UNIVERSITY OF SOUTHERN CALIFORNIADepartment of Economics
ECON 414 Introduction to Econometrics
Prof. Safarzadeh
HW #4 Student Name: ________________
1- Consider the one-variable regression model Y
i = ß
o + ß
1X
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 = ß
o + ß
1X
1i+ V
i, where V
i is the regression error using the measurement error in dependent variable, Y
i.
- Show that Vi = Ui + wi.
- Show that the regression Yi = ßo + ß1X1i+ Vi satisfies the assumptions of the classical regression.
- Are the OLS estimators consistent?
- Can confidence intervals be constructed in the usual way?
- 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 = ß
o + ß
1P
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 = ?
o + ?
1P
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 s
2u, s
2v, respectively and are mutually uncorrelated.
- Solve the two equations for Q and P to show how Q and P depend on U and V.
- Derive the means of P and Q.
- Derive the variance of P, the variance of Q, and the covariance between Q and p.
- 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.
- Use your answers to (b) and (c) to derive values of the regression coefficients.
- A researcher uses the slope of this regression as an estimate of the slope of the demand function (ß). Is the estimated slope too large or too small?
3- Suppose we have a regression model of Y = ßo + ß
1X + 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 + ß
1X
1 + ß
2X
2 + U, where E(X
2U) ? 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
1P + ß
2Y + U
Qs = ao + a
1P + a
2W + 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)
b1te
utb) y
t = e
box1
b1x2
b2 + u
tc) y
t = b
o + b
1x
b2 + u
td) y
t = b
1 + b
2(x
2t - b
3x
3t) + b
4(x
4t - b
3x
5t) + e
t7- 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) = X2 – 3XY + 2Y2, at xo = 1, yo = .5.
8- Linearize y
t = b
o + b
1x
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
1x
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?