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The slope a short regression finds

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The true model is $Y = 3 X - 2 Z + ϵ$ with $Z = 0.25 X + U$ , where $U$ and $ϵ$ are independent of $X$ and of each other. You regress $Y$ on $X$ alone and leave $Z$ out. What slope does the regression find in a very large sample? Give a decimal.

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Hint

Substitute $Z$ into the model for $Y$ and read the coefficient on $X$ .

Solution

Substituting gives $Y = (3 - 2 \times 0.25) X - 2 U + ϵ$ .
The terms in $U$ and $ϵ$ are independent of $X$ , so they do not change the slope.
The slope is $3 - 2 \times 0.25 = 2.5$ .

Answer: $2.5$

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