Exam 9: Assessing Studies Based on Multiple Regression

Errors-in-variables bias

In macroeconomics, you studied the equilibrium in the goods and money market under the assumption of prices being fixed in the very short run. The goods market equilibrium was described by the so-called IS equation R_i = β₀ - β₁Y_i + u_i where R represented the nominal interest rate and Y was real GDP. β₀ contained variables determined outside the system, such as government expenditures, taxes, and inflationary expectations. The money market equilibrium was given by the so-called LM equation R_i = $\gamma _ { 0 }$ + $\gamma _ { 1 }$ Y_i + v_i and $\gamma _ { 0 }$ contained the real money supply and the intercept from the money demand equation. Show that there is simultaneous causality bias in this situation.

Your textbook states that correlation of the error term across observations "will not happen if the data are obtained by sampling at random from the population." However, in one famous study of the electric utility industry, the observations were listed by the size of the output level, from smallest to largest. The pattern of the residuals was as shown in the figure. What does this pattern suggest to you?

To analyze the situation of simultaneous causality bias, consider the following system of equations: Y_i = β₀ + β₁X_i + u_i X_i = $\gamma _ { 0 }$ + $\gamma _ { 1 }$ Y_i + v_i Demonstrate the negative correlation between X_i and $\gamma _ { 1 }$ for $\gamma _ { 1 }$ < 0 , either through mathematics or by presenting an argument which starts as follows: "Imagine that u_i is negative."

Give at least three examples where you could envision errors-in-variables problems. For the case where the measurement error occurs only for the explanatory variable in the simple regression case, derive $\hat { \beta } _ { 1 } \stackrel { p } { \longrightarrow } \frac { \sigma _ { X } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { w } ^ { 2 } } \beta _ { 1 }$