FEB13011 Financial Methods and Techniques PART 2: Time Series Regression

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FEB13011 Financial Methodsand TechniquesPART 2
Time Series Regression
Dr. David J.C. Smant FEW, Room H14-26, Tel.
010-4081408E-mail smant_at_few.eur.nlHomepage ht
tp//people.few.eur.nl/smant
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Regression analysis (so far)

• Y b0 b1 X1 b2 X2
• Linear regression model
• Xi cross section regressions
• Xt time series regressions
• Xit panel regressions

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Time series regression analysis

• Yt b0 b1 X1t b2 X2t
• Estimation by Ordinary Least Squares (OLS)
• assumptions BLE1. linearity2. zero mean of
  residual3. constant variance of residual (vs.
  heteroscedasticity)4. zero covariance between
  pairs of residuals (vs. serial correlation)5.
  exogenous X variables (vs. endogeneity,
  errors-in-variables) 6. normal distribution (vs.
  dummy variables)
• Now stationary vs. non-stationary variables

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Time series variables
Typical time series variables
Stationarity? constant mean (constant/finite
variance) (constant autocorrelation)
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Time series variables
Warning Any modern empirical research using time
series regression is unreliable (i.e. worthless)
when no examination is done on 1 stationarity of
variables 2 cointegration properties of variables
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Other aspects of time processes

• For example,
• Pricing modelsWhat price for stock options?
  What long-term interest rate?
• Efficient market implicationsDo stock prices
  follow martingale (random walk) process? Are
  (excess/abnormal) returns predictable? PPP Are
  real exchange rates constant/stationary?Fisher
  effect Are real interest rates
  constant/stationary?
• Time-varying volatility, heteroscedasticityBias
  ed test statisticsTime-varying risk in financial
  models
• The nature of shockstemporary vs. permanent
  shocks

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COURSE OUTLINE

• 2 topics
• Non-stationarity and unit root tests
• Cointegration
• Practical econometricsFocus on the
• Why theory, motivation
• How estimation, diagnostics, tests
• Interpreting results
• Alternatives
• Basic principles, various methods, examples using
  EViews

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EVIEWS EXERCISE

• EViews Exercise 3, part of exam
• Tuesday Feb 19th
• Wednesday Feb 20th
• Friday Feb 22nd
• Also includes
• Unit root tests
• Cointegration (single equation methods)
• Warning
• Read literature in advanceReview EViews example
  from lecture slides, notes practice dataset

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FEB13011 FMTTIME SERIES REGRESSION
NONSTATIONARITY UNIT ROOTS

• NON-STATIONARITY, UNIT ROOT
• UNIT ROOT TESTS - Alternative tests -
  Example - Weaknesses

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EXAMPLE FTA 4 TIMES SERIES VARIABLES
Monthly observations January 1965 to December
1995 (372 months) FTA All Share index (LPRICE),
FTA Dividend index (LDIV), yield on 20 year UK
Gilts (R20), 91 day Treasury bills (RS)
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(NON-) STATIONARY TIME SERIES
A stochastic process Xt is said to be
(covariance, weakly) stationary if 1. EXt ?
for all t 2. VarXt ?2 lt ? for all t 3.
CovXt, Xtk ?k for all t and k Focus in
empirical work is on nr.1 - constant mean
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EXAMPLES OF SPURIOUS REGRESSION

• Egyptian infant mortality rate (Y), 1971-1990,
  annual data, on gross aggregate income of
  American farmers (I) and Total Honduran money
  supply (M)Y 179.9 0.2952 I 0.0439
  M R20.918, DW0.4752 (16.63) (2.32)
  (4.26)
• US Export Index (Y), 1960-1990, annual data, on
  Australian males life expectancy (X)Y -2943.
  45.7954 X R20.916, DW0.3599 (16.70)
  (17.76)
• US Defense Expenditure (Y), 1971-1990, annual
  data, on Population of South Africa (X)Y
  -368.99 0.0179 X R20.940, DW0.4069
  (11.34) (16.75)
• Total crime rates in the US (Y), 1971-19991,
  annual data, on Life expectancy of South Africa
  (X)Y -24569 628.9 X R20.811, DW0.5061
  (6.03) (9.04)
• Population of South Africa (Y), 1971-1990, annual
  data, on Total RD expenditure in the US (X)Y
  21698.7 111.58 X R20.974, DW0.3037
  (59.44) (26.40)
• Typical symptom high R2, high t-values or
  F-value, but low DW

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(NON-) STATIONARY TIME SERIES AND SPURIOUS
RELATIONSHIPS
Granger-Newbold (1974) introduced the notion of a
spurious regression. They said that macroeconomic
data was in general non-stationary (or
integrated) and that in regressions involving
levels of variables, the standard significance
tests were usually misleading. The conventional t
and F tests would tend to reject the hypothesis
of no relationship even if there was, in fact,
none.In a Monte-Carlo study Granger and Newbold
(1974) created two non-stationary random walks y
and x that were uncorrelated with each other.
They regressed y on x and found that in 75 per
cent of cases, the t test on thea2 rejected the
null that it was zero. So in 75 per cent of the
cases, a spurious significant relationship was
found. Brooks (2002) create 2 unrelated
non-stationary time seriesOLS regression
estimates biased (R2) (t-stat)
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NON-STATIONARY TIME SERIES SOLUTIONS

• Solutions to spurious regressions
• two types of non-stationarity
• Deterministic trend model ? detrending
• Stochastic trend model ? first-differencing
• cointegration modelingspecial case where
  first-differencing results in biased estimation
  results.
• Note Empirical results frequently very sensitive
  to the choice of detrending or first-differencing
  variables. Conclusions from results may be very
  different.

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DETRENDING
Linear trend model yt yDTt ut yDTt ?
? TRENDt TRENDt 1, 2, 3, 4, Detrending yt
- yDTt ut ? y-yDT is stationary ? variable
y is trend stationary
Note Other trend models quadratic trends
Hodrick-Prescott filter ( moving average)
others
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FIRST-DIFFERENCING, UNIT ROOTS
Stochastic trend model yt ySTt ut ySTt
ySTt-1 ? ?t trend yST is not fixed but
depends on its own history of random
shocks rewrite as yt yt-1 ? ?t where ?t
?t ut ut-1 First differencing yt - yt-1 ?
?t ? yt-yt-1 is stationary Variable y must be
differenced one time to become stationary
variable y is said to be integrated of order one
I(1), or difference stationary, or has a unit
rootDifferenced d times to become stationary,
variable is I(d) Stationary variable is I(0)
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UNIT ROOTS AND RANDOM WALKS
Random walk (with drift) is the simplest of unit
root time series yt yt-1 ? ?t ?t ? N(0,
?2?) A random walk cannot be detrended (not trend
stationary), but first difference of random walk
is stationary (difference stationary) ?yt yt -
yt-1 ? ?t Note Unit roots are closely
related to random walks/martingales, but not
identical! Unit roots allow serial correlation,
whereas random walks do not. Note Specific
tests for random walks autocorrelation tests
variance ratio tests other
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OBSERVATIONAL EQUIVALENCE?
Random walks Deterministic linear trend
Observational equivalenceRW process in this
case resembles deterministic quadratic trend
model. How do we know which one is correct? we
dont know!
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(NON-) STATIONARY TIME SERIES AND PREDICTABILITY
Stationary and trend-stationary time series are
predictable 1 yt1 ? yt ?t1, with
?lt1 Et(ytk) Et ?k yt ?k-1 ?t1 ?
?tk-1 ?tk ?k yt Et(ytk- yt) (?k1)
yt when ?1, random walk ? not predictable! 2
yt ? ? TRENDt ?t Et(ytk) Et ? ?
(tk) ?tk ? ? (tk) Et(ytk - yt ) ?
k when ?0, white noise ? not predictable
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UNIT ROOT TESTS

• Several alternative tests exist based on
  different econometric assumptions
• Tests for null-hypothesis unit root against
  alternative hypothesis of (trend-) stationarity,
  for example
• (Augmented) Dickey-Fuller test
• Phillips-Perron test
• Elliot-Rothenberg-Stock test
• Ng-Perron
• Other
• Tests for null-hypothesis (trend-) stationarity,
  for example
• KPSS test
• Leyborne-McCabe test
• Other

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(AUGMENTED) DICKEY-FULLER TEST
Test null hypothesis unit root yt yt-1 ?
?t against alternative hypothesis yt ? ? T
? yt-1 ?t i.e. trend-stationary time
series ADF test equation ?yt ? ? T (?-1)
yt-1 ?k1p ?k ?yt-k ?t Estimate using
OLS Incorporate lagged ?y to correct for possible
serial correlation Test H0 (?-1) 0 against
(?-1) lt 0 For example, t-statistic for (?-1)
non-standard distribution of test-statistics
and different distributions with/without
intercept and with/without trend Tables of
critical values specially calculated (Fuller,
1976 MacKinnon, 1991, 1996) See EViews
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ADF TEST (Cont)

• Include or exclude intercept ? and/or trend ?
  T?t-statistics are not reliable
• (most popular) show/discuss results for the
  different options
• special series of tests, use non-standard
  distribution test statistics (Perron, 1988)
• How to determine p of the lagged variables?Too
  few lags will leave autocorrelation in the errors
  and distort the test, but too many lags will
  reduce the power of the test.Alternative
  suggestions
• (most popular) information criteria use model
  specification tests such as AIC/BIC/SIC/F-test to
  determine optimal lag length
• recursive t-statistic procedure start with
  large p, reduce when longest lag is insignificant
  at 5 or 10 (Hall, 1994 Campbell-Perron, 1991)
• Also test that lags eliminate residual serial
  correlation

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PHILLIPS-PERRON TEST
The PP test has a different approach than ADF to
correcting the unit root test for possible serial
correlation. PP test equation ?yt ? ? T
(?-1) yt-1 ?t Estimate using OLS Test H0
(?-1) 0 against (?-1) lt 0 Adjusting the test
statistics (special formula), so that they
conform more closely to the standard DF
distribution. For example, Z-statistic (modified
t-statistic) calculation of the statistics is
somewhat complicated, EViews does it for
you non-standard distribution of test-statistics
and different distributions with/without
intercept and with/without trend Tables of
critical values specially calculated (Fuller,
1976) See EViews
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PP TEST (Cont)

• Include or exclude intercept ? and/or trend ? T?
• show/discuss results for the different options
• Calculation of the modified test statistics
  requires a choice of different methods
  spectral estimation method bandwith
• (too difficult, use EViews default options)

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Kwiatkowski, Phillips, Schmidt and Shin (KPSS)
ADF and PP tests, take a unit root as null
hypothesis and attempt to reject this hypothesis.
But it may be very difficult to reject a unit
root when in fact ? is close to one (that is,
tests may have low power). An alternative test,
KPSS, takes (trend-) stationarity as the null
hypothesis You may ignore the formula,
EViews will calculate the LM-statistic Too high
LM-value rejects the null hypothesis Critical
values from the distribution of the
KPSS-statisticare calculated by KPSS (1992,
Table 1, p.166) See EViews
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KPSS TEST (Cont)

• Include or exclude intercept ? and/or trend ? T?
• show/discuss results for the different options
• Calculation of the test statistic requires a
  choice of different methods spectral
  estimation method bandwith
• (too difficult, use EViews default options)

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ADF UNIT ROOT T-TEST
Variable FTA example is LPRICEADF t-test H0
(?-1)0, probability values in brackets using
McKinnon (1996) one-sided p-values
Conclusion? Unit root null-hypothesis rejected
in favor of (trend-)stationarity?
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ADF and PP unit root, KPSS stationarity tests
Conclusion? ADF, PP Unit root hypothesis
rejected in favor of (trend-)stationarity?KPSS
(trend-)stationarity hypothesis rejected in favor
of unit root?
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Culver/Papell (2006 J. Appl Econometrics ) Is
there a unit root in the inflation rate?...

• Economic relevance Inflation process related
  to alternative macro price/inflation models
  Inflation and real and nominal interest rate
• Unit root tests ADF lags selected with
  recursive t-statistic procedure, 10 level
  constant, no trend (using economic reasoning)
  KPSS
• Data IMF-IFS13 countriesCPI, monthly, January
  1957-September 1994
• Inflation ? ln(CPIt/CPIt-1)
• NOTE Also examined are ADF with structural
  break, panel unit root test. Panel-test results
  are very different.

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Culver/Papell (1997)

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UNIT ROOT TESTS COMMENT 1

• Unit root tests are known to have low power in
  finite samples
• I.e. it is very difficult to reject the unit root
  hypothesis, when the true coefficient actually is
  different from but close to 1.
• Some modified tests exist for improved results
  in finite samples
• Many economists also use theoretical
  considerations before they draw
  conclusions.Economic theory may suggest whether
  it is logical that variables have unit roots,
  trends, or some long-run average value

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UNIT ROOT TESTS COMMENT 2

• Unit root tests, as well as random walk tests,
  are known to be very sensitive to trend breaks or
  regime shifts. (Perron, 1989)
• Modified tests of unit roots and stationarity
  have been suggested to incorporate one or more
  trend breaks
• Trend breaks or regime shifts are very likely in
  economics

y
No reversion to full sample average ? unit root?
Full sample average
time
sample/regime 1
sample/regime 2
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CONCLUSIONS

• No empirical research using time series
  variables is acceptable without examining unit
  root hypothesis (and cointegration) first
• Various tests exist EViews shows which ones
  have become common Unfortunately, different
  tests do not always agree, because they rely
  on different econometric assumptions
• Unit root testing has some possible weaknesses,
  cannot be applied thoughtlessly in economic
  researchEconomic reasoning remains
  important(McCallum, 1993 Cochrane, 1991)

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EXAM ?

• Unit root tests
• Why?
• How? - particularly, knowing the result,
  reading output tables
• Issues, weaknesses?