Transcript FINANCIAL TIME-SERIES ECONOMETRICS

FINANCIAL TIME-SERIES
ECONOMETRICS
SUN LIJIAN
Feb 23,2001
CHAPTER 1
UNIVARIATE LINEAR STOCHASTIC PROCESS
Contents
1. BASIC CONCEPTS
Financial Economics and Uncertainty
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Stochastic Process, Stationarity and Autocorrelation
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Stochastic process (e.g., nondeterministic discrete time series)
two features: dependency and lack of replication
Realizations and statistics of probability distribution:
mean,variance,autocovariance
stationarity: a particular state of statistical equilibrium
strict stationarity: distribution properties unaffected by a change of time origin
weak stationarity: the first and second moments do not depend on time.
Ergodicity: the conditions about the consistency between sample statistics and
population statistics
Autocorrelation function (correlogram) and partial autocorrelation
ACF[  ( s)   ( s)  (0) ] and structure of the random process
PACF: “indirect” correlation eliminating the other past effects
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Stationary linear stochastic process
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White noise model
2
var(
u
)
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 ;
E(ut )  0 ;
t
cov(ut , ut s )  0
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Autoregressive model [AR(p)]
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p: lag order, :innovation(white noise process)
stationarity(characteristic roots of (1  i Li ) must lie outside of the unit
circle)
to calculate the second moments based on Yule=Walker equation
for an AR(p), there is no partial autocorrelation between yt and yt  sfor
s>p.
Moving average model[MA(q)]
yt    1 yt 1  2 yt 2     p yt  p  ut

yt    ut  1ut 1   2ut 2     qut q
stationary and non-deterministic process:
to calculate each statistics based on their definition
for an MA(q), there is no autocorrelation between yt and yt  s for s>p.
MA invertibility and AR stationarity
the PACF coefficients exhibit a geometrically decaying pattern.
 ARMA(p,q) model
p
q
i 1
j 0
yt     i yt i   j ut  j
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stationarity [same as AR(p)];invertibility[same as MA(q)]
the ACF will begin to decay at lag q,while PACF to decay at lag p.
Autoregressive integrated moving average Model[ARIMA(p,d,q)]
 ( L)yt      ( L)ut
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trend elimination
Wold’s decomposition theorem
2. Box-Jenkins Methodology
IdentificationEstimationDiagnostic Checking(Forecasting)
Principle of Parsimony
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Identification(Model Building)
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Plotting time series data
Pattern of the ACF and PACF
Test on Sample ACF and PACF (t, Q test)
Nonstationarity and seasonality adjustment (integrated process)
trend(mean by difference,variance by log transformation);
seasonality(regular by difference,irregular by additive or multiplicative
SARIMA)
Estimation
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General method
covariance matrixML function   2 Estimator(ML,QML,CML)
f y p  ,
long period needed
Special Method
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AR(p): OLS, Yule=Walker equation
MA(q) and ARMA(p,q): Gauss=Newton method(grid-search)
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Diagnostic Checking(Model Selection)
Residuals plot
 Information criteria[AIC(1969),SBIC(1978),etc.]
AIC= logˆ 2  2( p  q) / T ; ˆ 2 :estimator of var( ut)
SBIC= logˆ 2  ( p  q) log(T ) / T
They will be as small as possible(comparable with the same
period)
SBIC has superior large sample properties(asymptotically
consistent).
 Overfitting and splitting analysis
 Forecast adequacy
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3. FORECASTING
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Basic Concept
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Forecast Function
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Optimal forecast and prediction error
stationarity and convergency of forecast weight and error variance
The role of forecast model
AR(): make a forecast;MA():forecast error analysis
Significant level and confidence intervals
Iterative method
Solution methodology
Some Comments
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Efficacy of forecast(short period)
Conditional forecast (start period)
Large sample needed
4. SUMMARY AND CONCLUSIONS
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By definition, an ARMA model is weak stationary in that it has a finite and
time-invariant mean and covariances.For an ARMA model to be stationary,the
characteristic roots of the difference equation must lie inside the unit circle.
Moreover,the process must have started infinitely far in the past or the process
must always be in equilibrium.
A well estimated model (1) is parsimonious; (2) has coefficients that imply
stationarity and invertibility;(3) fits the data well;(4) has residula that
approximate a white-noise process; (5) has coefficients that do not change over
the sample period; and (6) has good out-of-sample forecasts.
Appendix : TSP Programs to Accompany Chapter 2
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BJIDENT (option) variables;
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Option: NDIFF, NSDIFF ( NSPAN), NLAG, NLAGP
Plot: series, ACF+PACF (20), Q (s-p-q-1)
Output value: ACF, PACF, Q
@AC, @PAC
BJEST (option) variables (start values);
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Option (unnecessary to specify if same):
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NBACK (start condition)---MA: small value(5), AR: large (10)
Start: value specification (order—AR,MA,CONST)previous
estimation results
Residuals: Q, p, periodogram (45 degree line)
AIC=2logL + 2(p+q)
BJFRCST (option) variables S start variables values;
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Option: CONBOUND(95%), NHORIZ, ORGBEG, ORGEND)