这个作业是计量经济学的测试题，需要回答或证明下列关于定量、计量的题目

ASSESSMENT : ECON0019A5UB/ECON0019A5UA/ECON2007A

A.1您在T≥2个时间段内随机抽取了n个追踪对象。对于

您观察到的单个i（= 1，…，n）（yit，xit），t = 1，…，T，满足

yit =β0+β1xit+ ai + uit，t = 1，…，T.（1）

（a）证明

∆yit = β1∆xit + ∆uit，i = 1，…，n，t = 2，…，T.（2）

详细讨论使用eq的优缺点。 （2）代替等式（1）为

估计和推论。

（b）给定您的样本，为（2）写出残差平方和（SSR）。假设这里和

在下面那个Pn

i = 1

PT

t = 2（∆xit）

样本中2> 0。证明

SSR是

βˆ

1 =

n

i = 1

PT

t = 2 ∆xit∆yit

n

i = 1

PT

t = 2（∆xit）

2

，

您在其中解释推导的每个步骤。

（c）假设

uit = uit-1 + eit

其中E [eit | xi1，….，xiT] = 0，t = 1，….，T。表明对于任何i，t，

E [∆uit | X1，…，Xn] = 0，

其中Xi =（∆xi2，….，∆xiT），i = 1，…，n。依次使用以表明βˆ

1是无偏的。如

证明的一部分，请仔细解释每个步骤。

(d) Suppose furthermore that

E

e

2

it|xi1, …., xiT

= σ

2

,

E [eiseit|xi1, …., xiT ] = 0, s 6= t.

Demonstrate that

Cov (∆uis, ∆uit|X1, …, Xn) =

σ

2

, s = t

0, s 6= t

.

Use this in turn to derive an expression of the conditional variance of βˆ

1, Var(βˆ

1|X1, …., Xn),

where you carefully explain each step of your derivation. Comment on the resulting variance

expression. In particular, how is the variability of the OLS estimator affected by the

variation of the error term and the regressor?

ECON0019 2 CONTINUED

(e) Assume that P r PT

t=2 (∆xit)

2 = 0

= 0. Show that this implies E

hPT

t=2 (∆xit)

2

i

>

0. Show consistency of βˆ

1 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.

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:

relinc \ = 0.046 − 5.869gpop + 0.738sk + 0.055educ, (3)

(0.079) (2.238) (0.294) (0.010)

with R2 = 0.775, standard error of residual SER = 0.1377, and heteroskedasticity-robust standard 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 technical assumptions.

(b) Interpret the above regression results and indicate whether or not the coefficients are significantly different from zero. Do the coefficients have the expected sign? Explain.

(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:

relinc \ = −0.068 − 0.063gpop + 0.719sk + 0.044educ (4)

(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)

ECON0019 3 TURN OVER

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.

(d) In order to test (3) against (4), 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.

(e) You decide to investigate further and estimate a restricted version of (4) 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.

(f) Next, you test the model described in (e) against (4). 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.

ECON0019 4 CONTINUED

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 Intergenerational 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 parents 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 RP

i

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 estimate the difference in the probability that Di = 1 between a household where RP

i = 10

and another one where RP

i = 0? Please explain your answer.

(b) Because risk attitudes for children (RC

i

) and parents (RP

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

R

C

i = α0 + α1R

P

i + Vi

,

using parental religion (Zi) as an instrumental variable for RP

i

. Describe in detail how you

would implement the TSLS estimator in this context. Discuss the validity of the instrumental variable suggested in this context. (Explain your intuition.)

ECON0019 5 TURN OVER

(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.

(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

expect the standard errors for the IV estimates to be larger than the standard errors for

the OLS estimates? Explain your answer.

(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.

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 αbP,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.

ECON0019 6 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.

(c) To estimate the regression using the IV described above, the researchers use Two-Stage

Least Squares and obtain an estimate for α, denoted αbP,TSLS, equal to -1.11 with a standard 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.

(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.

(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

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.

ECON0019 7 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 8 CONTINUED

NORMAL CUMULATIVE DISTRIBUTION FUNCTION (P rob(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 9 END OF PAPER