Time series assignment (HW4)

Do problems 3 and 6 from chapter 6 of Berndt (page 283 and 287.) No discussion necessary this time.

NEW NOTE: In Exercises 3 and 6, Berndt is looking for you to complete a series of analysis for both IE and IS structures. You all are only required to do one of them (IE/equipment prefered, but you can do either one). That means that things like 3c, you can skip the part which tells you to do everything you just did for IS for the IE variable.

NOTE: ignore Durbin-Watson in (3d).

Hints on the PDL stuff:

• Background: Suppose you had a regression of Y = alpha + beta₁ X₁ + beta₂ X₂ + noise. But you wanted to assume that beta₁ = beta₂. How would you do it? The easiers way would be to construct a new varible Z = X₁ + X₂ and do a regression of Y on Z, namely, Y = Z + noise.
• PDL: We want to assume that beta₁, beta₂, ..., beta₆ land on a polynomial when plotted verse their index number. In other words, we want beta_i = a + b*i + c*i² + d*i³. To do this we ahve to create the following variables:
  • Sum = X₁ + X₂ + X₃+ X₄+ X₅ + X₆
  • L = Linear = X₁ + 2 * X₂ + 3 * X₃ + 4 * X₄ + 5 * X₅ + 6 * X₆
  • Q = Quadratic = X₁ + 4 * X₂ + 9 * X₃ + 16 * X₄+ 25 * X₅+ 36 * X₆
  • C = Cubic = X₁ + 8 * X₂ + 27 * X₃ + 64 * X₄ + 125 * X₅ + 216 * X₆
  Now do a regression on Sum,L,Q and C.

  Finding out the coeficient of say X₂ can be constructed by using beta_SUM + 2*beta_L+4*beta_Q+8*beta_C.

• (PDL for part c) Now suppose that we want the coefficient of X₆ to be zero. We can create the following variables:
  • Sum' = Sum - C/216
  • L' = L - C/36
  • Q' = Q - C/6
  Now a regression on Sum', L' and Q' will generate coefficients beta_Sum' and beta_L' and beta_Q'.

  From these we can figure out what the coeficients are for the unprimed variables. To do this use:

  • beta_Sum = beta_Sum'
  • beta_L = beta_L'
  • beta_Q = beta_Q
  • beta_C = - beta_Sum'/216 - beta_L'/36 - beta_Q'/6
  Now we can figure out what the coefficients are for the orginal X's. With a bit of algebra, you should be able to show that the coefficient of X₆ is in fact zero.

I suggest you turn this in before homework 5, but I know you have various panics. So there is both a suggested due date and a hard deadline for this assignment. Which you follow is your choice. But the hard deadline will not have any extensions for any reasons at all.