# Regression marginal effects

$$\text{wage}=b_0+b_1\text{exper}+ b_2\text{exper}^2 + \textit{other variables}$$

How can I find the marginal effect of experience on wage? I thought it was just the derivative: $b_1$+$b_2$, but, for example, if I try plugging in numbers that are different by one year (experience 10 years => experience 11 years), that doesn't seem to be right:

$2.36\times 11-0.077\times 121-2.36\times 10+0.077\times 100=0.743$

Marginal: $2.36\times 1-0.077\times 1=2.283$

• What makes you feel that the answer is not correct? By the way, if you would like us to check it, please don't make us guess what you're doing: explain what the numbers "2.36", "11", etc. correspond to. BTW, I notice that some of your previous questions, such as stats.stackexchange.com/questions/43197, have been marked up in $\TeX$: it would be nice if you would mark this one up, too, to make it readable. – whuber Nov 10 '12 at 23:27

The marginal effect of experience on wage is the derivative $b1 + 2(b2)(exper)$, and varies with the number of years experience. For 10 years experience, the Marginal calculation should therefore be:

$2.36 - 2(0.077)(10) = 0.82$

Comparison of the wage at 10 and 11 years experience is only an approximation to the marginal effect at 10 years. A more accurate approximation can be obtained by comparing 10 and 10.1 years. This gives a difference in wage of:

$(2.36)(10.1) - (0.077)(102.01) - (2.36)(10) + (0.077)(100) = 0.081$

Multiplying by 10 to convert to per year units yields 0.81, very close to the above.

Finally, although I was able with some effort to make sense of your question, I suggest that you take note of the comment above by whuber - it would have been helpful, for example, to have stated that the estimated values of b1 and b2 were 2.36 and -0.077.

$wage=b_0+b_1(exper)+b_2(exper)^2+(others variables) = b_0 + 2.36(exper) - 0.077(exper)^2 + (other variables)$,

then the marginal effect of experience on wages is given by the partial derivative of wages wrt experience, given by $w_(exper) = 2.36 - 2*0.077(exper) = 2.36 - 0.154(exper).$ Hence the effect of adding an extra year of experience after 10 years of experience would be an increase of

$w_(exper) = 2.36 - 2*0.077(exper) = 2.36 - 0.154(10)= 0.82$

on wages at the 10th year. The exact increase in wages is given by

$2.36*11 - 0.077*11^2 - 2.36*10 + 0.077*10^2 = 0.743$

(which is not too far off \$0.82).

Remember the marginal effect is ONLY an approximation.