Econometric Methods, ECO341, Semester I, 2015-16

Homework IV (60 points)

Instructor: M.A. Rahman

Deadline: 3:00 pm, November 13, 2015.

Please read the instructions carefully and follow them while writing answers.

• Solutions to homework should be written in A4 size loose sheets. If you are not comfortable

writing on white sheets, please ask for biology paper in Tarun Book Store.

• Questions should be answered in order as they appear in the homework. Every new question

should begin in a new page. Please number all the pages of your homework solution.

• Please leave a margin of one inch from top and one inch from left. Staple the sheets on the

top-left.

• Matlab assignments: Simply answer the questions and present the codes in the appendix.

1. (4+10+6=20 points) A Sampling Experiment: Consider the model

Y = Xβ + ǫ, (1)

where, β=(10, 0.4, 0.6) and ǫ =√0.0625 u, with u ∼ N(0, 1). The design matrix X and first sample

values for u and y are given in file “sampledata.xls”.

(a) Find the covariance matrix of β using σ2 and the design matrix in “sampledata.xls”

(b) Draw 500 samples of size 20. The first sample should be same as that given in “sampledata.xls”

(technically you would only draw 499 samples). Compute the five hundred estimates for β and

unbiased estimates of σ2. Make a histogram for each component of β and σ2, what do you

observe? Compute the average covariance matrix from the 500 draws. Print the first ten sets

of estimates.

(c) Suppose, we now have the non sample information

Rβ = (0 1 1)

β1

β2

β3

= β2 + β3 = 1 (2)

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and that this information is used with the sample information for the 500 replications. Compute

the β’s and plot the distribution. Also, compute the average covariance matrix from the 500

simulations. What do you observe? Print the average (X ′X)−1 and (R(X ′X)−1R′)−1.

2. (3 × 6 = 18 points. Full points will be awarded only if answers are properly

simplified.) Consider the generalized regression model y = Xβ+u, where E(u) = 0 and E(uu′) =

σ2Ω with fixed regressors and some positive definite matrix Ω. Let β and βGLS denote the OLS

and GLS estimators, respectively. Define the OLS and GLS residuals as u = y −Xβ and uGLS =

y−XβGLS , respectively. Let β = (X ′Σ−1X)−1(X ′Σ−1y) and u = y−Xβ for some positive definite

matrix Σ. Based on the above setting answer the following. (Please use the notation as given

in the problem.)

(a) Find Var(β).

(b) Find Cov(β, β).

(c) Find Cov(βGLS , β).

(d) Find Var(u).

(e) Find Cov(uGLS , u).

(f) Find Cov(u, u).

3. (6+6+10=22 points) Consider the file ‘hsbdemo.xlsx’, which contains a sample of 200

observations taken from the High School and Beyond (HSB) longitudinal study of American youth

conducted by the National Opinion Research Center on behalf of the National Center for Educations

Statistics (NCES). A description of the variables, along with certain guidelines is present in excel

sheet ‘description’, but note that all the variables will not be used in this study. Based on the

data do the following.

(a) Regress read on intercept, gender, ses, schtyp, prog and socst. Report the estimates and

standard error of the regression coefficients. Calculate the R2.

(b) Regress math on intercept, gender, ses, schtyp, prog and science. Report the estimates and

standard error of the regression coefficients. Calculate the R2.

(c) However, the read and math scores correspond to the same individual, so it is natural to expect

that the errors from the two equations to be correlated. A set of equations that has contem-

poraneous cross-equation error correlation (i.e. the error terms in the regression equations are

correlated) is called a seemingly unrelated regression (SUR) model. Estimate the regression

equations from part(a) and part(b) assuming the errors are correlated. Report the estimates

and standard error of the regression coefficients for the SUR model. Compare the standard

errors of the coefficients from the OLS and SUR model. What do you observe?

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