Nonlinear Regression Functions

General Nonlinear Population Regression Function
1. Assumptions
  1. 1. Zero conditional mean
  2. 2. (X1,...,Xk,Y) are i.i.d.
  3. 3. "Enough" moments exist
  4. 4. No perfect multicolinearity
Polynomials in X
1. How to interpret coefficients?
  1. Plot estimated regression function
  2. Calculate the estimated effect on Y associated with a change in X
2. How many degrees (r)?
  1. Hypothesis Testing
    1. If I don't reject Ho: coefficient out of regression
  2. Increase the degree (r)
    1. More flexibility into regression function
    2. Reduction in precision of the estimated coefficients
Logarithmic functions of Y and/or X
1. Linear - Log
  1. Formula
  2. Interpretation of coefficient "after and before rule"
  3. 1% change in X is associated with a change in Y of 0.01beta1
2. Log - Linear
  1. Formula
  2. Interpretation of coefficient "after and before rule"
  3. A change in X by 1 unit is associated with 100beta1% change in Y
3. Log - Log
  1. Formula
  2. Interpretation of coefficients "after and before rule"
  3. 1% change in X is associated with a beta1% change in Y; beta1 is the elasticity of Y with respect to X
Linear vs. Nonlinear
Interactions between variable
1. Between 2 binary variables
  1. Include D1xD2 as a regressor (so that the effect of D1 depends on D2)
2. Between Continous and Binary variables
  1. Include DxX as a regressor (so that the effect of X to depend on D)
3. Between continous variables
  1. Include the interaction term X1xX2 as a regressor (so that you allow the effect of X1 to depend on X2)
4. We want to allow the effect of a variable to depend on the other
Questions that might arise
1. Are there nonlinear effects?
2. Are there nonlinear interactions?

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Nonlinear Regression Functions

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