Consider the Following Model Y_t = β₀ $X _ { t } ^ { e }$

Consider the following model Y_t = β₀ + $X _ { t } ^ { e }$ + u_t where the superscript "e" indicates expected values. This may represent an example where consumption depends on expected, or "permanent," income. Furthermore, let expected income be formed as follows: $X _ { t } ^ { e }$ = $x _ { t - 1 } ^ { e }$ + λ(X_t - $x _ { t - 1 } ^ { e }$ ); 0 < λ < 1
(a)In the above expectation formation hypothesis, expectations are formed at the end of the period, say the 31^st of December, if you had annual data. Give an intuitive explanation for this process.
(b)Rewrite the expectations equation in the following form: $X _ { t } ^ { e }$ = (1 - λ)
$x _ { t - 1 } ^ { e }$ + λX_t
Next, following the method used in your textbook, lag both sides of the equation and replace $x _ { t - 1 } ^ { e }$ Repeat this process by repeatedly substituting expression for $x _ { t - 2 } ^ { e }$ , $x _ { t - 3 } ^ { e }$ , and so forth. Show that this results in the following equation: $X _ { t } ^ { e }$ = λX_t + λ(1-λ)X_t-1 + λ(1- λ)² X_t-2 + ... + λ(1- λ)ⁿ X_t-n + (1 - λ)ⁿ⁺¹
$X _ { t + 1 } ^ { e }$ Explain why it is reasonable to drop the last right hand side term as n becomes large.
(c)Substitute the above expression into the original model that related Y to $X _ { t } ^ { e }$ Although you now have right hand side variables that are all observable, what do you perceive as a potential problem here if you wanted to estimate this distributed lag model without further restrictions?
(d)Lag both sides of the equation, multiply through by (1- λ), and subtract this equation from the equation found in (c). This is called a "Koyck transformation." What does the resulting equation look like? What is the error process? What is the impact effect (zero-period dynamic multiplier)of a unit change in X, and how does it differ from long run cumulative dynamic multiplier?