1. Chapter 4 Basic Estimation Techniques

2. Learning Objectives
Set up and interpret simple linear regression equations
Estimate intercept and slope parameters of a regression line using the method of least‐squares
Determine statistical significance using either t‐tests or p values associated with parameter estimates
Evaluate the “fit” of a regression equation to the data using the R2 statistic and test for statistical significance of the whole regression equation using an F‐test
Set up and interpret multiple regression models
Use linear regression techniques to estimate the parameters of two common nonlinear models: quadratic and log‐linear regression models

3. Basic Estimation Parameters Parameter estimation
The coefficients in an equation that determine the exact mathematical relation among the variables
Parameter estimation
The process of finding estimates of the numerical values of the parameters of an equation

4. Regression Analysis Regression analysis Dependent variable
A statistical technique for estimating the parameters of an equation and testing for statistical significance
Dependent variable
Variable whose variation is to be explained
Explanatory variables
Variables that are thought to cause the dependent variable to take on different values

5. Simple Linear Regression
True regression line relates dependent variable Y to one explanatory (or independent) variable X
Intercept parameter (a) gives value of Y where regression line crosses Y-axis (value of Y when X is zero)
Slope parameter (b) gives the change in Y associated with a one-unit change in X:

6. Simple Linear Regression
Regression line shows the average or expected value of Y for each level of X
True (or actual) underlying relation between Y and X is unknown to the researcher but is to be discovered by analyzing the sample data
Random error term
Unobservable term added to a regression model to capture the effects of all the minor, unpredictable factors that affect Y but cannot reasonably by included as explanatory variables

7. Fitting a Regression Line
Time series
A data set in which the data for the dependent and explanatory variables are collected over time for a single firm
Cross-sectional
A data set in which the data for the dependent and explanatory variables are collected from many different firms or industries at a given point in time

8. Fitting a Regression Line
Method of least squares
A method of estimating the parameters of a linear regression equation by finding the line that minimizes the sum of the squared distances from each sample data point to the sample regression line

9. Fitting a Regression Line
Parameter estimates are obtained by choosing values of a & b that minimize the sum of squared residuals
The residual is the difference between the actual and fitted values of Y: Yi – Ŷi
Equivalent to fitting a line through a scatter diagram of the sample data points

10. Fitting a Regression Line
The sample regression line is an estimate of the true (or population) regression line
Where and are least squares estimates of the true (population) parameters a and b

11. Sample Regression Line (Figure 4.2)
70,000
Sample regression line Ŝi = 11, A ei
Si = 60,000
Sales (dollars) •
60,000 •
50,000 •
Ŝi = 46,376
40,000 • •
30,000 •
20,000 •
10,000 A
2,000
4,000
6,000
8,000
10,000
Advertising expenditures (dollars)

12. Unbiased Estimators The estimates & do not generally
equal the true values of a & b
& are random variables computed using data from a random sample
The distribution of values the estimates might take is centered around the true value of the parameter
An estimator is unbiased if its average value (or expected value) is equal to the true value of the parameter

13. Relative Frequency Distribution* (Figure 4.3)1 1 2 3 4 5 6 7 8 9 10
*Also called a probability density function (pdf)

14. Statistical Significance
There is sufficient evidence from the sample to indicate that the true value of the coefficient is not zero
Hypothesis testing
A statistical technique for making a probabilistic statement about the true value of a parameter

15. Statistical Significance
Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e., b  0)
Even if b = 0, it is possible that the sample will produce an estimate that is different from zero
Test for statistical significance using t-tests or p-values

16. Statistical Significance
First determine the level of significance
Probability of finding a parameter estimate to be statistically different from zero when, in fact, it is zero
Probability of a Type I Error
1 – level of significance = level of confidence
Level of confidence is the probability of correctly failing to reject the true hypothesis that b = 0

17. Performing a t-Test t-ratio is computed as
Use t-table to choose critical t-value with n – k degrees of freedom for the chosen level of significance
n = number of observations
k = number of parameters estimated

18. Performing a t-Test t-statistic
Numerical value of the t-ratio
If the absolute value of t-statistic is greater than the critical t, the parameter estimate is statistically significant at the given level of significance

19. Using p-Values p-value gives exact level of significance
Treat as statistically significant only those parameter estimates with p-values smaller than the maximum acceptable significance level
p-value gives exact level of significance
Also the probability of finding significance when none exists

20. Coefficient of Determination
R2 measures the fraction of total variation in the dependent variable (Y) that is explained by the variation in X
Ranges from 0 to 1
High R2 indicates Y and X are highly correlated, and does not prove that Y and X are causally related

21. F-Test Used to test for significance of overall regression equation
Compare F-statistic to critical F-value from F-table
Two degrees of freedom, n – k & k – 1
Level of significance
If F-statistic exceeds the critical F, the regression equation overall is statistically significant at the specified level of significance

22. Multiple Regression Uses more than one explanatory variable
Coefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable, all else constant

23. Quadratic Regression Models
Use when curve fitting scatter plot is U-shaped or ∩-shaped
Y = a + bX + cX2
For linear transformation compute new variable Z = X2
Estimate Y = a + bX + cZ

24. Log-Linear Regression Models
Use when relation takes the form: Y = aXbZc
Transform by taking natural logarithms:
Percentage change in Y
Percentage change in X
b =
Percentage change in Y
Percentage change in Z
c =
b and c are elasticities

25. Summary
A simple linear regression model relates a dependent variable Y to a single explanatory variable X
The regression equation is correctly interpreted as providing the average value (expected value) of Y for a given value of X
Parameter estimates are obtained by choosing values of a and b that create the best-fitting line that passes through the scatter diagram of the sample data points
If the absolute value of the t-ratio is greater (less) than the critical t-value, then is (is not) statistically significant
Exact level of significance associated with a t-statistic is its p-value
A high R2 indicates Y and X are highly correlated and the data tightly fit the sample regression line

26. Summary
If the F-statistic exceeds the critical F-value, the regression equation is statistically significant
In multiple regression, the coefficients measure the change in Y associated with a one-unit change in that explanatory variable
Quadratic regression models are appropriate when the curve fitting the scatter plot is U-shaped or ∩-shaped (Y = a + bX + cX2)
Log-linear regression models are appropriate when the relation is in multiplicative exponential form (Y = aXbZc)
The equation is transformed by taking natural logarithms