# Consider the one-variable regression model Yi = βo + β1X1i+ Ui, and suppose

UNIVERSITY OF SOUTHERN CALIFORNIA

Department ofEconomics

ECON 414 Introduction to Econometrics

Prof. Safarzadeh

HW #4 Student Name: ________________

1- Consider the one-variable regression model Yi = βo + β1X1i+ Ui, and suppose that it satisfies the classical regression assumptions. Suppose that Yi is measured with error, so that the data are Yi = Yi + wi, where wi, is the measurement error which is i.i.d. and independent of Xi and Ui. Consider the population regression Yi = βo + β1X1i+ Vi, where Vi is the regression error using the measurement error in dependent variable, Yi.

a. Show that Vi = Ui + wi.

b. Show that the regression Yi = βo + β1X1i+ Vi satisfies the assumptions of the classical regression.

c. Are the OLS estimators consistent?

d. Can confidence intervals be constructed in the usual way?

e. Evaluate these statements: “Measurement error in the X’s is a serious problem. Measurement error in Y is not.”

2-The demand for a commodity is given by Qt = βo + β1Pt + Ut, where Q denotes quantity, P denotes price, and U denotes factors other than price that determine demand. Supply for the commodity is given by Qt = γo + γ1Pt+ Vt, where V denotes factors other than price that determine supply. Suppose that U and V both have a mean of zero, have variances σ2u, σ2v, respectively and are mutually uncorrelated.

a. Solve the two equations for Q and P to show how Q and P depend on U and V.

b. Derive the means of P and Q.

c. Derive the variance of P, the variance of Q, and the covariance between Q and p.

d. A random sample of observations of (Q, P) is collected, and Q is regressed on P. (That is, Q is the regressand and Pi is the regressor.) Suppose that the sample is very large.

i. Use your answers to (b) and (c) to derive values of the regression coefficients.

ii. A researcher uses the slope of this regression as an estimate of the slope of the demand function (β). Is the estimated slope too large or too small?

3- Suppose we have a regression model of Y = ßo + ß1X + U, where E(XU) ≠ 0. Suppose Z is a varible that is highly correlated with X and no correlation with U.

Do an OLS estimate of the ß1 coefficient and analyze the statistical properties of the estimated coefficient.

Do an IV estimate of the ß1 coefficient and analyze the statistical properties of the estimated coefficient.

4. Suppose we have a regression model of Y = ßo + ß1X1 + ß2X2 + U, where E(X2U) ≠ 0. Suppose Z is a varible that is highly correlated with X2 and no correlation with U.

Do an OLS estimate of the ß coefficients and analyze the statistical properties of the estimated coefficient.

Do an IV estimate of the ß coefficient and analyze the statistical properties of the estimated coefficient.

5. Demand and supply for a product is expressed as

Qd = ßo + ß1P + ß2Y + U

Qs = αo + α1P + α2W + V

Where, Q is the quantity, P is the price, Y is income, and W is the average wage in the industry.

Given that in the market P and quantity depend on each other, a two-way causality, the E(PU) ≠ 0 and E(PV) ≠ 0. What is the implication of the two-way causality to the estimated coefficients of the system of equations above?

How could you find a consistent eatimate of the coefficient?

6- For the following regression models discuss the estimation techniques (linear, log-linear, non linear). Linearize the models if needed.

a) yt = bo(1 + x)b1teut

b) yt = ebox1b1x2b2+ ut

c) yt = bo+ b1xb2 + ut

d) yt = b1 + b2(x2t – b3x3t) + b4(x4t – b3x5t) + et

7- Find the linear approximation of the following functions at the given points.

a) y = f(x) = X3 + 3X2 – 5X + 3, at xo = 1.

b) Z = f(x, y) = X2 – 3XY + 2Y2, at xo = 1,yo = .5.

8- Linearize yt = bo+ b1xb2 atbo = 0,b1 = .9, andb2 = 1.

9- Use the Data Set 1 in eee to run a linear regression of consumption on income. Use the estimated coefficients as the initial values for running a non-linear regressionyt = bo+ b1xb2 + U. Estimate the b coefficints of the non-linear regression and do a statistical analysis of the coefficients.

b. What is the implication of the non-liniear regerssion to MPC?

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