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# Multiple regression

#### aligardezi

##### New Member
Hi,
This might be a really simple question but I am trying to understand the following statement

The multiple regression model permits estimating the effect on a dependent variable of changing on regressor while holding the other regressors constant.

What does it mean "hold the other regressors constant"?

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
For e.g. if the regression equation is y=b0+b1*x1+b2*x2+b3*x3 ,
suppose initial values of regressor be x1=a,x2=b and x3=c where a,b,c are constants so that the value of dependent variable y is y= b0+b1*a+b2*b+b3*c ...(1)
Now suppose we change the regressor x1 to new value x1=a' while holding the other regressors x2,x3 at their previous values so that x2=b and x3=c are constant at their previous values , a new value of y results say y'=b0+b1*a'+b2*b+b3*c ...(2)
(2)-(1) => y'-y = b0+b1*a'+b2*b+b3*c - (b0+b1*a+b2*b+b3*c) = b1*(a'-a) thus we have estimated the effect of changing the regressor value x1 on the dependent variable y by holding other regressors constant.

thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @aligardezi It's a good question! In the assigned Stock & Watson, one of the multiple regression models (with three explanatory or independent variables; aka, three regressors) is given by:

TestScore = 649.6 - 0.29*STR + 3.87*Expn - 0.656*PctEL, where STR is the student-teacher ratio and Expn is total spending per student.

The -0.29 coefficient suggests that an increase of one student per class (STR = STR + 1 or ΔSTR = +1) is associated with a drop (due to the negative), on average, of test scores by 0.29. The qualifier--i.e., "hold the other regressors constant"--means that the expected drop of 0.29 in the test score, if STR increases by one unit, is conditional on holding the value of (Expn) and (PctEL) constant; that is, they do not change simultaneously, it is the "partial effect" of changing changing only the STR regressor rather than some combination of regressors. I hope that's helpful!

@ShaktiRathore Thank you! sorry for the cross-post #### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
David Harper CFA FRM your example really clarifies a lot better .

thanks