Non-linear relationships

Question 1

How to find x-value that maximizes y-hat

Answer 1

Take derivative of quadratic and set it equal to 0, then solve for x

Question 2

Slope of quadratic

Answer 2

Take derivative:

Beta 1 hat + 2 Beta 2 hat*x

Question 3

B1 with quadratic

Answer 3

estimated slope when x = 0

Question 4

B2 with quadratic

Answer 4

magnitude of the change in slope

for every one unit change in x, slope is expected to change by 2B-2hat

Question 5

Weakness of quadratic function

Answer 5

sign of the slope will flip flop from positive to negative which is harmful when measure some things, e.g. the optimal level of cholesterol to maximize health, or the relationship between happiness and wages

Question 6

Log functions

Answer 6

slope is always positive but it decreases as x increases, it will eventually get flat

Question 7

Level-level

Answer 7

^y = B0 + B1x

every 1 unit increase in x is predicted to change y by B1 units.

Question 8

Level-log

Answer 8

^y = B0 + log(x)B1

every 1% increase in x is predicted to increase y by B1 / 100 units

Question 9

Log-level

Answer 9

log(^y) = B0 + B1x

Every 1 unit increase in x is predicted to increase y by B1*100 percent

Question 10

Log-log

Answer 10

log(^y) = B0 + log(x)B1

Every 1% increase in x is predicted to increase y by B1%

Question 11

When to use logs

Answer 11

If y and x have a positive relationship but at some point there’s a diminishing marginal return.