ETC1000 Chap.6 Multiple Regression, Dummies and Interactions
Multiple Regression, Dummies and Interactions
Every coefficient carries a clause
A multiple regression coefficient answers a narrower question than a simple one. It gives the difference in the outcome between cases that differ by one unit in that variable and are alike on every other variable in the model.
That clause is the reason the technique exists, and it explains why a coefficient can shrink or even change sign when a second variable is added: part of what the first was measuring has been given to the variable it belonged to.
A dummy coefficient is always a comparison
A categorical explanation enters as a set of zero one variables with one category deliberately omitted.
The omitted category is the baseline, and every dummy coefficient is a difference against it. Including a dummy for every category duplicates the intercept and breaks the model, which software often hides by silently dropping a column the analyst never chose.
Interactions let a slope depend on a group
Multiplying two explanatory variables together lets the effect of one depend on the value of the other.
Reading the model group by group is the reliable method: for the baseline group both the dummy and the product term vanish, and for the comparison group the intercept gains the dummy coefficient while the slope gains the interaction coefficient.
Judging a larger model
Adding a variable can never lower the ordinary fit measure, so it cannot be used to choose between models.
The adjusted version charges a penalty per variable, which makes it comparable across specifications. Neither figure says anything about whether an individual coefficient is precisely estimated or free of omitted variable contamination.
What this chapter covers
- 01
Reading a coefficient with its holding fixed clause attached
- 02
Why adding a variable changes the coefficient on the first one
- 03
Dummy variables, the omitted baseline, and the count of dummies you need
- 04
Interaction terms and the slope difference identity
- 05
Main effects that stop being main effects once an interaction is present
- 06
The adjusted fit measure and what a high value does not certify
Reconstruct two fitted lines from four coefficients
- 1Write the fitted line for non-members and for members.
- 1Give the difference in slopes and check it against the interaction coefficient.
- 1Predict spend for a member with 10 hours of exposure.
Key terms
- Multiple regression
- A model with several explanatory variables, each coefficient read with the others held fixed.
- Holding fixed
- The clause meaning the arithmetic has removed the other variables that are in the model.
- Dummy variable
- A variable taking one inside a category and zero outside it.
- Baseline category
- The category left without a dummy, against which every other coefficient is measured.
- Interaction term
- The product of two explanatory variables, letting one variable's effect depend on the other.
- Adjusted fit measure
- A fit statistic penalised for the number of explanatory variables, so models of different sizes compare.
- Omitted variable
- A determinant of the outcome left out of the model, whose influence a correlated included variable carries.
Multiple Regression, Dummies and Interactions FAQ
Why did my coefficient change when I added another variable?
Because it now answers a different question. In the simple model it measured everything that travelled with that variable, including whatever the second variable represents. In the larger model that share has been given to the variable it belongs to. Neither estimate is wrong; the larger one answers the question a reader usually means, which is a comparison between cases that are alike in other respects.
How many dummy variables do I need for four categories?
Three. Four dummies for four categories add to a column of ones, which is what the intercept already is, so the model cannot separate them and either fails or silently drops one. With k categories the rule is k minus one dummies plus an omitted baseline, and changing which category is omitted changes every coefficient while leaving the predictions identical.
What does the interaction coefficient actually equal?
The difference between the two groups' slopes. If the baseline slope is 0.24 and the interaction coefficient is 0.66, the comparison group's slope is 0.90, and 0.90 minus 0.24 returns 0.66. Using that identity as a check catches the commonest slip, which is adding the dummy coefficient into a slope when it belongs to the intercept.
Does a high adjusted fit measure make the coefficients trustworthy?
No. Fit describes how close the predictions sit to the outcomes across the sample. It says nothing about whether an individual coefficient is precisely estimated, whether its sign is stable, or whether an omitted variable is contaminating it. Report the fit for the model and the standard error or interval for each coefficient you intend to interpret, because they are separate claims.
Exam move
Rehearse the interpretation sentence as a fixed template: the outcome and its units, the variable and its units, the holding fixed clause with the other variables named, and then a full stop before the word cause. Most of the marks in this topic are for producing that sentence, and most of the losses are for a clause left out of it.
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