Assignment1_2014Sping

Intermediate Econometrics

Assignment 1

(Due on 17:40 pm, 23rd Mar. 2014)

Q1.Judge if the following statements are True or False and provide a short explanation for your answer in each case.

a)If an estimator is unbiased, then it must be consistent; but not vice versa.

b)The sample covariance between the regressor and the error Cov x i,u i is 0.

c)Suppose the result of sample regression function is

log(salary)=8.34+0.095log(libvol), in which salary is the annual salary for new school graduates, libvol is the number of volumes in the school library, the slope coefficient shows that an increase in libvol by one percent increases salary by 9.5%.

d)If Cov x,u=0, then E u x does not depend on x.

e)One should always include more explanatory variables on the right-hand-side as the R2

will get larger.

f)Consider the regression model y=β0+β1x1+β2x2+u. If x1is positively correlated

with x2and the partial effect of x2on y is positive, then the omission of x2causes an upward bias in the OLS estimate of β1.

g)The OLS estimator will be biased if there exists heteroskedasticity in the model.

h)In the following model log(price)=9.49+0.312log dist,R2=0.162, where price is

measured in dollars. If its unit of measurement changes, then R2 will change as well.

i)If individuals who applied for participation in further training are divided randomly into

participants and non-participants, then the causal effect of the training measure may be identified without a doubt.

Q2.(Y i,X1i,X2i)satisfy the Gauss-Markov assumptions; in addition, var(u i| X1i,X2i)= 4 and var(X1i)=6. A random sample of size n=400 is drawn from the population.

a)Assume that X1and X2are uncorrelated. Compute the variance of β1.

b)Assume that corr X1,X2=0.5. Compute the variance of β1.

c)Comment on the following statements: “When X1and X2are correlated, the variance of

β1is larger than it would be if X1and X2were uncorrelated. Thus, if you are interested in β1, it is best to leave X2out of the regression if it is correlated with X1.”

Q3.Instead of estimating the coefficients β

1and β

2

from the model

y=α+β

1x1+β

2

x2+u

It is decided to use ordinary coefficients β

1and β

2

from the model

y=α+β

1x1?+β

2

x2+v

where x1?is the residual from a regression of x1on x2and v is the disturbance term.

a)Show that the resulting estimator of β

2

is identical to the regression coefficient of y on x2.

b)Prove that the estimators of β

1

obtained from each of the two equations are identical.

Q4. A researcher plans to study the causal effect of police on crime using data from a random sample of Chinese c ities. He plans to regress the city’s crime rate on the (per capita) size of the county’s police force. Explain why this regression is likely to suffer from omitted variable bias. What are the consequences of omitted variable bias? Which variables would you add to the regression to control for important omitted variables? And judge that whether the regression will likely over- or underestimate the effect of police on the crime rate when omitted those variables?

Q5.Prove that in the simple regression model y=β0+β1x+u,σ2 is an unbiased estimator for σ2under assumptions SLR.1 through SLR.5, where

σ2=1

n?2

u i2

n

i=1

=SSR/(n?2).

Computer exercise (please provide the STATA do file and log file along with your answers)

https://www.sodocs.net/doc/df7503710.html,e the data set in WAGE2.DTA for this problem.

a)Find the average years of education and generate a histogram chart to describe the

distribution of years of education in the sample.

b)Generate a new variable named deduc which categorizes the education levels into four

groups: junior middle school (9 years of education), senior high school (10-12 years of education), university (13-16 years of education), above university (years of

education>16).

c)Calculate the average wage for each education category (i.e., deduc). Claim that how the

average wage varies with the increase of education level and whether the variance of wage for each education level is constant for all education categories.

d)Run the simple regression of log(wage)(i.e., lwage) on educ, and obtain the slope

coefficient β1, Report the results as the following format and provide an interpretation for the slope coefficient and R2.

log(wage)=constant+coef.?educ

(std.err.) (std.err.)

e)And then run the multiple regression of log(wage) on educ and IQ, and obtain the slope

coefficient β1and β2, respectively. Recode the regression results following the above format. Interpret the meaning for all the estimated coefficients. With compare to the results obtained in (d), claim that how the slope coefficients for educ change when IQ are controlled in the regression and if the change is just like what you expected.

f)Please run a regression to verify that how IQ varies with educ, and obtain the slope

coefficient, say δ1. Find the relationship among β1, β1, β2and δ1; then explain this relationship intuitively.

g)Run the multiple regression of log(wage) on educ, IQ and exper, married, and black.

Recode the regression results in an equation as in (d). Claim that how the coefficient in front of educ varies when more regressors are included in the model.