# State whether the following models are linear regression models:

November 24, 2016

Chapter 2 homework

Problems:

2.8

State whether the following models are linear regression models:

a. Yi = B1 +B2 (1/Xi)

b. Yi = B1 + B2 ln Xi + ui

c. ln Yi = B1 + B2Xi + ui

d. ln Yi = B1 + B2 ln Xi + ui

e. Yi = B1 + B2 B3 Xi + ui

f. Yi = B1 + B3/2 Xi + ui

Note: ln stands for the natural log, that is, log to the base (More onthis in

Chapter 4.)

2.10

From the data given in the preceding problem, a random sample of

Y was drawn against each X. The result was as follows:

Y 70 65 90 95 110 115 120 140 155 150

X 80 100 120 140 160 180 200 220 240 260

a. Draw the scatter gram with Y on the verticalaxis and X on the horizontal axis.

b. What can you say about the relationship betweenY and X?

c. What is the SRF for this example? Show all yourcalculations in the manner of Table 2-4.

d. On the same diagram, show the SRF and PRF.

e. Are the PRF and SRF identical? Why or why not?

2.13

Table 2-10 gives data on the nominal interest rate (Y) and theinflation rate (X) for the year 1988 for nine industrial countries.

NOMINAL INTEREST RATE (Y) AND INFLATION (X) IN NINE INDUSTRIALCOUNTRIES FOR THE YEAR 1988

Country Y (%) X (%)

Australia 11.9 7.7

France 7.5 3.1

Germany 4.0 1.6

Italy 11.3 4.8

Mexico 66.3 51.7

Switzerland 2.2 2.0

United Kingdom 10.3 6.8

United States 7.6 4.4

Source: Rudiger Dornbusch and Stanley Fischer, Macroeconomics, 5th ed.,McGraw-Hill, New York, 1990, p. 652. The original data are from various issuesof International

a. Plot these data with the interest rate on thevertical axis and the inflation rate on the horizontal axis. What does thescatter gram reveal?

b. Do an OLS regression of Y on X. Present all yourcalculations.

c. If the real interest rate is to remain constant,what must be the relationship between the nominal interest rate and theinflation rate? That is, what must be the value of the slope coefficient in theregression of Y on X and that of the intercept? Do your results suggest thatthis is the case? For a theoretical discussion of the relationship among thenominal interest rate, the inflation rate, and the real interest rate, see anytextbook on macroeconomics and look up the topic of the Fisher equation, namedafter the famous American economist, Irving Fisher.

2.22

Table 2-16 (on the textbook’s Web site) gives data on investment rate(ipergdp) and savings rate (spergdp), both measured as percent of GDP, for across- section of countries. These rates are averages for the period1960–1974.*

a. Plot the investment rate on the vertical axisand the savings rate on the horizontal axis.

b. Eyeball a suitable curve from the scatterdiagram in (a).

c. Now estimate the following model

Ipergdpi = B1+B2 spergdpi + ui

d. Interpretthe estimated coefficients.

e. Whatgeneral conclusion do you draw from your analysis?

Note:Save your results for further analysis in the next chapter.

Chapter 3 homework

Question:

3.7

Based on the data for the years 1962 to 1977 forthe United States, Dale Bails and Larry Peppers

18 obtained the following demand function forautomobiles:

Yt = 5807 + 3.24Xt r2 =0.22

se = (1.634)

where Y = retail sales of passenger cars(thousands) and X = the real disposable income (billions of 1972 dollars).

Note: The se for b1 is not given.

a. Establisha 95% confidence interval for B2.

b. Testthe hypothesis that this interval includes B2=0. If not, would you acceptthis null hypothesis?

c. Computethe t value under H0:B2 =0. Is it statisticallysignificant at the 5 percentlevel? Which t test do you use, one-tailedor two-tailed, and why?

3.9

You are given the following databased on 10 pairs of observations on Y and X.

∑yi = 1110 ∑Xi = 1680 ∑XiYi = 204,200

∑X2i = 315,400 ∑Y2i = 133,300

Assuming all the assumptions of CLRMare fulfilled, obtain

a. b1 and b2

b. standard errors of these estimators.

c. r2.

d. Establish 95% confidence intervalsfor B1 and B2.

e. On the basis of the confidenceintervals established in (d), can you accept the hypothesis that B2 = 0?

3.10

Based on data for the United Statesfor the period 1965 to 2006 (found in Table3-4 on the textbook’s Web site), thefollowing regression results were obtained:

GNPt= -995.5183 + 8.7503M1t r2= 0.9488

se =( ) (0.3214)

t = (-3.8258) ( )

where GNP is the gross nationalproduct (\$, in billions) and M1 is the money supply (\$, in billions).

Note: M1 includes currency, demand deposits,traveler’s checks, and other checkable deposits.

a. Fill in the blank parentheses.

b. The monetarists maintain that moneysupply has a significant positive impact on GNP. How would you test thishypothesis?

c. What is the meaning of the negativeintercept?

d. Suppose M1 for 2007 is \$750 billion. What isthe mean forecast value of GNP for that year?

3.18

Refer to Example 2.1 on years ofschooling and average hourly earnings. The data for this example are given inTable 2-5 and the regression results are presented in Eq. (2.21). For thisregression

a. Obtain the standard errors of theintercept and slope coefficients and

r2.

b. Test the hypothesis that schoolinghas no effect on average hourly earnings.

Which test did you use and why?

c. If you reject the null hypothesis in(b), would you also reject the hypothesis that the slope coefficient in Eq.(2.21) is not different from 1? Show the necessary calculations

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