Use the data in CEOSAL. 2 to answer this question. (i)...

Use the data in CEOSAL. 2 to answer this question. (i) Estimate the model $$\ lsalary = beta_{0}+\beta_{1} \text { lsales }+\beta_{2} \text {lmktval}+\beta_{3} \text { ceoten }+\beta_{4} \text {ceoten}^{2}+u$$ by OLS using all of the observations, where lsalary, lsales, and lmktvale are all natural logarithms. Report the results in the usual form with the usual OLS standard errors. (You may verify that the heteroskedasticity-robust standard errors are similar.) (ii) In the regression from part (i) obtain the studentized residuals; call these stri. How many studentized residuals are above 1.96 in absolute value? If the studentized residuals were independent draws from a standard normal distribution, about how many would you expect to be above two in absolute value with 177 draws? (iii) Reestimate the equation in part (i) by OLS using only the observations with $\left|s t r_{i}\right| \leq 1.96 .$ How do the coefficients compare with those in part (i)? (iv) Estimate the equation in part (i) by LAD, using all of the data. Is the estimate of $\beta_{1}$ closer to the OLS estimate using the full sample or the restricted sample? What about for $\beta_{3} ?$ (v) Evaluate the following statement: "Dropping outliers based on extreme values of studentized residuals makes the resulting OLS estimates closer to the LAD estimates on the full sample."

Use the data in CEOSAL. 2 to answer this question.
(i) Estimate the model
$$\ lsalary = beta_{0}+\beta_{1} \text { lsales }+\beta_{2} \text {lmktval}+\beta_{3} \text { ceoten }+\beta_{4} \text {ceoten}^{2}+u$$
by OLS using all of the observations, where lsalary, lsales, and lmktvale are all natural
logarithms. Report the results in the usual form with the usual OLS standard errors. (You may
verify that the heteroskedasticity-robust standard errors are similar.)
(ii) In the regression from part (i) obtain the studentized residuals; call these stri. How many
studentized residuals are above 1.96 in absolute value? If the studentized residuals were
independent draws from a standard normal distribution, about how many would you expect to
be above two in absolute value with 177 draws?
(iii) Reestimate the equation in part (i) by OLS using only the observations with $\left|s t r_{i}\right| \leq 1.96 .$ How do the coefficients compare with those in part (i)?
(iv) Estimate the equation in part (i) by LAD, using all of the data. Is the estimate of $\beta_{1}$ closer to the OLS estimate using the full sample or the restricted sample? What about for $\beta_{3} ?$
(v) Evaluate the following statement: "Dropping outliers based on extreme values of studentized
residuals makes the resulting OLS estimates closer to the LAD estimates on the full sample."

Key Concepts

Impact of Sample Restriction on Regression Estimates

Restricting the sample, for example by excluding observations identified as outliers, can significantly affect the estimated regression coefficients. By comparing estimates from the full sample with those from a restricted sample that excludes extreme values, researchers can assess the sensitivity of their results to outliers and better understand the robustness of the estimated relationships.

Least Absolute Deviations (LAD) Estimation

LAD estimation is an alternative regression technique that minimizes the sum of the absolute differences between the observed values and the predicted values, rather than the sum of squared differences. This method is more robust to outliers compared to OLS, as it less aggressively penalizes large deviations, and provides a useful comparison to OLS estimates when the sample may contain influential observations.

Outlier Detection in Regression

Outlier detection involves identifying data points that deviate markedly from the pattern established by the majority of observations. In regression analysis, metrics like studentized residuals are used to flag potentially problematic observations by comparing them to a threshold (commonly 2 in absolute value), thereby enabling researchers to assess the impact of these outliers on the estimated regression parameters.

Ordinary Least Squares (OLS) Estimation

OLS is a statistical method used to estimate the parameters of a linear regression model by minimizing the sum of the squared differences between the observed dependent variable values and those predicted by the linear function of the explanatory variables. This approach produces best linear unbiased estimators under the classical regression assumptions and serves as the standard method for determining relationships between variables in empirical research.

Heteroskedasticity and Robust Standard Errors

Heteroskedasticity occurs when the variance of the errors in a regression model is not constant across observations, which can lead to biased standard error estimates and unreliable inference. Robust standard errors are adjusted for this non-constant variance, providing more reliable inference even when the error variance is heterogeneous, and are crucial for ensuring the validity of hypothesis tests in applied work.

Studentized Residuals

Studentized residuals are the residuals from an OLS regression divided by an estimate of their standard deviation. They standardize the residuals, allowing for the assessment of whether an individual observation's deviation from the regression line is unusually large, which is helpful in detecting outliers or influential points in the data analysis process.

Transcript

00:02 This is the result for part 1.

00:09 We use the full sample, which has 177 observations.

00:21 From this estimation, we obtain the studentized residuals, and we call them str sub i.

00:36 The number of studentized residuals, which are above 1 .3 .5 .5.

00:42 0 .96 in absolute value is 9.

00:54 If the studentized residuals were independently drawn from a standard normal distribution, we would expect about 5 % of our sample.

01:17 So 177 times 5 % you will get a number between 8 and 9.

01:29 It is 8 .56 something.

01:34 We would expect between 8 and 9 cases of student tides residuals to be above 2.

01:47 It is so because in a standard normal distribution, about 95 .5 % of the observations are within two standard errors or standard deviation are within two standard deviation and in a standard normal distribution the standard deviation is one so 95 .5 % of the observation are within two.

02:52 Equivalently it means 5 % up to 5 % of the observations are either above 2 or less than minus 2.

03:18 That's why we have the 5 % number here.

03:23 And i should say right here to be above 2 in absolute value.

03:35 You can check, you can fact check this statement.

03:39 And you will find that there are eight observations with studentized residuals above two in absolute value.

04:07 Part 3.

04:10 The studentized residuals are used to detect outliers.

04:16 We will drop their alliers, which are defined as observation with studentized residuals above 1 .96 in absolute value.

04:28 So we will drop the 9 cases we find out in part 2.

04:38 We will re -estimate the model in part 1 again using 169 observations.

04:50 This is the result.

04:55 Compare with the regression in part 1, we find that the main coefficient becomes significant at the 1 % level.