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程序代写案例-TERM 2020
时间：2021-05-06
SUMMER TERM 2020
ONLINE 24-HOUR EXAMINATION
ECON0019: QUANTITATIVE ECONOMICS AND ECONOMETRICS
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points, but the penalised mark will not be reduced below the pass mark: marks already at or below
the pass mark will not be reduced.
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.
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cheating, and falsification of extenuating circumstances.
ECON0019 1 TURN OVER
PART A
Answer all questions from this section.
A.1 You have randomly sampled n individuals whom you follow over T ≥ 2 time periods. For
individual i (= 1, ..., n) you observe (yit, xit), t = 1, ..., T , which satisfies
yit = β0 + β1xit + ai + uit, t = 1, ..., T. (1)
(a) Show that
∆yit = β1∆xit + ∆uit, i = 1, ..., n, t = 2, ..., T. (2)
Discuss in detail the advantages and disadvantages of using eq. (2) instead of eq. (1) for
estimation and inference.
ANSWER:
• Advantages of OLS based on (2): No potential biases due to xit being correlated with
ai.
• Dis-advantages of OLS based on (2): Generally bigger variance due to fewer observa-
tions after transformation In particular, (2) requires variation in xit over time while
(1) does not. However, depending on the autocorrelation structure of ∆uit vs the one
of uit, OLS based on (2) may still perform better even in terms of variance.
• For both regressions, clustered standard errors will generally be needed. So in this sense,
neither dominates the other.
(b) Write up the sum of squared residuals (SSR) for (2) given your sample. Suppose here and
in the following that
∑n
i=1
∑T
t=2 (∆xit)
2 > 0 in your sample. Show that the minimizer of
SSR is
βˆ1 =
∑n
i=1
∑T
t=2 ∆xit∆yit∑n
i=1
∑T
t=2 (∆xit)
2
,
where you explain each step of your derivation.
ANSWER:
SSR =
n∑
i=1
T∑
t=2
(∆yit − β1∆xit)2 .
The minimizer βˆ1 satisfies
0 = −1
2
∂SSR
∂β1
=
n∑
i=1
T∑
t=2
(
∆yit − βˆ1∆xit
)
∆xit
=
n∑
i=1
T∑
t=2
∆yit∆xit − βˆ1
n∑
i=1
T∑
t=2
(∆xit)
2
ECON0019 2 CONTINUED
⇒ βˆ1 =
∑n
i=1
∑T
t=2 ∆xit∆yit∑n
i=1
∑T
t=2 (∆xit)
2
.
(c) Suppose that
uit = uit−1 + eit
where E [eit|xi1, ...., xiT ] = 0, t = 1, ...., T . Show that for any i, t,
E [∆uit|X1, ..., Xn] = 0,
where Xi = (∆xi2, ....,∆xiT ), i = 1, ..., n. Use this in turn to show that βˆ1 is unbiased. As
part of your proof, carefully explain each step.
ANSWER:
E [∆uit|X1, ..., Xn] = E [eit|X1, ..., Xn]
=
(*)
E [eit|∆Xi]
= E [E [eit|xi1, ...., xiT ] |Xi]
= 0,
where (*) uses that data is randomly sampled. Write
βˆ1 = β1 +
∑n
i=1
∑T
t=2 ∆xit∆uit∑n
i=1
∑T
t=2 (∆xit)
2
, (3)
and take conditional expectations on both sides,
E[βˆ1|X1, ...., Xn] = β1 +
∑n
i=1
∑T
t=2 ∆xitE[∆uit|X1, ...., Xn]∑n
i=1
∑T
t=2 (∆xit)
2
= β1
(d) Suppose furthermore that
E
[
e2it|xi1, ...., xiT
]
= σ2,
E [eiseit|xi1, ...., xiT ] = 0, s 6= t.
Demonstrate that
Cov (∆uis,∆uit|X1, ..., Xn) =
{
σ2, s = t
0, s 6= t .
Use this in turn to derive an expression of the conditional variance of βˆ1, Var(βˆ1|X1, ...., Xn),
where you carefully explain each step of your derivation. Comment on the resulting variance
ECON0019 3 TURN OVER
expression. In particular, how is the variability of the OLS estimator affected by the
variation of the error term and the regressor?
ANSWER:
Cov (∆uis,∆uit|X1, ..., Xn) = E [eiseit|X1, ..., Xn]
=
(*)
E [eiseit|Xi]
= E [E [eiseit|xi1, ...., xiT ] |Xi]
=
{
σ2, s = t
0, s 6= t ,
where (*) uses that data is randomly sampled. Thus,
Var(βˆ1|X1, ...., Xn) = Var
(∑n
i=1
∑T
t=2 ∆xit∆uit∑n
i=1
∑T
t=2 (∆xit)
2
∣∣∣∣∣X1, ...., Xn
)
=
1[∑n
i=1
∑T
t=2 (∆xit)
2
]2Var
(
n∑
i=1
T∑
t=1
∆xit∆uit
∣∣∣∣∣X1, ...., Xn
)
=
1[∑n
i=1
∑T
t=2 (∆xit)
2
]2 n∑
i,j=1
T∑
s,t=2
∆xis∆xitVar (∆uis∆ujt|X1, ...., Xn)
=
1[∑n
i=1
∑T
t=2 (∆xit)
2
]2 n∑
i=1
T∑
t=2
(∆xit)
2 σ2
=
σ2∑n
i=1
∑T
t=2 (∆xit)
2
.
We see that the variance of the OLS estimator increases as a function of σ2, which makes
sense: More noise in data makes it harder to learn about β1. Reversely, higher realised
variation of ∆xit provides a stronger signal in data about the value of β1 and so decreases
the variance of the OLS estimator
(e) Assume that Pr
(∑T
t=2 (∆xit)
2 = 0
)
= 0. Show that this implies E
[∑T
t=2 (∆xit)
2
]
>
0. Show consistency of βˆ1 under this assumption, where you clearly explain each step,
including which assumptions and limit results that you employ. In particular, explain why
the assumption stated at the beginning of this question is needed.
ECON0019 4 CONTINUED
ANSWER: By the LLN for i.i.d. data,
1
n
n∑
i=1
T∑
t=2
∆xit∆uit → p
T∑
t=2
E [∆xit∆uit] =
T∑
t=2
E [∆xitE [∆uit|X1, ..., Xn]] = 0,
1
n
n∑
i=1
T∑
t=2
(∆xit)
2 → p
T∑
t=2
E
[
(∆xit)
2
]
> 0.
Consistency now follows from (3) together with the continuous mapping theorem. We can
apply the continuous mapping theorem since f(a, b) = a/b is continuous everywhere except
at b = 0 and b = 0 is ruled out due to the assumption at the beginning of the question.
A.2 An extension of the Solow growth model, that includes human capital in addition to physical
capital, suggests that investment in human capital (education) will increase the wealth of a
nation (per capita income). To test this hypothesis, you collect data for 104 countries and
perform the following regression:
r̂elinc = 0.046 − 5.869gpop+ 0.738sk + 0.055educ, (4)
(0.079) (2.238) (0.294) (0.010)
with R2 = 0.775, standard error of residual SER = 0.1377, and heteroskedasticity-robust stan-
dard errors reported in parentheses. Here, relinc is GDP per worker relative to the United
States, gpop is the average population growth rate, 1980 to 1990, sk is the average investment
share of GDP from 1960 to 1990, and educ is the average educational attainment in years for
1985.
(a) Discuss the implications and validity of each of the following assumptions in the context of
the above regression:
i. Data is i.i.d.
ii. E [u|gpop, sk, educ] = 0 where u is the regression error.
In the following we will assume that (i)-(ii) are satisfied together with other relevant tech-
nical assumptions.
ANSWER:
• Re. (i): First, data needs to come from the same population in order for us to set
up a model for this population. Second, random sampling is used in the analysis of
the variance of the OLS estimators, incl computation of standard errors. In terms of
validity, first note that each observational unit is a country. Thus, if we define the
population as all countries in the world, the sample is by definition drawn from this
population and so identically distributed. Data would then be i.i.d. if we had randomly
ECON0019 5 TURN OVER
sampled the 108 countries. But the question does not specify how data is collected so the
answer to this part is open ended. However, it seems unlikely that they are randomly
sampled - there are only around 200 countries in the world and many of these do not
publish reliable statistics. If they are not randomly sampled, then the independence
assumption is most likely violated due to spill-over effects (trade, migration, etc): For
example, a given country’s income level will likely depend on the income levels of its
trading partners.
• Re. (ii): This assumption is needed to ensure that OLS is unbiased. In our application,
the error term here contains other factors affecting GDP per worker, incl institutional
and geographical differences. These are most likely correlated with gpop, sk and educ
in which case (ii) is violated.
(b) Interpret the above regression results and indicate whether or not the coefficients are sig-
nificantly different from zero. Do the coefficients have the expected sign? Explain.
ANSWER:
• A one percentage point decrease in the population growth rate increases GDP per worker
relative to the United States by roughly 0.06.
• An increase in the investment share of 0.1 results in an increase of GDP per worker
relative to the United States by approximately 0.07.
• For every additional year of average educational attainment, the increase is 0.055.
• The regression explains 77.5 percent of the variation in relative productivity.
• All coefficients are significantly different from zero at conventional levels.
• All coefficients carry the expected sign.
(c) To test for equality of the coefficients between the OECD and other countries, you introduce
a binary variable (oecd), which takes on the value of one for the OECD countries and is
zero otherwise. You obtain the following regression estimates:
r̂elinc = −0.068− 0.063gpop+ 0.719sk + 0.044educ (5)
(0.072) (2.271) (0.365) (0.012)
+0.381oecd− 8.038(oecd× gpop)− 0.430(oecd× sk)
(0.184) (5.366) (0.768)
+0.003(oecd× educ)
(0.018)
where R2 = 0.845 and SER = 0.116. Write down the two regression functions, one for the
OECD countries, the other for the non-OECD countries. Explain. Interpret any differences.
ANSWER:
non-OECD : r̂elinc = −0.068− 0.063gpop+ 0.719sk + 0.044educ.
ECON0019 6 CONTINUED
OECD : r̂elinc = 0.313− 8.101gpop+ 0.289sk + 0.047educ.
We see that relinc in non-OECD countries tend to be much less negatively affected by
population growth while increases in investment has a much bigger impact.
(d) In order to test (4) against (5), you compute the corresponding F -statistic which takes the
value 6.76 in your sample. Write up the null hypothesis and its alternative that you are
testing in terms of the population regression coefficients. What do you conclude? Explain.
ANSWER:
H0 : βoecd = βoecd×gpop = βoecd×sk = βoecd×educ = 0
versus
HA : At least one of the above coefficients is non-zero.
We have 4 restrictions and so F -statistic follows the F4,∞ distribution in large samples.
The critical value is 2.37 at the 5% level and hence we can reject the null hypothesis that
the coefficients are equal. We conclude that data supports the hypothesis that there are
differences in relinc between OECD and non-OECD countries after controlling for gpop,
sk and educ.
(e) You decide to investigate further and estimate a restricted version of (5) where you enforce
the same slopes across OECD and non-OECD countries, but allow their intercepts to differ.
In this new regression, the t-statistic for oecd is 3.17. What is the p-value of the t-statistic?
What do you conclude? Explain your answer.
ANSWER: t-statistic follows N (0, 1) distribution in large samples and so its p-value is
p = Pr (|t| > 3.17) = 2 (1− Pr (t ≤ 3.17)) = 2 (1− 0.9992) = 0.16%.
That is, there’s only 0.16% probability of observing a more extreme outcome under the null.
In particular, we reject at a 1% level. We conclude that, under the maintained hypothesis
that the effect of gpop, sk and educ are the same between the two groups, the level of relinc
between OECD and non-OECD countries is different.
(f) Next, you test the model described in (e) against (5). The value of the corresponding
F -statistic is 1.05. Do you accept or reject the null?
Looking at the tests in this and two previous questions, what is your overall conclusion?
Explain your answer.
ANSWER: We here test
H0 : βoecd×gpop = βoecd×sk = βoecd×educ = 0,
and so so F -statistic follows the F3,∞ distribution in large samples with corresponding
critical value of 2.68 at the 5% level. Thus, we cannot reject the null hypothesis, and so
there appears to be no differences in the impact of gpop, sk and educ on relinc between
OECD and non-OECD countries. The over-all conclusion is that there only seems to be a
level difference between the two groups of countries.
ECON0019 7 TURN OVER
PART B
Answer ONE question from this section.
B.1 Intergenerational mobility is related to several aspects. For example, theoretical studies have
examined the repercussions of the transmission of preferences and attitudes from parents to
children. Thomas Dohmen, Armin Falk, David Huffman and Uwe Sunde (“The Intergenera-
tional Transmission of Risk and Trust Attitudes”) use the German Socio-Economic Panel Study
(SOEP) to empirically examine, among other things, the transmission of attitudes from par-
ents to children and potential mechanisms for such transmission. Aside from comprehensive
demographic information on all individuals in a given household, the survey contains a set of
individual questions regarding risk attitudes (in 2004). (The authors also look at trust.) People
were asked questions eliciting their willingness to take risks on an eleven-point scale. For these
variables, zero (0) would correspond to ‘completely unwilling to take risks’ and the value ten
(10) means that the person is ‘completely willing to take risks.’
(a) One possible way to investigate the transmission of risk attitudes is to examine how parental
characteristics (including their risk attitudes) relate to the probability that a child has a
high score in terms of the risk attitude measure elicited on an 11-point scale as indicated
above. To do this, generate a variable Di = 1 if the child in household i has risk attitude
measure equal to 6 or above and Di = 0, otherwise. (While separate measures are available
for both parents, to keep matters simple we focus here on a single measure for parents.)
Taking RPi to be the parental score for that same measure in the household, suppose you
are interested in the model:
Di = 1(β0 + β1R
P
i + Ui ≥ 0).
Assuming that Ui follows a standard logistic distribution, write down the log-likelihood
function for this estimation problem when you have N observations. How would you es-
timate the difference in the probability that Di = 1 between a household where R
P
i = 10
and another one where RPi = 0? Please explain your answer.
ANSWER: The log-likelihood function is:
n∑
j=1
{−b0 − b1RPi − ln[1 + exp(−b0 − b1RPi )]} × (1−Di)− ln[1 + exp(−b0 − b1RPi )]×Di
or equivalently
n∑
j=1
{b0 + b1RPi − ln[1 + exp(b0 + b1RPi )]} ×Di − ln[1 + exp(b0 + b1RPi )]× (1−Di).
ECON0019 8 CONTINUED
Once estimates are obtained, the estimated difference in probabilities is given by:
exp(−βˆ0)/(1 + exp(−βˆ0))− exp(−βˆ0 − βˆ110)/(1 + exp(−βˆ0 − βˆ110))
or, equivalently,
exp(βˆ0 + βˆ110)/(1 + exp(βˆ0 + βˆ110))− exp(βˆ0)/(1 + exp(βˆ0)).
(b) Because risk attitudes for children (RCi ) and parents (R
P
i ) are measured contemporaneously,
the authors worry about ‘reverse causality’ where children’s attitudes may be at least partly
shaping parents’ attitudes. To address this issue they estimate
RCi = α0 + α1R
P
i + Vi,
using parental religion (Zi) as an instrumental variable for R
P
i . Describe in detail how you
would implement the TSLS estimator in this context. Discuss the validity of the instru-
mental variable suggested in this context. (Explain your intuition.)
ANSWER: TSLS: ( 1 ) Regress RPi on Zj. ( 2 ) Regress R
C
i on Rˆ
P
i . The instrumental
variable is valid if cov(Zi, Vi) = 0. This means that any unobserved variables that affect the
child’s risk attitudes should be uncorrelated with parental religion.
(c) The F-statistic for the first stage regression using the mother’s risk attitudes as covariate
in the main equation of interest and her religion as instrumental variable is 9.99. (The
F-statistic when using father’s risk attitudes and religion is 7.32.) Discuss in detail the
relevance of the instrumental variable.
ANSWER: The values for the F-statistics suggest that the instrument may not be suffi-
ciently strong and thus present substantial bias.
(d) In a regression where the risk attitude for both mother and father are included individually
as covariates in a multiple linear regression model, both coefficients on those variables are
around 0.15 with standard errors at around 0.02 for each one of them. The TSLS estimates
on the other hand, produce estimates for the coefficient on the mother’s risk attitude at
about 0.23 and for the coefficient on the father’s risk attitude at about 0.02. (Religion for
each parent is available as an intrumental variable for each of their risk attitude variables.)
The standard error for those estimates are, in both cases, around 0.10. Why would you
ECON0019 9 TURN OVER
expect the standard errors for the IV estimates to be larger than the standard errors for
the OLS estimates? Explain your answer.
ANSWER: Under homoscedasticity,
v̂ar(βˆIV ) =
σˆ2
SSTxR2x,z
>
σˆ2
SSTx
= v̂ar(βˆOLS)
(e) Imagine you had data on the risk attitude for successive generations of a single household
and you want to estimate the regression
RG+1 = α0 + α1RG + VG+1,
where RG+1 and RG are, once again, the risk attitudes in generation G+ 1 (child) and in
generation G (parent). Assuming these are not measured contemporaneously so that the
issues raised in item (b) are not present, are there conditions under which an OLS estimator
is unbiased? Elaborate on your answer.
ANSWER: The estimator would be biased since the model does not satisfy strict exo-
geneity.
B.2 In “Excess Capacity and Policy Interventions: Evidence from the Cement Industry,” Tetsuji
Okazaki, Ken Onishi and Naoki Wakamori estimate the demand for cement in Japan using data
on different regions across years. Their specification for the demand function is
ln(Qmt) = αP ln(Pmt) + α
>
XXmt + Umt,
where Qmt is the quantity of cement demanded in region m and year t (from 1970 to 1995), Pmt
is the price in that region and year and Xmt are year- and region-specific demand shifters. The
Ordinary Least Squares (OLS) estimate for α, denoted by α̂P,OLS, equals -0.07 with a standard
error equal to 0.16.
(a) Explain in detail why the above estimate for the slope coefficient (−0.07) cannot be directly
interpreted as the price-elasticity of demand for cement.
ANSWER: Simultaneity bias.
ECON0019 10 CONTINUED
(b) To produce cement, crushed limestone, cray and other minerals are mixed and put into a
kiln to be heated. This process yields clinker, which is an intermediate cement product.
In a final stage, the grinded clinker is mixed with gypsum, another intermediate input, to
produce cement. The researchers then use the (log) price of gypsum as an instrumental
variable for the (log) price of cement to estimate the price-elasticity of demand. The OLS
regression of (log) cement prices on (log) gypsum prices (and X) yields a coefficient of 0.06
and the F-test statistic for the first stage equals 17.0. Discuss in detail the exogeneity and
relevance of this instrumental variable.
ANSWER: First stage F − stat indicates that the IV is sufficiently correlated with the
endogenous variable. Validity holds if the price of gypsum is not correlated with other un-
observed determinants of demand for cement.
(c) To estimate the regression using the IV described above, the researchers use Two-Stage
Least Squares and obtain an estimate for α, denoted α̂P,TSLS, equal to -1.11 with a stan-
dard error equal to 0.58. Describe in detail the TSLS procedure. Is it possible to test
whether the IV is exogenous? Explain in detail. What if there were two instrumental
variables? Explain in detail.
ANSWER: Textbook. With two IVs one could use overidentification restrictions to test
exogeneity, but not with only one IV.
(d) Suppose the researchers were also interested in examining the time series behaviour for the
quantity of cement sold in a particular region in Japan on a given year, ln(Qt). To do so,
they obtain estimates for the following autoregressive model using data over various years
for this region of Japan:
ln(Qt) = α0 + α1 ln(Qt−1) + ηt.
Would the OLS estimator be unbiased in this case? Under what assumptions would it be
consistent? Explain your answers in detail.
ANSWER: The OLS estimator is biased, but consistent.
(e) Suppose the researchers only observe whether Qmt is larger or smaller than a given fixed
threshold Q in a given year but otherwise observe prices and X. Let Dmt record whether
Qmt > Q (Dmt = 1) or not (Dmt = 0). While the regression
ln(Qmt) = αP ln(Pmt) + α
>
XXmt + Umt
ECON0019 11 TURN OVER
is no longer estimable, they are still able to estimate the model given by
Dmt =
{
1 if βP ln(Pmt) + β
>
XXmt + Vmt > ln(Q)
0 if βP ln(Pmt) + β
>
XXmt + Vmt ≤ ln(Q)
Assume that the error term follows a standard normal distribution (i.e., Vmt ∼ N (0, 1))
and write down the log-likelihood function for this model assuming that the data comprises
of a random sample. If Umt ∼ N (0, σ2) how are βP and αP related? Explain your answer
in detail.
ANSWER: Log-likelihood function for the probit. βP = αP /σ.
ECON0019 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
ECON0019 13 TURN OVER
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 14 END OF PAPER
学霸联盟
ONLINE 24-HOUR EXAMINATION
ECON0019: QUANTITATIVE ECONOMICS AND ECONOMETRICS
All work must be submitted anonymously. Please ensure that you add your candidate number and the
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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
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cheating, and falsification of extenuating circumstances.
ECON0019 1 TURN OVER
PART A
Answer all questions from this section.
A.1 You have randomly sampled n individuals whom you follow over T ≥ 2 time periods. For
individual i (= 1, ..., n) you observe (yit, xit), t = 1, ..., T , which satisfies
yit = β0 + β1xit + ai + uit, t = 1, ..., T. (1)
(a) Show that
∆yit = β1∆xit + ∆uit, i = 1, ..., n, t = 2, ..., T. (2)
Discuss in detail the advantages and disadvantages of using eq. (2) instead of eq. (1) for
estimation and inference.
ANSWER:
• Advantages of OLS based on (2): No potential biases due to xit being correlated with
ai.
• Dis-advantages of OLS based on (2): Generally bigger variance due to fewer observa-
tions after transformation In particular, (2) requires variation in xit over time while
(1) does not. However, depending on the autocorrelation structure of ∆uit vs the one
of uit, OLS based on (2) may still perform better even in terms of variance.
• For both regressions, clustered standard errors will generally be needed. So in this sense,
neither dominates the other.
(b) Write up the sum of squared residuals (SSR) for (2) given your sample. Suppose here and
in the following that
∑n
i=1
∑T
t=2 (∆xit)
2 > 0 in your sample. Show that the minimizer of
SSR is
βˆ1 =
∑n
i=1
∑T
t=2 ∆xit∆yit∑n
i=1
∑T
t=2 (∆xit)
2
,
where you explain each step of your derivation.
ANSWER:
SSR =
n∑
i=1
T∑
t=2
(∆yit − β1∆xit)2 .
The minimizer βˆ1 satisfies
0 = −1
2
∂SSR
∂β1
=
n∑
i=1
T∑
t=2
(
∆yit − βˆ1∆xit
)
∆xit
=
n∑
i=1
T∑
t=2
∆yit∆xit − βˆ1
n∑
i=1
T∑
t=2
(∆xit)
2
ECON0019 2 CONTINUED
⇒ βˆ1 =
∑n
i=1
∑T
t=2 ∆xit∆yit∑n
i=1
∑T
t=2 (∆xit)
2
.
(c) Suppose that
uit = uit−1 + eit
where E [eit|xi1, ...., xiT ] = 0, t = 1, ...., T . Show that for any i, t,
E [∆uit|X1, ..., Xn] = 0,
where Xi = (∆xi2, ....,∆xiT ), i = 1, ..., n. Use this in turn to show that βˆ1 is unbiased. As
part of your proof, carefully explain each step.
ANSWER:
E [∆uit|X1, ..., Xn] = E [eit|X1, ..., Xn]
=
(*)
E [eit|∆Xi]
= E [E [eit|xi1, ...., xiT ] |Xi]
= 0,
where (*) uses that data is randomly sampled. Write
βˆ1 = β1 +
∑n
i=1
∑T
t=2 ∆xit∆uit∑n
i=1
∑T
t=2 (∆xit)
2
, (3)
and take conditional expectations on both sides,
E[βˆ1|X1, ...., Xn] = β1 +
∑n
i=1
∑T
t=2 ∆xitE[∆uit|X1, ...., Xn]∑n
i=1
∑T
t=2 (∆xit)
2
= β1
(d) Suppose furthermore that
E
[
e2it|xi1, ...., xiT
]
= σ2,
E [eiseit|xi1, ...., xiT ] = 0, s 6= t.
Demonstrate that
Cov (∆uis,∆uit|X1, ..., Xn) =
{
σ2, s = t
0, s 6= t .
Use this in turn to derive an expression of the conditional variance of βˆ1, Var(βˆ1|X1, ...., Xn),
where you carefully explain each step of your derivation. Comment on the resulting variance
ECON0019 3 TURN OVER
expression. In particular, how is the variability of the OLS estimator affected by the
variation of the error term and the regressor?
ANSWER:
Cov (∆uis,∆uit|X1, ..., Xn) = E [eiseit|X1, ..., Xn]
=
(*)
E [eiseit|Xi]
= E [E [eiseit|xi1, ...., xiT ] |Xi]
=
{
σ2, s = t
0, s 6= t ,
where (*) uses that data is randomly sampled. Thus,
Var(βˆ1|X1, ...., Xn) = Var
(∑n
i=1
∑T
t=2 ∆xit∆uit∑n
i=1
∑T
t=2 (∆xit)
2
∣∣∣∣∣X1, ...., Xn
)
=
1[∑n
i=1
∑T
t=2 (∆xit)
2
]2Var
(
n∑
i=1
T∑
t=1
∆xit∆uit
∣∣∣∣∣X1, ...., Xn
)
=
1[∑n
i=1
∑T
t=2 (∆xit)
2
]2 n∑
i,j=1
T∑
s,t=2
∆xis∆xitVar (∆uis∆ujt|X1, ...., Xn)
=
1[∑n
i=1
∑T
t=2 (∆xit)
2
]2 n∑
i=1
T∑
t=2
(∆xit)
2 σ2
=
σ2∑n
i=1
∑T
t=2 (∆xit)
2
.
We see that the variance of the OLS estimator increases as a function of σ2, which makes
sense: More noise in data makes it harder to learn about β1. Reversely, higher realised
variation of ∆xit provides a stronger signal in data about the value of β1 and so decreases
the variance of the OLS estimator
(e) Assume that Pr
(∑T
t=2 (∆xit)
2 = 0
)
= 0. Show that this implies E
[∑T
t=2 (∆xit)
2
]
>
0. Show consistency of βˆ1 under this assumption, where you clearly explain each step,
including which assumptions and limit results that you employ. In particular, explain why
the assumption stated at the beginning of this question is needed.
ECON0019 4 CONTINUED
ANSWER: By the LLN for i.i.d. data,
1
n
n∑
i=1
T∑
t=2
∆xit∆uit → p
T∑
t=2
E [∆xit∆uit] =
T∑
t=2
E [∆xitE [∆uit|X1, ..., Xn]] = 0,
1
n
n∑
i=1
T∑
t=2
(∆xit)
2 → p
T∑
t=2
E
[
(∆xit)
2
]
> 0.
Consistency now follows from (3) together with the continuous mapping theorem. We can
apply the continuous mapping theorem since f(a, b) = a/b is continuous everywhere except
at b = 0 and b = 0 is ruled out due to the assumption at the beginning of the question.
A.2 An extension of the Solow growth model, that includes human capital in addition to physical
capital, suggests that investment in human capital (education) will increase the wealth of a
nation (per capita income). To test this hypothesis, you collect data for 104 countries and
perform the following regression:
r̂elinc = 0.046 − 5.869gpop+ 0.738sk + 0.055educ, (4)
(0.079) (2.238) (0.294) (0.010)
with R2 = 0.775, standard error of residual SER = 0.1377, and heteroskedasticity-robust stan-
dard errors reported in parentheses. Here, relinc is GDP per worker relative to the United
States, gpop is the average population growth rate, 1980 to 1990, sk is the average investment
share of GDP from 1960 to 1990, and educ is the average educational attainment in years for
1985.
(a) Discuss the implications and validity of each of the following assumptions in the context of
the above regression:
i. Data is i.i.d.
ii. E [u|gpop, sk, educ] = 0 where u is the regression error.
In the following we will assume that (i)-(ii) are satisfied together with other relevant tech-
nical assumptions.
ANSWER:
• Re. (i): First, data needs to come from the same population in order for us to set
up a model for this population. Second, random sampling is used in the analysis of
the variance of the OLS estimators, incl computation of standard errors. In terms of
validity, first note that each observational unit is a country. Thus, if we define the
population as all countries in the world, the sample is by definition drawn from this
population and so identically distributed. Data would then be i.i.d. if we had randomly
ECON0019 5 TURN OVER
sampled the 108 countries. But the question does not specify how data is collected so the
answer to this part is open ended. However, it seems unlikely that they are randomly
sampled - there are only around 200 countries in the world and many of these do not
publish reliable statistics. If they are not randomly sampled, then the independence
assumption is most likely violated due to spill-over effects (trade, migration, etc): For
example, a given country’s income level will likely depend on the income levels of its
trading partners.
• Re. (ii): This assumption is needed to ensure that OLS is unbiased. In our application,
the error term here contains other factors affecting GDP per worker, incl institutional
and geographical differences. These are most likely correlated with gpop, sk and educ
in which case (ii) is violated.
(b) Interpret the above regression results and indicate whether or not the coefficients are sig-
nificantly different from zero. Do the coefficients have the expected sign? Explain.
ANSWER:
• A one percentage point decrease in the population growth rate increases GDP per worker
relative to the United States by roughly 0.06.
• An increase in the investment share of 0.1 results in an increase of GDP per worker
relative to the United States by approximately 0.07.
• For every additional year of average educational attainment, the increase is 0.055.
• The regression explains 77.5 percent of the variation in relative productivity.
• All coefficients are significantly different from zero at conventional levels.
• All coefficients carry the expected sign.
(c) To test for equality of the coefficients between the OECD and other countries, you introduce
a binary variable (oecd), which takes on the value of one for the OECD countries and is
zero otherwise. You obtain the following regression estimates:
r̂elinc = −0.068− 0.063gpop+ 0.719sk + 0.044educ (5)
(0.072) (2.271) (0.365) (0.012)
+0.381oecd− 8.038(oecd× gpop)− 0.430(oecd× sk)
(0.184) (5.366) (0.768)
+0.003(oecd× educ)
(0.018)
where R2 = 0.845 and SER = 0.116. Write down the two regression functions, one for the
OECD countries, the other for the non-OECD countries. Explain. Interpret any differences.
ANSWER:
non-OECD : r̂elinc = −0.068− 0.063gpop+ 0.719sk + 0.044educ.
ECON0019 6 CONTINUED
OECD : r̂elinc = 0.313− 8.101gpop+ 0.289sk + 0.047educ.
We see that relinc in non-OECD countries tend to be much less negatively affected by
population growth while increases in investment has a much bigger impact.
(d) In order to test (4) against (5), you compute the corresponding F -statistic which takes the
value 6.76 in your sample. Write up the null hypothesis and its alternative that you are
testing in terms of the population regression coefficients. What do you conclude? Explain.
ANSWER:
H0 : βoecd = βoecd×gpop = βoecd×sk = βoecd×educ = 0
versus
HA : At least one of the above coefficients is non-zero.
We have 4 restrictions and so F -statistic follows the F4,∞ distribution in large samples.
The critical value is 2.37 at the 5% level and hence we can reject the null hypothesis that
the coefficients are equal. We conclude that data supports the hypothesis that there are
differences in relinc between OECD and non-OECD countries after controlling for gpop,
sk and educ.
(e) You decide to investigate further and estimate a restricted version of (5) where you enforce
the same slopes across OECD and non-OECD countries, but allow their intercepts to differ.
In this new regression, the t-statistic for oecd is 3.17. What is the p-value of the t-statistic?
What do you conclude? Explain your answer.
ANSWER: t-statistic follows N (0, 1) distribution in large samples and so its p-value is
p = Pr (|t| > 3.17) = 2 (1− Pr (t ≤ 3.17)) = 2 (1− 0.9992) = 0.16%.
That is, there’s only 0.16% probability of observing a more extreme outcome under the null.
In particular, we reject at a 1% level. We conclude that, under the maintained hypothesis
that the effect of gpop, sk and educ are the same between the two groups, the level of relinc
between OECD and non-OECD countries is different.
(f) Next, you test the model described in (e) against (5). The value of the corresponding
F -statistic is 1.05. Do you accept or reject the null?
Looking at the tests in this and two previous questions, what is your overall conclusion?
Explain your answer.
ANSWER: We here test
H0 : βoecd×gpop = βoecd×sk = βoecd×educ = 0,
and so so F -statistic follows the F3,∞ distribution in large samples with corresponding
critical value of 2.68 at the 5% level. Thus, we cannot reject the null hypothesis, and so
there appears to be no differences in the impact of gpop, sk and educ on relinc between
OECD and non-OECD countries. The over-all conclusion is that there only seems to be a
level difference between the two groups of countries.
ECON0019 7 TURN OVER
PART B
Answer ONE question from this section.
B.1 Intergenerational mobility is related to several aspects. For example, theoretical studies have
examined the repercussions of the transmission of preferences and attitudes from parents to
children. Thomas Dohmen, Armin Falk, David Huffman and Uwe Sunde (“The Intergenera-
tional Transmission of Risk and Trust Attitudes”) use the German Socio-Economic Panel Study
(SOEP) to empirically examine, among other things, the transmission of attitudes from par-
ents to children and potential mechanisms for such transmission. Aside from comprehensive
demographic information on all individuals in a given household, the survey contains a set of
individual questions regarding risk attitudes (in 2004). (The authors also look at trust.) People
were asked questions eliciting their willingness to take risks on an eleven-point scale. For these
variables, zero (0) would correspond to ‘completely unwilling to take risks’ and the value ten
(10) means that the person is ‘completely willing to take risks.’
(a) One possible way to investigate the transmission of risk attitudes is to examine how parental
characteristics (including their risk attitudes) relate to the probability that a child has a
high score in terms of the risk attitude measure elicited on an 11-point scale as indicated
above. To do this, generate a variable Di = 1 if the child in household i has risk attitude
measure equal to 6 or above and Di = 0, otherwise. (While separate measures are available
for both parents, to keep matters simple we focus here on a single measure for parents.)
Taking RPi to be the parental score for that same measure in the household, suppose you
are interested in the model:
Di = 1(β0 + β1R
P
i + Ui ≥ 0).
Assuming that Ui follows a standard logistic distribution, write down the log-likelihood
function for this estimation problem when you have N observations. How would you es-
timate the difference in the probability that Di = 1 between a household where R
P
i = 10
and another one where RPi = 0? Please explain your answer.
ANSWER: The log-likelihood function is:
n∑
j=1
{−b0 − b1RPi − ln[1 + exp(−b0 − b1RPi )]} × (1−Di)− ln[1 + exp(−b0 − b1RPi )]×Di
or equivalently
n∑
j=1
{b0 + b1RPi − ln[1 + exp(b0 + b1RPi )]} ×Di − ln[1 + exp(b0 + b1RPi )]× (1−Di).
ECON0019 8 CONTINUED
Once estimates are obtained, the estimated difference in probabilities is given by:
exp(−βˆ0)/(1 + exp(−βˆ0))− exp(−βˆ0 − βˆ110)/(1 + exp(−βˆ0 − βˆ110))
or, equivalently,
exp(βˆ0 + βˆ110)/(1 + exp(βˆ0 + βˆ110))− exp(βˆ0)/(1 + exp(βˆ0)).
(b) Because risk attitudes for children (RCi ) and parents (R
P
i ) are measured contemporaneously,
the authors worry about ‘reverse causality’ where children’s attitudes may be at least partly
shaping parents’ attitudes. To address this issue they estimate
RCi = α0 + α1R
P
i + Vi,
using parental religion (Zi) as an instrumental variable for R
P
i . Describe in detail how you
would implement the TSLS estimator in this context. Discuss the validity of the instru-
mental variable suggested in this context. (Explain your intuition.)
ANSWER: TSLS: ( 1 ) Regress RPi on Zj. ( 2 ) Regress R
C
i on Rˆ
P
i . The instrumental
variable is valid if cov(Zi, Vi) = 0. This means that any unobserved variables that affect the
child’s risk attitudes should be uncorrelated with parental religion.
(c) The F-statistic for the first stage regression using the mother’s risk attitudes as covariate
in the main equation of interest and her religion as instrumental variable is 9.99. (The
F-statistic when using father’s risk attitudes and religion is 7.32.) Discuss in detail the
relevance of the instrumental variable.
ANSWER: The values for the F-statistics suggest that the instrument may not be suffi-
ciently strong and thus present substantial bias.
(d) In a regression where the risk attitude for both mother and father are included individually
as covariates in a multiple linear regression model, both coefficients on those variables are
around 0.15 with standard errors at around 0.02 for each one of them. The TSLS estimates
on the other hand, produce estimates for the coefficient on the mother’s risk attitude at
about 0.23 and for the coefficient on the father’s risk attitude at about 0.02. (Religion for
each parent is available as an intrumental variable for each of their risk attitude variables.)
The standard error for those estimates are, in both cases, around 0.10. Why would you
ECON0019 9 TURN OVER
expect the standard errors for the IV estimates to be larger than the standard errors for
the OLS estimates? Explain your answer.
ANSWER: Under homoscedasticity,
v̂ar(βˆIV ) =
σˆ2
SSTxR2x,z
>
σˆ2
SSTx
= v̂ar(βˆOLS)
(e) Imagine you had data on the risk attitude for successive generations of a single household
and you want to estimate the regression
RG+1 = α0 + α1RG + VG+1,
where RG+1 and RG are, once again, the risk attitudes in generation G+ 1 (child) and in
generation G (parent). Assuming these are not measured contemporaneously so that the
issues raised in item (b) are not present, are there conditions under which an OLS estimator
is unbiased? Elaborate on your answer.
ANSWER: The estimator would be biased since the model does not satisfy strict exo-
geneity.
B.2 In “Excess Capacity and Policy Interventions: Evidence from the Cement Industry,” Tetsuji
Okazaki, Ken Onishi and Naoki Wakamori estimate the demand for cement in Japan using data
on different regions across years. Their specification for the demand function is
ln(Qmt) = αP ln(Pmt) + α
>
XXmt + Umt,
where Qmt is the quantity of cement demanded in region m and year t (from 1970 to 1995), Pmt
is the price in that region and year and Xmt are year- and region-specific demand shifters. The
Ordinary Least Squares (OLS) estimate for α, denoted by α̂P,OLS, equals -0.07 with a standard
error equal to 0.16.
(a) Explain in detail why the above estimate for the slope coefficient (−0.07) cannot be directly
interpreted as the price-elasticity of demand for cement.
ANSWER: Simultaneity bias.
ECON0019 10 CONTINUED
(b) To produce cement, crushed limestone, cray and other minerals are mixed and put into a
kiln to be heated. This process yields clinker, which is an intermediate cement product.
In a final stage, the grinded clinker is mixed with gypsum, another intermediate input, to
produce cement. The researchers then use the (log) price of gypsum as an instrumental
variable for the (log) price of cement to estimate the price-elasticity of demand. The OLS
regression of (log) cement prices on (log) gypsum prices (and X) yields a coefficient of 0.06
and the F-test statistic for the first stage equals 17.0. Discuss in detail the exogeneity and
relevance of this instrumental variable.
ANSWER: First stage F − stat indicates that the IV is sufficiently correlated with the
endogenous variable. Validity holds if the price of gypsum is not correlated with other un-
observed determinants of demand for cement.
(c) To estimate the regression using the IV described above, the researchers use Two-Stage
Least Squares and obtain an estimate for α, denoted α̂P,TSLS, equal to -1.11 with a stan-
dard error equal to 0.58. Describe in detail the TSLS procedure. Is it possible to test
whether the IV is exogenous? Explain in detail. What if there were two instrumental
variables? Explain in detail.
ANSWER: Textbook. With two IVs one could use overidentification restrictions to test
exogeneity, but not with only one IV.
(d) Suppose the researchers were also interested in examining the time series behaviour for the
quantity of cement sold in a particular region in Japan on a given year, ln(Qt). To do so,
they obtain estimates for the following autoregressive model using data over various years
for this region of Japan:
ln(Qt) = α0 + α1 ln(Qt−1) + ηt.
Would the OLS estimator be unbiased in this case? Under what assumptions would it be
consistent? Explain your answers in detail.
ANSWER: The OLS estimator is biased, but consistent.
(e) Suppose the researchers only observe whether Qmt is larger or smaller than a given fixed
threshold Q in a given year but otherwise observe prices and X. Let Dmt record whether
Qmt > Q (Dmt = 1) or not (Dmt = 0). While the regression
ln(Qmt) = αP ln(Pmt) + α
>
XXmt + Umt
ECON0019 11 TURN OVER
is no longer estimable, they are still able to estimate the model given by
Dmt =
{
1 if βP ln(Pmt) + β
>
XXmt + Vmt > ln(Q)
0 if βP ln(Pmt) + β
>
XXmt + Vmt ≤ ln(Q)
Assume that the error term follows a standard normal distribution (i.e., Vmt ∼ N (0, 1))
and write down the log-likelihood function for this model assuming that the data comprises
of a random sample. If Umt ∼ N (0, σ2) how are βP and αP related? Explain your answer
in detail.
ANSWER: Log-likelihood function for the probit. βP = αP /σ.
ECON0019 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
ECON0019 13 TURN OVER
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 14 END OF PAPER
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