EMET8002 Chap.2 Simple and Multiple Linear Regression
Simple and Multiple Linear Regression
The linear model
This chapter covers the linear regression model as the course presents it: an outcome split into a signal explained by the regressors and a noise term, estimated by ordinary least squares, which minimises squared residuals.
It separates the unobserved population line from the sample line you estimate.
Interpreting many regressors
Multiple regression reads each coefficient as the effect of its variable holding the others constant.
The chapter applies the Y, X, Z approach to choosing variables and explains why adjusted R-squared is a fairer measure of fit than R-squared.
Credit in the SAT score regressions
The Week 3 lab regresses state SAT scores on per-pupil spending, first alone and then with the share of students sitting the test, median income and the share of adults with a degree.
Credit goes to describing every variable and its correlations before estimating, to comparing results with and without outliers, to testing the residuals for normality, and to a quantile regression at the 10th, 50th and 90th percentiles that shows whether one slope fits the whole distribution.
What this chapter covers
- 01
Dependent and independent variables
- 02
Intercept and slope
- 03
Ordinary least squares and residuals
- 04
Population versus sample regression line
- 05
Ceteris paribus interpretation
- 06
Outcomes, explanatory variables and controls
- 07
R-squared and adjusted R-squared
Worked example · free
Fitting a wage line to five workers
- 1The means are 5 years and $28 an hour.
- 1Deviation cross-products sum to 64 and squared experience deviations to 40, so the slope is 64/40 = 1.6.
- 1The intercept is 28 minus 1.6 times 5, which is 20.
- 1Each extra year of experience is associated with an hourly wage $1.60 higher on average.
Key terms
- Residual
- The gap between an observed outcome and the value the fitted line predicts for it.
- Control Variable
- A regressor held constant so that the relationship between the main variable and the outcome is isolated.
- Adjusted R-Squared
- A version of R-squared that penalises extra regressors and need not rise when one is added.
- Maximum Likelihood
- An estimation method that picks the parameters making the observed data most likely.
Simple and Multiple Linear Regression FAQ
What does holding other factors constant mean in a multiple regression?
Each slope measures the change in the outcome for a one-unit change in its own regressor while every other included regressor stays fixed. It does not hold fixed anything left out of the model, which is why omitted variables remain a concern.
Why is adjusted R-squared preferred to R-squared?
R-squared never falls when a regressor is added, so long models always look better. Adjusted R-squared corrects for the number of predictors and can fall when a weak regressor enters, which makes it a fairer basis for comparing specifications.
What is the Y, X, Z approach?
It is the course's way of choosing variables: Y is the outcome you expect to be influenced, X is the main explanatory variable, often one a policy can move, and Z collects the controls held constant to isolate the relationship between X and Y.
What is the difference between the error term and a residual?
The error is the unobserved gap between an outcome and the population regression line. A residual is its sample counterpart, the gap between an observed outcome and the fitted line you estimated. Residuals can be computed and plotted; errors cannot, which is why diagnostics work with residuals.
Why is the intercept often not interpreted?
The intercept is the predicted outcome when every regressor equals zero. In many applications, such as state test scores with spending and income both at zero, that point lies far outside the data, so the intercept anchors the line but carries no useful economic meaning.
Assessment move
Write an interpretation sentence for every coefficient you estimate, with units and the phrase holding other factors constant. Then estimate the same model with and without outliers and say which you prefer and why. Then add a control to a simple model and explain in two sentences why the coefficient of interest moved, naming the variable it was previously picking up.
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