Question

Question 4 (15p): Nonlinear estimation a. In a nonlinear BoBiXi+B2X?+ b. Consider the model Y = AK"Lb. How can this model be estimated consistently given data Y.Ki c. How can we detect nonlinearities in Stata using regressions or plots? How can a study of residuals r regression Y,-A +AX + β2x? + th, how do we interpret the coefficients? What is the marginal effect of an increase in X,? and L,? indicate nonlinearities? Show if possible using relevant formulas.

Answer 1

a) In such a scenario, we cannot fit a linear regression line to our model. The effect of a marginal change in X on Y is not constant. It depends on the value of X. Hence the slope coefficients do not have their usual interpretation.

Now, for a continuous independent variable X, the marginal effect (given by its coefficient $\small \beta$) is the first derivative of the response function Y with respect to X. However, if X is a categorical variable (say a binary variable that can take two values 0 and 1), then the marginal effect $\small \beta$ is the difference in the adjusted probabilities of these two groups.

b) We can use the double log transformation to consistently estimate this production function.

$\small Y= AK^{\alpha }L^{\beta }$

$\small ln Y= ln A + \alpha lnK + \beta lnL + \epsilon$

where we add $\small \epsilon$, a random disturbance term to the model .

This equation can be estimated via regression analysis given data on Y, K and L.

c) Detecting non-linearity

i) Use scatter plots in stata to find a non linear relation between the dependent and independent variables. Command in stata- scatter . Further use lfit command to show a linear fit.

ii) Plot residuals versus each of the independent variable. Finding a non-random pattern is indicative of non-linearity.