1. The Multiple Regression Model
Onon Khanoi
Berhanu Anbesa

2. 2. Estimating a Multiple Regression Model
Content
1. Introduction
2. Estimating a Multiple Regression Model
3. Forecasting from a Multiple Regression Model
4. Interval Estimation
5. Hypothesis Testing
6. Polynomial Equations
7. Interaction Variables
8. Goodness-of-fit
9. Summary

3. Introduction 1
Multiple regression model – econometric model with more than one explanatory variable.
Our model is: SALES = ß1+ß2PRICE + ß3 ADVERT+e , where
SALES – monthly sales revenue in a given city (1000USD)
PRICE – price in that city
ADVERT – monthly advertising expenditure in that city (1000USD)
ß2 - the change in monthly SALES (1000USD) when price index PRICE is increased by 1 unit ($1) and advertising expenditure ADVERT is held constant
ß3 - the change in monthly SALES (1000USD) when advertising expenditure ADVERT is increased by 1 unit ($1000) and price index PRICE is held constant.
e – random error term

4. Opening Eviews work file1
1. Open Eviews work file - Andy
Number of observation
Data series
Page name
2. Create objects as a group: Sales, Price and Advert
3. Select the name table5_1

5. Spreadsheet 1
1. Click View → Descriptive Stats → Common Sample

6. Saving 1 1. Click Freeze to save 2. Click Name
2. Write name: Andy_statistics
Table object

7. Estimating a Multiple Regression Model2
Two ways: through Quick menu, Object menu and equation Command 1. Using the Quick menu write: sales c price advert
Explanatory variables
Constant term
Dependent variable
Click Quick → Estimation Equation
Our equation is:
SALES = ß1+ß2PRICE + ß3 ADVERT+e

8. Regression output 2 Name it burger_eqn Coefficient:
The negative coefficient on PRICE suggests that demand is price elastic. With advertising held constant, an increase in price of $1 will lead to a fall in monthly revenue of $7,0908; or reduction in price of $1 will lead to an increase in revenue of $7,908.
The coefficient on advertising is positive. With price held constant, an increase in advertising expenditure of $1000 will lead to an increase in sales revenue of $1,863.
C is not directly interpretable. It is not possible that if both proce and advertising expenditure were zero the sales revenue would be $118,914.

9. Forecasting from a Multiple Regression Model3
In addition to providing information about how sales change when price or advertising change, the estimated equation can be used for prediction.
We will now forecast hamburger sales revenue for PRICE=5.5 and ADVERT=1.2
b1=C(1)
b2=C(2)
b3=C(3)
Objective is to get Eviews to perform the calculation:
SALES=b1 + b2x5.5+b3x1.2

10. Forecasting from a Multiple Regression Model: A Simple Forecasting Procedure3
Eviews command is:
scalar sales_f=c(1)+c(2)*5.5+c(3)*1.2
You will see:
Double click sales_f in work file
Forecast result:

11. Forecasting from a Multiple Regression Model: Using the Forecast option3
We need to extend the size of work file to accommodate the observations for which we want forecast.
Click Proc → Structure/Resize Current Page
Change range of data to 76
Range and sample changed to 76
Double click
Click Yes

12. Forecasting from a Multiple Regression Model: Using the Forecast option3
Open PRICE and change 76 to 5.5
Same steps for ADVERT change to 1.2
Click to open for editing
Change NA to 5.5
Click to open for editing
Change NA to 1.2
Click to close for editing
New value
Click to close for editing
New value

13. Forecasting from a Multiple Regression Model: Using the Forecast option3
Open burger_eqn equation and click Forecast
Open salesf and sef as a group
Change:
Forecast value
Add name for standard error
Forecast for observation 76
SALES= and se(f)=4.942
Can be used to compute a forecast
Interval as SALES ±t(1-α/2,72) x se(f).
Click OK

14. The least squares covariance matrix
Interval Estimation:
The least squares covariance matrix 4
The variances and covariances of the least squares estimators give us information about the reliability of the estimators. The smaller their variances, the higher the probability that they will produce estimates “near” the true parameter values.
Open burger_eqn and click View → Covariance Matrix
Var(b1)
Cov(b1, b2)
Cov(b1, b3)
Var(b2)
Cov(b2, b3)
Var(b3)

15. Computing interval estimates
Interval Estimation:
Computing interval estimates 4
To view upper and lower bounds of the interval estimates
Type the command one by one : scalar scalar scalar scalar scalar
Value
BETA2_LOW
BETA2_UP
Easy way to get a same result:
Open burger_eqn
Click View → Coefficient Diagnostics → Confidence Intervals → choose .95 → OK

16. Interval estimates for linear combinations of coefficients
Interval Estimation:
Interval estimates for linear combinations of coefficients 4
Big Andy plans to increase advertising expenditure by $800 and drop the price by 40 cents.
λ=E(SALES1)-E(SALES0)= -0.4ß2+0.8ß3
1. Open Burger_eqn → click Coefficient Diagnostics → Wald Test – Coefficient Restrictions…
2. Type -0.4*c(2)+0.8*c(3)=0
Estimate

17. Two-tail tests of significance
Hypothesis Testing :
Two-tail tests of significance 5
Ho =ß2=0 (no price effect) H1 =ß2≠0 (there is a price effect )
Ho =ß3=0 (no advertising effect) H1 =ß3≠0 (there is an advertising effect )
T-values and p-values for two-tail tests of significance

18. Two-tail tests of significance
Hypothesis Testing :
Two-tail tests of significance 5
P-value for ADVERT is We can confirm the p-value giving the command: scalar Open pval_advert
In the case of advertising expenditure we reject Ho =ß3=0 at a 5% significance level because p-value of is less than 0.05.
If we want to make a decision about Ho = by comparing the calculated value t=2.726 to a 5% critical value.
Give command: scalar
The answer is tc=1.993, a value that leads us to reject Ho =ß3=0 because > 1.993

19. One-tail tests of significance
Hypothesis Testing :
One-tail tests of significance 5
Ho =ß2≥0
H1 =ß2<0
Eview command:
scalar
Thus, making the test decision by reference to the critical value, we reject Ho =ß2≥0 in favor of H1 =ß2<0 because <

20. Testing nonzero values
Hypothesis Testing :
Testing nonzero values 5
One tail: For advertising to be effective, ß2 must be greater than 1. Thus we test:
Ho =ß3≤1
H1 =ß3>1.
Eview command:
scalar result is t=1.263
scalar resultis p=0.105
Choosing α=0.05 as our significance level, relevant critical value is t(o.95,72)= Since 1.263<1.666, we don’t reject Ho. There is insufficient evidence in our sample conclude that advertising will be cost effective.

21. Testing nonzero values
Hypothesis Testing :
Testing nonzero values 5
We will now test two-tails test:
Ho =ß3=1
H1 =ß3≠1
Open burger_eqn → Coefficient Diagnostics → Wald Test – Coefficient Restrictions
Type null hypothesis c(3)=1
Or you can use this command: burger_eqn.wald c(3)

22. Testing nonzero values
Hypothesis Testing :
Testing nonzero values 5
T-value denominator
T-value numerator
t=b3-1/se(b3)

23. Testing linear combinations of coefficients
Hypothesis Testing :
Testing linear combinations of coefficients 5
Big Andy’s marketing advisor claims that dropping the price by 20 cents will be more effective for increasing sales revenue than increasing advertising expenditure by $500.
-0.2ß2>0.5ß3.
Hypothesis:
Ho =-0.2ß2-o.5ß3≤ 0
H1 =-0.2ß2-o.5ß3>0
Open burger_eqn → Coefficient Diagnostics → Wald Test – Coefficient Restrictions
Write -0.2*c(2)-0.5*c(3)=0
Two-tail p-value
As 1.263<1.666, we do not reject Ho. At a 5% significance level, there is not enough evidence to support marketing advisor.
You can also use command: burger_eqn.wald -0.2*c(2)-0.5*c(3)=0

24. Polynomial Equations 6
To take into account the fact that the marginal effect of advertising is most likely a diminishing function of advertising, the squared value of advertising is added to Andy’s SALES equation. SALES = ß1+ß2PRICE + ß3 ADVERT+ß4 ADVERT2+e
Click Quick →Equation Estimation → write sales c price advert advert^2
Name it andy_quad

25. Polynomial Equations 6
Example: the marginal effect of advertising on SALES when ADVERT=2 ($2000 per week). Open ANDY_QUAD → View →Coefficient Diagnostics → Wald Test-Coefficient Restrictions Or use command → andy_quad.wald c(3)+2*c(4)*2=0
Marginal effect when ADVERT=2
If ADVERT=2, the marginal effect of an additional $1000 of advertising expenditure is $1,079

26. Polynomial Equations 6
The Wald test also reports the t –statistics for the hypothesis that the estimated marginal effect is zero:
We cannot reject the null hypothesis that the marginal effect is zero at the 5% level of significance.
From the figure you can see that 2 is near to the maximum point where the slope of the fitted quadratic function is zero.

27. The optimal level of advertising
Polynomial Equations
The optimal level of advertising 6
Andy’s objective is maximize profit, but not SALES. Thus, advertising should be increased to the point where
ß3+2ß4ADVERT0=1
Enter command:
andy_quad.wald (1-c(3))/(2*c(4))=0

28. Interaction Variables7
Open new work file – pizza4 Our equation is PIZZA = ß1+ß2AGE + ß3 INCOME+e As age might moderate the income effect: PIZZA = ß1+ß2AGE + ß3 INCOME+ß4(AGExINCOME)+e Command: equation pizza_eq.Is pizza c age income age*income

29. Interaction Variables7
For individuals of ages 20 and 50, the marginal effect are given by
pizza_eq.wald c(3)+c(4)*20=0
pizza_eq.wald c(3)+c(4)*50=0
The results are:
The marginal effect of income for the 50-year-old is much smaller than the marginal effect of income for the 20-year-old.

30. Goodness-of-fit 8
Below the parameter estimates are various summary measures including the R2=
Omitted the intercept:

31. Summary 9
Multiple regression model – econometric model with more than one explanatory variable.
We did regression SALES = ß1+ß2PRICE + ß3 ADVERT+e
We forecasted hamburger sales revenue for PRICE=5.5 and ADVERT=1.2 with SALES=b1 + b2x5.5+b3x1.2
Interval estimation:
Computed Confidence Intervals with .95 significance level.
Checked how Sales will change when advertising expenditure increased by $800 and the price dropped by 40 cents.
Checked hypothesis:
1. Ho =ß2=0 H1 =ß2≠0
Ho =ß3=0 H1 =ß3≠0
3. Ho =ß3≤1
H1 =ß3>1.
Polynominal equation with SALES = ß1+ß2PRICE + ß3 ADVERT+ß4 ADVERT2+e
Interaction variables with PIZZA = ß1+ß2AGE + ß3INCOME+ß4(AGExINCOME)+e
2. Ho =ß2≥0
H1 =ß2<0
4. Ho =ß3=1
H1 =ß3≠1
5. Ho =-0.2ß2-o.5ß3≤ 0
H1 =-0.2ß2-o.5ß3>0