Reference 3

Dummy Variable Trap

Author

Byeong-Hak Choe

Published

January 29, 2026

Modified

January 29, 2026

🚫 Dummy Variable Trap (Perfect Multicollinearity)

Suppose our categorical variable is orange juice brand with 3 categories:

Dominick’s (D)
Minute Maid (M)
Tropicana (T)

Create three dummy variables:

\(D_D = 1\) if Dominick’s, 0 otherwise
\(D_M = 1\) if Minute Maid, 0 otherwise
\(D_T = 1\) if Tropicana, 0 otherwise

Because every purchase/week observation belongs to exactly one brand,

\[ D_D + D_M + D_T = 1 \quad \text{for every observation.} \]

✅ What goes wrong if we include an intercept + all dummies?

Consider the regression model (e.g., for log sales):

\[ y_i = \beta_0 + \beta_D D_{D,i} + \beta_M D_{M,i} + \beta_T D_{T,i} + \varepsilon_i \]

But from \(D_D + D_M + D_T = 1\), we can solve for \(D_T\):

\[ D_{T,i} = 1 - D_{D,i} - D_{M,i} \]

Substitute into the regression:

\[ \begin{aligned} y_i &= \beta_0 + \beta_D D_{D,i} + \beta_M D_{M,i} + \beta_T(1 - D_{D,i} - D_{M,i}) + \varepsilon_i \\ &= (\beta_0 + \beta_T) + (\beta_D - \beta_T)D_{D,i} + (\beta_M - \beta_T)D_{M,i} + \varepsilon_i \end{aligned} \]

🔥 Key point

The model depends only on:

\((\beta_0 + \beta_T)\)
\((\beta_D - \beta_T)\)
\((\beta_M - \beta_T)\)

So \(\beta_0, \beta_D, \beta_M, \beta_T\) are not uniquely identified (infinitely many coefficient sets give the same fitted values).

That is the dummy variable trap:
intercept + all brand dummies ⇒ perfect multicollinearity.

✅ Fix: drop one dummy (choose a reference brand)

Drop \(D_T\) (Tropicana becomes the baseline/reference):

\[ y_i = \beta_0 + \beta_D D_{D,i} + \beta_M D_{M,i} + \varepsilon_i \]

Now:

\(\beta_0\) = expected \(y\) for Tropicana (reference brand)
\(\beta_D\) = difference (Dominick’s − Tropicana)
\(\beta_M\) = difference (Minute Maid − Tropicana)

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