Time series dynamics of financial monetary variables

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Time series dynamics of financial / monetary
variables

• VARIOUS TIME SERIES PROCESSES discrete
  time continuous time
• TESTS FOR RANDOM WALKS 3 different RW types
  (RW1, RW2, RW3)

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THE IMPORTANCE OF STOCHASTIC TIME PROCESSES

• For example,
• Pricing modelsWhat price for stock options?
  What long-term interest rate?
• Efficient market implicationsDo stock prices
  follow martingale process? Are (excess/abnormal)
  returns predictable? PPP Are real exchange
  rates constant/stationary?Fisher effect Are
  real interest rates constant/stationary?
• Non-stationarity and statistical inference
  spurious (nonsensical) regression estimates
• Time-varying volatility, heteroscedasticity
  biased test statistics

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TIME SERIES PROPERTIES
Let us start by looking at the time series plot
UK FT-All Share Index, monthly from January 1965
to December 1995
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TIME SERIES, STOCHASTIC PROCESSES

• Random walks
• Martingales
• Fair game
• Gaussian white noise process
• Brownian motion / Wiener process / Ito process
• Poisson process
• Markov process
• ARCH/GARCH, stochastic volatility

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MARTINGALES AND FAIR GAMES
Martingale model A stochastic process Xt is a
martingale if EXt1 ?t Xt or Et Xt1
Xt Submartingale if EXt1 ?t ? Xt, and
supermartingale if EXt1 ?t ? Xt Fair game
model A stochastic process yt is a fair game
if Eyt1 ?t 0 Follows that if Xt is a
martingale or pure random walk, (Xt1 Xt) is a
fair game Note Only referring to expected
values! Nothing about variances of shocks.
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RANDOM WALKS (type 1, 2, 3)
A stochastic process Xt is a (pure) random walk
if Xt1 Xt ?t1 ?t1 ? i.i.d. (0, ?2)
Et?t1 0, Et?2t1 ?2 random walk with
drift if Xt1 Xt k ?t1 EXt1 ?t Et
Xt ?t1 Et Xt Et ?t1 Xt Random
walk is a martingale with constant variance of
the innovations Campbell/Lo/MacKinlay
(1997)Random walk type 2 independent
increments, but not identically distributed (e.g.
random heteroscedasticity)Random walk type 3
uncorrelated increments, but not independent, not
identically distributed (e.g. (G)ARCH cov(?t,
?t-k)0, cov(?2t, ?2t-k)?0)
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GAUSSIAN WHITE NOISE PROCESS
A stochastic process Xt is a Gaussian white noise
process if Xt1 ?t1 ?t1 ? i.i.d. normal(0,
?2)
From continuous time modelsBROWNIAN MOTION /
WIENER PROCESSDerived from the simple random
walk, when time intervals become smaller and
approach zero. Random walk with drift ?x(t) a
?t b ? ??t ?t?0 ? ? N(0, 1)ITO
PROCESSSimilar to Wiener process but parameters
are functions of x, t ?x(t) a(x,t) ?t b(x,t)
? ??t ?t?0 ? ? N(0, 1)
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RANDOM WALKS AND(NON-) STATIONARY TIME SERIES

• A stochastic process Xt is said to be
  (covariance, weakly) stationary if
• EXt ? for all t
• VarXt ?2 lt ? for all t
• CovXt, Xtk ?k for all t and k
• Two models of non-stationarity
• Time trends e.g. yt ? ? T utor
  deterministic non-stationarity
• Unit roots, random walk (with drift) e.g. yt
  yt-1 ? utor stochastic non-stationarityNote
  unit root and random walk closely related but
  not the sameUnit root allows autocorrelation and
  random walk does not.

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(NON-) STATIONARY TIME SERIES
Random walks Deterministic linear trend
Observational equivalenceIs this RW process
perhaps a deterministic quadratic trend model?
You cant tell
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MEAN REVERSION
Mean reversion in contrast to a random walk
describes the phenomenon that a variable appears
to be pulled back to some long-run average level
over time (Hull, 2000) E.g. Variables
returning to deterministic trends E.g. Market
prices returning to fundamental values In other
words, transitory/temporary deviations from some
long-run level. Changes in the variable (growth
rates, returns) must be negatively serially
correlated at some frequency for the correction
to occur ? predictable.
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RANDOM WALK TESTS

• Campbell/Lo/MacKinlay (1997)
• Random walk type 1 I.I.D. increments
• sequences and reversals, runs,
• Allowing time-varying variances of shocks, i.e.
  heteroscedasticityHeteroscedasticity is
  important in financial time series
• Random walk type 2 independent increments
• filter rules, technical analysis
• Random walk type 3 uncorrelated increments
• Autocorrelation coefficients
• Portmanteau statistics
• Return regressions
• Variance ratios

Standard tests for RW1, but adjustable for
time-varying variances of shocks RW3
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AUTOCORRELATION COEFFICIENTS
Random walk implies increments (i.e. returns in
financial markets using log prices) are i.i.d.
(RW1), independent (RW2), or uncorrelated (RW3)
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AUTOCORRELATION COEFFICIENTS (Cont)
When (log) prices are assumed to follow RW3
(heteroscedasticity)
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PORTMANTEAU STATISTICS
The Q-statistic due to Box-Pierce (1970) and
Ljung-Box (1978) tests whether a range of
autocorrelations are all zero.

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PORTMANTEAU STATISTICS (Cont)
When (log) prices are assumed to follow RW3
(heteroscedasticity)

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REGRESSION-BASED APPROACH

• Fama/French (1988) suggest a test of mean
  reversion in stock prices by running the
  following regression
• rt,tk ak bk rt-k,t ?tk
• where r is the continuously compounded (log)
  return, i.e. rt,tk log(ptk) log(pt).
• If returns are mean reverting (averting)
  coefficient bk must be negative (positive).
  Random walk in prices suggests coefficient bk
  must be zero.
• Overlapping observations problem- using return
  horizon of k periods, rt,tk and rt1,t1k have
  k-1 single-period returns in common overlapping
  observations- the residuals ?tk of the
  regression equation are serially correlated and
  standard error of coefficient bk is biased.-
  Methods exist to correct regression st.errors for
  autocorrelation (e.g. Heteroscedasticity and
  Autocorrelation Consistent (HAC) standard errors
  using the Newey-West method)

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VARIANCE RATIO TESTS
Developed by Cochrane (1988), Lo-MacKinlay (1988,
1989)Since the variance of a random walk series
increases linear with time, the variance of a
k-period change must be k times the variance of
the 1-period change.

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VARIANCE RATIO TESTS (Cont)
Campbell/Lo/MacKinlay (1997, Ch.2)
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VARIANCE RATIO TESTS (Cont)
Campbell/Lo/MacKinlay (1997, Ch.2)
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RANDOM WALK TESTS
3112 372- 1 for 1st diff
Autocorrelations lnFTAprice1st differences
Ljung-Box Q
Sample 1965M01 1995M12 Included observations
371 Autocorrelation Partial
Correlation AC   PAC  Q-Stat  Prob        .
       . 1 0.123 0.123 5.6185 0.018
       .        .
2 -0.108 -0.124 9.9524 0.007        .
       . 3 0.080 0.114 12.386 0.006
       ..        .. 4 0.062
0.021 13.826 0.008        .        .
5 -0.100 -0.093 17.607 0.003        ..
       .. 6 -0.044 -0.014 18.330 0.005
       ..        .. 7 0.033
0.010 18.739 0.009        ..        ..
8 -0.048 -0.049 19.610 0.012        .
       . 9 0.082
0.122 22.172 0.008        ..        ..
10 0.026 -0.028 22.432 0.013        ..
       .. 11 -0.043 -0.021 23.139 0.0
17        ..        .. 12
0.004 0.008 23.144 0.027
Critical values N(0,1) two sided test1
2.575 1.9610 1.64
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RANDOM WALK TESTS
Variance ratio test lnFTAprice
Critical values N(0,1) two sided test1
2.575 1.9610 1.64
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REGRESSION-BASED MEAN REVERSION TESTS
Fama-French return regressions lnFTAprice(tk)-ln
FTAprice(t) ak ßk lnFTAprice(t)-lnFTAprice(t
-k)
Critical values t? two sided test1
2.5765 1.96010 1.645
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Liu He (1991) A Variance-Ratio Test of Random
Walks in Foreign Exchange Rates
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Liu He (1991)
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Liu He (1991)
Critical values ?2(15)1 30.585 25.0010 22.31