Nonstationary Time Series

1
Lecture 4

• Non-stationary Time Series
• Unit Root Tests

2
Spurious Regression

• Regression with I(1) variables
• yt ? ? Xt ut
• Phillips (1986, JEC)
• t O(T1/2), F O(T1/2) t and F diverge as T
  ?
• DW O(T-1) DW ? 0 as T ?
• plim R2 1 R2 ? 1 as T ?
• Nelson Plosser (1982, JME)
• 13 out of 14 macro time series are I(1).

3
Distinctions

• Trend stationary process
• Including a trend function (t) is enough.
• I(d) process
• Differencing d times is required to make it
  stationary.
• (broadly) non-stationary
• difference stationary
• Estimators have a non-standard distribution.
• Note A random walk process is one example.

4
Trend stationary vs. Difference stationary

• Property Comparison
• ACF dies out vs. slowly decaying
• Dynamic multiplier dies out vs. persistent
• MSE of forecast error converges vs diverges
• Mean reversion vs aversion
• Returning to vs. divergent from an equilibrium
  path
• Long-run variance of Dyt zero vs. non-zero

5
Why is it important?

• Numerous papers have tested if their data are
  difference stationary.
• Determines if your theory is supported.
• Property of the data, PPP, Gibson Paradox,
• Preliminary examination for econometric
  estimation.
• To see if differencing is necessary.
• Cointegration

6
Non-standard Distribution

• FCLT (Functional Central limit Theorem)
• T-1/2SrT ? ? W(r)
• where W(r) is a wiener process (Brownian motion)
• W(r) N(0, r)
• ?2 is the longrun variance.
• Continuous mapping theorem
• If SrT ? S(.) and g is continuous, then
  g(SrT ) ? g(S(.)).

7
Unit Root Tests

• Dickey Fuller Unit root tests
• yt ?yt-1 ut (two more equations)
• H0 ? 1 Ha ? lt 1
• Two statistics
• T(?_hat - 1) ? .5W(1)2- ?u2 /?2/? W(r)2 dw
• t-statistics for ? 1 ? something random
• Under iid errors, ?u2 /?2 1. Use DF critical
  values, not usual t distribution.

8
Three ways to handle autocorrelation

• ADF tests
• Add k augmented terms
• How to determine k ? ? 11 methods (t, AIC,)
• Phillips-Perron tests
• Estimate ?u2 /?2 and transform the statistics.
• ?2 is the long-run variance (spec. density at 0)
  whose calculation involves bandwidth selections.
• How to determine the bandwidth ?
• ? automatic procedure by Andrews (1991)

9

• IV tests
• Find the max. order q for MA(q) model, and use
  yt-k where kgtq.
• Others
• LM tests
• Augmented, transformation, and IV versions

10
Unit Root tests with Structural Breaks

• Perron (1989, Econ) DF type
• showed that ignoring an exogenous break results
  in loss of power.
• Reversed the result most macro series are I(0).
• A Lee (1995, ET) LM extension
• Reversed Perrons finding due to a problem in the
  DF regressions.

11
Tests with Endogenous breaks

• Minimum statistics
• Zivot Andrews (1992) one break
• Reversed Perrons result.
• Lumsdaine Papell (1997) Two breaks
• Tend to reject the null.
• Lees claim (1998a) Not invariant to the
  magnitude of the breaks under the null.

12

• LM tests
• Min LM tests (one or two breaks) Lee (1998b)
• Invariant
• Multiple breaks Lee (1998c)
• Using invariance property

13
Stationarity Tests

• KPSS tests
• Swap the null and alternative
• level stationarity
• trend stationarity

14
Generic Tests

• Impulse response measure
• Example RATS program
• Variance Ratio Test
• popular in finance

15
Beveridge-Nelson decomposition

• Decompose a time series as two parts.
• Permanent
• Temporary
• see handout
• Example bn.pgm (RATS)

16
Fractional Integration

• ARIFIMA(p, d, q)
• where d is a real number
• Longrun memory process
• Estimate d using the spectral density function
  (periodogram)