Questions: Cointegration and Error Correction Models

Question 1

Two series y_t and x_t are both I(1). Which finding would indicate they are cointegrated?

• A. The regression of y on x has R² above 0.9
• B. The residuals from regressing y_t on x_t are stationary
• C. Both Δy_t and Δx_t are stationary
• D. The correlation of the levels exceeds 0.8

Answer: (B)

Cointegration means a linear combination of the I(1) series is I(0). The residual from the levels regression estimates that combination, so testing it for stationarity is the Engle-Granger test. Stationary first differences hold for any pair of I(1) series, and high R² or correlation often appears even between unrelated random walks, so neither shows cointegration.

Question 2

In the error correction model Δy_t = α + γΔx_t + δ(y_{t−1} − βx_{t−1}) + u_t, an estimate δ = −0.25 means about one quarter of last period's deviation from the long-run relationship is corrected in the current period.

• T. True
• F. False

Answer: True

If y was above its long-run level βx last period, the negative δ pulls Δy down this period by 25% of the gap. A negative δ is required for the system to move back toward equilibrium.

Question 3

Why does the Engle-Granger residual-based test use different critical values from a standard Dickey-Fuller test?

• A. Because the residuals are always normally distributed
• B. Because OLS chooses β to minimize residual variance, which makes the residuals look more stationary than a series that was not estimated
• C. Because the test is two-sided
• D. Because the residuals have a deterministic trend

Answer: (B)

OLS picks the combination of y and x with the smallest residual variance. That fitting biases the test toward finding stationarity, so stricter, more negative critical values are needed to keep the test's size correct.

Question 4

Explain why a regression in first differences alone can be misspecified when two series are cointegrated.

Model answer: Differencing removes the unit roots but also removes the information that the levels are tied together in the long run. If the series are cointegrated, past deviations from the long-run relationship help predict changes, so the differenced model omits a relevant variable, the lagged error correction term.

The Granger representation theorem states that cointegrated series have an error correction representation, so the error correction term belongs in the model.