Overview
Module 9 introduces the idea that the OLS estimator is BLUE: It is the best, linear, unbiased estimator available. But this requires that some important assumptions hold. Module 9 thus lays out these assumptions as well as methods for checking for potential violations. Module 9 then lays out some common OLS extensions, including dummy and categorical independent variables and interaction terms.
Objectives
- Explain what we mean when we say that the OLS estimator is BLUE.
- Grasp the intuition behind the core OLS assumptions.
- Examine and begin checking assumptions in R; explain what violations of different assumptions mean for statistical inference.
- Incorporate and interpret dummy and categorical independent variables as well as interaction terms in linear regression.
Assignment
Assignment 4 is due by 11:59pm CST on Wednesday, 3/17. Be sure to download the Assignment 4 RMD and review it early in the week so that you can plan your time accordingly.
The Module
As you work your way through the video lectures, note that I again draw many examples from the “CASchools” dataset that is available upon installing and loading the “AER” package in R. You should feel free to follow along with these examples in R, pausing the video as necessary. To do so, start by running this code, which loads the required package and creates our main IV and DV:
# Load the data
require(AER)
data(CASchools)
# Dependent variable
CASchools$score <- (CASchools$read + CASchools$math)/2
# Independent variable
CASchools$STR <- CASchools$students/CASchools$teachers
# Our simple model
model <- lm(score ~ STR, data = CASchools)OLS Assumptions: BLUE
First read OIS Section 8.3 on “Checking Model Assumptions Using Graphs.” Then watch the video lecture series on “OLS Assumptions.”
OLS Assumptions: The Core Assumptions
OLS Assumptions: Checking Assumptions
OLS Extensions
We have so far mainly included numeric variables in our regression models. These are by far the easiest to interpret. But we might also wish include other variable types in our models, including dummy and categorical variables. The main challenges here concern interpretation and procedure. In terms of interpretation, we can’t straightforwardly interpret the coefficient for a dummy independent variable as “a change in Y is linked to a change in Y.” In terms of procedure, we need to be somewhat careful in our syntax whenever we run a regression that includes a categorical independent variable.
Start by reading Kellstedt and Whitten, the Fundamentals of Political Science Research, pp. 202-212. Then watch the brief video series on “OLS Extensions.”
Extensions: Dummy independent variables
As you watch the next video, note that there is an error on my last, which I neglected to correct during recording. Specifically, when you come to the last slide, note the final bullet point: STR should equal 0. Thus, the correct interpretation should read “On average, when STR = 0, we expect schools with Hi_ELL to have test scores around 692.361 – 19.533 = 672.8.”
Extensions: Categorical independent variables
Extensions: Interaction terms
Standard error and confidence intervals
Now read OIS, Sections 4.1 through 4.2 (pp. 169-179). Note that the Syllabus incorrectly states that you should also read Section 4.3. This is an error; we will read Section 4.3 next week when we begin performing hypothesis tests.
Now watch the videos on “Standard error” and “Confidence intervals”.
Extending confidence intervals to indicator variables
Let’s now see how we might extend the logic of CIS to indicator variables. We have already encountered indicator variables while working with the British Election Study (BES) data in Exercise 1 and Assignment 1: In particular, we created the indicator variable female, which took the value 1 if the respondent was female and 0 if the respondent was male. Can we apply CIs to this sort of variable?
Think about this for a moment: In Assignment 1, many of you correctly noted that female is a (nominal) categorical variable, and accordingly, it seems strange to calculate its mean and standard deviation. But many of you also noted that the mean does supply us with some useful information about the variable: namely, the proportion of respondents in the data that are female. This is a good intuition. When we deal with indicator variables, we are – strictly speaking – interested in a proportion rather than an average or central tendency (which are captured by the arithmetic mean). Accordingly, we have to shift our thinking and terminology somewhat. But don’t be thrown by this shift! Although the next video introduces some new parameters (π) and estimators (p-hat) – as well as some math – the concepts themselves are fairly straightforward. For instance, you will find that the formula for calculating the standard deviation of the sample proportion (p-hat) initially looks very similar to the formula for calculating the standard deviation of a numeric variable.
With this in mind, watch the video on “Confidence Intervals: Extensions”.
Note that we did not read the sections from OIS that deal with the sample proportion. If you would like to do so, the relevant section is Section 3.3 (“Geometric distribution”) on pages 141-145. (The following section on the “Binomial distribution” is also a useful.) We will read the section on the sampling distribution of the sample proportion next week (Section 4.5).
Practice questions
Now complete the practice questions, which can be downloaded at the “Files/Practice Questions” section at Canvas. You’ll note that the functions that you used to complete Exercise 2 will be very handy here. In fact, we are going to “prove” the logic of CIs in the very same way that we “proved” the CLT: experimentally, through simulation. The real payoff in using R to simulate our data and build CIs is that we can actually visualize what we mean when we interpret our CIs by saying that “we constructed this CI using a method that produces CIs that contain the true population parameter 95 of 100 times.” Let’s attach some real meaning to this phrase!
Assignment 2
Once you have completed the practice questions, complete Assignment 2. You can download the RMD at the “Files/Exercises and Assignments” section at Canvas. Submit your completed PDF via Canvas by 5pm on Monday, 2/15. I repeat my standing admonition that you collaborate with your fellow students to complete the practice questions, exercises, and assignments. Of, course, this does not mean copy-pasting answers or code; it does mean that you should talk about the questions, discuss your answers, and help each other to better understand both the concepts and code.