Chapter 7 practice questions
2. The estimated market demand for good X is
Q̂ = 16,784 - 232.43P + 0.225M - 895.3PZ
where Q̂ is the estimated number of units of good X demanded, P is the price of the good, M
is income, and P is the price of related good Z. (All parameter estimates are statistically
significant at the 1 percent level.)
Z
a. Is X a normal or an inferior good? Explain.
b. Are X and Z substitutes or complements? Explain.
c. At P = $22.50, M = 43,499, and P = $12.50, compute estimates for the price ( Ê ), income
( Ê M ), and cross-price elasticities ( Ê XZ ).
Z
3. The empirical demand function for good X is estimated in log-linear form as
ln Q̂ = 16.56727 - 2.85ln P - 0.05152ln M + 0.35ln PY
where Q̂ is the estimated number of units of good X demanded, P is the price of X, M is
income, and P is the price of related good Y. (All parameter estimates are statistically
significantly different from zero at the 5 percent level.)
Y
a. Is X a normal or an inferior good? Explain.
b. Are X and Y substitutes or complements? Explain.
c. Express the empirical demand function in the alternative (nonlogarithmic) form:
d. At P = $10, M = $55,600, and P = $12, what are the estimated price ( Ê ), income ( Ê M ),
Y
and cross-price elasticities ( Ê XZ )? What is the predicted number of units of good X
demanded?
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6. A linear trend equation for sales of the form
Qt = a + bt
was estimated for the period 2003–2017 (i.e., t = 2003, 2004, . . . , 2017). The results of the
regression are as follows:
a. Evaluate the statistical significance of the estimated coefficients. (Use 5 percent for the
significance level.) Does this estimation indicate a significant trend?
b. Using this equation, forecast sales in 2020 and 2025.
c . Comment on the precision of these two forecasts.
7. Consider a firm subject to quarter-to-quarter variation in its sales. Suppose that the following equation was estimated using quarterly data for the period 2011–2018 (the time
variable goes from 1 to 32). The variables D1, D2, and D3 are, respectively, dummy variables for the first, second, and third quarters (e.g., D1 is equal to 1 in the first quarter and
0 otherwise).
Qt = a + bt + c1 D1 + c2 D2 + c3 D3
The results of the estimation are presented here:
a. At the 5 percent level of significance, perform t- and F-tests to check for statistical
significance of the coefficients and the equation. Discuss also the significance of the
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coefficients and equation in terms of p-values.
b. Calculate the intercept in each of the four quarters. What do these values imply?
c. Use this estimated equation to forecast sales in the four quarters of 2019.
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Answers:
2. a. X is a normal good. A positive parameter estimate for income (0.225) means the quantity
of X demanded increases (decreases) when income increases (decreases), all other factors
constant, and this is the definition of a normal good.
b. X and Z are complements. A negative parameter estimate for the price of Z (–895.3)
means that the quantity of X demanded increases (decreases) when the price of Z
decreases (increases), which is the definition of two goods that are complements in
consumption.
c. Q̂ = 16,784 - 232.43(22.50) + 0.225(43,499) - 895.3(12.50) = 10,150.4
Ê = b̂(P / Q) = -232.43(22.50 / 10,150.4) = -0.5152
ÊM = ĉ(M / Q) = 0.225(43,499 /10,150.4) = 0.964
Ê XZ = d̂(PZ / Q) = -895.3(12.50 / 10,150.4) = -1.10
3. a. X is an inferior good since the estimated coefficient on ln M, the income elasticity, is
negative (–0.05152).
b. X and Y are substitutes since the estimated coefficient on ln PY , the cross-price elasticity,
is positive (0.35).
c. Q̂ = 15,670,175P-2.85 M -0.05152 PY 0.35 (Note: 15,670,175 = e16.56727.)
d. For all values of P, M, PY, the elasticity estimates are constant and equal to Ê = –2.85,
Ê M = –0.05152, and Ê XZ = 0.35. Estimated Q:
Q̂ = 30,083 = 15,670,175(10)-2.85 (55,600)-0.05152 (12)0.35 .
6. a. â : p-value is 0.0123, so â is easily significant at the 5 percent level of significance.
b̂ : p-value is 0.0135, so b̂ is easily significant at the 5 percent level of significance.
Sales exhibit a statistically significant negative trend over time (i.e., b̂ < 0 and p-value is
small). The model as a whole, as indicated by the extremely small p-value on the Fstatistic, explains a statistically significant amount of the variation in sales.
b. Q2020 = 4,430.0 –1.77(2020) = 855
Q2025 = 4,430.0 –1.77(2025) = 846
c. The farther the values of the variables in the forecast are from the mean values of the
regression, the less precise the forecast will be. Thus the forecast for 2025 will be less
precise than the forecast for 2020.
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7. a. For a 5 percent significance level with 27 (32 – 5) degrees of freedom, the critical value
of t is 2.052. All of the parameter estimates are statistically significant at the 5 percent
level (and also at the 1 percent level, as you can see by looking at the p-values).
For â : t = 5.90 > 2.052
For b̂ : | t | = |–3.73| > 2.052
For ĉ1 : t = 4.03 > 2.052
For ĉ2 : t = 4.24 > 2.052
For ĉ3 : | t | = |–4.60| > 2.052
For a 5 percent significance level with 4 (= 5 – 1) and 27 (= 32 – 5) degrees of freedom,
the critical value of F is 2.73. The regression equation is statistically significant because
the F-ratio (87.89) is easily greater than the critical value of F.
b. The intercepts are 254.203 (= 245.251 + 8.952) for the first quarter, 250.709 (= 245.251 +
5.458) for the second quarter, 236.552 (= 245.251 – 8.699) for the third quarter, and
245.251 for the fourth quarter. These intercept values imply that, after accounting for
trend, sales will be greatest in quarter 1, then quarter 2, then quarter 4, and smallest in
quarter 3.
c. Q̂2019 I = 245.251- 2.359 ´ 33+ 8.952 = 176.4
Q̂2019II = 245.251- 2.359 ´ 34 + 5.458 = 170.50
Q̂2019 III = 245.251- 2.359 ´ 35- 8.699 = 153.99
Q̂2019 IV = 245.251- 2.359 ´ 36 = 160.327
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