Solutions to Exercises in Chapter 9

Transcription

1 in 9. (a) When a GPA is increased by one unit, and other variables are held constant, average starting salary will increase by the amount $643. Students who take econometrics will have a starting salary which is $5033 higher, on average, than the starting salary of those who did not take econometrics. The intercept suggests the starting salary for someone with zero GPA and no econometrics is $24,200. However, this figure is likely to be unreliable since there would be no one with a zero GPA. (b) A suitably modified equation is (c) SAL = β + β 2GPA + β 3METRICS + β 4SEX + e To see if the value of econometrics is the same for men and women, we change the model to SAL = β + β 2GPA + β 3METRICS + β 4SEX + β 5METRICS SEX + e The estimated models, with standard errors in parentheses below the estimated coefficients, are SAL SAL = GPA METRICS 205 SEX (09) (356) (460) (420) = GPA METRICS 280 SEX METRICS SEX (04) (365) (582) (500) (966) The estimated equation for part (b) suggests the starting salary for females is $205 lower than that for males. However, this estimated coefficient is less than its standard error. The hypothesis that males and females have the same starting salary would not be rejected. The estimated equation for part (c) suggests that: Value of econometrics for men = 4924 Value of econometrics for women = = 599 That is, econometrics is more valuable for women than men. However, the estimated coefficient for METRICS SEX is not significantly different from zero. The hypothesis that econometrics is equally valuable for men and women would not be rejected. 9.2 (a) Considering each of the coefficients in turn, we have the following interpretations. Intercept: At the beginning of the time period over which observations were taken, on a day which is not Friday, Saturday or a holiday, and a day which has neither a full moon nor a half moon, the average number of emergency room cases was 94. T: The average number of emergency room cases has been increasing by per day.

2 2 (b) (c) HOLIDAY: The average number of emergency room cases goes up by 3.9 on holidays. FRI and SAT: The average number of emergency room cases goes up by 6.9 and 0.6 on Fridays and Saturdays, respectively. FULLMOON: The average number of emergency room cases goes up by 2.45 on days when there is a full moon. However, a null hypothesis stating that a full moon has no influence on the number of emergency room cases would not be rejected. NEWMOON: The average number of emergency room cases goes up by 6.4 on days when there is a new moon. However, a null hypothesis stating that a new moon has no influence on the number of emergency room cases would not be rejected. See files xr9-2.xls, xr9-2.sas, xr9-2.szm and xr9-2.wf. The null and alternative hypotheses are H0: β 6 = β 7 = 0 H : β6 or β7 is nonzero. The test statistic is ( SSER SSEU) 2 F = SSE (229 7) U where SSE R = is the sum of squared errors from the estimated equation with FULLMOON and NEWMOON omitted and SSE U = is the sum of squared errors from the estimated equation with these variables included. The calculated value of the F statistic is.29 with corresponding p-value of Thus, we do not reject the null hypothesis that new and full moons have no impact on the number of emergency room cases. 9.3 (a) The estimated equation (with standard errors in parentheses) is PRICE = SQFT AGE 4393 D92 (839) (.000) (7.0) (27) 0435 D D D D96 (232) (2) (233) (95) (b) (c) The estimated coefficient for SQFT suggests that an additional square foot of floor space will increase the price of the house by $73. The positive sign is as expected, and the estimated coefficient is significantly different from zero. The estimated coefficient for AGE implies the house price is $79 less for each year the house is older. The negative sign implies older houses cost less, other things being equal. The coefficient is significantly different from zero. The estimated coefficients for the dummy variables are all negative and they become increasingly negative as we move from D92 to D96. Thus, house prices have been steadily declining in Stockton over the period Including a dummy variable for 99 would have introduced exact collinearity unless the intercept was omitted. The collinearity arises because the sum of the dummy variables equals, the value of the constant term.

3 3 9.4 (a) The estimated equation, with standard errors in parentheses, is ln( PRICE ) = UTOWN SQFT SQFT UTOWN (0.0264) (0.0359) (0.0000) ( ) AGE POOL FPLACE ( ) (0.005) (0.004) (b) In the log-linear functional form ln( y) = β + β 2x+ e, we have dy dy dx y =β 2 dx y = β or 2 Thus, a unit change in x leads to a percentage change in y equal to 00 β 2. Using this result for the coefficients of SQFT and AGE, we find that an additional square foot of floor space increases price by 0.036%; a house which is a year older leads to a reduction in price of %. Both estimated coefficients are significantly different from zero. PRICEpool (c) ln( PRICEpool ) ln( PRICEnopool ) = ln =δ 3 PRICE nopool PRICEpool δ3 Thus, = e PRICE nopool PRICE pool PRICEnopool δ3 And = e PRICE nopool (e) (f) Using the result in part (c), the percentage change in price due to the presence of a pool is ( ) e 00 =.92% Applying the result in part (c) to the presence of a fireplace gives, as the percentage change in price attributable to a fireplace, ( ) e 00 = 0.66% In this case the difference in log-prices is given by ( PRICE ) ( PRICE ) ln ln = = utown noutown and the percentage change in price attributable to being near the university, for a 2500 square-feet home, is ( ) e 00 = 28.%.

4 4 9.5 The sums of squared residuals from separate estimation of the two equations are SSE G = SSE W = Adding these two values together gives which is the value given in (9.7.7). 9.6 (a) The estimated equation, with standard errors in parentheses, is ln( Sal ) = Apr Apr Apr3 (0.6464) (0.5765) (0.4486) (0.6053) Disp DispAd (0.052) (0.562) (b) The estimates of β 2, β 3 and β 4 are all significant and have the expected signs. That is, the sign of β 2 is negative, while the signs of the other two coefficients are positive. These signs imply that Brands 2 and 3 are substitutes for Brand. If the price of Brand rises, then sales of Brand will fall, but a price rise for Brand 2 or 3 will increase sales of Brand. Furthermore, with the log-linear function, the coefficients are interpreted as proportional changes in quantity from a -unit change in price. For example, a one-unit increase in the price of Brand will lead to a 375% decline in sales; a one-unit increase in the price of Brand 2 will lead to a 5% increase in sales. These percentages are large because prices are measured in dollar units. If we wish to consider a cent change in price a change more realistic than a -dollar change then the percentages 375 and 5 become 3.75% and.5%, respectively. (c) There are three situations that are of interest. (i) No display and no advertisement (ii) (iii) { } Sal = exp β + β Apr + β Apr2 + β Apr3 = Q A display but no advertisement { } exp{ } Sal = exp β + β Apr + β Apr2 + β Apr3 + β = Q β A display and an advertisement { } exp{ } Sal = exp β + β Apr + β Apr2 + β Apr3 + β = Q β The estimated percentage increase in sales from a display but no advertisement is Sal ˆ 2 Sal ˆ b5 00 = ( e ) 00 = 52.8% Sal ˆ The estimated percentage increase in sales from a display and an advertisement is Sal ˆ 3 Sal ˆ b6 00 = ( e ) 00 = 38% Sal ˆ The signs and relative magnitudes of 5 b and 6 b are consistent with economic logic. A display increases sales; a display and an advertisement increase sales by an even bigger amount.

5 5 The results of these tests appear in the table below. They suggest that both a store display and a newspaper advertisement will increase sales, and that both forms of advertising will increase sales by more than a store display by itself. Part H 0 Test Value Degrees of Freedom 5% Critical Value Decision (i) β 5 = 0 t = Reject H 0 (ii) β 6 = 0 t = Reject H 0 (iii) β 5 = β 6 = 0 F = 42.0 (2,46) 3.20 Reject H 0 (iv) β 6 β 5 t = Reject H (a) The estimated equation, with standard errors in parentheses, is price = age 2.358net (0.25) (0.0868) (0.263) All estimated coefficients are significantly different from zero. The intercept suggests that the average price of CDs that have a 999 copyright and are not sold on the internet is $5.46. For every year the copyright date is earlier than 999, the price increases by 29 cents. For CDs sold through the internet, the price is $2.36 cheaper. The positive coefficient of age supports Mixon and Ressler s hypothesis. (b) The estimated equation, with standard errors in parentheses, is price = old 2.357net (0.24) (0.257) (0.263) Again, all estimated coefficients are significantly different from zero. They suggest that the average price of new releases, not sold on the internet, is $5.53. If the CD is not a new release, the price is 79 cents higher. If it is purchased over the internet, the price is $2.36 less. The positive coefficient of old supports Mixon and Ressler s hypothesis.

6 6 9.8 (a) The estimated coefficient of the price of alcohol suggests that, if the price of pure alcohol goes up by $ per liter, the average number of days (out of 3) that alcohol is consumed will fall by (b) The price elasticity at the means is given by qp = = pq 3.49 (c) To compute this elasticity, we need q for married black males in the 2-30 age range. It is given by q = = Thus, the price elasticity is qp = = pq The coefficient of income suggests that a $ increase in income will increase the average number of days on which alcohol is consumed by If income was measured in terms of thousand-dollar units, which would be a sensible thing to do, the estimated coefficient would change to (e) The effect of gender suggests that, on average, males consume alcohol on.637 more days than women. On average, married people consume alcohol on less days than single people. Those in the 2-20 age range consume alcohol on.53 less days than those who are over 30. Those in the 2-30 age range consume alcohol on more days than those who are over 30. This last estimate is not significantly different from zero, however. Thus, two age ranges instead of three (2-20 and more than 20), is likely to be adequate. Black and Hispanic individuals consume alcohol on and less days, respectively, than individuals from other races. All coefficients are significantly different from zero, except that for the dummy variable for the 2-30 age range.