Assignment Help >> Microeconomics
1- Consider the one-variable regression model Y_{i} = β_{o} + β_{1}X_{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} + β_{1}X_{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} + β_{1}X_{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} + β_{1}P_{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} + γ_{1}P_{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 σ^{2}u, σ^{2}v, 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 + ß_{1}X + 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 ß_{1} coefficient and analyze the statistical properties of the estimated coefficient.
- Do an IV estimate of the ß_{1} coefficient and analyze the statistical properties of the estimated coefficient.
4. Suppose we have a regression model of Y = ßo + ß_{1}X_{1} + ß_{2}X_{2} + U, where E(X_{2}U) ≠ 0. Suppose Z is a varible that is highly correlated with X_{2} 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 + ß_{1}P + ß_{2}Y + U
Qs = αo + α_{1}P + α_{2}W + 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) y_{t} = b_{o}(1 + x)^{b1t}e^{ut}
b) y_{t} = e^{bo}x1^{b1}x2^{b2 }+ u_{t}
c) y_{t} = b_{o }+ b_{1}x^{b2} + u_{t}
d) y_{t} = b_{1} + b_{2}(x_{2t} - b_{3}x_{3t}) + b_{4}(x_{4t} - b_{3}x_{5t}) + e_{t}
7- Find the linear approximation of the following functions at the given points.
a) y = f(x) = X^{3} + 3X^{2} - 5X + 3, at xo = 1.
b) Z = f(x, y) = X^{2} - 3XY + 2Y^{2}, at xo = 1, yo = .5.
8- Linearize y_{t} = b_{o }+ b_{1}x^{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 }+ b_{1}x^{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?