1. Chapter 10
Simple Regression

2. Null Hypothesis
The analysis of business and economic processes makes extensive use of relationships between variables.

3. Correlation Analysis
The correlation coefficient is a quantitative measure of the strength of the linear relationship between two variables.

4. Correlation Analysis
The sample correlation coefficient:
where:

5. follows a Student’s t Distribution with (n-2) degrees of freedom
Correlation Analysis
The null hypothesis of no linear association:
where the random variable:
follows a Student’s t Distribution with (n-2) degrees of freedom

6. Tests for Zero Population Correlation
Let r be the sample correlation coefficient, calculated from a random sample of n pairs of observation from a joint normal distribution. The following tests of the null hypothesis
have a significance value :
1. To test H0 against the alternative
the decision rule is

7. Tests for Zero Population Correlation (continued)
2. To test H0 against the alternative
the decision rule is

8. Tests for Zero Population Correlation (continued)
3. To test H0 against the two-sided alternative
the decision rule is
Here, t n-2, is the number for which
Where the random variable tn-2 follows a Student’s t distribution with (n – 2) degrees of freedom.

9. Linear Regression Model (Example 10.2)

10. Linear Regression Model (Figure 10.1)

11. Linear Regression Model
LINEAR REGRESSION POPULATION EQUATION MODEL
Where 0 and 1 are the population model coefficients and  is a random error term.

12. Linear Regression Outcomes
Linear regression provides two important results:
Predicted values of the dependent or endogenous variable as a function of an independent or exogenous variable.
Estimated marginal change in the endogenous variable that results from a one unit change in the independent or exogenous variable.

13. Least Squares Procedure
The Least-squares procedure obtains estimates of the linear equation coefficients b0 and b1, in the model
by minimizing the sum of the squared residuals ei
This results in a procedure stated as
Choose b0 and b1 so that the quantity
is minimized. We use differential calculus to obtain the coefficient estimators that minimize SSE..

14. Least-Squares Derived Coefficient Estimators
The slope coefficient estimator is
And the constant or intercept indicator is
We also note that the regression line always goes through the mean X, Y.

15. Standard Assumptions for the Linear Regression Model
The following assumptions are used to make inferences about the population linear model by using the estimated coefficients:
The x’s are fixed numbers, or they are realizations of random variable, X that are independent of the error terms, i’s. In the latter case, inference is carried out conditionally on the observed values of the x’s.
The error terms are random variables with mean 0 and the same variance, 2. The later is called homoscedasticity or uniform variance.
The random error terms, I, are not correlated with one another, so that

16. Regression Analysis for Retail Sales Analysis (Figure 10.5)
The regression equation is
Y Retail Sales = X Income b0 b1

17. Analysis of Variance
The total variability in a regression analysis, SST, can be partitioned into a component explained by the regression, SSR, and a component due to unexplained error, SSE
With the components defined as,
Total sum of squares
Error sum of squares
Regression sum of squares

18. Regression Analysis for Retail Sales Analysis (Figure 10.7)
The regression equation is
Y Retail Sales = X Income

19. Coefficient of Determination, R2
The Coefficient of Determination for a regression equation is defined as
This quantity varies from 0 to 1 and higher values indicate a better regression. Caution should be used in making general interpretations of R2 because a high value can result from either a small SSE or a large SST or both.

20. Correlation and R2
The multiple coefficient of determination, R2, for a simple regression is equal to the simple correlation squared:

21. Estimation of Model Error Variance
The quantity SSE is a measure of the total squared deviation about the estimated regression line, and ei is the residual. An estimator for the variance of the population model error is
Division by n – 2 instead of n – 1 results because the simple regression model uses two estimated parameters, b0 and b1, instead of one.

22. Sampling Distribution of the Least Squares Coefficient Estimator
If the standard least squares assumptions hold, then b1 is an unbiased estimator of 1 and has a population variance
and an unbiased sample variance estimator

23. Basis for Inference About the Population Regression Slope
Let 1 be a population regression slope and b1 its least squares estimate based on n pairs of sample observations. Then, if the standard regression assumptions hold and it can also be assumed that the errors i are normally distributed, the random variable
is distributed as Student’s t with (n – 2) degrees of freedom. In addition the central limit theorem enables us to conclude that this result is approximately valid for a wide range of non-normal distributions and large sample sizes, n.

24. Excel Output for Retail Sales Model (Figure 10.9)
The regression equation is
Y Retail Sales = X Income se
SSR
SSE
SST
MSR
MSE b0 b1
sb1
tb1

25. Tests of the Population Regression Slope
If the regression errors i are normally distributed and the standard least squares assumptions hold (or if the distribution of b1 is approximately normal), the following tests have significance value :
To test either null hypothesis
against the alternative
the decision rule is

26. Tests of the Population Regression Slope (continued)
2. To test either null hypothesis
against the alternative
the decision rule is

27. Tests of the Population Regression Slope (continued)
3. To test the null hypothesis
Against the two-sided alternative
the decision rule is

28. Confidence Intervals for the Population Regression Slope 1
If the regression errors i , are normally distributed and the standard regression assumptions hold, a 100(1 - )% confidence interval for the population regression slope 1 is given by
Where t(n – 2, /2) is the number for which
And the random variable t(n – 2) follows a Student’s t distribution with (n – 2) degrees of freedom.

29. F test for Simple Regression Coefficient
We can test the hypothesis
against the alternative
By using the F statistic
The decision rule is
We can also show that the F statistic is
For any simple regression analysis.

30. Key Words Analysis of Variance
Assumptions for the Least Squares Coefficient Estimators
Basis for Inference About the Population Regression Slope
Coefficient of Determination, R2
Confidence Intervals for Predictions
Confidence Intervals for the Population Regression Slope b1
Correlation and R2
Estimation of Model Error Variance
F test for Simple Regression Coefficient
Least-Squares Procedure
Linear Regression Outcomes

31. Key Words (continued) Linear Regression Population Equation Model
Population Model
Sampling Distribution of the Least Squares Coefficient Estimator
Tests for Zero Population Correlation
Tests of the Population Regression Slope