Errors and Residuals in Statistics and Linear Regression Model
Economics 141 (Intro to Econometrics) Professor Yang
Spring 2001
Answers to Midterm Test No. 1
1. Consider a regression model of relating Y (the dependent variable) to X (the independent variable) Yi = (0 + (1Xi+ (i where (i is the stochastic or error term. Suppose that the estimated regression equation is stated as Yi = (0 + (1Xi and ei is the residual error term.
A. What is ei and define it precisely. Explain how it is related to (i.
ei is the residual error term in the sample regression function and is defined as eI hat = Y – Y hat. ei is the estimated error term of the population function.
B. What is (i and define it precisely. What are the four reasons for the inclusion of this error term in the population regression function (model)?
(i is the stochastic term in the population regression function. The four reasons for its existence are: 1. Omitted variable 2. Measurement error 3. Different functional form 4. to account for purely randomness in the human behavior.
C. Draw a graph where you can clearly show E(Yi(XI) = (( + ((XI and Yi = (0 + (1Xi. Show also in your graph (( and e6 for the X6. This graph graph will show true and estimated regression lines together with their respective error terms.
See Figure 2.1 on pages 18 (& 39) of the textbook for the graph.
D. Distinguish or make contrast between an estimator and an estimate.
An estimator is a formula such as the OLS formula that tells us how to compute beta hat, and an estimate is the value of beta computed by that formula.
2. In a study of fertility patterns a random sample of ten newly married couples were asked the number of children they desire to have (X). Twenty years later all ten couples were asked the number of children they actually had (Y). The following table contains the data for X and Y.
Actual and Desired Number of Children of Ten Randomly Selected
Spring 2001
Answers to Midterm Test No. 1
1. Consider a regression model of relating Y (the dependent variable) to X (the independent variable) Yi = (0 + (1Xi+ (i where (i is the stochastic or error term. Suppose that the estimated regression equation is stated as Yi = (0 + (1Xi and ei is the residual error term.
A. What is ei and define it precisely. Explain how it is related to (i.
ei is the residual error term in the sample regression function and is defined as eI hat = Y – Y hat. ei is the estimated error term of the population function.
B. What is (i and define it precisely. What are the four reasons for the inclusion of this error term in the population regression function (model)?
(i is the stochastic term in the population regression function. The four reasons for its existence are: 1. Omitted variable 2. Measurement error 3. Different functional form 4. to account for purely randomness in the human behavior.
C. Draw a graph where you can clearly show E(Yi(XI) = (( + ((XI and Yi = (0 + (1Xi. Show also in your graph (( and e6 for the X6. This graph graph will show true and estimated regression lines together with their respective error terms.
See Figure 2.1 on pages 18 (& 39) of the textbook for the graph.
D. Distinguish or make contrast between an estimator and an estimate.
An estimator is a formula such as the OLS formula that tells us how to compute beta hat, and an estimate is the value of beta computed by that formula.
2. In a study of fertility patterns a random sample of ten newly married couples were asked the number of children they desire to have (X). Twenty years later all ten couples were asked the number of children they actually had (Y). The following table contains the data for X and Y.
Actual and Desired Number of Children of Ten Randomly Selected