Name:_____________________________
Student #:__________________________
BUEC 333 FINAL EXAM
There is a formula sheet attached at the end of the exam and some blank pages for your rough work.
Use the bubble sheet to record your answers for the multiple choice questions and write your answers to the short answer questions in the space provided.
Multiple Choice (1 point each)
1.) Suppose that on any given trading day, a stock’s price can increase, decrease, or remain unchanged. Likewise, on any given day, the weather can be sunny or cloudy. You have the following information about the joint distribution of weather and stock price changes. It is sunny today. What is the probability of a price increase?
Sunny
Cloudy
Decrease
3/20
1/10
Unchanged Increase
3/10
1/5 1/10 a.) 1/4 b.) 3/10 c.) 3/16 d.) 3/20 e.) none of the above
2.) Suppose you have a random sample of 10 observations from a normal distribution with mean = 10 and variance = 2. The sample mean (x-bar) is 8 and the sample variance is 3. The sampling distribution of x-bar has a.) mean 8 and variance 0.2 b.) mean 10 and variance 2 c.) mean 8 and variance 3 d.) mean 10 and variance 0.3 e.) none of the above
3.) If q is an unbiased estimator of Q , then: a.) Q is the mean of the sampling distribution of q b.) q is the mean of the sampling distribution of Q c.) Var[ q ] = Var[ Q ] / n where n = the sample size d.) q = Q e.) a and c
4.) Suppose you want to test the following hypothesis at the 5% level of significance:
H
0
:
μ
=
μ
0
H
1
: μ ≠ μ
0
Which of the following is/are true? a.) the probability of a Type I error is 0.05 b.) the probability of a Type I error is 0.025 c.) the t statistic for this test has a t distribution with n-1 degrees of freedom d.) a and c e.) b and c
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5.) The significance level of a test is the probability that you: a.) reject the null when it is true b.) fail to reject the null when it is false c.) reject the null when it is false d.) fail to reject the null when it is true e.) none of the above
6.) Suppose that Y i
= α + βX i
+ ε i
and ε i
= u i
+ v i
+ X i
. Assume that u , v , and X are independently distributed as standard normals . In this case, the OLS estimator, beta-hat, is: a.) unbiased and efficient b.) unbiased and not efficient c.) biased and efficient d.) biased and not efficient e.) none of the above
7.) For a model to be correctly specified under the CLRM, it must: a.) be linear in the coefficients b.) include all relevant independent variables and their associated transformations c.) have an additive error term d.) all of the above e.) none of the above
8.) The Gauss-Markov Theorem says that when the 6 classical assumptions are satisfied: a.) The least squares estimator is unbiased b.) The least squares estimator has the smallest variance of all linear estimators c.) The least squares estimator has an approximately normal sampling distribution d.) The least squares estimator is consistent e.) None of the above
9.) The OLS estimator is said to be unbiased when: a.) Assumptions 1 through 2 are satisfied b.) Assumptions 1 through 3 are satisfied c.) Assumptions 1 through 4 are satisfied d.) Assumptions 1 through 5 are satisfied e.) Assumptions 1 through 6 are satisfied
10.) The sampling variance of the slope coefficient in the regression model with one independent variable: a.) will be smaller when there is more variation in ε b.) will be larger when there is less variation in ε c.) will be smaller when there is more variation in X d.) will be larger when there is more co-variation in ε and X e.) none of the above
11.) Omitting a relevant explanatory variable that is uncorrelated with the other independent variables causes: a.) bias and no change in variance b.) bias and an increase in variance c.) bias and a decrease in variance d.) no bias e.) none of the above
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12.) Consider that the correct model is Y i
= α + β
1
X
1i
+ β
2
X
2i
+ ε i
. Instead, you base your OLS estimates on the following model where Y i
= α + β
1
X
1i
+ ε i
. The correlation between X
1i
and X
2i
is positive. Furthermore, the population regression coefficient
β
2
is positive. The bias for the OLS estimator of
β
1
: a.) is positive b.) is negative c.) is zero d.) cannot be determined with the information provided. e.) none of the above.
13.) In a regression explaining earnings, you include one independent variable: an individuals' number of years of education as an independent variable but nothing else. You know that more educated people earn more. You also know that more educated people drink more. In this case, the bias for the
OLS estimate of the effect of education on earnings: a.) is positive b.) is negative c.) is zero d.) cannot be determined with the information provided. e.) none of the above.
14.) Suppose we estimate the linear regression Y
 wage, X i
is years of employment experience, and D i

X i

5 D i where Y i
is person i ’s hourly
equals one for men and zero for women. If we define a new variable F i
equal to one for men and zero for women instead, the least squares estimates would be: a.) Y i
 
X i

5 F i b.) Y i
X i

F i c.) Y i
 
X i

5 F i d.) Y i
 
X i

5 F i e.) none of the above
15.) Apart from inflating the F statistic and R
2
, there is another potential problem with excluding the constant term. By doing so, we are likely to make the OLS estimates of our included independent variables: a.) unbiased b.) inefficient c.) inconsistent d.) unstable e.) none of the above
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Short Answer (10 points each – show your work!)
1.) An oil exploration firm has recently acquired the drilling rights to two sites, one in Alberta and one in Texas. The exact quantity of oil that can be extracted from each site is unknown. A geophysicist estimates that the quantity of oil (in millions of barrels) at Site A is independent of the quantity of oil at Site B. Based on the available evidence, he assigns the following probability distribution to the quantities that can be extracted from each site:
Site A quantity
Probability
0
1/4
1
1/2
2
1/4
Site B quantity
Probability
0
1/6
1
1/2
2
1/3 a.) What is the probability that the total quantity of oil that can be extracted from the two sites is exactly 2 million barrels? b.) After purchasing the drilling rights, the firm conducts a test drilling at site A. The test drilling indicates that 1 million barrels can be extracted from that site. What is the expected total quantity of oil that can be extracted from the two sites? a) Since these fields are geographically separated from one another, we can assume that they are independent of one another. In that case,
Quantity 0 1 2 3 4
Probability (1/4)*(1/6) =
1/24
(1/4)*(1/2) +
(1/6)*(1/2) =
5/24
(1/4)*(1/3) +
(1/2)*(1/2) +
(1/6)*(1/4) =
9/24
(1/2)*(1/3) +
(1/4)*(1/2) =
7/24
(1/4)*(1/3) =
2/24
The probability that total production is equal to two million barrels is then 9/24 or 37.5%. b.) The expected value of production would simply be 1 million barrels plus the expected value of Site
B production, or 1 million + (0 million)*1/6 + (1 million)*1/2 + (2 million)*1/3 = 2.167 million barrels.
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2.) Suppose we have a linear regression model with one independent variable and no intercept :
Y i
= βX i
+ ε i
Suppose also that
ε i
satisfies the six classical assumptions. a.) Verbally, explain the steps necessary to derive the least squares estimator. b.) Formally, derive a mathematical expression for this estimator given your answer in part a).
For part a)
1.) Thus, we first have to define our residual as the difference between that which is observed and that which is predicted by the regression. In this way, the residual is best thought of as a prediction error, that is, something we would like to make as small as possible.
2.) Next, we need to define a minimization problem. Because our residuals will likely be both positive and negative, simply considering their sum is unsatisfactory as these will tend to cancel one another out. Additionally, minimizing the sum of residuals does not generally yield a unique answer. A better way forward is to minimize the sum of the squared “prediction errors” which will definitely yield a unique answer and which will penalize us for making big errors.
3.) We must take the derivatives of the sum of squared residuals with respect to the beta-hats and set them equal to zero. These first order conditions establish the values of beta-hat for which the sum of squared residuals “bottoms out” and is, thus, minimized.
4.) Finally, we must solve for the values of the beta-hats which are consistent with these first order conditions, thus, yielding our least squares estimators.
For part b) e i
  i

X i
Min
 ˆ n   i

1 e i
2  i n

1

Y i
 
X i
2

 i n

1
Y i
 n 
2
 i

1

Y i

X i n    i

1 i
2
This allows us to derive the following first order condition:
 n  i

1
  ˆ e i
2
 
2 i n 

1

Y X i i
 
2
 ˆ i n 

1
X i

0
 i n 

1

Y X i i
   ˆ i n 

1
X i
 ˆ i n 

1
 
 i n 

1

Y X i i

 ˆ  n  i

1
Y X i i n  i

1

0
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3.) Consider the following equation for a sample of divorced men who failed to make at least one child support payment in the last four years (standard errors in parentheses):
P i

2.0 0.50
M i

25.0
Y i

0.80
A i

3.0
B i

0.15
C
(0.10) (20.0) (1.00) (3.0) (0.05) i
Where: P is the number of missed child support payments for the i -th divorced man
Y is the share of income taken up by child support payments for the i -th divorced man
M
A
is the number of months of unemployment for the
is the age of the i -th divorced man i -th divorced man
B is an index of how strongly the i -th divorced man expresses his religious beliefs
C is the number of children for the i -th divorced man a.) What is the unit of measurement of a standard error? b.) Why is the standard error of Y so much larger than that of C ? Hint: think of the Venn diagrams developed in Lecture 12. c.) You expected that the coefficients on M and Y are positive. Test these two hypotheses at the 5% level of significance, assuming that N is so large that t converges on a normal distribution. d.) Test the hypothesis that the coefficient on A is not zero at the 1% level of significance. e.) Are there any potential violations of the CLRM in the equation above? If so, what are they and what do they imply about your answers to c) and d) above? a) Standard errors are measured in the same units as the regressor whose coefficient they correspond to, e.g., the 0.10 number attached to M is measured in months as well. This allows us to form a tstatistic by dividing the coefficient value by the standard error to arrive at a unit-less measure of how far t is dispersed from the central tendency of the t-distribution. b) The standard error vitally depends on the amount of variation in the independent variable. In particular, this is the amount of unique variation corresponding to the areas of X and Y which overlap in the Venn diagram. If that area is small because of multicollinearity, then the standard error is large and the t-statistics is consequently low. c) From the table of critical values, the critical value for a one-sided test at the 5% level of significance is 1.645. In that case,
Coefficient: β
M
Hypothesized + sign:
β
Y
+
Calculated t-statistic:
5.0 1.25
Significance: Significant Insignificant
Thus, we reject the null hypothesis that the coefficient on M is less than or equal to zero. Likewise, we fail to reject the null hypothesis that the coefficient on Y is less than or equal to zero. d) From the table of critical values, the critical value for a two-sided test at the 1% level of significance is 2.576. In that case, a t-value of 0.80 does not clear the hurdle, so we fail to reject the null hypothesis that coefficient is equal to zero. e) C and Y are certainly correlated (perhaps even perfectly if there are strict and binding formulas for the payment of child support). This would be most relevant for our conclusions in part c) as little unique variation in Y could contribute to its insignificance (as the standard error is large and the power is low).
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4.) Suppose the data generating process is: ln( WAGE )
 
0
 
1
( EXPERIENCE )
 
2
( EDUCATION )
 
3
( MALE )
 
4
( AGE )
  where: WAGE is the wage earned in the labour market;
EXPERIENCE is number of years of labour market experience;
EDUCATION is years of formal education;
MALE is a dummy variable that equals 1 if a person is male, and 0 otherwise;
AGE is a person’s age. a.) If you omit EXPERIENCE from the model, how do you expect it to affect your estimate of β
4
?
Be precise, and explain your reasoning. b.) What if you omit MALE instead? Be precise, and explain your reasoning. c.) Now suppose that

1

0 . Does that change your answer to part a? Explain. d.) Now suppose you want to test whether the marginal effect of AGE is the same for men and for women. How would you test this hypothesis? a) EXPERIENCE and AGE will almost certainly be positively correlated. We would also expect the
“true” coefficient for EXPERIENCE to be positive in sign (more experience generates higher wages).
Thus, by excluding EXPERIENCE but including AGE, we are likely to induce an upwards bias on the coefficient attached to AGE. b.) In this case, the effects are less clear. Unless there is a compositional difference in the work force whereby male workers are systematically older than female workers, we may expect that no bias will result as the correlation between MALE and AGE is probably zero (or at least, very small and of a non-obvious sign). c) Absolutely. Even though EXPERIENCE and AGE are positively correlated, the zero coefficient suggests that EXPERIENCE has no effect on WAGE. Thus, EXPERIENCE is an irrelevant variable and, thus, should be excluded from the regression. d) This would come from interacting MALE and AGE. Thus, you would test whether the coefficient attached to this new variable, MALE*AGE, was equal to zero.
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5.) You have been hired by a hedge fund as a market analyst. Your first day on the job, your boss tells you that oil prices have her “worried about excess portfolio volatility” and that she is “looking for investments that hedge against future oil price increases”. She asks you to analyze the relationship between oil prices and the rate of return on several assets, hoping that you can find one asset whose returns are unrelated to oil prices and one asset whose returns are negatively related to oil prices. You spend the better part of the day running least squares regressions and settle on this one (standard errors in parentheses): ln t
 
A t

0.025
B t
(0.45) (0.00001) (0.001) n

43 R
2 
0.35 F

10.8 r
AB

0.02
where: O t
is the price of one barrel of crude oil in month t
A t
is the monthly rate of return on shares of Apple of California
B t
is the monthly rate of return on shares of the Potash Corporation of Saskatchewan r
AB
is the sample correlation between A t
and B t a.) Interpret the value of the R
2 and the F statistic (assume a critical value of 3.23) b.) Interpret the estimated coefficients on A t
and B t
. c.) One of your co-workers thinks B t
satisfies your boss’ request for an asset “whose returns are negatively related to oil prices”. Another co-worker thinks you should test this to be sure.
State the appropriate null hypothesis, test it, and interpret the result. d.) One of your co-workers thinks A t
satisfies your boss’ request for an asset “whose returns are unrelated to oil prices”. Another co-worker disagrees because “the coefficient on
A t
is statistically significant”. Which of them is correct? Explain. a.) The R
2
represents the proportion of the total variation in ln(O) around its mean which is explained by variation in the independent variables. In this regression, we are, thus, explaining 35% of the variation in logged oil prices….good, but not great. The F statistic is the value of the test statistic under the null hypothesis that all the true coefficients attached to the independent variables are equal to zero. Given a value of 10.8 relative to a critical value of 3.23), we are therefore going to reject this particular null hypothesis on the joint insignificance of our independent variables. b.) These suggest that if the shares of Apple rise by 1 percentage point this is associated with a .041% rise in the price of a barrel of oil. Likewise, if the shares of the Potash Corporation rise by 1 percentage point this is associated with a 2.5% fall in the price of a barrel of oil. c.) This would amount to a one-sided test of the null hypothesis that the beta-hat is greater than or equal to zero. At the 5% level, the critical value is 1.684. The value of t-statistic in this case is 25.0.
Therefore, you can easily reject the null hypothesis as the t-statistic is very far away from values which would be consistent with a zero or positive coefficient. d.) True, the coefficient on A is statistically significant with a massive t-statistic of 41.0 (compared to a 1% critical value on a two-sided test of 2.704). At the same time, there are a lot of zeros to the right of the decimal point. Thus, we have to contrast statistical and economic significance. In this case, a coefficient value of 0.00041 suggests that shares of ABC Corporation would have to rise by 25 percentage points before oil prices budged even 1%. This is about as unrelated as we can expect in real world data.
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6.) The first half of the course was dedicated to developing the least squares estimator. The rest of the course was dedicated to considering those instances when problems with the least squares estimator arise. Underlying the discussion, there were the six assumptions of the classical linear model. a.) Name the six assumptions and explain what each of them mean. b.) Some of these assumptions are necessary for the OLS estimator to be unbiased. Some of these assumptions are necessary for the OLS estimator to be “best”. Explain the distinction between these two concepts. c.) Indicate which of the six assumptions are necessary for the OLS estimator to be unbiased and which of the six assumptions are necessary for the OLS estimator to be “best”. d.) In general, would you prefer your estimates to be biased but efficient or unbiased but not efficient?
Explain your answer. a) The regression model is: a.) linear in the coefficients, b.) is correctly specified, and c.) has an additive error term.
The error term has zero population mean or E(ε i
) = 0.
All independent variables are uncorrelated with the error term, or Cov(X i
,ε i
) = 0 for each independent variable X i
(we say there is no endogeneity).
Errors are uncorrelated across observations, or Cov(ε i
,ε j
) = 0 for two observations i and j (we say there is no serial correlation).
The error term has a constant variance, or Var(ε i
) = σ
2
for every i (we say there is no heteroskedasticity).
No independent variable is a perfect linear function of any other independent variable (we say there is no perfect collinearity). b) Unbiasedness relates to the property whereby the expected value of an estimator is equal to the population parameter of interest.
“Best” relates to the size of the sampling variance of any such estimator with the lower, the better.
Blah, blah, blah… c) Of the assumption listed above the first three are required for unbiasedness. Four through six are necessary for the OLS estimator to be “best”. d) Itis probably better to have an indication that your estimator is centered on the population parameter “on average” rather than be “wrong” but very precisely estimated. Thus, bias is the greater sin than inefficiency (although I am open to students persuasively arguing the opposite if we think of a small bias versus large variance case).
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