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 at bo = 0, b1 = .9, and b2 = 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 regression yt = 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?