In the model y_t = 2 + 0.5z_t + 0.3z_{t−1} + 0.1z_{t−2} + u_t, z rises by one unit in period 0 and returns to its old level in period 1. By how much does y differ from its old level in period 2?
A0.5
B0.3
C0.1
D0.9
A temporary change affects y_t through the lag that corresponds to period 0. In period 2, the period-0 rise enters through z_{t−2}, whose coefficient is 0.1. By period 3, y is back at its old level.
Question 2 Multiple Choice
In the same model, what is the long-run propensity?
A0.5
B0.8
C0.9
D2.9
The LRP is δ₀ + δ₁ + δ₂ = 0.5 + 0.3 + 0.1 = 0.9. It is the change in y, after two periods, from a permanent one-unit increase in z.
Question 3 True / False
It is possible for each lag coefficient in an FDL model to be individually insignificant while the long-run propensity is significant.
TTrue
FFalse
Answer: True
Current and lagged z are often highly correlated, which inflates the standard errors of individual δ's. Their sum can still be estimated precisely, because the regression can tell the combined effect apart even when it cannot apportion it among the lags.
Question 4 Short Answer
Describe how to obtain the standard error of the LRP θ = δ₀ + δ₁ + δ₂ directly from a single OLS regression.
Think about your answer, then reveal below.
Model answer: Substitute δ₀ = θ − δ₁ − δ₂ into the model. It becomes y_t = α₀ + θz_t + δ₁(z_{t−1} − z_t) + δ₂(z_{t−2} − z_t) + u_t. Regress y_t on z_t, (z_{t−1} − z_t), and (z_{t−2} − z_t); the coefficient on z_t is θ̂ and its reported standard error is the standard error of the LRP.
This reparameterization is a general trick for getting the standard error of a linear combination of coefficients without computing covariances by hand.