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 e↵ect 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 = 0 + 1SMi + 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 1 obtained by regressing AM on SM .

ANSWER: With u˜ = u v,

AMi = 0 + 1SMi + 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) = 2u + 2v .

Thus, (2) also satisfies SLR.1-SLR.5 and we obtain

E

h

ˆ1|SM1, ..., SMn

i

= 1, Var(ˆ1|SM1, ..., SMn) =

2

nˆ2SM

,

where 2 =Var(u˜i|xi).

ECON0019 1 TURN OVER

(b) You use the following estimator of the variance of the OLS estimator ˆ1 as described in (a),

dVar(ˆ1) = ˆ2

nˆ2SM

, ˆ2 =

1

n 2

nX

i=1

uˆ2i , ˆ

2

SM =

1

n

nX

i=1

SMi SM

2

,

where uˆi = AMi ˆ0 ˆ1SMi, i = 1, ..., n. Is this a consistent estimator of the variance of

ˆ1? 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 gSM i = SMi + vi where vi still satisfies the assumptions stated in (a),

i = 1, ..., n. Derive the probability limit of the OLS estimator of 1 obtained by regressing

AM⇤ on gSM i.

ANSWER: With u˜ = u 1v,

AM⇤i = 0 + 1

⇣gSM i vi⌘+ ui = 0 + 1gSM i + u˜i, (3)

where SLR.1-SLR.5 combined with E [vi] = 0 and (SMi, ui) ? vi yield

E

h

u˜igSM ii = 12v

Thus, by the LLN,

ˆ1 = 1 +

1

n

Pn

i=1

⇣gSM i gSM⌘ u˜i

ˆ2gSM !

p 1

1

2

v

2gSM

!

.

(d) You obtain a consistent estimator ˆ2v of

2

v =Var(v). Use ˆ

2

v to develop a consistent esti-

mator of 1.

ANSWER: First compute the OLS estimator in (c), ˆ1, and then

ˆ⇤1 = ˆ1

1 ˆ

2

v

ˆ2gSM

!1

.

Combining the answer to (c) with ˆ2v !p 2v , we obtain ˆ⇤1 !p 1.

(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 e↵ect 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,

\math = 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

\math ⇡ 11.13

100

(%spend)

and so

\math ⇡ 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 1 denoting the coecient for log (spend), we wish to test H0 : 1 = 0 vs

HA : 1 > 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 a↵ects students’ math performance.

ECON0019 3 TURN OVER

(c) You conjecture that family background has an e↵ect 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

2 = 3 = 0 to hold in the following regression for meal to be a valid proxy for poverty,

poverty = 0 + 1meal + 2 log(spend) + 3 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 2 is not too big, the bias will be

negiglible

(d) Including meal you obtain the following results,

\math = 23.14 + 7.75 log(spend) 1.26 log (enroll) .324meal, R2 = .1893.

(24.99) (3.04) (.580) (.036)

Explain why the e↵ect 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 coecient on meal is negative (.324). From Wooldridge we then

find that omitting meal from the regression produces an upward biased estimator of 1 [ig-

noring the presence of log (enroll) in the model]. So when we control for the poverty rate,

the e↵ect of spending falls.

(e) Interpret the coecients on log (enroll) and meal.

ANSWER: Once we control for meal, the coecient on log (enroll) becomes negative with

t-stat of –2.17, which is significant at the 5% level against a two-sided alternative. The

coecient implies that

\math ⇡ 1.26

100

(%enroll)

Therefore, a 10% increase in enrollment leads to a drop in math of .126 percentage points,

a small e↵ect. 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 e↵ect.

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 E↵ects of Entry on

Incumbent Innovation and Productivity,” (The Review of Economics and Statistics, Vol.91,

No.1, 2009), Philippe Aghion, Richard Blundell, Rachel Grith, Peter Howitt and Susanne

Prantl study the e↵ects 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(0 + 1EFj + 2Dj + 3Dj ⇥ EFj ),

where Pj is the count of patents for firm j in a given year, EFj 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 EFj 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 2 to be between 0.582 and 0.852 (depending on the specification used).

Does this imply that the partial e↵ect 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 > 0, its probability mass function

is

P(Y = k) =

k exp()

k!

for k = 0, 1, 2, . . . .

ANSWER: The log-likelihood function is:

nX

j=1

{pj(0 + 1eFj + 2dj + 3dj ⇥ eFj ) exp(0 + 1eFj + 2dj + 3dj ⇥ eFj )}

ECON0019 6 CONTINUED

(omitting terms that do not depend on teh coecients of interest). The PEA with respect

to Dj is:

(ˆ2 + ˆ3EFj )⇥ exp(ˆ0 + ˆ1EFj + ˆ2Dj + ˆ3Dj ⇥ EFj ).

Its sign is thus that of (ˆ2 + ˆ3EFj ) which may di↵er from the sign of ˆ2.

(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 e↵ected entry di↵erentially 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, EFj :

LPj = ↵0 + ↵1E

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 a↵ect 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 a↵ect 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 = ⇡ˆ0 + ⇡ˆ1Zj , where ⇡ˆ0 and ⇡ˆ1 are OLS estimates from a regression of E

F

j on a

constant and Zj . If one uses EˆFj as an instrumental variable instead of Zj how would the

estimates compare with those obtained in the previous item? Elaborate.

Hint: Since ⇡ˆ0 and ⇡ˆ1 are obtained by OLS, EFj = Eˆ

F

j + Vj = ⇡ˆ0 + ⇡ˆ1Zj + Vj and

ECON0019 7 TURN OVER

Pn

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 coecient of this regression is given byPn

j=1(Eˆ

F

j EˆFj )(EFj EFj )Pn

j=1(Eˆ

F

j EˆFj )2

= 1 +

Pn

j=1(Eˆ

F

j EˆFj )VjPn

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 = ↵0 + ↵1E

F

t + ↵2LPt1 + 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 EFt or LPt1.

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 + ✏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 o↵-trade (supermarkets and o↵-licences).

(a) Assume that ✏ ⇠ N (0,2). Provide the (log-)likelihood function for the above model.

Hint: The cummulative distribution function for ✏ is F✏(e) = (e/) and its probability

density function f✏(e) = (e/)/ where (·) and (·) 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 / (where is the

standard deviation of ✏j) to estimates of the coecients 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 coecients correspond to /.

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 coecients

from a probit regression to those from the Tobit model. The estimated Tobit coecients,

ˆ, must be divided by the estimated parameter ˆ to make this comparison possible. As we

saw in section 4, whilst this parameter does not a↵ect the sign of the estimated marginal

e↵ect, it does impact its magnitude. If the assumptions of the Tobit model are valid, then

the probit coecients should be largely equivalent to the modified Tobit coecients ˆ/ˆ.”

(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 = k(x>i /) which

can be estimated by ˆk(x>i ˆ/) once ML estimates are produced. One can then estimate

the “average elasticity” (simlar to APE) as:

N1

NX

i=1

ˆk(x

>

i ˆ/ˆ)/yi

or the “elasticity at the average” (similar to PEA) as:

ˆk(x

>ˆ/ˆ)/y.

(d) Suppose that instead of individual data, you have access to data on the market shares for

o↵-trade beer (i.e., beer bought in supermarkets and o↵-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 o↵-trade beer:

logSm = 0 + 1 logEm + 2 logPm + ✏m (5)

where Sm is the market share for o↵-trade beer in market m, Em is the expenditure on

alcohol in market m and Pm is the price for o↵-trade beer in market m. (Assume that o↵-

trade beer prices are uniform within a market.) Since the market share for o↵-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 o↵-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 = 0 + 1 logEt + 2 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 F⌫1,⌫2 distribution

⌫2\⌫1 1 2 3 4 5 6 7 8 10 12 15 20 30 50 1

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

1 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

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