Demand Estimation - Regression Flashcards
What is regression analysis?
Regression analysis is a statistical technique used to estimate the relationships among variables, often to determine the demand function using data on quantity, price, income, and other variables.
Regression Line and Least Squares Regression
What is it?
Formula
(3)
The econometrician believes that, on average, there is a linear relation between Y and X, but there is also some random variation in the relationship.
Mathematically, the true (or population) regression model
𝑌=𝑎+𝑏𝑋+𝑒
𝑎 unknown population intercept parameter.
𝑏 unknown population slope parameter.
𝑒 random error term with mean zero and standard deviation 𝜎.
fitting the best line through the points
Regression Line and Least Squares Regression
Picture
Evaluating Statistical Significance
Why may the true line be different?
What do we use for t-statistics?
What do we use for p-value? (3)
Regression lines are usually estimated using a sample of data points so ‘true’ line may be a bit different. How can we be sure that a slope really is non-zero?
t-statistics (crude rule of thumb)
- When |𝑡|>2, we are 95 percent confident the true parameter in the regression is not zero.
p-value (a precise indication of confidence)
When 𝑝<0.05,
- significant at 5% (most popular test) = 95% confident that coefficient is not zero.
- Confidence level = (1 – P-value) so the smaller the p-value, the better the confidence level.
- If p-value is 0.03 then coefficient is significant at a 3% level and we are 97% confident that the coefficient is different from zero.
What is R-Square in regression analysis?
R-Square, or the coefficient of determination, measures the proportion of the variation in the dependent variable that is explained by the regression model. It ranges from 0 (no association) to 1 (perfect fit).
What does the F-statistic indicate in regression analysis?
The F-statistic assesses whether the regression model has statistically significant explanatory power. If the significance F is less than 0.05, the model is considered significant.
What is multiple regression?
Multiple regression is a regression technique that involves more than one independent variable. It can be used to model nonlinear relationships and interactions between variables.
How do you interpret the regression results?
By examining the R-square, F-statistic, and the significance of individual coefficients. For example, a high R-square indicates a good fit, and significant coefficients suggest meaningful relationships between variables.
Regression for Nonlinear Functions and Multiple Regression
Formulas
An example
Interpret the regression results (7)
- The R-square of 0.79: the regression model explains 79% indicates that the regression explains 79 percent of the variation Y variable (i.e. demand).
- The F-statistic suggests that the regression is significant at the 1.82 percent level, so the manager can be reasonably confident that the equation has explanatory power.
- All the estimated parameters are statistically significant at the 5 percent level, except for the coefficient of advertising.
- Thus, it does not appear that advertising has a statistically significant effect on the demand for the rental units.
- Distance from campus appears to be a very significant negative determinant of the demand for apartments. The t-statistic for this coefficient is in excess of 4! For every mile an apartment is away from campus, the agent loses 5.78 customers, on average.
- The coefficient of price is -0.14 and significant at 5%. A £100 increase in price reduces the quantity demanded at an apartment building by 14 units.
Since the agent cannot relocate its apartment closer to campus, and the advertising does not have a statistically impact on units rented, it would appear the agent can do to maximise revenue is to alter rents (i.e. price).
