Interactions Flashcards
Explain a non-linear function
The effect one unit x has on y depends on the value of x. in other words, its marginal effect is not constant.
Regressions where one unit change in x’s impact on y will differ dependent on the value of x. in other words, it does not have a constant marginal cost. the relation between x and y is not linear. One unit change from 5 to 6 will have a different impact than 10-11.
What types of interactions between independent variables do we have?
Interaction between two binary variables
Interaction between binary and continuous variable
Interaction between two continuous variables
Explain interactions between two binary variables
(lag ny) D1 and D2 is dummy variables. B1 is the effect of changing D=0 to D=1. In this specification, this effect does not depend on the value of D2.
To allow the effect of changing D1 to depend on D2, include the “interaction term” D1 * D2 as a regressor
So if our regression is y = b0 + b1D1 + b2D2 + b3 * D1 * D2 + u:
B3 is the incremental effect o changing D1 from 0 to 1 when D2 = 1
In short terms: if we have y = b0 + b1D1 + b2D2, the value of b1 is only dependent on the value og D1. What if it in reality also depends on the value of D2? That is why we fix it to:
y = b0 + b1D1 + b2D2 + b3 * D1 * D2 + u:
b3 will now capture the last effect, which is how D1 and D2 will interfer.
Interactions between binary and continuous variables
y = b0 + b1x + b2D + u
y = b0 + b1x + b2D + b3D*x + u
To allow the effect of changing x to depend on D, include the “interaction term” D * x as a regressor
Binary and continuous interactions: two regression lines
y = β0 + β1x + β2D + β3D × x + u
When Di = 0: y = β0 + β1x + u.
When Di = 1: y = (β0 + β2) + (β1 + β3)x + u
By playing with specification of the regression equation we can have regression lines that are different: Both in slopes and intercepts (current specification). In slopes only (drop β2D from regression equation). In intercepts only (drop β3D × x from regression equation).
We are now allowing x to depend on D. The value of x might be different based of D. we get the different alternatives:
B0 + b1 + b2D: this allows for a different intercept, but has the same slope
B0 + b1x + d2D + b3(x * D): allows for different intercept and slope
B0 + b1x + b2 (x*D): same intercept, allows for different slope
Steps:
1.Test if the two lines in fact are the same. This is done by F-statistical testing with joint hypothesis. Test if b2 and b3 are 0. If so, that leaves us with b0 + b1x which is the same slope and same intercept as the original
- Test if the two lines have the same slope
If so, b3(x + d) = 0, because b3 = 0.
So divide b3 on its standard error and find its t-statistics.
3.Test id they have the same intercept
y = b0 + b1x + b2D + b3(x * D)
y = b0 + b1X + B2D
To have the same intercept, b2D must equal 0, leaving us with intercept b0 and not b0 + b2. Find b2’s t-stat by dividing its value on its standard error.
Explain the interpretation of the different log equations
Lin-log; 1% increase in X gives a 0,01 * b1 increase in y
Log-lin: one unit increase in x gives 100% * b1 increase in y
Log-Log: 1 % increase in x gives b1% increase in y.
What is a general strategy for modelling a nonlinear function
1.use your economic knowledge
2.estimate a regression using OLS
3. test if non-linear is better than linear. Can be done by t-stat and f-stat
4-plot the nonlinear function. Does the regressor describe the data well? Does the regression fit the scatterplot?
5.estimate effect on Y of a change in x
What is polynomials
Polynomials is made by having x with powers. Most usually 2, 3 or 4 powers. A power of 2 makes a quadratic function while power of 3 makes a cubic function. The amount of degrees/powers can be denoted as r. r degrees gives r-1 bends.
What is the steps with polynomials
Find out how many powers that is suitable for your regression by hypothesis testing. Some times you might know what powers that are most suitable by economic theory or by just looking at the scatterplot.
- pick a maximum value for r, lets say 3
- test for non-linear versus linear. Can test for several powers by using F-stat. The nullhypothesis will be that b2 = 0, b3 = 0 etc. The alternative hypothesis is that at least one is not 0.
- can then test if power 3 is suitable. Use t-stat to see if the beta coefficient is 0. H0: b = 0, HA: b is not 0.
- if you do not reject h0, eliminate one power from the beta
- continue until you find the highest power that is statistically significant.
How to test cubic vs. linear and cubic vs quadratic?
Cubic versus linear: F-test
Cubic versus quadratic: t-test
Explain the quadratic model
The quadratic model has power of 2. That means that it will have a turning point. This leaves us with a miximum value. All results after this maximum value should not be interpreted as they are not reliable. So we are given a specific range to exploit.
