the Excel regression output.

  11. Confidence and Significance Levels • 99% Confidence = 1% Significance • P-value of 0.01 or less • 95% Confidence = 5% Significance • P-value of 0.05 or less • 90% Confidence = 10% Significance • P-value of 0.10 or less • Smaller Significance Levels Are Better • Find P-values for t and F statistics on Excel regression output

  12. Multiple & Nonlinear Regression • Multiple Regression • Y= a + bX + cW + dZ • Nonlinear Regression • Quadratic: Y = a + bX + cX2 • Log-Linear: Y = aXbZc • Or Ln Y = (ln a) + b(ln X) + c(ln Z)

  13. Multiple Regression: Milkwh

  14. Quadratic Regression: Milkwh

  15. Log Linear Regression: LnMilkwh

  16. Demand Regression Project DATA: ElecDemandData2012.xls under Project Materials on D2L 1. Using the data file above, run a linear regression of Dependent Variable: Milkwh Explanatory Variables: Pop, Pkwh, PGas, Income. Which coefficients (including the constant) are statistically significant at the 10% level or better? Which are not significant? How much of the variation in the dependent variable is explained by the estimated equation? Is the equation as a whole statistically significant? At what level?

  17. Finding the Marginal Revenue Equation: Overview • Evaluate estimated demand at means of all explanatory variables except price • Calculate average effect of non-price variables to get demand equation in this form • Q = A - b(P) • Rearrange to find Inverse Demand equation • P = (A/b) - (1/b)Q • MR has twice the slope of inverse demand • MR = (A/b) – (2/b)Q • The end result is an equation, not a number

  18. Finding the Marginal Revenue Equation: Example • Write your regression equation in this form • Milkwh = 11,000 – 3600Pkwh + 0.0041Pop + 2150PGas • 11,000 is the intercept or constant coefficient • 3600, 0.0041, and 2150 are estimated coefficients • These are made-up numbers for this example • Use the mean values of the non-electricity-price variables • Pop=5,756,577 PGas=11.4 • Substitute into your regression equation and simplify • Milkwh = 11,000 – 3600Pkwh + 0.0041(5,756,577) + 2150(11.4) • Milkwh= [11000 + 23,601.97 + 24,510] – 3600(Pkwh) • Milkwh = 59,111.97 – 3600(Pkwh) • This is Q = A – bP from the previous slide

  19. Finding Marginal Revenue Example, Continued • Milkwh = 59,111.97 – 3600(Pkwh) • From end of previous slide • Rearrange to find Inverse Demand • P = (A/b) - (1/b)Q = (A – Q)/b • Pkwh = (59,111.97 – Milkwh)/(3600) • Pkwh = 16.42 – 0.00028(Milkwh) • This is the inverse demand equation • Marginal Revenue has twice the slope • MR = 16.42 – 0.00056(Milkwh)

More Related