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FRM Part I · FRM Exam Part I · Nonstationary Time Series

An analyst fits the regression Y_t = δ·Y_{t-1} + ε_t to a monthly series of a bond spread and wants to test for a unit root using the Dickey-Fuller approach. Which of the following correctly states the null hypothesis and the usual way the test is run?

The null hypothesis is a unit root, δ = 1. The regression is rewritten as ΔY_t = γY_{t-1} + ε_t with γ = δ − 1, so the null becomes γ = 0, tested using Dickey-Fuller critical values rather than standard t values.

  1. AH0: δ = 1 (unit root); the model is rewritten as ΔY_t = γY_{t-1} + ε_t and H0 is γ = 0Correct
  2. BH0: δ = 0; the model is rewritten in levels and tested with a standard normal critical value
  3. CH0: |δ| < 1 (stationary); the test uses the ordinary t-distribution
  4. DH0: γ < 0 in the differenced regression, with a one-sided normal test

Explanation

The Dickey-Fuller test subtracts Y_{t-1} from both sides so that γ = δ − 1. The null of a unit root (δ = 1) becomes γ = 0. Under the null the t-statistic does not follow a standard t-distribution, so special DF critical values are required, which rules out the other options.

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