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d be able to explain:
Long memory processes
• In particular, Fractional White Noise (FWN). The role of the long memory parameterdin this process:
o The process is mean reverting for1o The process is covariance stationary for5.0• The ACF of a long memory process has a hyperbolic decay in contrast to the exponential/geometric decay of the ACF of a classical stationary process. (Be able to sketch the two types of ACFs for comparison). (0=d
• The spectrum of a time series and the spectral shapes of the following processes:
o White noise ()0=d
o Random walk/martingale process ()1=d
o FWN (is some fractional value/real number). d
o (Be able to sketch the above spectra for comparison)
• The Geweke and Porter-Hudak (GPH) test for long memory:
o The rationale for the test (i.e., based on the shape of the spectrum close to frequency zero corresponding to the long run periodic components in the time series).
o The form of the spectral regression:
􀂃 How a cut-off for the number of frequencies in the regression can be chosen (and why this cut-off is important).
􀂃 How an estimate of is obtained from the regression (i.e., as the negative of the slope coefficient). d
• Applications of the GPH test. For example:
o Testing for long memory in the forward premium in the foreign exchange market (see Lecture 7).
o Testing for long memory in the real exchange rate (see handout for Seminar 6).
Unit root testing
• The distinction between a difference stationary (DS)/I(1) process and a trend stationary (TS)/I(0) process. Understand that:
o Shocks to an I(1) processes have permanent effects on the level of the series.
o Classical inferences (t and F tests) are generally inapplicable in models with I(1) processes.
o There are spurious regression problems with I(1) processes (see also below).
o Consequently testing for unit roots in financial time series is very important.
• Testing for (auto-regressive) unit roots:
o That the distribution of the Dickey-Fuller (DF) t-statistic (τ statistic) for testing the null of a unit root is non-standard (sketch the classical t-distribution and τdistributions for comparison).
o That there are 3 test regressions for the DF test corresponding to different assumptions about the mean under the alternative of stationarity:
􀂃 Specifically there are models with: a trend and intercept; an intercept only; and neither a trend nor intercept.
􀂃 Explain how you might choose which test regression to use (see Seminar 6 handout).
o Testing for unit roots when there are higher order dependencies in the data:
􀂃 The Augmented Dickey Fuller (ADF) test.
􀂃 The Phillips-Perron test (see handout for Seminar 6).
o The circumstances in which unit root tests have low power.
o The KPSS test (test of the null of stationarity – see Seminar 6 handout).
Cointegration
• That there are spurious regression problems with non-stationary time series. Regressions involving independent I(1) series will typically display:
o Significant t-statistics.
o A high R-squared.
o A highly autocorrelated (I(1)) error term.
• That there is an important exception to the spurious regression problem: cointegrating relationships:
o Define cointegration: i.e., a linear combination of I(1) variables which is I(0) – CI(1,1).
o More generally: a linear