Non-linear Regression Flashcards
What is the test conducted in testing non-linear relationships?
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.
Equation for the change in Y
= function with the new value substituted in - function with the old value substituted in
How do you find the F-statistic for the predicted change in the dependent variable?
Conduct a joint test of the equation found of the change in Y
How do you compute the standard error for a change in Y?
Sub in the change in Y equation = |change in Y|/sqrt(n)
General approach to modelling non-linear relationships using multiple linear regression
- Investigate scatter plots of dependent variable and variable of interest (as well as key possible control variables)
- Specify different non-linear functions and estimate their parameters by OLD
- 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
- Plot the estimated non-linear regression function
- 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.
What test should be conducted to determine whether the population regression is linear or not?
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
What is the purpose of a sequential hypothesis test?
To determine the number power to use
Process of sequential hypothesis testing
- Pick a max value for r (the power) - usually 4
- 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.
- 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
- You continue this process until you reject the hypothesis and the level that you reject is the order of the regression.
Linear-log model interpretation
1% change in Xi is associated with a 0.01xBeta (unit) change in Y
Log-linear model interpretation
1 unit change in Xi is associated with a 100Beta% change in Y
Log-log model interpretation
1% change in Xi is associated with a Beta% change in Y
Beta is the elasticity of Y with respect to X
Comparing logarithmic specifications
Which models can you compare and which ones can you not?
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
Why are interaction terms added?
In order to determine interactions between independent variables
Binary-Binary interaction model
Determining the predicted effect of having one dummy depending on the outcome of another dummy variable.
Partial effects of binary-binary interaction models
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
Partial effect of a specific dummy (not a change)
Just sub in this dummy to the regression and this is the change in Y
Continuous-binary interaction models
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.
Continuous-continuous interaction models
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.
Partial effect of continuous-continuous model
change in Y/Change in X(1 or 2) = function(changed variables) - function(original variable)
Steps for analysing rich models
Models combining polynomials, logarithms and interactions
- Data visualisation - scatter plots
- Model estimation - is there a non-linear relationship? differences between different dummy cases
- Partial effects estimation
- Partial effects testing
1) Data visualisation
Steps for analysing rich models
Create scatter plot + analuse it
- dependent on Y
- Independent on X
2) Model estimation
Steps for analysing rich models
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?
3) Partial effects estimation
Steps for analysing rich models
find change in Y function
4) Partial effects testing
Steps for analysing rich models
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
Set up for difference-in-differences model
Treatment group - individuals exposed to the policy
Control group - individuals not exposed to the policy
Process of difference-in-differences models
- Create two dummy variables
* Treati =1 if received the policy
* Posti = 1 post the implementation of the policy - Run the difference-in-difference regression:
Yi=Beta0 + Beta1(Treat) + Beta2(Post) + Beta3(Treat x Post) + u
Interpretation of a difference-in=-differences model
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.
