Non-linear Regression

Question 1

What is the test conducted in testing non-linear relationships?

Answer 1

H0: Beta2=0 vs. H1: Beta2=!0
in this the null is interpreted as the variable having to significance in the model ie. it cannot predict any of the dependent variable.

Question 2

Equation for the change in Y

Answer 2

= function with the new value substituted in - function with the old value substituted in

Question 3

How do you find the F-statistic for the predicted change in the dependent variable?

Answer 3

Conduct a joint test of the equation found of the change in Y

Question 4

How do you compute the standard error for a change in Y?

Answer 4

Sub in the change in Y equation = |change in Y|/sqrt(n)

Question 5

General approach to modelling non-linear relationships using multiple linear regression

Answer 5

1. Investigate scatter plots of dependent variable and variable of interest (as well as key possible control variables)
2. Specify different non-linear functions and estimate their parameters by OLD
3. Determine whether the non-linear model improves upon the linear model using adjusted R-squared - test the null of no linear relationship against the alternative that it is non-linear
4. Plot the estimated non-linear regression function
5. Estimat the effect on Y of a change in Xm for different X values then, compute the SE of change in U and the 95% CI using an appropriate F-statistic which depend on the size of the non-linear regression model.

Question 6

What test should be conducted to determine whether the population regression is linear or not?

Answer 6

Joint hypoethisis test of all the non-linear coefficients equalling zero or not (if rejected then they are significant and the regression is non-linear)
done using the F-statistic

Question 7

What is the purpose of a sequential hypothesis test?

Answer 7

To determine the number power to use

Question 8

Process of sequential hypothesis testing

Answer 8

1. Pick a max value for r (the power) - usually 4
2. Using the t-statistic test Beta(r)=0 - if you fail to reject the hypothesis this means that the regressor isn’t statistically significant to the model.
3. If you fail to reject then use the t-stat to test Beta(r-1)=0 - if you reject the hypotheisis you STOP as this means the regressor is statistically significant to the model
4. You continue this process until you reject the hypothesis and the level that you reject is the order of the regression.

Question 9

Linear-log model interpretation

Answer 9

1% change in Xi is associated with a 0.01xBeta (unit) change in Y

Question 10

Log-linear model interpretation

Answer 10

1 unit change in Xi is associated with a 100Beta% change in Y

Question 11

Log-log model interpretation

Answer 11

1% change in Xi is associated with a Beta% change in Y

Beta is the elasticity of Y with respect to X

Question 12

Comparing logarithmic specifications

Which models can you compare and which ones can you not?

Answer 12

You can only compare the models with the same dependent variable e.g. log-linear and log-log or linear-log and linear models BUT you cannot compare linear-log and log-linear or linear-log and log-log

Question 13

Why are interaction terms added?

Answer 13

In order to determine interactions between independent variables

Question 14

Binary-Binary interaction model

Answer 14

Determining the predicted effect of having one dummy depending on the outcome of another dummy variable.

Question 15

Partial effects of binary-binary interaction models

Answer 15

Change in Y = function (with first dummy condition) - function (with second dummy condition)
only one dummy can change so to compare do two sperate equations and compare the results of each

Question 16

Partial effect of a specific dummy (not a change)

Answer 16

Just sub in this dummy to the regression and this is the change in Y

Question 17

Continuous-binary interaction models

Answer 17

Allows us to estimate the interaction coefficient which tells us how an additional unit of the continuous variable impacts the dpendent variable depending on the case of the dummy variable.
e.g. one extra year in the country is associated with $1, 020 earnings increase per year if the immigrant is male.

Question 18

Continuous-continuous interaction models

Answer 18

Beta3 is a change in the partial effect of 1 unit increase on one continuous variable if the other continuous variable also increases 1 unit.

Question 19

Partial effect of continuous-continuous model

Answer 19

change in Y/Change in X(1 or 2) = function(changed variables) - function(original variable)

Question 20

Steps for analysing rich models

Models combining polynomials, logarithms and interactions

Answer 20

1. Data visualisation - scatter plots
2. Model estimation - is there a non-linear relationship? differences between different dummy cases
3. Partial effects estimation
4. Partial effects testing

Question 21

1) Data visualisation

Steps for analysing rich models

Answer 21

Create scatter plot + analuse it
- dependent on Y
- Independent on X

Question 22

2) Model estimation

Steps for analysing rich models

Answer 22

Conduct different tests to determine different questions e.g. is there a non-linear relationship? what are the differences for different dummies? What is the interaction between variables?

Question 23

3) Partial effects estimation

Steps for analysing rich models

Answer 23

find change in Y function

Question 24

4) Partial effects testing

Steps for analysing rich models

Answer 24

Process of determining whether the partial effects are statistically significant and what are their confidence intervals.
H0: change in Y =0 vs. H1: change in Y =!0
If you fail to reject the hypothesis it means that the partial effect is not statistically significant.

remember to use the f-statisitc for this test

Question 25

Set up for difference-in-differences model

Answer 25

Treatment group - individuals exposed to the policy
Control group - individuals not exposed to the policy

Question 26

Process of difference-in-differences models

Answer 26

1. Create two dummy variables
   * Treati =1 if received the policy
   * Posti = 1 post the implementation of the policy
2. Run the difference-in-difference regression:
   Yi=Beta0 + Beta1(Treat) + Beta2(Post) + Beta3(Treat x Post) + u

Question 27

Interpretation of a difference-in=-differences model

Answer 27

The policy yields a (statistically or not significant) Beta3 increase in the dependent variable compared to a Beta0 baseline earnings level in the conrol group during the pre-policy period.