Question

Exercise 8. - Q 6.24, Page 253 - Data on the weekly sales of a major brand of canned tuna bya supermarket chain in a large midwestern U.S. city during a mid-1990s calendar year are contained in the file tuna. dta. There are 52 observations on the variables. Variables are described as: SALI = unit sales of brand no. 1 canned tuna, APRI price per can of brand no. 1 canned tuna, APR2 price per can of brand no. 2 canned tuna APR3 price per can of brands no. 3 of canned tuna. Consider the following equation: which generates the following Stata output: . gen InSal log(sal1) . gen 1nApr1 log (apr1) . gen 1nApr2 log(apr2) . reg InSall InApr1 lnApr2 1nApr3 df Number of obs Source MS 52 - Model |18.3051488 Residual15.4585237 F(3, 48) 3 6.10171627 Prob>F 18.95 48 .322052576 R-squared 0.5422 Total |33.7636725 51 .662032793 Root MSE 5675 Insal1 | p>It! Coef. [95% Conf. Interval] Std. Err. t 1nApr-4.624576 .6383371 7.24 0.0005.908039 3.341113 1nApr 9903785 .5338406 1.86 0.070 lnApr3 1.68714 .7468207 2.26 0.028 1871649 3.187115 cons7.889381 251446 31.38 0.00 7.383815 8.394947 2.063737 .08298 a) Interpret the coefficients of the estimated equation. What are their expected signs? (b) Do the estimates have the expected signs? Are they significantly different from zero at a 5% significance level? (c) The marketing manager for no. 1 brand of tuna claims that it is the price of brand 1 relative to the prices of brands 2 and 3 that is important. She suggests the model: AP APRI Show that this model is a restricted version of the original model where +As + β4-0, with (d) Using a 10% significance level, test whether the data supports the marketing manager's claim (e) Estimate the restricted model given in part (c). Report the results. Interpret the estimates Are the estimates significantly different from zero? (f) Which brand, no. 2 or no. 3, is the strongest competitor to brand no. 1? Why? g) Does a hypothesis test confirm your answer to part (f)? Do the test twice: once using the model in part (a) and once using the model in part (c)

Show that this model is a restricted version of the original model where β2 + β3 + β4 = 0, withα2 = β3 and α3 = β4.
(d) Using a 10% significance level, test whether the data supports the marketing manager’s claim. (e) Estimate the restricted model given in part (c). Report the results. Interpret the estimates. Are the estimates significantly different from zero?

(f) Which brand, no. 2 or no. 3, is the strongest competitor to brand no. 1? Why?
(g) Does a hypothesis test confirm your answer to part (f)? Do the test twice: once using the model in part (a) and once using the model in part (c).

Answer 1

Answer d:-

You need to do an F-test! gretk linear restrictions Specify restrictions Please refer to Help for guidance) Use bootstrap Cancel QK Help Restriction: Test statistic: E1, 48) -3.84136, with p-value-0.0558224 Restricted estimates: error t-ratio p-value 1.02e-058 -6.997 6.70e08**t 2.527 0.014 4.879 1.17et05*** 8.35679 -4.01785 0.0819806 101.9 0574208 0521531 0553366 1 apr1 1_apr2 l apr3 1.31775 2.70010 Standard error of the regfession - 0.583719 The F-stat restriction is 3.84136. The P-value for the F-stat= 0.0558. Since this value is less than 10%, then

SUMMARY OUTPUT unre stricted) oaression Stotistics Multiple R R Square 0.736812 0.542155 Adjusted R Square 0.51354 Sta ndard Error 0.567497 Observations 52 ANOVA MS Regression 8. 30 610171718.94534 3.024945-08 Residual 48 52126 0.322053 51 33.76367 134 Coofficant. Stondord Error StP-valua Lower 95% 7.889381 0251445999 3137604 1225-337.38881484 Intercept nAPRI APRZ) 4.62458 0.638337047-7.24472 3.1E-09-590аз8785 0.990379 0533840579 1855195 0069715 0082979771 168714 0746020613 2.261519 002829 0187164799 SUMMARY OUTPUT restricted aression Stotistics Multiple R 0.710996 R Square Adjusted R Square 0.505515 0.435332 Sta ndard Error 0.585719 Observations 52 ANOVA dif sS MS Regression 17.0680908534015 25.0543.21188E-08 Residual 49 564226 0.340727 Tota 51 33.76367134 Coofficant. Stondord ErrortStotP-valua Lower 95% 8.356787 0.081960588 101.9362 102E-58 8192040672 nterept nAPR1/2) し31775 0.521 531057-252669 Q014794-2365802121

e. Estimate the restricted model given in part (c). Report the results. Interpret the estimates. Are the estimates significantly different from zero? The restricted Excel regression above shows the results. We see below that Gretl yields the same results Model 2: OLS, using observations 1-52 Dependent variable:1 sall Coefficient Std. Errort-ratio p-value 8.35679 0.0819806 101.9362 <0.0000 0.0 const 1 aprlto2 1 aprlto3 1.31775 0.521531 2.7001 0.553366 4.8794 2.5267 0.00001*** 8.437187 S. Mean depend Sum squared resid R-squared F(2, 49) Log-likelihood D. dependent var 0.813654 ent yar 16.69564 S.Е. ofregression 0.505515Adjusted R-squared 25.04646 P-value(F) 44.24631 Akaike criterion 0.583719 0.485332 3.21e-08 94.49262 96.73680 Schwarz criterion 100.3463 Hannan-Quinn The t-stats are both significant to at least the 5% level. nterpretation If the relative price of brand 1 to brand 2 increases by 1%, quantity demand falls by 1.31896. If the relative price of brand i to brand 3 increases by 1%, quantity demand falls by 2.700%

f. Which brand, no. 2 or no. 3, is the strongest competitor to brand no. 1? Why? In the unrestricted model in part (a) the coefficient for In(APR3) is greater than the coefficient for În(ARP2): β,-0.99<馬ー1.69. lfbrand 3 dropsits price by 196, we lose more demand (-1.69%) than when brand 2 drops its price by 1%. So, brand 1 is more pnce sensitive to changes in price of brand 3 than to changes in price of brand 2. Brand 3 is the stronger competitor. Using the restricted model in part (c): I would, once again, consider brand 3 as the stronger competitor to brand 1. The demand for brand 1 is more sensitive to relative price changes to brand3 (a2--2.7) thanto brand 2 (a3) Therefore, the price competition between brands 1 and 3 is fiercer than the price competition between brands 1 and 2.