multiple regression with two explanatory variables: example 1 this sequence provides a geometrical...

MULTIPLE REGRESSION WITH TWO EXPLANATORY VARIABLES: EXAMPLE

1

This sequence provides a geometrical interpretation of a multiple regression model with two explanatory variables.

EARNINGS = b1 + b2S + b3EXP + u

S

b1

EARNINGS

EXP

2

Specifically, we will look at an earnings function model where hourly earnings, EARNINGS, depend on years of schooling (highest grade completed), S, and years of work experience, EXP.



S

b1

EARNINGS

EXP

3

The model has three dimensions, one each for EARNINGS, S, and EXP. The starting point for investigating the determination of EARNINGS is the intercept, b1.



S

b1

EARNINGS

EXP

4

Literally the intercept gives EARNINGS for those respondents who have no schooling and no work experience. However, there were no respondents with less than 6 years of schooling. Hence a literal interpretation of b1 would be unwise.



S

b1

EARNINGS

EXP

pure S effect

5

S

b1

EARNINGS

b1 + b2S



The next term on the right side of the equation gives the effect of variations in S. A one year increase in S causes EARNINGS to increase by b2 dollars, holding EXP constant.

EXP

pure EXP effect

6

S

b1

b1 + b3EXP

EARNINGS

EXP



Similarly, the third term gives the effect of variations in EXP. A one year increase in EXP causes earnings to increase by b3 dollars, holding S constant.

pure EXP effect

7

S

b1

b1 + b3EXP

b1 + b2S + b3EXP

EARNINGS

EXP

b1 + b2S

combined effect of S and EXP



Different combinations of S and EXP give rise to values of EARNINGS which lie on the plane shown in the diagram, defined by the equation EARNINGS = b1 + b2S + b3EXP. This is the nonstochastic (nonrandom) component of the model.

pure S effect

8



The final element of the model is the disturbance term, u. This causes the actual values of EARNINGS to deviate from the plane. In this observation, u happens to have a positive value.

pure EXP effect

S

b1

b1 + b3EXP

EARNINGS

EXP

u

b1 + b2S

pure S effect

b1 + b2S + b3EXP


b1 + b2S + b3EXP + u

9



A sample consists of a number of observations generated in this way. Note that the interpretation of the model does not depend on whether S and EXP are correlated or not.

pure EXP effect

S

b1

b1 + b3EXP

EARNINGS

EXP

u

b1 + b2S

pure S effect

b1 + b2S + b3EXP



pure EXP effect

10

S

b1

b1 + b3EXP


EARNINGS

EXP

u

However we do assume that the effects of S and EXP on EARNINGS are additive. The impact of a difference in S on EARNINGS is not affected by the value of EXP, or vice versa.



b1 + b2S

pure S effect

b1 + b2S + b3EXP


The regression coefficients are derived using the same least squares principle used in simple regression analysis. The fitted value of Y in observation i depends on our choice of b1, b2, and b3.

11


uXXY 33221 33221ˆ XbXbbY

Fitted model

True model

The residual ei in observation i is the difference between the actual and fitted values of Y.

12


iiiiii XbXbbYYYe 33221ˆ


Fitted model

True model

We define RSS, the sum of the squares of the residuals, and choose b1, b2, and b3 so as to minimize it.

13




Fitted model

True model

233221

2 )( iiii XbXbbYeRSS

First, we expand RSS as shown.

14




Fitted model

True model

233221

2 )( iiii XbXbbYeRSS

)2222

22(

323233122133

22123

23

22

22

21

2

iiiiii

iiiiii

XXbbXbbXbbYXb

YXbYbXbXbbY

iii

iiiii

iiii

XXbbXbb

XbbYXbYXb

YbXbXbnbY

3232331

2213322

123

23

22

22

21

2

22

222

2

15


iii

iiiii

iiii

XXbbXbb

XbbYXbYXb

YbXbXbnbYRSS

3232331

2213322

123

23

22

22

21

2

22

222

2

Then we use the first order conditions for minimizing it.

01

bRSS

0

2

bRSS

0

3

bRSS



Fitted model

True model

33221 XbXbYb

We thus obtain three equations in three unknowns. Solving for b1, b2, and b3, we obtain the expressions shown above. (The expression for b3 is the same as that for b2, with the subscripts 2 and 3 interchanged everywhere.)

16

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii


23322 XXYYXX iii


Fitted model

True model

17

The expression for b1 is a straightforward extension of the expression for it in simple regression analysis.



Fitted model

True model

33221 XbXbYb

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

23322 XXYYXX iii

18

However, the expressions for the slope coefficients are considerably more complex than that for the slope coefficient in simple regression analysis.



Fitted model

True model

33221 XbXbYb

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

23322 XXYYXX iii

19

For the general case when there are many explanatory variables, ordinary algebra is inadequate. It is necessary to switch to matrix algebra.



Fitted model

True model

33221 XbXbYb

23322

233

222

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

23322 XXYYXX iii

. reg EARNINGS S EXP

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 67.54 Model | 22513.6473 2 11256.8237 Prob > F = 0.0000 Residual | 89496.5838 537 166.660305 R-squared = 0.2010-------------+------------------------------ Adj R-squared = 0.1980 Total | 112010.231 539 207.811189 Root MSE = 12.91

------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.678125 .2336497 11.46 0.000 2.219146 3.137105 EXP | .5624326 .1285136 4.38 0.000 .3099816 .8148837 _cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213------------------------------------------------------------------------------

Here is the regression output for the earnings function using Data Set 21.

20

EXPSINGSNEAR 56.068.249.26ˆ


21

It indicates that earnings increase by $2.68 for every extra year of schooling and by $0.56 for every extra year of work experience.






22

Literally, the intercept indicates that an individual who had no schooling or work experience would have hourly earnings of –$26.49.






23

Obviously, this is impossible. The lowest value of S in the sample was 6. We have obtained a nonsense estimate because we have extrapolated too far from the data range.






Copyright Christopher Dougherty 2012.

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2012.10.28

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multiple regression with two explanatory variables: example 1 this sequence provides a geometrical...

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