Unit Roots and Cointegration

Question 1

What is a unit root?

Answer 1

A unit root in a time series means the series is non-stationary, it implies that shocks have a permanent effect

Question 2

Stationarity tests

Answer 2

• Dickey-Fuller
• Augmented Dickey-Fuller
• Phillips-Perron
• KPSS Test

Question 3

Dickey-Fuller

Answer 3

• Null Hypothesis H₀: The series has a unit root (p=1)
• Reject H₀ if δ (p-1) significantly less than zero

Question 4

Augmented Dickey-Fuller

Answer 4

• Extends the DF test to include lagged differences to account for autocorrelation

Question 5

Phillips-Perron

Answer 5

Addresses limitations of ADF, particularly when error terms are heteroscedastic.
* adjusts the test statistic and its standard error using the Newey-West estimator

Question 6

KPSS Test

Answer 6

Reverse of DF and ADF. Tests for stationarity
* H₀: The series is stationary
* H₁: The series is non-stationary

Question 7

What to do if a series has a unit root?

Answer 7

• Differencing: Apply the difference operator ΔX_t = X_t - X_t-1 to make the series stationary
• Use models like ARIMA which explicitly handle differencing

Question 8

Cointegration

Answer 8

When two or more non-stationary time series are linked by a long-term equilibrium relationship.

Question 9

Testing for cointegration

Answer 9

• Engle-Granger Two-Step Method
• Johansen Test

Question 10

Engle-Granger Two-Step Method

Answer 10

1. Regress Y_t on X_t to estimate β
2. Test the residuals Z_t = Y_t - βX_t for stationarity using ADF

Question 11

Johansen Test

Answer 11

A multivariate test for cointegration
* uses the eigenvalues of a matrix to determine the number of cointegrating relationships