SUMMER TERM 2018
ECON2007: QUANTITATIVE ECONOMICS AND ECONOMETRICS
TIME ALLOWANCE: 3 hours
Answer ALL TWO questions from Part A and answer ONE question from Part B.
Questions in Part A carry 60 per cent of the total mark and questions in Part B carry 40 per cent of
the total. Tables for the normal and F-distribution are at the end of the examination paper.
In cases where a student answers more questions than requested by the examination rubric, the policy
of the Economics Department is that the student’s first set of answers up to the required number will
be the ones that count (not the best answers). All remaining answers will be ignored.
PART A
Answer all questions from this section.
A.1 Consider the following regression model without an intercept,
yi = βxi + ui, (1)
where we have observed (yi, xi) while ui is unobserved, i = 1, ..., n. We are interested in doing
inference regarding the unknown parameter β.
(a) Write up the sum of squared residuals for the model, and derive the first-order condition
that the OLS estimator, βˆ, of β has to satify. Show that βˆ is given by
βˆ =
∑n
i=1 xiyi∑n
i=1 x
2
i
. (2)
ANSWER: The sum of squared residuals (SSR) takes the form
SSR (β) =
n∑
i=1
(yi − βxi)2
with first-order condition
0 =
∂SSR(βˆ)
∂β
= 2
n∑
i=1
(yi − βˆxi)xi = 2
n∑
i=1
yixi − 2βˆ
n∑
i=1
x2i
Solving for βˆ yields (2).
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(b) State conditions on the model and the data under which the OLS estimator will be consis-
tent.
ANSWER: Under the following conditions the OLS estimator is consistent:
i. (yi, xi), i = 1, ..., n, are i.i.d. (and all relevant moments exist)
ii. (yi, xi) satisfies eq. (1), i = 1, ..., n.
iii. E [u|x] = 0.
iv. E
[
x2
]
> 0.
(c) Derive the large-sample distribution of the OLS estimator under the conditions you provided
in (b). In doing so, (i) state which asymptotic limit results you rely on and (ii) where in
your derivations you employ these limit results together with the conditions you provided
in (b).
βˆ =
∑n
i=1 xiyi∑n
i=1 x
2
i
=
∑n
i=1 xi (βxi + ui)∑n
i=1 x
2
i
= β +
1
n
∑n
i=1 xiui
1
n
∑n
i=1 x
2
i
. (3)
In large samples, using that E [uixi] = 0 (due to (iii)),
1
n
n∑
i=1
xiui ∼ N
(
0,
Var (ux)
n
)
,
1
n
n∑
i=1
x2i ' E
[
x2
]
, (4)
where we have employed the CLT and LLN for i.i.d. data (and so using (i)), respectively.
Since E
[
x2
]
> 0, as imposed in (iv), (3)-(4) yield
βˆ ∼ β +
N
(
0, Var(ux)n
)
E [x2]
= N
(
β,
1
n
× Var (ux)
E [x2]2
)
.
(d) Provide a consistent estimator of the standard errors of the OLS estimator.
ANSWER: From (c), Var(βˆ) ' 1n Var(ux)E[x2]2 which is consistently estimated by V̂ar(βˆ) =
1
n
V̂ar(ux)
Eˆ[x2]2
, where
V̂ar (ux) =
1
n
n∑
i=1
x2i uˆ
2
i , Eˆ
[
x2
]
=
1
n
n∑
i=1
x2i ,
and uˆi = yi − βˆxi, i = 1, ..., n.
(e) Suppose that data is in fact generated by the following regression model,
yi = β0 + β1xi + ui,
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where β0 6= 0. Will the βˆ in eq. (2) be a biased estimator of β1? If so, derive the bias of
the estimator.
ANSWER: Yes, it is a biased estimator unless
∑n
i=1 xi = 0:
βˆ =
∑n
i=1 xi (β0 + β1xi + ui)∑n
i=1 x
2
i
= β1 +
1
n
∑n
i=1 xiui
1
n
∑n
i=1 x
2
i
+ β0
∑n
i=1 xi∑n
i=1 x
2
i
,
and so using that E [xiui|x1, ..., xn] = 0,
E[βˆ|x1, ..., xn] = β1 + β0
∑n
i=1 xi∑n
i=1 x
2
i
.
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A.2 In Table 10.1 on the next page you find seven different estimated regressions numbered (1)-(7)
for the effect of drunk driving laws on traffic deaths (reprinted from Stock and Watson, 2007,
Introdution to Econometrics, Pearson Education). The data are for the “lower 48” U.S. states,
for 1982-88. The traffic fatality rate is the number of traffic deaths in a given state in a given
year, per 10,000 people living in that state in that year. The beer tax is the tax on a case of
beer in \$. Drinking age variables are binary indicating whether the legal drinking age is 18, 19,
or 20. Drinking age 21 is the excluded dummy variable.
(a) New Jersey has a population of 8.1 million people. Suppose that New Jersey increased the
tax on a case of beer by \$1. Use the results in column (4) to predict the number of lives
that would be saved over the next year in New Jersey. Construct a 99% confidence interval
ANSWER: (−.45× 810) = 364.5 less fatalities in the state of New Jersey. (Since fatality
rate measured in deaths per 10,000 people living in that state in that year, and since New
Jersey’s population in 1998 is 8.1 million). A 99% confidence interval is
[−0.45± (2.58× 0.22)]× 810 = [−1.0176,−0.1176]× 810 = [−824.256, 95.256].
(b) Explain what is meant by “time effects” which are included in the regression in column (5).
Do they seem to matter? Explain.
ANSWER: Time effects are time dummies for each year (1982-88) of the sample, ex-
cluding one. The time dummies seem to be weakly significant: The joint test for their
significance has a p-value of p = 3.7%. Thus, if we test at a 5% level, we reject the null
of no time effects, while if we test at 1% level, we accept the null and conclude that time
effects are jointly not significant.
(c) A researcher conjectures that the unemployment rate has a different effect on traffic fatali-
ties in the Western states compared to the other states. How would you test this hypothesis?
(Be clear about the specification of the regression and the statistical test you would use.)
ANSWER: We need to introduce a dummy variable for Western states and generate an
interaction variable between this dummy variable and unemployment rate. The null hy-
pothesis of no difference between Western and Eastern states would be that the regression
coefficient associated with this interaction variable is zero. This could be test using the
corresponding t-statistic.
(d) Write up the two regression models whose estimates are reported in columns (1) and (2).
Explain which type of omitted variable bias the model in column (2) controls for relative
to the model in column (1). Which type of omitted variable bias does the model in column
(2) not control for?
column (1) : deathsit = β0 + β1beer taxit + uit,
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column (2) : deathsit = β0 + β1beer taxit + ai + uit,
where deathsit and beer taxit are #traffic deaths and beer tax in state i in year t, respec-
tively, while ai is the state effect. Regression model (2) controls for omitted factors that
are time-invariant/constant over times. It does not control for omitted variables that are
varying over time.
(e) Based on the estimates reported in columns (1) and (2), does the aforementioned omitted
variable bias seem to be a concern in this application?
ANSWER: Comparing the two estimates of β1, βˆ1 = 0.36 and βˆ1 = −0.66, we see that
the introduction of state effects has a sizable impact on the estimate. So yes, controlling
for state effects is important – without these the resulting estimate βˆ1 = 0.36 suffers from
a severe bias.
(f) Provide two different estimators of the regression in column (2). Is one preferable to the
other?
ANSWER: One can either use first-differencing or within estimators:
βˆ1 =
∑n
i=1
∑T
t=2 ∆deathsit∆beer taxit∑n
i=1
∑T
t=2 ∆beer tax
2
it
β˜1 =
∑n
i=1
∑T
t=1 deathsitbeer taxit∑n
i=1
∑T
t=1 beer tax
2
it
,
where ∆deathsit = deathsit−deathsit−1, ∆beer taxit = beer taxit−∆beer taxit−1, deathsit =
deathsit − 1T
∑T
t=1 deathsit, and beer taxit = beer taxit − 1T
∑T
t=1 beer taxit. There is no
clear ranking of the two estimators: Depending on the autocorrrelation structure in the
errors, one will be more efficient than the other.
(g) What is meant by “Clustered standard errors?”? Do they seem to matter in this applica-
tion?
ANSWER: Clustered standard errors are standard errors that are robust to potential
autocorrelation in the regression errors over time. The use of these do have an impact on
our inference: For example, comparing column (6) and (7), we see that the effect of beer
tax becomes insignificant when using clustered standard errors.
PART B
Answer ONE question from this section.
B.1 In “Virtual Classrooms: How Online College Courses Affect Students” (American Economic
Review, Vol.107(9), 2017), Eric Bettinger, Lindsay Fox, Susanna Loeb and Eric Taylor explore
the effects of taking a college course online, instead of in-person, on outcomes such as student
ECON2007 6 CONTINUED
achievement and progress in college. In their data, each course is offered both online and
in-person, and each student enrolls in either an online section or an in-person section. Both
sections are identical otherwise: they follow the same syllabus and textbook and class sizes are
approximately the same. Evaluations and marking standards are also homogeneous.
(a) Let Di denote whether a student enrols in an online course (Di = 1) or in an in-person
section (Di = 0). The authors have information on whether the in-person section is offered
at the student’s home campus (Availi ) and how far the student is to his/her local campus
(Disti). The former variable is a dummy = 1 if the student’s local campus offers an in-
person section and = 0, otherwise. The latter variable records distances in tens of miles.
They allow those variables to affect a student’s decision to take an online or in-person
section. More precisely, the authors consider the following (Linear Probability) model for
the decision to take an online or in-person section:
P(Di = 1|Avail,Dist) = β0 + βAvailAvaili + βDistDisti + βAvail×DistiAvaili ×Disti. (5)
(The authors also allow other student characteristics to influence their decision, but I omit
those here for simplicity.) Write down the expression for the Partial Average Effect (PAE)
for a marginal change in Disti. Explain how the computation of the Partial Average Effect
(PAE) compares with the computation of the Partial Effect at the Average (PEA) for this
particular model.
ANSWER: The PAE is the average of βˆDist + βˆAvail×DistiAvaili. The PEA is βˆDist +
βˆAvail×Disti
∑N
i=1 Availi/N , so PAE = PEA.
(b) The model above can be represented as:
Di = 1, if β0 + βAvailAvaili + βDistDisti + βAvail×DistiAvaili ×Disti − Ui ≥ 0
= 0, otherwise.
If the error term Ui is uniformly distributed on the interval [0, 1], one obtains the Linear
Probability Model above. Suppose instead that the error term Ui follows a normal distribu-
tion with zero mean and unit variance (i.e., Ui ∼ N (0, 1)). Write down the (log-)likelihood
for this model and the expression for the PAE of a marginal change in Disti.
HINT: Let Φ(·) and φ(·) denote the CDF and PDF of a standard normal random variable,
respectively.
∑N
i=1
[
Di ln Φ(β0 + βAvailAvaili + βDistDisti +
βAvail×DistiAvaili×Disti)+(1−Di) ln(1−Φ(β0+βAvailAvaili+βDistDisti+βAvail×DistiAvaili×
ECON2007 7 TURN OVER
Disti))
]
. The PAE is the average of φ(βˆ0 + βˆAvailAvaili + βDistDisti + βˆAvail×DistiAvaili ×
Disti)× (βˆDist + βˆAvail×DistiAvaili)
(c) The main goal of the paper is to estimate the effect of online courses on student outcomes
(Yi) (for example, achievement or progress in college). To address this question, the authors
focus on a model similar to:
Yi = δ0 + δDDi + δAvailAvaili + δDistDisti + Vi,
where Di,Availi and Disti are defined as above and Vi is an unobservable error term.
(Again, the authors also allow other student characteristics to influence the outcome Yi,
but I omit those here for simplicity.) The authors worry that the decision to enrol in
an online course may be correlated with unobservable factors encoded in Vi and use the
interaction Availi × Disti as an instrumental variable for Di. Describe how you would
implement the TSLS estimator in this context. The table below presents OLS and TSLS
estimates (and their standard errors) for δD. (Yi here is the Course Grade ranging from
A = 4 to F = 0 and standard errors are in parenthesis.)
Yi = Course Grade (A = 4, . . . , F = 0)
OLS TSLS
Took course online (Di) -0.381 -0.440
(0.012) (0.049)
Sample mean for 2.821
dependent variable
Observations 2,323,023
How do you interpret the results? Describe how you would test whether Di is endogenous.
ANSWER: TSLS: ( 1 ) Regress Di on Availi,Disti and Availi × Disti. ( 2 ) Regress Yi
on Availi,Disti and Dˆi. Taking an online course leads to a point estimate reduction of 0.44
points out of a 2.821 average in the Course Grade. The endogeneity of Di could be tested
used Hausman test: regress outcome on covariates and residual from first-stage regression.
(d) The F -statistic for the hypothesis that βAvail×Disti = 0 in the Linear Probability Model
represented by equation (5) is approximately 100. Explain why this is important in this
context. The authors also write that it is important that “(i) any mechanism through which
ECON2007 8 CONTINUED
students’ distance from campus affects course grades (i.e., Yi) is constant across terms with
and without an in-person class option; and (ii) any mechanism causing grades (i.e., Yi) to
differ between terms with and without an in-person class option affects students homoge-
neously with respect to their distance from campus.” Why is this important?
ANSWER: The F -statistic indicates that the IV is associated with the endogenous vari-
able (relevance). The following discussion is concerned with exogeneity of the instrumental
variable (validity).
(e) Let Y t now denote the average Course Grade in the college for year t. Suppose you are
interested in modelling the dynamic evolution of Y t using the following autoregressive
model:
Y t = ρ0 + ρ1Y t−1 + ηt, |ρ1| < 1.
If you estimate the model by OLS, is the estimator unbiased and consistent if ηt is serially
correlated? Explain.
ANSWER: Biased since strict exogeneity is violated. The estimator may be inconsistent
if the serial correlation leads to a violation of contemporaneous exogeneity. For example, if
ηt = γηt−1 + Vt, then cov(Y t−1, ηt) = cov(ρ0 + ρ1Y t−2 + ηt−1, γηt−1 + Vt) 6= 0.
(f) How would you test whether there is serial correlation in ηt?
ANSWER: Explain Durbin’s h test procedure.
B.2 This question is based on “Capital Accumulation and Growth: A New Look at the Empirical
Evidence”, by Steve Bond, Asli Leblebicioglu and Fabio Schiantarelli (Journal of Applied Econo-
metrics, Vol.25, 2010). In this article, the authors are interested in a regression model for the
(logarithm of) output-per-capita yt in a given country and time period t similar to:
yt = α+ ρyt−1 + βxt + γt+ t, (6)
where xt is (the logarithm of) investment-per-output. Assume for items (a)-(c) that t is not
serially correlated.
(a) Imagine that investment rates are also affected by current output-per-capita, so that
xt = ψ0 + ψyyt + ηt. (7)
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Assume ηt is not serially correlated. Equations (6) and (7) then form a simultaneous equa-
tion system. Obtain the reduced form equations for yt and xt. Is equation (6) identified?
ANSWER: The reduced form equations are
yt = pi10 + pi11yt−1 + pi12t+ ν1
xt = pi20 + pi21yt−1 + pi22t+ ν2
where
pi10 =
α+ βψ0
1− βψy
pi11 =
ρ
1− βψy
pi12 =
γ
1− βψy
pi20 =
ψ0 + αψy
1− βψy
pi21 =
ψyρ
1− βψy
pi22 =
γψy
1− βψy
The first equation contains two endogenous variables and two exogenous variables. It can-
not be identified.
(b) The second equation has two endogenous variables (yt and xt) and two (contemporaneously)
exogenous variables (yt−1 and t). It can be identified using yt−1 and t as instrumental vari-
ables for yt. Explain how you would implement the TSLS estimator in this context. How
would you assess relevance?
ANSWER: TSLS: ( 1 ) Regress yt on yt−1 and t. ( 2 ) Regress xt on yˆt. To assess
relevance, look at the F -statistic for the hypothesis that pi11 = pi12 = 0.
(c) A classmate suggests that you can perform an over-identification test since there are two
instrumental variables and only one endogenous regressor. Describe how to implement such
ECON2007 10 CONTINUED
a test (under homoskedasticity). Would you be able to perform this test if there were only
one instrumental variable?
ANSWER: Overidentification test: 1. Estimate the coefficients by TSLS and obtain
residuals ηˆt. 2. Regress ηˆt on all exogenous variables. Record the R
2. 3. Under the
null hypothesis that all IVs are exogenous NR2 ∼ χ21. If there were only one instrumental
variable the statistic would be degenerate and the test would not be feasible.
(d) Assume that xt = ψ0 + ηt (ψy = 0), ρ = 0 and γ = 0. Furthermore, suppose you only ob-
serve yt if yt ≥ 10 and whether this occurs or not. Under the assumption that t ∼ N (0, σ2),
write down the (log-)likelihood for the model.
HINT: Let Φ(·) and φ(·) denote the CDF and PDF of a standard normal random variable,
respectively. The CDF for Ui is F (u) = Φ(u/σ) and its PDF is f(u) = φ(u/σ)/σ.
ANSWER: Write the log-likelihood for the Tobit model. The likelihood is for a single
observation is:
Φ
(
10− α− βxt
σ
)1(yt<10)
×
(
1
σ
φ
(
yt − α− βxt
σ
))1(yi≥10)
.
The likelihood function is the product over all observations. The log-likelihood function is
its log. Note that symmetry of the normal distribution implies that the expression above can
also be written as:(
1− Φ
(
α+ βxt − 10
σ
))1(yt<10)
×
(
1
σ
φ
(
yt − α− βxt
σ
))1(yt≥10)
.
(e) Assume that xt = ψ0 + ηt (i.e., ψy = 0), ρ = 0 and γ = 0. How would you test whether t
is serially correlated? Explain.
(f) Assume that xt = ψ0 + ηt (i.e., ψy = 0), β = 0 and γ = 0. Take a country for which yt is
stationary (i.e., |ρ| < 1), so that
yt = α+ ρyt−1 + t.
ECON2007 11 TURN OVER
If t = λt−1 + νt, is the OLS estimator for α and ρ consistent?
ANSWER: No, in this case cov(yt−1, t) = λcov(yt−1, t−1) 6= 0.
ECON2007 12 CONTINUED
5 % Critical values for the Fν1,ν2 distribution
ν2\ν1 1 2 3 4 5 6 7 8 10 12 15 20 30 50 ∞
1 161 199. 216. 225. 230. 234. 237. 239. 242. 244. 246. 248. 250. 252. 254.
2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 19.5 19.5 19.5
3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.79 8.74 8.70 8.66 8.62 8.58 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 5.96 5.91 5.86 5.80 5.75 5.70 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.74 4.68 4.62 4.56 4.50 4.44 4.36
10 4.96 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.38 2.31 2.23 2.16 2.07 2.00 1.88
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.35 2.28 2.20 2.12 2.04 1.97 1.84
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.16 2.09 2.01 1.93 1.84 1.76 1.62
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 1.99 1.92 1.84 1.75 1.65 1.56 1.39
80 3.97 3.11 2.72 2.49 2.33 2.21 2.13 2.06 1.95 1.88 1.79 1.70 1.60 1.51 1.32
100 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.93 1.85 1.77 1.68 1.57 1.48 1.28
120 3.91 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.91 1.83 1.75 1.66 1.55 1.46 1.25
∞ 3.85 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.83 1.75 1.67 1.57 1.46 1.35 1.00
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NORMAL CUMULATIVE DISTRIBUTION FUNCTION (Prob(z < za) where z ∼ N(0, 1))
za 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7703 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
ECON2007 14 END OF PAPER