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Chapter 8 Further Inference in the Multiple Regression Model In this chapter we continue working with the hamburger chain data found in Table 7.1. Our task is to use EViews to conduct single and joint coefficient hypothesis tests and to incorporate nonsample information.

8. The F-test

8.1 Constructing the F-Statistic

The F-statistic given in equation (8.1.3).

To compute this statistic we need: i. = the sum of squared errors for the model being tested 11. ... = the for the restricted model, in which we assume the hypothesis is true

J is the number of hypotheses being tested. iv. T is the number of sample observations v. K is thenumber of parameters in the original model (including the intercept)

We will use the F-test to test the null hypothesis : = against H, : .

Open the which we used in Chapter 7 , and click on Equation and type in the total revenue equation making sure to include the intercept (C in notation). Click OK.

Method:

Sample:

coef(6) ftest(1 .@ssr SSE-U ftest(2)=result-7-1 .@regobs-result-7-1 .@ncoef

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04R1/00 1150

var

Further Inference in the Multiple Regression Model 87

From the stored results, we must save the sum of squared errors.

ftest coef vector )=result-7-1

T-K

Now estimate the "restricted" model, assuming the null hypothesis is true.

--

Method. Least Squares ILS and MA]

e the resulting object "Restrictl".

Dependent Variable: TR Method: Least Squares Date: Time:

Coefficient Std. Error t-Statistic Prob.

91.83057 1.871312 49.07282 0.0000 2.949066 0.171520 17.19372 0.0000

0.855334 Mean dependent 120.3231

6.268585 Akaike info criterion 6.546681 1964.758 Schwarz criterion

-168.2137 F-statistic

In

@cfdist(x,vl,v2) vl v2

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@qfdist(p,vl,v2) Fc v2

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ftest(5)=l -@cfdist(ftest(4), 1 ,ftest(2)) P-value ftest(6)=@qfdist(.95, I ,ftest(2))

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View/Coefficient TestsTWald-Coefficient

88 Chapter 8

Save the sum of squared errors and compute the F-statistic.

order to computep-values and critical values for the F-statistic we need to use two EViews functions.

computes the probability that the F-random variable with and degrees of

freedom falls to the left of the value x. That is, x ]

computes the critical value from the F-distribution with vl and degrees of

freedom such that the probability to the left of is p. That is, p =

Using these functions,

F-critical value

Freezing the result, and editing, we have the following:

SSEU 1805.168 T-K 49.00000

Note: We have "programmed" the calculations in EViews. Given the results of the unrestricted and restricted regressions, the F-statistic value could be computed on a calculator and the critical value looked up (approximately) in the F-tables at the end of The advantage of using EViews is that we can compute thep-value and the exact critical value.

8.1.2 Using EViews Coefficient Tests

EViews makes using the F-test very simple. Return to the regression output object Result-7-1.

Click on Restrictions,as shown in the next figure.

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A dialog box opens in which you in the hypotheses you wish to test. The test is stated in terms of the coefficient number, as shown in the representations view of the output

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Estimation Equation:

TR = + +

Coefficients:

is Enter the hypothesis and click OK.

commas:

L ila.

The result is shown in the next figure, which is Table 8.1 in Ignore the "Chi-square"value. It is an alternative testing procedure which we will not consider.

RESULT-7-1

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90 Chapter 8

Equation:

4.331940 Probability

8.2 Testing the Significance of the Model

The overall significance of the model is evaluated using an F-test. In the multiple regression model

the overall test of model significance is the joint test of the null hypothesis : = = .., against the alternative hypothesis at least one of the nonzero.

In this test is reported automatically as part of the regression output, in the lower right-hand corner. We are given the value of the test statistic and its p-value.

We can also create this value from the table output in Chapter 7.7.1 of this manual.

Alternatively, in the estimation object, we can specify the hypothesis as we did in the previous section. We now jointly test the null hypothesis that the and coefficients are zero. Multiple hypotheses are separated by commas in the Wald test dialog box.

Coefficie , ,

: C(2)=0

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The output is the same as reported in the EViews output.

Equation: RESULT-7-1

Null Hypothesis:

8.3 An Extended Model

The contains 78 weeks of observations on tr,p and a. To reproduce the results in Result 8.5

Click on Equation from the EViews main menu and specify equation including the advertising squared term AA2. Note the EViews exponentiation character is Click on OK.

. Dependent by list of and PDL terms, OR an

tr p a

Your results are as follows:

) i

1

var

Ho: P3 P4 0

View/Coefficient TeststWald - toolbar C(3)=0, 0

F(J,T-K)

92 Chapter 8

Included observations: 78

Coefficient Std. Error t-Statistic Prob.

110.4641 3.741410 29.52473 0.0000 -1 0.19792 1.581 822 -6.446944 0.0000

A 3.360999 0.421 708 7.969966 0.0000 -0.026755 0.015887 -1.604041 0.0964

0.878548 Mean dependent 122.6179

5.918707 Akaike info criterion 2592.301 Schwarz criterion

8.4 The Significance of Advertising

Next we generate the F-statistic for the joint test of the hypothesis: = 0, = to determine the significance of advertising in our model.

Click on Coefficient Restrictions from your equation's and enter the joint null hypothesis restrictions C(4) = in the coefficient restrictions field of the Wald Test dialog box. Click OK.

The resulting test statistic has J degrees of freedom in the numerator and T-K degrees of freedom in the denominator, where J = the number of coefficient restrictions, T = the number of observations in the sample, and K = the number of estimated coefficients. In this case, J= 2, T = 78, and K = 4.

Equat~on.

C(3)=0 C (4)=0

F -s ta t~s t~c Probabil~ty

F(2,74)

EViews

((eq-r.@ssr eq~u.@ssr)/2)/(eq~u.@ssr/(@regobs eq-u.@ncoef)) @qfdist(.95,2,74)

tr, pl Pzp, ( q r ) tr, = pl Pzpf P3af p4a2, el (equ) @ssr UE/2), @regobs @ncoef (K @qfdist(l-a,vl,v2) a. vl

freedom v2

(eq-r) (eq-u)

Advertising

ViewICoefficient TestsIWald toolbar, p3 80P4


RESULT-8-5

Null Hypothesis:

261.4110 0 000000

The F-statistic is 261.41 whereas the critical value is 3.12. As your text notes, our conclusion is to reject the null hypothesis that advertising has no statistically significant effect on total revenue in favor of the alternative hypothesis that at least one of the advertising coefficients is non-zero.

The foregoing F-test statistic and F-critical value could also be computed with the following commands typed into the command window:

scalar fstat = - -scalar fcrit =

where =

+ + + + is the unrestricted equation is the sum of squared errors (SSEin

is the number of regression observations (Tin your textbook)

+ is the restricted equation + e,

is the number of estimated coefficients in your textbook) is the upper percent right tail critical value for the F-distribution with

degrees of in the numerator and degrees of freedom in the denominator.

Note: To calculate fstat, you must have previously estimated and named both the restricted and unrestricted equations.

8.5 The Optimal Level of

To compute the F statistic regarding the hypothesis that $40,000 per week is the optimal level of advertising

Click on - Coefficient Restrictions from your equation's enter the restriction + = 1, and click OK.

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fcrit-ad

fcrit-ad =

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View/Coeficient TestsIWald - toolbar P3 80P4 2P2 + 40P3 1600P4

The test statistic follows the F distribution with degrees of freedom in the numerator equal to the number of restrictions (J= 1 in this case) and denominator degrees of freedom equal to the number of regression observations less the number of estimated coefficients (T = 74 in this case). The critical value for a = .05 can be computed by typing the following command into the command window:

scalar fcrit-ad =

where we name the critical value so as not to confuse it with the F-critical value calculated earlier.

Since the test statistic value 0.0637 is less than the critical value 3.970, we fail to reject the null hypothesis that the optimal level of advertising is $40,000. Note that this same result is forthcoming

a t-test of the optimal advertising hypothesis. For a single equality hypothesis there is an exact relationship between the F-test and t-test procedures: = F, as explained in Chapter 8.4.2 of Therefore, either test will produce the same decision. In the optimal advertising case, note that the square root of 0.063721 is 0.252, the t-test statistic reported in result R8.6 in

8.6 The Optimal Level of Advertising and Price

Next we conduct the F test regarding optimal advertising and price presented in section 8.4.3 of In

Click on Coefficient Restrictions from your equation result-8-5 and enter the restrictions + = 1 and + + = 175 and click OK.

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C(3)+80*C(4)=1 C[1)+2*C(2)+40*C(3)+1600*C(4)=175

F(2,74) EViews

fcrit-ad-price @qfdist(.95,2,74)


Coefficient

The results of this test are

Wald Test Equation: RESULT8-5

Null Hypothesis:

F-statistic 1.752979 Probability 0.180375

You can compute the critical value for a = .05 by typing the following command into the command window:

scalar =

Since the F-statistic = 1.75 is less than the F-critical value = 3.12, we fail to reject the null hypothesis that the sample data are consistent with the joint hypothesis that optimal weekly advertising expenditures are $40,000, and at a price of $2 per burger, total revenue will average $175,000 per week.

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