Download Problem Set 1 Econometria - MFEE - FGV Cecilia

Problem Set 1
Econometria - MFEE - FGV
Cecilia Machado
Due: 16 Abril de 2012
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) The conditional expectation of Y given X, E(Y X = x), is calculated as follows:
k
y i Pr(Y = y i X= x)
A)
B) E E(Y X)]
i=1
k
l
Yi Pr(X = x i Y= y)
E(Y X= x i ) Pr(X = xi)
C)
D)
i=1
i=1
2) Assume that Y is normally distributed N(µ, 2 ). To find Pr(c1
to calculate Pr(d 1
Z
A)
(1.96) -
(1.96)
C)
(d 2 ) - (1 -
d2) =
Y
c2 ), where c1 < c2 and di =
ci – µ
, you need
B) 1 - ( (d 2) - (d 1 ))
D) (d 2 ) - (d 1 )
(d 1 ))
^
3) An estimator µY of the population value µY is unbiased if
^
A) E(µY) = µY.
B) Y
^
C) µY = µ .
Y
p
µY.
D) Y has the smallest variance of all estimators.
^
4) An estimator µY of the population value µY is consistent if
A) its mean square error is the smallest possible.
B) Y is normally distributed.
^
C) µY p µ .
Y
D) Y
p
0.
~
^
5) An estimator µY of the population value µY is more efficient when compared to another estimator µY , if
A) its c.d.f. is flatter than that of the other estimator.
^
~
B) both estimators are unbiased, and var(µY) < var(µY).
^
~
C) E(µY) > E(µY).
D) it has a smaller variance.
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ESSAY.
6) The expectations augmented Phillips curve postulates
p=
– f (u – u),
where p is the actual inflation rate, is the expected inflation rate, and u is the unemployment rate, with "–"
indicating equilibrium (the NAIRU – Non-Accelerating Inflation Rate of Unemployment). Under the
assumption of static expectations ( = p –1), i.e., that you expect this period's inflation rate to hold for the next
period ("the sun shines today, it will shine tomorrow"), then the prediction is that inflation will accelerate if the
unemployment rate is below its equilibrium level. The accompanying table below displays information on
accelerating annual inflation and unemployment rate differences from the equilibrium rate (cyclical
unemployment), where the latter is approximated by a five-year moving average. You think of this data as a
population which you want to describe, rather than a sample from which you want to infer behavior of a larger
population. The data is collected from United States quarterly data for the period 1964:1 to 1995:4.
Joint Distribution of Accelerating Inflation and Cyclical Unemployment,
1964:1-1995:4
p–
p–1 > 0 (X = 0)
p–
p–1
0 (X = 1)
Total
(u – u) > 0
(Y = 0)
0.156
(u – u) 0
(Y = 1)
0.383
Total
0.297
0.164
0.461
0.453
0.547
1.00
0.539
(a) Compute E(Y) and E(X), and interpret both numbers.
(b) Calculate E(Y X= 1) and E(Y X= 0). If there was independence between cyclical unemployment and
acceleration in the inflation rate, what would you expect the relationship between the two expected values to
be? Given that the two means are different, is this sufficient to assume that the two variables are independent?
(c) What is the probability of inflation to increase if there is positive cyclical unemployment? Negative cyclical
unemployment?
(d) You randomly select one of the 59 quarters when there was positive cyclical unemployment ((u – u) > 0).
What is the probability there was decelerating inflation during that quarter?
7) In considering the purchase of a certain stock, you attach the following probabilities to possible changes in the
stock price over the next year.
Stock Price Change During
Next Twelve Months (%)
+15
+5
0
–5
–15
Probability
0.2
0.3
0.4
0.05
0.05
What is the expected value, the variance, and the standard deviation? Which is the most likely outcome? Sketch
the cumulative distribution function.
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8) IQs of individuals are normally distributed with a mean of 100 and a standard deviation of 16. If you sampled
students at your college and assumed, as the null hypothesis, that they had the same IQ as the population, then
in a random sample of size
(a) n = 25, find Pr(Y < 105).
(b) n = 100, find Pr(Y > 97).
(c) n = 144, find Pr(101 < Y < 103).
9) Consider the following alternative estimator for the population mean:
~ 1 1
7
1
7
1
7
Y= ( Y1 + Y2 + Y3 + Y4 + ... + Yn–1 + Yn)
n 4
4
4
4
4
4
~
Prove that Y is unbiased and consistent, but not efficient when compared to Y.
10) Some policy advisors have argued that education should be subsidized in developing countries to reduce
fertility rates. To investigate whether or not education and fertility are correlated, you collect data on population
growth rates (Y) and education (X) for 86 countries. Given the sums below, compute the sample correlation:
n
i=1
Yi = 1.594;
n
i=1
Xi = 449.6;
n
i=1
Yi Xi = 6.4697;
n
Y
i=1
2
= 0.03982;
i
n
i=1
X
2
= 3,022.76
i
11) The neoclassical growth model predicts that for identical savings rates and population growth rates, countries
should converge to the per capita income level. This is referred to as the convergence hypothesis. One way to
test for the presence of convergence is to compare the growth rates over time to the initial starting level.
(a) If you regressed the average growth rate over a time period (1960-1990) on the initial level of per capita
income, what would the sign of the slope have to be to indicate this type of convergence? Explain. Would this
result confirm or reject the prediction of the neoclassical growth model?
(b) The results of the regression for 104 countries were as follows:
g6090 = 0.019 – 0.0006 × RelProd60 , R2 = 0.00007, SER = 0.016,
where g6090 is the average annual growth rate of GDP per worker for the 1960-1990 sample period, and
RelProd60 is GDP per worker relative to the United States in 1960.
Interpret the results. Is there any evidence of unconditional convergence between the countries of the world? Is
this result surprising? What other concept could you think about to test for convergence between countries?
(c) You decide to restrict yourself to the 24 OECD countries in the sample. This changes your regression output
as follows:
g6090 = 0.048 – 0.0404 RelProd60 , R2 = 0.82 , SER = 0.0046
How does this result affect your conclusions from above?
12) Assume that there is a change in the units of measurement on both Y and X. The new variables are Y *= aY and
X* = bX. What effect will this change have on the regression slope?Testgen questions still do not copy to other
applications.
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13) At the Stock and Watson (http://www.pearsonhighered.com/stock_watson) website go to Student Resources
and select the option "Datasets for Replicating Empirical Results." Then select the "CPS Data Used in Chapter 8"
(ch8_cps.xls) and open it in Excel. This is a rather large data set to work with, so just copy the first 500
observations into a new Worksheet (these are rows 1 to 501).
In the newly created Worksheet, mark A1 to A501, then select the Data tab and click on "sort." A dialog box will
open. First select "Add level" from one of the options on the left. Then select "sort by" and choose "Northeast"
and "Largest to Smallest." Repeat the same for the "South" as a second option. Finally press "ok."
This should give you 209 observations for average hourly earnings for the Northeast region, followed by 205
observations for the South.
a. For each of the 209 average hourly earnings observations for the Northeast region and separately for the
South region, calculate the mean and sample standard deviation.
b Use the appropriate test to determine whether or not average hourly earnings in the Northeast region the
same as in the South region.
c Find the 1%, 5%, and 10% confidence interval for the differences between the two population means. Is your
conclusion consistent with the test in part (b)?
d In all three cases of using the confidence interval in (c), the power of the test is quite low (5%). What can you
do to increase the power of the test without reducing the size of the test?
14) The news-magazine The Economist regularly publishes data on the so called Big Mac index and exchange rates
between countries. The data for 30 countries from the April 29, 2000 issue is listed below:
Country
Currency
Indonesia
Italy
South Korea
Chile
Spain
Hungary
Japan
Taiwan
Thailand
Czech Rep.
Russia
Denmark
Sweden
Mexico
France
Israel
China
South Africa
Switzerland
Poland
Germany
Malaysia
New Zealand
Rupiah
Lira
Won
Peso
Peseta
Forint
Yen
Dollar
Baht
Crown
Ruble
Crown
Crown
Peso
Franc
Shekel
Yuan
Rand
Franc
Zloty
Mark
Dollar
Dollar
Price of Actual Exchange Rate
Big Mac per U.S. dollar
14,500
4,500
3,000
1,260
375
339
294
70
55
54.37
39.50
24.75
24.0
20.9
18.5
14.5
9.90
9.0
5.90
5.50
4.99
4.52
3.40
7,945
2,088
1,108
514
179
279
106
30.6
38.0
39.1
28.5
8.04
8.84
9.41
.07
4.05
8.28
6.72
1.70
4.30
2.11
3.80
2.01
4
Singapore
Brazil
Canada
Australia
Argentina
Britain
United States
Dollar
Real
Dollar
Dollar
Peso
Pound
Dollar
3.20
2.95
2.85
2.59
2.50
1.90
2.51
1.70
1.79
1.47
1.68
1.00
0.63
The concept of purchasing power parity or PPP ("the idea that similar foreign and domestic goods … should
have the same price in terms of the same currency," Abel, A. and B. Bernanke, Macroeconomics, 4th edition,
Boston: Addison Wesley, 476) suggests that the ratio of the Big Mac priced in the local currency to the U.S.
dollar price should equal the exchange rate between the two countries.
(a) Enter the data into your regression analysis program (EViews, Stata, Excel, SAS, etc.). Calculate the predicted
exchange rate per U.S. dollar by dividing the price of a Big Mac in local currency by the U.S. price of a Big Mac
($2.51).
(b) Run a regression of the actual exchange rate on the predicted exchange rate. If purchasing power parity held,
what would you expect the slope and the intercept of the regression to be? Is the value of the slope and the
intercept "far" from the values you would expect to hold under PPP?
(c) Plot the actual exchange rate against the predicted exchange rate. Include the 45 degree line in your graph.
Which observations might cause the slope and the intercept to differ from zero and one?
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