EC 595 Advanced Applied Econometrics

Homework 1

1. Consider the following camelback function:

   f(x,y) = -4x² + 2x⁴ - x⁶/3 - xy + 4y² - 4y⁴

   • Represent the function and evaluate its first and second derivatives
   • Find all the optima (maxima and minima) (x^*,y^*) of the function.

2. The classical set of data LONGLEY.TXT is known for its problem of multilinearity when fitting with a linear regression model. That is, all the 16 observations of 7 data series about the U.S. economy given in LONGLEY.TXT are highly correlated. These data series are

   YEAR	= 1947 to 1962
   PGNP	= GNP deflator, 1954=100
   GNP	= Gross national product, millions of dollars
   UEM	= Unemployed, thousands
   AF	= Armed Forces, thousands
   POP	= Population, thousands
   EM	= Employed Persons, thousands

   Consider the linear equation:

   f(β) = β₀+β₁PGNP+β₂GNP+β₃UEM+β₄AF+β₅POP+β₆YEAR.

   Where β = (β₀,β₁,β₂,β₃,β₄,β₅,β₆)' is a column vector of 7 unknown coefficients, and the corresponding variable PGNP, GNP, UEM, AF, POP, YEAR are the data series described above.

   Using the above linear equation f(β) to fit the employment data EM. Let's define the error

   ε = EM - f(β).

   Note that ε is a 16x1 vector. The least squares estimation is to find a vector of coefficients β^* so that the sum of error squared is minimized. That is,

   Minimize S(β) = ε'ε = ∑_i=1,...,16ε_i²

   1. Set up the objective function S for this least squares problem, and use MATA optimize() to find the solution β^*. Make sure that the corresponding gradient must be a zero vector, and the Hessian matrix must be positive definite so that β^* is indeed a least squares solution.

   2. Since the function f(β) is linear, it is better to solve for β^* using linear algebra of least squares. Do you find the same solutions from both linear and nonlinear least squares techniques? Why? Why not?

3. Assuming normal density for the error ε, formulate the log-likelihood function for the above problem. Find the maximum likelihood estimates of the model.