Use the data in RDCHEM to further examine the effects of...

Use the data in RDCHEM to further examine the effects of outliers on OLS estimates and to see how LAD is less sensitive to outliers. The model is $$=\beta_{0}+\beta_{1} \text { sales }+\beta_{2} \text { sales }^{2}+\beta_{3} \text { profmarg }+u$$ where you should first change sales to be in billions of dollars to make the estimates easier to interpret. (i) Estimate the above equation by OLS, both with and without the firm having annual sales of almost $40 billion. Discuss any notable differences in the estimated coefficients. (ii) Estimate the same equation by LAD, again with and without the largest firm. Discuss any important differences in estimated coefficients. (iii) Based on your findings in (i) and (ii), would you say OLS or LAD is more resilient to outliers?

Use the data in RDCHEM to further examine the effects of outliers on OLS estimates and to see how
LAD is less sensitive to outliers. The model is
$$=\beta_{0}+\beta_{1} \text { sales }+\beta_{2} \text { sales }^{2}+\beta_{3} \text { profmarg }+u$$
where you should first change sales to be in billions of dollars to make the estimates easier to
interpret.
(i) Estimate the above equation by OLS, both with and without the firm having annual sales of
almost $40 billion. Discuss any notable differences in the estimated coefficients.
(ii) Estimate the same equation by LAD, again with and without the largest firm. Discuss any
important differences in estimated coefficients.
(iii) Based on your findings in (i) and (ii), would you say OLS or LAD is more resilient to outliers?

Key Concepts

Robustness in Statistical Estimation

Robustness refers to the ability of an estimation method to remain reliable and unaffected by violations of assumptions or by aberrant data points such as outliers. In the context of regression, a robust estimator produces consistent and accurate estimates even when the dataset includes influential outliers, making it important for ensuring the validity of inferential statistics in applied research.

Least Absolute Deviations (LAD) Regression

LAD regression estimates model parameters by minimizing the sum of absolute deviations between the observed values and the predicted values. This approach is more robust to outliers compared to OLS, as it does not square the residuals, thereby reducing the influence of extreme values on the estimation process.

Data Transformation

Data transformation involves modifying the scale or distribution of variables to facilitate better interpretation or meet model assumptions. For example, rescaling a variable such as 'sales' from millions to billions can simplify the interpretation of coefficients and may help in reducing computational difficulties, though it does not inherently affect the influence of outliers.

Outliers

Outliers are observations in a dataset that lie far away from the bulk of the data. They can significantly influence statistical estimates, especially those based on methods that are sensitive to extreme values. Understanding how outliers affect model estimates is key in data analysis, as they may distort the estimated relationships between variables.

Ordinary Least Squares (OLS) Regression

OLS is a statistical method used to estimate the parameters in a linear regression model by minimizing the sum of squared residuals. Although it is widely used due to its optimality properties under classical assumptions, OLS can be highly sensitive to outliers because squared errors amplify the influence of extreme observations.

Transcript

00:01 All right, hello everybody.

00:02 Today we are going to be doing a fairly basic econometrics exercise, exploring the sensitivity of ols and lad estimates and how sensitive they are to outliers.

00:13 So let's get right into it.

00:14 First thing we want to do is install our woldridge package.

00:20 We don't already have it installed.

00:24 I should already have this installed, but the fact that i don't slightly worries me.

00:32 Okay, and there we go.

00:34 The package is now installed and i can open up the library.

00:38 I should, for this one, we're also going to be using the l1 pack package.

00:45 This package has the lad function, so it just makes it a little bit easier on us to actually perform the lad function.

00:57 So that's that and really click in a comment here.

01:02 I'm going to write out our formula.

01:04 Our formula for today is going to be r &rd in tens is equal to sales plus sales square.

01:12 Sales square is a column already in the dataset so we don't have to worry about manually squaring sales plus profmarch.

01:22 Okay, cool.

01:23 So now let's get straight into it.

01:27 So first we want to estimate the equation using ols.

01:31 But before we do that, actually, we need to change sales and sales square to be in billions of dollars.

01:37 Right now, they're in millions.

01:39 So this is pretty easy.

01:41 It would be if i had the dataset.

01:44 But let's look at this really quick, right? so we have our data set here.

01:49 You can see sales, 4 ,570.

01:53 This is millions.

01:54 All of these are millions.

01:56 And sales square is therefore millions squared.

01:59 So we have 2 .089 times 10 to the seventh, right? or the sixth, et cetera, et cetera, et cetera.

02:07 So what we're going to have to do is we are going to have to convert sales to billions and sales squares to billions.

02:17 So how do we do that? well, to change something, what we're going to want to do is we're going to say rd chem.

02:22 That's our table or data set.

02:25 And then we're going to select the sales column.

02:28 Then we're going to use our variable equality operator, right? so rdcm sales is going to be equal to sales divided by 1 ,000.

02:38 Pretty simple.

02:39 To do this for sales squared, we can't divide by 1 ,000.

02:43 In fact, we have to divide by 1 ,000 squared or 1 million.

02:52 So doing that, you'll now see we come up to here.

02:56 Our sales is now 4 .570, right? that's 1 ,000 times less.

03:01 And our sales, the coefficient here went from 10 to the 7th to 10 to the 1st.

03:06 That's a difference of 10 to the 6th or 1 million.

03:10 So you can see that this all worked out fine.

03:13 All right, now we need to perform our ols estimates...

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