In a lagged dependent variable model with β₁ = 0.7 and β₂ = 0.3, a permanent one-unit increase in X occurs at time t. What is the total long-run effect on Y?
A0.3 — only the immediate impact coefficient matters
B0.7 — the persistence coefficient captures all dynamic effects
C1.0 — the long-run effect is β₂/(1 − β₁) = 0.3/0.3
D∞ — the effects accumulate indefinitely and never stop growing
The long-run multiplier is β₂/(1 − β₁) = 0.3/(1 − 0.7) = 0.3/0.3 = 1.0. The mechanism: the immediate effect is β₂ = 0.3. Next period, Y is higher by 0.3, which feeds into Yₜ₋₁, adding β₁ × 0.3 = 0.21. The following period adds β₁² × 0.3 = 0.147, and so on. The sum of this geometric series is β₂/(1 − β₁). Option A misses all the dynamic propagation. Option D would require β₁ ≥ 1 (non-stationary process) — with β₁ = 0.7 < 1, the process is stationary and the series converges.
Question 2 Multiple Choice
OLS estimates in a lagged dependent variable model are inconsistent when...
AThe sample size is small — OLS requires at least 100 observations with lagged variables
BThe errors uₜ are serially autocorrelated, because Yₜ₋₁ and uₜ are then correlated through past errors
CThe coefficient β₁ is close to 1, making the model nearly nonstationary
DThe model includes more than one lag of Y, requiring instrumental variables
The consistency of OLS in the lagged dependent variable model requires that Yₜ₋₁ be uncorrelated with the current error uₜ. If errors are serially autocorrelated (e.g., uₜ = ρuₜ₋₁ + εₜ), then uₜ₋₁ enters uₜ, but uₜ₋₁ also determined Yₜ₋₁ (since Yₜ₋₁ = ... + uₜ₋₁). This creates a correlation between the regressor Yₜ₋₁ and the error uₜ — a violation of the exogeneity condition — causing OLS to be biased and inconsistent. Sample size (Option A) affects precision, not consistency. β₁ near 1 (Option C) is a stationarity concern, not a bias concern per se.
Question 3 True / False
In a lagged dependent variable model, a coefficient β₁ = 0.9 implies that shocks to Y dissipate quickly because 0.9 is less than 1.
TTrue
FFalse
Answer: False
β₁ = 0.9 implies HIGH persistence, not quick dissipation. After a shock, the deviation in Y decays as 0.9^t: after 5 periods it is still 59% of the original size (0.9⁵ ≈ 0.59); after 20 periods it is still 12% (0.9²⁰ ≈ 0.12). The process is stationary only because 0.9 < 1 — it does eventually revert to the mean — but it does so slowly. Quick dissipation would require β₁ close to 0. A coefficient close to 1 is often described as 'near unit root' behavior, where shocks are extremely persistent.
Question 4 True / False
When residuals from a lagged dependent variable regression show serial autocorrelation, this is a signal that the exogeneity assumption for Yₜ₋₁ may be violated, making OLS estimates biased.
TTrue
FFalse
Answer: True
Serial autocorrelation in the residuals means uₜ is correlated with uₜ₋₁. But Yₜ₋₁ depends on uₜ₋₁ (since Yₜ₋₁ = β₀ + β₁Yₜ₋₂ + β₂Xₜ₋₁ + uₜ₋₁), so Yₜ₋₁ is correlated with uₜ. This violates the OLS exogeneity condition, causing bias. This is why testing for serial correlation (Durbin-Watson, Breusch-Godfrey) is essential in LDV models — it is not just a specification nicety but a direct check on whether OLS is consistent.
Question 5 Short Answer
Explain the difference between the short-run and long-run effects of X on Y in a lagged dependent variable model, and derive why the long-run multiplier is β₂/(1 − β₁).
Think about your answer, then reveal below.
Model answer: The short-run effect is β₂: a one-unit increase in X raises Y immediately by β₂. But Y also feeds back on itself through the lagged term. In the next period, Y is higher by β₂, which via Yₜ₋₁ raises Y again by β₁β₂. The period after that adds β₁²β₂, and so on. The total long-run effect is the sum of this geometric series: β₂(1 + β₁ + β₁² + ...) = β₂ × 1/(1 − β₁) = β₂/(1 − β₁), valid when |β₁| < 1. This multiplier is always larger than the short-run effect β₂ when β₁ > 0, and it can be substantially larger when β₁ is close to 1 — a permanent change in X has outsized cumulative consequences in highly persistent processes.
The long-run vs. short-run distinction is the core practical contribution of the LDV model. A policy that raises X by 1 unit produces an immediate effect of β₂, but the full reckoning takes many periods to play out. A researcher who only reports the contemporaneous coefficient β₂ is dramatically underestimating the impact of a permanent policy change when β₁ is substantial.