SUMMER TERM 2019
ECON0019: 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 You wish to quantify the effect of cannabis consumption on student performance. You carry out
a survey asking a random sample of your fellow students about their average mark after two
years of studies and number of times they have consumed cannabis in the last 30 days. Let AMi
and SMi be student i’s self-reported average mark and number of times used, i = 1, ..., n, where
n is the number of students in the sample.
(a) Suppose that AM is observed with measurement error while SM is observed without. That
is, AMi = AM
∗
i +vi, where AM
∗
i is the actual average mark and vi is the measurement error.
The measurement error is assumed to be fully independent of (SMi, ui) with E [vi] = 0,
i = 1, ..., n. Suppose that the actual average mark satisfies
AM∗i = β0 + β1SMi + ui, (1)
and that SLR.1-SLR.5 are satisfied in the above model. Derive the (conditional on SM1, ..., SMn)
mean and variance of the OLS estimator of β1 obtained by regressing AM on SM .
ANSWER: With u˜ = u− v,
AMi = β0 + β1SMi + u˜i. (2)
SLR.1-SLR.5 combined with E [vi] = 0 and (SMi, ui) ⊥ vi yield
E [u˜i|SMi] = 0 and Var (u˜i|SMi) = σ2u + σ2v .
Thus, (2) also satisfies SLR.1-SLR.5 and we obtain
E
[
βˆ1|SM1, ..., SMn
]
= β1, Var(βˆ1|SM1, ..., SMn) = σ
2
nσˆ2SM
,
where σ2 =Var(u˜i|xi).
ECON0019 1 TURN OVER
(b) You use the following estimator of the variance of the OLS estimator βˆ1 as described in (a),
V̂ar(βˆ1) =
σˆ2
nσˆ2SM
, σˆ2 =
1
n− 2
n∑
i=1
uˆ2i , σˆ
2
SM =
1
n
n∑
i=1
(
SMi − SM
)2
,
where uˆi = AMi− βˆ0− βˆ1SMi, i = 1, ..., n. Is this a consistent estimator of the variance of
βˆ1? Explain.
ANSWER: Since (2) satisfies SLR.1-SLR.5, we know from the lectures/Wooldridge that the
above variance estimator is consistent.
(c) Consider the reverse situation: You observe the actual mark average AM∗ but now instead
of SM you observe S˜M i = SMi + vi where vi still satisfies the assumptions stated in (a),
i = 1, ..., n. Derive the probability limit of the OLS estimator of β1 obtained by regressing
AM∗ on S˜M i.
ANSWER: With u˜ = u− β1v,
AM∗i = β0 + β1
(
S˜M i − vi
)
+ ui = β0 + β1S˜M i + u˜i, (3)
where SLR.1-SLR.5 combined with E [vi] = 0 and (SMi, ui) ⊥ vi yield
E
[
u˜iS˜M i
]
= −β1σ2v
Thus, by the LLN,
βˆ1 = β1 +
1
n
∑n
i=1
(
S˜M i − S˜M
)
u˜i
σˆ2
S˜M
→p β1
(
1− σ
2
v
σ2
S˜M
)
.
(d) You obtain a consistent estimator σˆ2v of σ
2
v =Var(v). Use σˆ
2
v to develop a consistent esti-
mator of β1.
ANSWER: First compute the OLS estimator in (c), βˆ1, and then
βˆ∗1 = βˆ1
(
1− σˆ
2
v
σˆ2
S˜M
)−1
.
Combining the answer to (c) with σˆ2v →p σ2v, we obtain βˆ∗1 →p β1.
(e) Still considering the scenario in (c), discuss how realistic the following two assumptions
are, E [vi] = 0 and vi fully independent of (SMi, ui), when the measurement error is due to
incorrect reporting of cannabis consumption.
ECON0019 2 CONTINUED
ANSWER: First, students who smoke may well likely understate their true consumption
and so E [v] < 0. Second, even if they try to tell the truth, then most likely students who
don’t smoke (SM∗ = 0) are very likely to report SM = 0; but students who do smoke
(SM∗ > 0) are more likely to miscount number of times they smoked. This implies that v
and SM∗ are likely dependent.
(f) Suppose that you observe SM and AM without measurement error. However, some of the
students that you asked to participate in the survey refused. Is this a concern regarding
the validity of SLR.1-SLR.5?
ANSWER: SLR.4 may be violated in the sample. If the selection (reason for not partici-
pating) is mean-independent of u so that E[u|s,M ] = 0, where s is the selection dummy
variable, SLR.4 will still hold for the selected sample. However, if the sample selection
non-random (dependent on u) SLR.4 will fail to hold.
A.2 You are interested in estimating the effect of per-student spending on math performance. For
that purpose, you use a data set on 408 schools in the UK. For each school, the data set contains
math, the percentage of students receving a passing mark in a standardized math test, together
with spend, per-student spending, and enroll, number of students enrolled.
(a) You obtain the following regression results,
m̂ath = −69.24 + 11.13 log(spend) + 0.22 log (enroll) , R2 = .0297.
(26.72) (3.30) (.615)
If spend increases by 10% what is the (approximate) estimated percentage change in math?
ANSWER: We have
∆m̂ath ≈ 11.13
100
(%∆spend)
and so
∆m̂ath ≈ 11.13
100
× 10 = 1.113.
That is, we expect 1.113% more students to pass the math test if we increase spending by
10%.
(b) Test the hypothesis that math does not change with spend against the alternative that it
does increase with spend. Perform the test at a 5% and 1% level. Conclude.
ANSWER: With β1 denoting the coefficient for log (spend), we wish to test H0 : β1 = 0 vs
HA : β1 > 0. The t-stat is tobs = 11.13/3.30 = 3.37 which we then compare with critical val-
ues 1.645 (5%) and 2.326 (1%). We reject the null at both levels and so conclude that there
is strong statistical evidence of that school spending affects students’ math performance.
ECON0019 3 TURN OVER
(c) You conjecture that family background has an effect on student performance and would
like to include poverty, the percentage of students in a given school that live in poverty,
in your regression. However, this variable is not in the data set and you instead decide
to include meal, the percentage of students eligible for free school meals, as an additional
regressor. Is this a sensible strategy? Explain.
ANSWER: The usual proxy variable argument should be employed: First, eligibility for
the free school meals is very tightly linked to being economically disadvantaged. Therefore,
the percentage of students eligible for free school meals is very similar to the percentage of
students living in poverty. Thus, we expect it to be a good proxy. Formally, we need that
δ2 = δ3 = 0 to hold in the following regression for meal to be a valid proxy for poverty,
poverty = δ0 + δ1meal + δ2 log(spend) + δ3 log(enroll) + v.
Even after controlling for meal, school spending may predict level of poverty in which case
using meal as a proxy will lead to biased results. If δ2 is not too big, the bias will be
negiglible
(d) Including meal you obtain the following results,
m̂ath = −23.14 + 7.75 log(spend)− 1.26 log (enroll)− .324meal, R2 = .1893.
(24.99) (3.04) (.580) (.036)
Explain why the effect of spending on math is lower in this new regression compared to the
one in (a).
ANSWER: First note that meal is found to be relevant. We can then use our usual reason-
ing on omitting important variables from a regression equation. The variables log(spend)and
meal are negatively correlated: school districts with poorer children spend, on average, less
on schools. Further, the coefficient on meal is negative (−.324). From Wooldridge we then
find that omitting meal from the regression produces an upward biased estimator of β1 [ig-
noring the presence of log (enroll) in the model]. So when we control for the poverty rate,
the effect of spending falls.
(e) Interpret the coefficients on log (enroll) and meal.
ANSWER: Once we control for meal, the coefficient on log (enroll) becomes negative with
t-stat of –2.17, which is significant at the 5% level against a two-sided alternative. The
coefficient implies that
∆m̂ath ≈ 1.26
100
(%∆enroll)
Therefore, a 10% increase in enrollment leads to a drop in math of .126 percentage points,
a small effect. Both math and meal are percentages. Therefore, a ten percentage point
increase in meal leads to about a 3.23 percentage point fall in math, a sizeable effect.
ECON0019 4 CONTINUED
(f) What do you make of the increase in R2 from the regression in (a) to the regression in (d)?
ANSWER: The regression in (a) explains less than 3% of the variation in math while the
one in (d) explains almost 19%. Most of the (explained) variation in math must therefore
be due to meal. This seems to indicate that family income (or related factors, such as living
in poverty) are much more important in explaining student performance than are spending
per student or other school characteristics.
ECON0019 5 TURN OVER
PART B
Answer ONE question from this section.
B.1 Schumpeterian growth theory implies that the threat of technologically advanced entry spurs
innovation incentives in sectors close to the technology frontier, where successful innovation
allows incumbents to survive the threat, but discourages innovation in laggard sectors, where
the threat reduces incumbents’ expected rents from innovating. In “The Effects of Entry on
Incumbent Innovation and Productivity,” (The Review of Economics and Statistics, Vol.91,
No.1, 2009), Philippe Aghion, Richard Blundell, Rachel Griffith, Peter Howitt and Susanne
Prantl study the effects of firm entry on labour productivity — more specifically, the real output
per employee in the firm — and innovation — more specifically, the count of patents issued to the
firm — taking into account how far the industry of interest is from the technological frontier. The
authors use data from the United Kingdom and measure distance to the technological frontier by
comparing the labour productivity in the industry in the United Kingdom to labour productivity
in the same industry in the United States.
(a) To study the relationship between entry, distance to the frontier and patent counts, the
authors use a Poisson model. Suppose you decide to estimate a similar (i.e., Poisson model)
where the expected number of patents is given by:
E(Pj |Dj , EFj ) = exp(β0 + β1EFj + β2Dj + β3Dj × EFj ),
where Pj is the count of patents for firm j in a given year, E
F
j measures the entry rate of
foreign firms in firm j’s industry in the previous year and Dj measures the distance from
the technological frontier. Both Dj and E
F
j are continuous. Write down the expression
for the (log-)likelihood used to compute the Maximum Likelihood Estimator. In their
estimates (which uses a somewhat more sophisticated version of the model above), the
authors estimate β2 to be between 0.582 and 0.852 (depending on the specification used).
Does this imply that the partial effect at the average (PEA) for distance to the technological
frontier is positive? Please elaborate on your answer.
Hint: If Y follows a Poisson distribution with parameter λ > 0, its probability mass function
is
P(Y = k) =
λk exp(−λ)
k!
for k = 0, 1, 2, . . . .
ANSWER: The log-likelihood function is:
n∑
j=1
{pj(β0 + β1eFj + β2dj + β3dj × eFj )− exp(β0 + β1eFj + β2dj + β3dj × eFj )}
ECON0019 6 CONTINUED
(omitting terms that do not depend on teh coefficients of interest). The PEA with respect
to Dj is:
(βˆ2 + βˆ3EFj )× exp(βˆ0 + βˆ1EFj + βˆ2Dj + βˆ3Dj × EFj ).
Its sign is thus that of (βˆ2 + βˆ3EFj ) which may differ from the sign of βˆ2.
(b) The authors note that “entry can be endogenous to innovation and productivity growth”
and consider a set of instrumental variables related to policy reforms related to entry:
“reforms at the European level and reforms at the U.K. level that changed the entry costs
and effected entry differentially across industries and time.” The European reforms were
undertaken as part of the Single Market Programme and deemed to reduce medium or high
entry barriers. The U.K. reforms include, for instance, privatization cases which resulted
in opening up markets to firm entry. Consider then the following simple linear regression
model for labour productivity growth, ∆LPj , as it relates to entry, E
F
j :
∆LPj = α0 + α1E
F
j + Uj , (4)
where Uj is an unobserved error. Suppose you have at your disposal one instrumental vari-
able Zj that consolidates information about the implementation of the reforms alluded to
above. Describe how you would implement the TSLS estimator in this context. How would
you argue for the validity of this instrument?
ANSWER: TSLS: ( 1 ) Regress EFj on Zj. ( 2 ) Regress ∆LPj on Eˆ
F
j . The in-
strumental variable is valid if cov(zj , uj) = 0. This means that any unobserved variables
that affect the number of patents do not vary systematically with this variable. This will be
the case if Thatcher era privatisations and the EU Single Market Programme only influence
ilabour productivity through entry and do not affect it directly.
(c) How can you use the estimates from (4) above to test whether EFj is endogenous?
ANSWER: Describe Hausman regression-based test for endogeneity.
(d) Let EˆFj = pˆi0 + pˆi1Zj , where pˆi0 and pˆi1 are OLS estimates from a regression of E
F
j on a
constant and Zj . If one uses Eˆ
F
j as an instrumental variable instead of Zj how would the
estimates compare with those obtained in the previous item? Elaborate.
Hint: Since pˆi0 and pˆi1 are obtained by OLS, E
F
j = Eˆ
F
j + Vj = pˆi0 + pˆi1Zj + Vj and
ECON0019 7 TURN OVER
∑n
j=1(Eˆ
F
j − EˆFj )Vj = 0. Furthermore, EFj = EˆFj .
ANSWER: The first stage using EˆFj is obtained by the OLS estimates of a regression of
EFj on Eˆ
F
j . The slope coefficient of this regression is given by∑n
j=1(Eˆ
F
j − EˆFj )(EFj − EFj )∑n
j=1(Eˆ
F
j − EˆFj )2
= 1 +
∑n
j=1(Eˆ
F
j − EˆFj )Vj∑n
j=1(Eˆ
F
j − EˆFj )2
= 1
and the intercept is given by EFj − EˆFj = 0. Consequently the forecast for EFj used in the
second stage in the item above is exactly the same.
(e) Imagine you have time series data for a single firm and estimate the following time-series
regression by OLS:
∆LPt = α0 + α1E
F
t + α2∆LPt−1 + Ut,
Would the estimator be unbiased? Under what conditions would it be consistent? Elabo-
rate on your answers.
ANSWER: The estimator would be biased since the model does not satisfy strict exo-
geneity. It will be consistent if Ut is not correlated with E
F
t or ∆LPt−1.
B.2 To study alcohol consumption in the UK, James Collis, Andrew Grayson and Surjinder Johal
(“Econometric Analysis of Alcohol Consumption in the UK”, HRMC Working Paper 10, Decem-
ber 2010) use data from the Expenditure and Food Survey (2001-2006) to estimate the following
model:
Y ∗j = X
>
j β + j
Yj = max{Y ∗j , 0}
where Yj is the proportion of total expenditure on a particular category of alcohol by household j
and the explanatory variables Xj include (log) prices for all alcohol categories, (log) income and
other controls. The alcohol categories analyzed were beer, wine, spirits, cider and ready-to-drink
(RTDs, also known as ‘alcopops’). Each category was also subdivided into on-trade (pubs and
restaurants) and off-trade (supermarkets and off-licences).
(a) Assume that ∼ N (0, σ2). Provide the (log-)likelihood function for the above model.
Hint: The cummulative distribution function for is F(e) = Φ(e/σ) and its probability
density function f(e) = φ(e/σ)/σ where Φ(·) and φ(·) are, respectively, the cummulative
ECON0019 8 CONTINUED
distribution function and the probability density function for the standard normal distribu-
tion.
ANSWER: TOBIT likelihood.
(b) To assess the adequacy of the Tobit, the authors compare estimates of β/σ (where σ is the
standard deviation of j) to estimates of the coefficients from a Probit where the dependent
variable is whether expenditure on alcohol (for particular categories) is zero or positive.
Part of the table is reproduced below (for the purposes of the exam, it is irrelevant whether
the table cells or numbers are shaded or not):
Explain why this comparison might be useful.
ANSWER: The model for y = 0 or 6= 0 is a Probit and its coefficients correspond to β/σ.
As indicated by the authors: “Wooldridge proposes an informal evaluation of the general ap-
propriateness of the Tobit model. This is conducted by comparing the estimated coefficients
from a probit regression to those from the Tobit model. The estimated Tobit coefficients,
βˆ, must be divided by the estimated parameter σˆ to make this comparison possible. As we
saw in section 4, whilst this parameter does not affect the sign of the estimated marginal
effect, it does impact its magnitude. If the assumptions of the Tobit model are valid, then
the probit coefficients should be largely equivalent to the modified Tobit coefficients βˆ/σˆ.”
(c) The researchers are ultimately interested in the elasticities with respect to prices (own-
and cross-) and to income. Since those variables are entered as logarithms, you decide to
ECON0019 9 TURN OVER
estimate those as:
= ∂E(Y |X = x)/∂xk/y
where xk is the relevant variable for the elasticity of interest (i.e., log of own price, log of
substitute category or log of income). Explain how you would estimate the elasticity for a
particular household. Suggest a measure of elasticity for the general population and explain
how you would estimate it.
ANSWER: Given the model with normal residuals, ∂E(y|x)/∂xk = βkΦ(x>i β/σ) which
can be estimated by βˆkΦ(x
>
i βˆ/σ) once ML estimates are produced. One can then estimate
the “average elasticity” (simlar to APE) as:
N−1
N∑
i=1
βˆkΦ(x
>
i βˆ/σˆ)/yi
or the “elasticity at the average” (similar to PEA) as:
βˆkΦ(x
>βˆ/σˆ)/y.
(d) Suppose that instead of individual data, you have access to data on the market shares for
off-trade beer (i.e., beer bought in supermarkets and off-licences) and prices for each of
the alcohol categories in several local markets in the United Kingdom. Consider then the
following model for the market share for off-trade beer:
logSm = β0 + β1 logEm + β2 logPm + m (5)
where Sm is the market share for off-trade beer in market m, Em is the expenditure on
alcohol in market m and Pm is the price for off-trade beer in market m. (Assume that off-
trade beer prices are uniform within a market.) Since the market share for off-trade beer
depends not only on the variables above, but also on other variables not included in the
model (e.g., prices for other alcohol categories), you decide to use a variable Zm encoding
the distribution costs of supermarkets or off-licences for beer (e.g., average distance to beer
producers) as an instrumental variable for logPm. (Assume that logEm is uncorrelated
with m.) Describe how you would implement the TSLS estimator in this context. How
would you argue for the validity of this instrument?
ANSWER: TSLS: ( 1 ) Regress logPm on Zm and logEm. ( 2 ) Regress logSm on
̂logPm and logEm. The instrumental variable is valid if cov(Zm, m) = 0. This means
that distribution costs for beer are not correlated with the unobservable m. If on-trade beer
prices are also related to Zm, the validity may be threatened.
ECON0019 10 CONTINUED
(e) Consider now equation (5) for a single market but across many periods t and suppose there
are no endogeneity issues:
logSt = β0 + β1 logEt + β2 logPt + t
Explain how you would test whether there is serial correlation in t. Would serial correla-
tion imply that OLS is inconsistent?
ANSWER: Explain Durbin-Watson. Serial correlation would not necessarily imply incon-
sistency.
ECON0019 11 TURN OVER
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
ECON0019 12 CONTINUED
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
ECON0019 13 END OF PAPER