Question

Consider the following model: Y = β₁ + β₂X_2t + β₃X_3t + γ₄Y_t-1. Using a sample of 36 months, we estimate this model and obtain the following results:
y_t = 1.33 + 17.6x_2t + 0.94x_3t + 0.39Y_t-1
(0.02) (2.3) (3.35) (0.015)
R² = 0.89 DW = 2.86 (Durbin Watson statistic)

If X₃ were to increase by 1-unit in time t, by how much would we expect Y to change overall, including current and future time periods, as a result of this increase?

• A.

  45.13

• B.

  1.54

• C.

  28.85

• D.

  6.5

Answer 1

The correct option is B.) 1.54

Coefficient of x3t is 0.94 in the above regression model. If X3 increases by 1 unit Yt will chnage by 0.94 units.

It is beacuse,

yt = 1.33 + 17.6x2t + 0.94(x3t +1)+ 0.39Yt-1

yt = 1.33 + 17.6x2t + 0.94x3t+0.94 + 0.39Yt-1

Thus, Yt changes by 0.94 units.

Also coefficient of yt-1 is 0.39. Thus, a unit change in yt will change yt+1 by 0.39 units.

An increase of x3 by 1 unit at time t will lead yt to increase by 0.94 units, yt+1 will increase by 0.94*0.39 , yt+2 will increase by 0.94*0.39*0.39 ----till infinity.

Overall change in Y = 0.94 + 0.94*0.39 + 0.94*0.39*0.39 + 0.94*0.39*0.39*0.39 -----

Whenever series is of the form a + ax + ax2 + ax3 -------

when absolute value of x < 1 then: a + ax + ax2 + ax3 ----------------------------------- = a/(1 - x)

Hence,  0.94 + 0.94*0.39 + 0.94*0.39*0.39 + 0.94*0.39*0.39*0.39 -------------------------------- = 0.94/(1 - 0.39) = 1.54

Hence, Overall change in Y including current and future time periods is 1.54

Hence, the correct answer is B.) 1.54