1- consider the one-variable regression model

Question

1-  Consider the one-variable regression model Y_i = β_o + β₁X_1i+ U_i, and suppose that it satisfies the classical regression assumptions.  Suppose that Y_i is measured with error, so that the data are  Y_i = Y_i + w_i, where w_i, is the measurement error which is  i.i.d. and independent of X_i and U_i.  Consider the population regression Y_i = β_o + β₁X_1i+ V_i, where V_i is the regression error using the measurement error in dependent variable, Y_i.

a. Show that V_i = U_i + w_i.

b. Show that the regression Y_i = β_o + β₁X_1i+ V_i 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 Q_t = β_o + β₁P_t + U_t, where Q denotes quantity, P denotes price, and U denotes factors other than price that determine demand.  Supply for the commodity is given by Q_t = γ_o + γ₁P_t+ V_t, 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.

1.  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.

1.  Do an OLS estimate of the ß coefficients and analyze the statistical properties of the estimated coefficient.
2.  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.

1. 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?
2. 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) y_t = b_o(1 + x)^b1te^ut

b) y_t = e^box1^b1x2^b2 + u_t

c) y_t = b_o + b₁x^b2 + u_t

d) y_t = b₁ + b₂(x_2t - b₃x_3t) + b₄(x_4t - b₃x_5t) + e_t

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 y_t = b_o + b₁x^b2  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 y_t = b_o + b₁x^b2 + 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?