LECTURE NOTE, DAY 17
CH 6. MULTIPLE REGRESSION MODEL
ECO 3007, 2016 SPRING INSTRUCTOR : JUNGMO YOON
HANYANG UNIVERSITY
6.2. Interpretation of Multiple Regression Coefficients Read pages 186 - 189. In a multiple regression model,
(0.1) Yi = β0+ β1X1i+ β2X2i+ ui
the interpretation of the coefficient β1 is different than it was for a simple regression.
Now β1 is the effect on Y of a unit change in X1, holding X2 constant (or controlling for X2).
Remark. Often we say that β1 is the partial effect on Y of X1, holding X2 fixed.
Example 1) Let Y be the selling price of houses, X1 the number of bedrooms, and X2 the number of bathrooms. Consider changes in prices when we change the number of bedrooms by ∆X1, while hold X2 constant. Then the new price will be
(0.2) Y + ∆Y = β0+ β1(X1+ ∆X1) + β2X2 while the expected value of the old price would be described by
(0.3) Y = β0+ β1X1+ β2X2. Subtracting (0.3) from (0.2) yields
∆Y = β1X1
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2 ECO 3007, 2016 SPRING INSTRUCTOR : JUNGMO YOON HANYANG UNIVERSITY
therefore,
β1 = ∆Y
∆X1
, while holding X2 constant.
The point of this example is the followings. In reality, X1 and X2 are highly correlated, so when we see houses with many bedrooms, they tend to have many bathrooms, too. So if you see higher prices for houses with many bedrooms, you do not exactly know whether it is because it has more bedrooms or perhaps more bathrooms. The bottom line is that it is not easy to observe the effect on Y of X1 while hold X2 constant. We need a good way to tell this partial effect (the impact of bedrooms on the selling price, holding the number of bathrooms constant). That is what regression does. And it is one of reasons why regression is so useful.
Remark 0.1. The discussion here is not rigorous. The precise meaning of ‘holding X2 constant’ can be understood through the lens of the (Gauss-)Frish-Waugh theorem.
If you are interested, read Appendix 6.3.
Constant Regressor
Another way to write the equation (0.1) is
(0.4) Yi = β0X0i+ β1X1i+ β2X2i+ ui
where X0i = 1 for every i. X0i is called the constant regressor. In equation (0.1) we call β0 the intercept. In equation (0.4) β0 is called the slope for the constant regressor.
