UNIVERSITY OF SOUTHERN CALIFORNIA Department of Economics ECON 414 Introduction to Econometrics Prof. Safarzadeh HW #4 Student Name: 1- Consider the one-variable regression model Yi = ?o + ?1X1i+ 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. Document Preview: UNIVERSITY OF SOUTHERN CALIFORNIA Department of Economics ECON 414 Introduction to Econometrics Prof. Safarzadeh HW #4 Student Name: 1- Consider the one-variable regression model Yi = ?o + ?1X1i+ 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 = ?o + ?1X1i+ Vi, where Vi is the regression error using the measurement error in dependent variable, Yi. Show that Vi = Ui + wi. Show that the regression Yi = ?o + ?1X1i+ 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 Xs is a serious problem. Measurement error in Y is not. 2-The demand for a commodity is given by Qt = ?o + ?1Pt + 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 = ?o + ?1Pt+ Vt, where V denotes factors other than price that determine supply. Suppose that U and V both have a mean of zero, have variances s2u, s2v, 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 (?). Is the estimated slope too large or too small? 3- Suppose we have a regression model of Attachments: HW4-N-1.doc; UNIVERSITY OF SOUTHERN CALIFORNIA Department of Economics ECON 414 Introduction to Econometrics Prof. Safarzadeh HW #4 Student Name: 1- Consider the one-variable regression model Yi = ?o + ?1X1i+ 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. Document Preview: UNIVERSITY OF SOUTHERN CALIFORNIA Department of Economics ECON 414 Introduction to Econometrics Prof. Safarzadeh HW #4 Student Name: 1- Consider the one-variable regression model Yi = ?o + ?1X1i+ 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 = ?o + ?1X1i+ Vi, where Vi is the regression error using the measurement error in dependent variable, Yi. Show that Vi = Ui + wi. Show that the regression Yi = ?o + ?1X1i+ 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 Xs is a serious problem. Measurement error in Y is not. 2-The demand for a commodity is given by Qt = ?o + ?1Pt + 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 = ?o + ?1Pt+ Vt, where V denotes factors other than price that determine supply. Suppose that U and V both have a mean of zero, have variances s2u, s2v, 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 (?). Is the estimated slope too large or too small? 3- Suppose we have a regression model of Attachments: HW4-N-1.doc
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