Final Exam: EC 83A | Complete Solution

Final Exam: EC 83A
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1 ) The probability of an event A or B (Pr(A or B)) to occur equals 1 ) A) Pr(A) × Pr(B). B) Pr(A)
Pr(B)
.
C) Pr(A) + Pr(B) if A and B are mutually exclusive. D ) Pr(A) + Pr(B) even if A and B are not mutually exclusive. 2 ) The cumulative probability distribution function (CDF) shows the probability 2 ) A) that a random variable takes on a particular value given that another event has happened. B) of two or more events occurring at once. C) of all possible events occurring. D ) that a random variable is less than or equal to a particular value. 3 ) Let Y be a random variable. Then var(Y) equals 3 ) A) E (Y - µY) . B) E (Y -µY)2 . C) E[Y - µY)2]. D ) E (Y - µY) .
4 ) The conditional distribution of Y given X = x, Pr(Y = yX=x), is 4 ) A) Pr(X = x, Y = y)
Pr(Y = y) B) Pr(Y = y)
Pr(X = x)
.
C) Pr(X = x, Y = y)
Pr(X = x) . D ) _
l
i=1
Pr(X = xi, Y = y).
5 ) Two random variables X and Y are independently distributed if all of the following conditions 5 ) hold, with the exception of A) knowing the value of one of the variables provides no information about the other. B) E(Y) = E[E(Y X)]. C) if the conditional distribution of Y given X equals the marginal distribution of Y.
D ) Pr(Y = y X = x) = Pr(Y = y). 6 ) Two variables are uncorrelated in all of the cases below, with the exception of 6 ) A) having a zero covariance. B) ΗX Y K Η 2
X Η 2
Y.
C) being independent. D ) E(Y X) = 0. 1
7 ) The sample average is a random variable and 7 ) A) has a probability distribution that is the same as for the Y1,..., Yn i.i.d. variables. B) is a single number and as a result cannot have a distribution. C) has a probability distribution called its sampling distribution. D ) has a probability distribution called the standard normal distribution. 8 ) An estimator is 8 ) A) a random variable. B) a nonrandom number. C) a formula that gives an efficient guess of the true population value. D ) an estimate. 9 ) With i.i.d. sampling each of the following is true except 9 ) A) Y is a random variable. B) var(Y) = Η 2
Y/n.
C) E(Y) < E(Y). D ) E(Y) = µY.
10) ^ The standard error of Y, S E(Y) = ΗY is given by the following formula: 10) A) _1
n
n
i=1
(Yi – Y)2. B)
S 2
Y
n . C) SY
n
. D ) SY.
11) A type I error is 11) A) always 5%. B) the error you make when rejecting the null hypothesis when it is true. C) always the same as (1-type II) error. D ) the error you make when rejecting the alternative hypothesis when it is true. 12) Among all unbiased estimators that are weighted averages of Y1,..., Yn Y, is 12) A) the only consistent estimator of µY.
B) a number which, by definition, cannot have a variance. C) the most unbiased estimator of µY.
D ) the most efficient estimator of µY.
2
13) The formula for the sample variance is 13) A) S _ 2
Y = 1
n– 1
n
i=1
(Yi – Y). B) S _ 2
Y = 1
n– 1
n – 1
i=1
(Yi – Y)2.
C) S _ 2
Y = 1
n– 1
n
i=1
(Yi – Y)2. D ) S _ 2
Y = 1
n– 1
n
i=1
(Yi – µY)2.
14) Degrees of freedom 14) A) is something that certain undergraduate majors at your university/college other than
economics seem to have an Q amount of. B) in the context of the sample variance formula means that estimating the mean uses up some
of the information in the data. C) ensure that S 2
Y = Η 2
Y .
D ) are (n-2) when replacing the population mean by the sample mean. 15) A type II error 15) A) cannot be calculated when the alternative hypothesis contains an "=" . B) is typically smaller than the type I error. C) is the error you make when not rejecting the null hypothesis when it is false. D ) is the error you make when choosing type II or type I. 16) The t-statistic is defined as follows: 16) A) 1.96. B) t =
Y – µY,H0
S E(Y) .
C) t =
Y – µY,H0
Η 2
Y
n
. D ) t =
(Y – µY,H0)2
S E(Y) .
17) The standard error for the difference in means if two random variables M and W , when the two 17) population variances are different, is A) SM
nM
+
SW
nW
. B)
S 2
M
nM
+
S 2
W
nW
.
C) 1
2
(
S 2
M
nM
+
S 2
W
nW
). D )
S 2
M+ S 2
W
nM + nW
.
3
18) The sample covariance can be calculated in any of the following ways, with the exception of: 18) A) _1
n
n
i=1
(Xi – µX)(Yi – µY).
B) _ 1
n– 1
n
i=1
XiYi – n
n – 1
X Y.
C) _ 1
n– 1
n
i=1
(Xi – X)(Yi – Y).
D ) rX YSYSY, where rX Y is the correlation coefficient. 19) ^ When the estimated slope coefficient in the simple regression model, Ά1, is zero, then 19) A) R2 = Y . B) R2 > (SSR/TSS). C) 0 < R2 < 1. D ) R2 = 0. 20) The regression R2 is defined as follows where ESS is Explained Sum of Squares, RSS is Residual
Sum of Squares, TSS is total sum of squares, and SSR Sum of Squared Residuals. 20) A) ESS
TSS B) SSR
n-2
C) RSS
TSS D )
_
_ _
n
i=1
(Yi - Y)(Xi - X)
n
i=1
(Yi - Y)2
n
i=1
(Xi - X)2
21) The standard error of the regression (SER) is defined as follows 21) A) ^ _ 1
n-2
 n
i=1
u 2
i B) ^ _ 1
n-1
 n
i=1
u 2
i C) 1-R2 D ) SSR 22) Which of the following statements is correct? 22) A) ESS > TSS B) TSS = ESS + SSR C) ESS = SSR + TSS D ) R2 = 1 - (ESS/TSS)
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23) Binary variables 23) A) exclude certain individuals from your sample. B) can take on only two values. C) are generally used to control for outliers in your sample. D ) can take on more than two values. 24) The reason why estimators have a sampling distribution is that 24) A) individuals respond differently to incentives. B) in real life you typically get to sample many times. C) economics is not a precise science. D ) the values of the explanatory variable and the error term differ across samples. 25) The OLS estimator is derived by 25) A) minimizing the sum of squared residuals. B) minimizing the sum of absolute residuals. C) connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the
highest Xi observation. D ) making sure that the standard error of the regression equals the standard error of the slope
estimator. 26) Interpreting the intercept in a sample regression function is 26) A) not reasonable because economists are interested in the effect of a change in X on the change
in Y.
B) reasonable if your sample contains values of Xi around the origin. C) not reasonable because you never observe values of the explanatory variables around the
origin. D ) reasonable because under certain conditions the estimator is BLUE. 27) The variance of Y 27) i is given by A) the variance of ui. B) Ά 2
1 var(Xi) + var(ui).
C) Ά 2
0 + Ά 2
1 var(Xi) + var(ui). D ) the variance of the residuals. 28) ^ The OLS residuals, ui, are defined as follows: 28) A) Yi - Ά0 - Ά1Xi B) ^ Yi - Yi C) ^ ^ ^ Yi - Ά0 - Ά1Xi D ) (Yi - Y)2
29) The sample average of the OLS residuals is 29) A) unobservable since the population regression function is unknown. B) some positive number since OLS uses squares. C) dependent on whether the explanatory variable is mostly positive or negative. D ) zero. 5
30) The slope estimator, Ά1, has a smaller standard error, other things equal, if 30) A) the sample size is smaller. B) the intercept, Ά0, is small. C) there is a large variance of the error term, u.
D ) there is more variation in the explanatory variable, X.
31) The sample regression line estimated by OLS 31) A) cannot have a slope of zero. B) will always run through the point (X, Y). C) will always have a slope smaller than the intercept. D ) is exactly the same as the population regression line. 32) The OLS residuals 32) A) can be calculated by subtracting the fitted values from the actual values. B) are unknown since we do not know the population regression function. C) should not be used in practice since they indicate that your regression does not run through
all your observations. D ) can be calculated using the errors from the regression function. 33) ^ If the three least squares assumptions hold, then the large sample normal distribution of Ά1 is 33) A) N(0, 1
n
var[Xi - µX)ui]
[var(Xi)]2 ). B) N(Ά1, 1
n
var(ui)]
[var(Xi)]2
).
C)
_
N(Ά1,
Η 2
u
n
i=1
(Xi - X)2
. D ) N(Ά1, 1
n
var(ui)]2
[var(Xi)]2
).
34) In the simple linear regression model Y 34) i = Ά0 + Ά1Xi + ui,
A) Ά0 + Ά1Xi represents the population regression function. B) the absolute value of the slope is typically between 0 and 1. C) the intercept is typically small and unimportant. D ) Ά0 + Ά1Xi represents the sample regression function. 35) To obtain the slope estimator using the least squares principle, you divide the 35) A) sample covariance of X and Y by the sample variance of X.
B) sample variance of X by the sample covariance of X and Y.
C) sample covariance of X and Y by the sample variance of Y.
D ) sample variance of X by the sample variance of Y.
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36) To decide whether or not the slope coefficient is large or small, 36) A) the slope coefficient must be larger than one. B) the slope coefficient must be statistically significant. C) you should change the scale of the X variable if the coefficient appears to be too small. D ) you should analyze the economic importance of a given increase in X.
37) E(u 37) i Xi) = 0 says that A) dividing the error by the explanatory variable results in a zero (on average). B) the conditional distribution of the error given the explanatory variable has a zero mean. C) the sample mean of the Xs is much larger than the sample mean of the errors. D ) the sample regression function residuals are unrelated to the explanatory variable. 38) Multiplying the dependent variable by 100 and the explanatory variable by 100,000 leaves the 38) A) OLS estimate of the slope the same. B) OLS estimate of the intercept the same. C) variance of the OLS estimators the same. D ) regression R2 the same. 39) When estimating a demand function for a good where quantity demanded is a linear function of 39) the price, you should A) reject the idea that price determines demand unless the coefficient is at least 1.96. B) use a one-sided alternative hypothesis to check the influence of price on quantity. C) use a two-sided alternative hypothesis to check the influence of price on quantity. D ) not include an intercept because the price of the good is never zero. 40) The confidence interval for the sample regression function slope 40) A) allows you to make statements about the economic importance of your estimate. B) adds and subtracts 1.96 from the slope. C) can be used to conduct a test about a hypothesized population regression function slope. D ) can be used to compare the value of the slope relative to that of the intercept. 41) Under the least squares assumptions (zero conditional mean for the error term, Xi and Yi being 41) i.i.d., and Xi and ui having finite fourth moments), the OLS estimator for the slope and intercept A) has a normal distribution even in small samples. B) has an exact normal distribution for n > 15. C) is unbiased. D ) is BLUE. 42) Consider the following regression line: TestScore = 698.9 – 2.28 × STR. You are told that the
t-statistic on the slope coefficient is 4.38. What is the standard error of the slope coefficient? 42) A) 1.96 B) -1.96 C) 0.52 D ) 4.38 7
43) Imagine that you were told that the t-statistic for the slope coefficient of the regression line 43) TestScore = 698.9 – 2.28 × STR was 4.38. What are the units of measurement for the t-statistic? A) points of the test score B) standard deviations C) TestScore
STR D ) number of students per teacher 44) The construction of the t-statistic for a one- and a two-sided hypothesis 44) A) is the same. B) uses ±1.96 for the two-sided test, but only +1.96 for the one-sided test. C) depends on the critical value from the appropriate distribution. D ) is different since the critical value must be 1.645 for the one-sided hypothesis, but 1.96 for the
two-sided hypothesis (using a 5% probability for the Type I error). 45) The 95% confidence interval for Ά1 is the interval 45) A) (Ά1 - 1.96S E)(Ά1) , Ά1 + 1.96S E(Ά1)). B) ^ ^ (Ά1 - 1.96, Ά1 + 1.96). C) ^ ^ ^ ^ (Ά1 - 1.96S E)(Ά1) , Ά1 + 1.96S E(Ά1)). D ) ^ ^ ^ ^ (Ά1 - 1.645S E)(Ά1) , Ά1 + 1.645S E(Ά1)). 46) The 95% confidence interval for the predicted effect of a general change in X is 46) A) ^ ^ ^ ^ (Ά1Px - 1.96S E(Ά1) × Px,ȱΆ1Px + 1.96S E(Ά1) × Px). B) ^ ^ ^ ^ (Ά1Px - 1.645S E(Ά1) × Px,ȱΆ1Px + 1.645S E(Ά1) × Px). C) ^ ^ (Ά1Px - 1.96, Ά1Px + 1.96). D ) (Ά1Px - 1.96S E(Ά1) × Px, Ά1Px + 1.96S E(Ά1) × Px). 47) ^ The estimator of the variance of Ά1 is 47) A)
^
_
S
2
u
n
i=1
X 2
i - X
. B)
^
_
Su
n
i=1
Xi - X 2
.
C)
^
_
S
2
u
n
i=1
Xi - X 2
. D )
_ ^
_
1
n
×
1
n-2
n
i=1
Xi - X 2 u 2
i
1
n
n
i=1
Xi - X 2
2 .
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48) One of the following steps is not required as a step to test for the null hypothesis: 48) A) compute the p-value. B) ^ compute the standard error of Ά1.
C) compute the t-statistic. D ) test for the errors to be normally distributed. 49) The only difference between a one- and two-sided hypothesis test is 49) A) the sign of the slope coefficient. B) dependent on the sample size n . C) how you interpret the t-statistic. D ) the null hypothesis. 50) The proof that OLS is BLUE requires all of the following assumptions with the exception of: 50) A) the errors are normally distributed. B) the errors are homoskedastic. C) large outliers are unlikely. D ) E(ui Xi) = 0.