1. Analysis of Economic Data
Dr. Ka-fu Wong
ECON1003
Analysis of Economic Data

2. Multiple Regression and Correlation Analysis
Chapter Twelve
Multiple Regression and Correlation Analysis
GOALS
Describe the relationship between two or more independent variables and the dependent variable using a multiple regression equation.
Compute and interpret the multiple standard error of estimate and the coefficient of determination.
Interpret a correlation matrix.
Setup and interpret an ANOVA table.
Conduct a test of hypothesis to determine if any of the set of regression coefficients differ from zero.
Conduct a test of hypothesis on each of the regression coefficients. l

3. Multiple Regression Analysis
For two independent variables, the general form of the multiple regression equation is:
Yi = E(Yi | X1i , X2i )+ ei
= b0 + b1 X1i + b2 X2i + ei
X1i and X2i are the i-th observation of independent variables.
b0 is the Y-intercept.
b1 is the net change in Y for each unit change in X1 holding X2 constant. It is called a partial regression coefficient, a net regression coefficient, or just a regression coefficient.

4. Visualize the multiple linear regression in a plot
The simple linear regression model
allows for one independent variable, “x”
y =b0 + b1x + e y
y = b0 + b1x
Note how the straight line
becomes a plain, and...
y = b0 + b1x1+ b2x2 X 1
The multiple linear regression model
allows for more than one independent variable.
Y = b0 + b1x1 + b2x2 + e X2

5. Visualize the multiple non-linear regression in a ploty
y= b0+ b1x2 b0 X 1

6. Visualize the multiple non-linear regression in a ploty
y= b0+ b1x2 b0 X 1
y = b0 + b1x12 + b2x2
… a parabola becomes a parabolic surface X2

7. Multiple Regression Analysis
The general multiple regression with k independent variables is given by:
Yi = E(Yi |X1i , X2i , …, Xki) + ei
= b0 + b1 X1i + b2 X2i +…+ bk Xki+ ei
When k>2, it is impossible to visualize the regression equation in a plot.
The least squares criterion is used to develop an estimation of this equation.
Because determining b1, b2, etc. is very tedious, a software package such as Excel or other statistical software is recommended to estimate them.

8. Choosing the line that fits best Ordinary Least Squares (OLS) Principle
Straight lines can be describe generally by Yi = b0 + b1 X1i + b2 X2i +…+ bk Xki
Finding the best line with smallest sum of squared difference is the same as
Let b0* , b1* , b2* … bk* be the solution of the above problem.
Y* = b0* + b1*X1+ b2*X2+ …+bk*Xk is known as the “average predicted value” (or simply “predicted value”) of y for any vector of (X1, X2, …, Xk).

9. Coefficient estimates from the ordinary least squares (OLS) principle
Solving the minimization problem implies the first order conditions:

10. Coefficient estimates from the ordinary least squares (OLS) principle
Solving the first order conditions implies
The solution of b0* , b1* , b2* … bk*
Y* = b0* + b1*X1+ b2*X2+ …+bk*Xk is known as the “average predicted value” (or simply “predicted value”) of y for any vector of (X1, X2, …, Xk).

11. Multiple Linear Regression Equations
An estimation of the coefficients is too complicated by hand, let me use some computational software packages, such as Excel. 16

12. Interpretation of Estimated Coefficients
Slope (k)
Estimated Y Changes by k for Each 1 Unit Increase in Xk Holding All Other Variables Constant
Note Yi = E(Yi |X1i , X2i , …, Xki) + ei
= b0 + b1 X1i + b2 X2i +…+ bk Xki+ ei
E(Yi | X1i , X2i , …, Xki=g+1) - E(Yi | X1i , X2i , …, Xki=g)
= b0 + b1 X1i + b2 X2i +…+ bk (g+1) – [b0 + b1 X1i + b2 X2i +…+ bk g ]
= k
Example: If 1 = 2, then sales (Y) is expected to increase by 2 for each 1 unit increase in advertising (X1) given the number of sales rep’s (X2)
Y-Intercept (0)
Average value of Y when all Xk = 0 17

13. Parameter Estimation Example
You work in advertising for the New York Times. You want to find the effect of ad size (sq. in.) & newspaper circulation (000) on the number of ad responses (00).
You’ve collected the following data:
Resp Size Circ
Is this model specified correctly?
What other variables could be used (color, photo’s etc.)? 18

14. Parameter Estimation Computer Output
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Param=0 Prob>|T|
INTERCEP
ADSIZE
CIRC b0 b2 b1

15. Interpretation of Coefficients Solution
Slope (b1)
# Responses to Ad is expected to increase by (20.49) for each 1 sq. in. increase in Ad Size Holding Circulation Constant
Slope (b2)
# Responses to Ad is expected to increase by (28.05) for each 1 unit (1,000) increase in circulation Holding Ad Size Constant
Y-intercept is difficult to interpret. How can you have any responses with no circulation?

16. Multiple Standard Error of Estimate
The multiple standard error of estimate is a measure of the effectiveness of the regression equation.
It is measured in the same units as the dependent variable.
It is difficult to determine what is a large value and what is a small value of the standard error.

17. Multiple Standard Error of Estimate
The formula is:
Because in computing y*, k+1 parameters must be fixed. Hence, we lost k+1 degree of freedom.
Interpretation is similar to that in simple linear regression.

18. Multiple Regression and Correlation Assumptions
The independent variables and the dependent variable have a linear relationship.
The dependent variable must be continuous and at least interval-scale.
The variation in (Y-Y*) or residual must be the same for all values of Y. When this is the case, we say the difference exhibits homoscedasticity.
The residuals should follow the normal distributed with mean 0.
Successive values of the dependent variable must be uncorrelated.

19. The ANOVA Table
The ANOVA table reports the variation in the dependent variable. The variation is divided into two components.
The Explained Variation is that accounted for by the set of independent variable.
The Unexplained or Random Variation is not accounted for by the independent variables.

20. Correlation Matrix
A correlation matrix is used to show all possible simple correlation coefficients among the variables.
See which xj are most correlated with y, and which xj are strongly correlated with each other.

21. Multicollinearity High correlation between X variables
Multicollinearity makes it difficult to separate effect of x1 on y from the effect of x2 on y. Leads to unstable coefficients depending on X variables in model
Always exists -- matter of degree
Example: using both age & height as explanatory variables in same model

22. Detecting Multicollinearity
Examine correlation matrix
Correlations between pairs of X variables are more than with Y variable
Few remedies
Obtain new sample data
Eliminate one correlated X variable

23. Correlation Matrix Computer Output
Correlation Analysis
Pearson Corr Coeff /Prob>|R| under HO:Rho=0/ N=6
RESPONSE ADSIZE CIRC
RESPONSE
ADSIZE
CIRC
All 1’s
rY1: correlation between response and ADSIZE
rY2 : correlation between response and CIRC
r12: correlation between ADSIZE and CIRC

24. Global Test
The global test is used to investigate whether any of the independent variables have significant coefficients. The hypotheses are:
The test statistic follows an F distribution with k (number of independent variables) and n-(k+1) degrees of freedom, where n is the sample size.

25. Test for Individual Variables
This test is used to determine which independent variables have nonzero regression coefficients.
The variables that have zero regression coefficients are usually dropped from the analysis.
The test statistic is the t distribution with n-(k+1) degrees of freedom.

26. EXAMPLE 1
A market researcher for Super Dollar Super Markets is studying the yearly amount families of four or more spend on food. Three independent variables are thought to be related to yearly food expenditures (Food). Those variables are: total family income (Income) in $00, size of family (Size), and whether the family has children in college (College).

27. Example 1 continued
Note the following regarding the regression equation.
The variable college is called a dummy or indicator variable. It can take only one of two possible outcomes. That is a child is a college student or not.
Other examples of dummy variables include
gender,
the part is acceptable or unacceptable,
the voter will or will not vote for the incumbent governor.
We usually code one value of the dummy variable as “1” and the other “0.”

28. EXAMPLE 1 continued Family Food Income Size Student 1 3900 376 4 22
5300
515 5 3
4300
516
4900
468
6400
538 6
7300
626 7
543 8
437 9
6100
608 10
513 11
7400
493 12
5800
563

29. EXAMPLE 1 continued
Use a computer software package, such as Excel, to develop a correlation matrix.
From the analysis provided by Excel, write out the regression equation:
Y*= X X X3
What food expenditure would you estimate for a family of 4, with no college students, and an income of $50,000 (which is input as 500)?

30. EXAMPLE 1 continued The regression equation is
Food = Income Size Student
Predictor Coef SE Coef T P
Constant
Income
Size
Student
S = R-Sq = 80.4% R-Sq(adj) = 73.1%
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total

31. EXAMPLE 1 continued From the regression output we note:
The coefficient of determination is 80.4 percent. This means that more than 80 percent of the variation in the amount spent on food is accounted for by the variables income, family size, and student.
Each additional $100 dollars of income per year will increase the amount spent on food by $109 per year.
An additional family member will increase the amount spent per year on food by $748.
A family with a college student will spend $565 more per year on food than those without a college student.

32. EXAMPLE 1 continued The correlation matrix is as follows:
Food Income Size
Income
Size
Student
The strongest correlation between the dependent variable and an independent variable is between family size and amount spent on food.
None of the correlations among the independent variables should cause problems. All are between –.70 and .70.

33. EXAMPLE 1 continued
The estimated food expenditure for a family of 4 with a $500 (that is $50,000) income and no college student is $4,491.
Y* = (500) + 748(4) (0)
= 4491

34. EXAMPLE 1 continued
Conduct a global test of hypothesis to determine if any of the regression coefficients are not zero.
H0 is rejected if F>4.07.
From the computer output, the computed value of F is
Decision: H0 is rejected. Not all the regression coefficients are zero

35. EXAMPLE 1 continued
Conduct an individual test to determine which coefficients are not zero. This is the hypotheses for the independent variable family size.
From the computer output, the only significant variable is SIZE (family size) using the p-values. The other variables can be omitted from the model.
Thus, using the 5% level of significance, reject H0 if the p-value<.05

36. EXAMPLE 1 continued
We rerun the analysis using only the significant independent family size.
The new regression equation is:
Y* = X2
The coefficient of determination is 76.8 percent. We dropped two independent variables, and the R-square term was reduced by only 3.6 percent.

37. Example 1 continued Regression Analysis: Food versus Size
The regression equation is
Food = Size
Predictor Coef SE Coef T P
Constant
Size
S = R-Sq = 76.8% R-Sq(adj) = 74.4%
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total

38. Evaluating the Model yi* = b0 +b1x1i+…+bkxki Evaluating the Model
Most of the procedures to evaluate the multiple regression model are the same as those discussed in the chapter of simple regression models.
Residual analysis
Test for linearity
Global F-test
Test for the coefficient of correlation irrelevant.
Test for individual slope of regression line not enough.
Non-independence of error variables.
Durbin-Watson Statistics
Outliers
Evaluating the Model

39. Chapter Twelve
Multiple Regression and Correlation Analysis
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