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Explain how a time series is checked for the presence of a unit root.

FRM® Part I · Quantitative Analysis · Non-Stationary Time Series · FRM1.QA.11.e

The idea

A unit root is tested with the Dickey-Fuller test, which writes the AR(1) model in differences: Change = γ × yt−1 + εt. Here the change is yt − yt−1 and γ = φ − 1. The null hypothesis is a unit root (γ = 0) and the alternative is stationarity (γ < 0), so the test is one-sided. The t-ratio of the estimate of γ is compared not with normal or t critical values but with Dickey-Fuller critical values, which are more negative. At 5% they are about −1.94 with no constant, −2.86 with a constant and −3.41 with a constant and a time trend. The augmented test (ADF) adds lagged differences of y so that the errors are serially uncorrelated, with the number of lags chosen by an information criterion. Failing to reject does not prove a unit root, since the test has low power. The usual mistake is to use standard critical values.

Formulas

Dickey-Fuller test statistic

DF statistic = ĝ ÷ SE(ĝ)

ĝ
the estimated coefficient on y(t − 1) in the regression of the change in y on lagged y, equal to φ − 1
SE(ĝ)
its standard error

A one-sided test of a unit root (g = 0) against stationarity (g < 0). Compare with Dickey-Fuller critical values, about −2.86 at 5% with a constant, never with normal ones.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the Dickey-Fuller test, what are the null and alternative hypotheses?

  1. Null: the series has a unit root; alternative: the series is stationary - correct
  2. Null: the series is stationary; alternative: the series has a unit rootWhy not: Reverses the hypotheses; stationarity is the alternative.
  3. Null: the series has no serial correlation; alternative: it has serial correlationWhy not: This is the Ljung-Box or Durbin-Watson type of null, not the unit-root null.
  4. Null: the series has a deterministic trend; alternative: it has a stochastic trendWhy not: Mixes up trend types; the test is about g = phi - 1 = 0 versus g < 0.

The test regresses the change in y on lagged y; the null is g = 0 (phi = 1, a unit root) and the one-sided alternative is g < 0.

A harder one

An analyst tests the log of an equity index that has risen steadily for 30 years. Which ADF specification and 5% critical value are appropriate?

  1. Include no constant or trend; the 5% critical value is about -1.94Why not: Appropriate only for a series with no drift or nonzero mean, not a steadily rising index.
  2. Include a constant only; the 5% critical value is about -2.86Why not: Appropriate for a series fluctuating around a nonzero level, not one with a steady trend.
  3. Include a constant and a deterministic time trend; the 5% critical value is about -3.41 - correct
  4. Include a constant and a time trend, but use the one-sided normal critical value of -1.645Why not: Uses the standard normal rather than the Dickey-Fuller distribution.

A visibly trending series calls for the constant-plus-trend version, which has the most negative critical values (about -3.41 at 5%).

The hardest tier

An ADF regression with a constant has g-hat = -0.085 and standard error 0.03. The Dickey-Fuller critical values are -2.86 at 5% and -2.57 at 10%. What is the correct conclusion?

  1. The t-ratio is -2.83: do not reject at 5%, but reject at 10% - correct
  2. The t-ratio is -2.83: reject at 5% and at 10%Why not: Rounds the statistic past the 5% critical value; -2.83 is above (less negative than) -2.86.
  3. The t-ratio is -2.83: do not reject at 5% or at 10%Why not: Fails to see that -2.83 is below the 10% critical value of -2.57.
  4. The t-ratio is -2.83: reject at 5%, do not reject at 10%Why not: Mixes up the order of significance levels: a rejection at 5% would imply a rejection at 10%.

t = -0.085 / 0.03 = -2.83. It is above -2.86 (no rejection at 5%) but below -2.57 (rejection at 10%). The evidence is borderline.

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