The Properties of Time Series: Lecture 4

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The Properties of Time Series Lecture 4
Previously introduced AR(1) model Xt fXt-1
ut (1) (a) White Noise (stationary/no
unit root) Xt ut i.e. f 0 in AR(1)
equation (1) (b) Random Walk
(non-stationary/unit root) Xt Xt-1
ut i.e. f 1 in AR(1) equation (1) (Unit root
since equation solves for f equal to one) (c)
Stationary Process / No Unit Root Xt
fXt-1 ut i.e. f lt 1 in AR(1) equation (1) In
Unit Root tests we test null hypothesis f1 in Xt
fXt-1 ut Or null f f-1 0 in ?Xt
fXt-1 ut
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Testing Strategy for Unit Roots
Three main aspects of Unit root testing -
Deterministic components (constant, time
trend). - ADF Augmented Dickey Fuller test -
lag length - use F-test or Schwarz Information
Criteria - In what sequence should we
tests? - Phi and tau tests
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Testing Strategy for Unit Roots
Formal Strategy (A) Set up Model (1) Use
informal tests eye ball data and
correlogram (2) Incorporate Time trend if data
is upwards trending (3) Specification of ADF
test how many lags should we incorporate to
avoid serial correlation?
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Example- Real GDP (2000 Prices) Seasonally
Adjusted
(1) Plot Time Series - Non-Stationary (i.e.
time varying mean and correlogram non-zero)
GDP
Time
r
k
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Unit Root Testing
(1) Plot First Difference of Time Series -
Stationary (i.e. constant mean and correlogram
zero)
Time
r
k
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Unit Root Testing
(2) Incorporate Linear Trend since data is
trending upwards
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Unit Root Testing
(3) Determine Lag length of ADF test Estimate
general model and test for serial correlation EQ
( 1) ?Yt aßtrend fYt-1 ?1?Yt-1 ?2?Yt-2
?3?Yt-3 ?4?Yt-4 ut EQ( 1) Modelling DY
by OLS (using Lab2.in7) The estimation
sample is 1956 (2) to 2003 (3) n 190
Coefficient Std.Error t-value
t-prob Part.R2 Constant 0.538887
0.3597 1.50 0.136 0.0121 Trend
0.00701814 0.004836 1.45
0.148 0.0114 Y_1 -0.0156708
0.01330 -1.18 0.240 0.0075 DY_1
-0.0191048 0.07395 -0.258
0.796 0.0004 DY_2 0.137352
0.07297 1.88 0.061 0.0190 DY_3
0.188071 0.07354 2.56
0.011 0.0345 DY_4 0.0474897
0.07473 0.635 0.526 0.0022 AR 1-5
test F(5,178) 1.7263 0.1308 Test
accepts null of no serial correlation.
Nevertheless we use F-test and Schwarz Criteria
to check model.
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Unit Root Testing
(3) Determine Lag length of ADF test Model EQ (
1) ?Yt aßtrend fYt-1 ?1?Yt-1 ?2?Yt-2
?3?Yt-3 ?4?Yt-4 ut EQ ( 2) ?Yt aßtrend
fYt-1 ?1?Yt-1 ?2?Yt-2 ?3?Yt-3 ut EQ ( 3)
?Yt aßtrend fYt-1 ?1?Yt-1 ?2?Yt-2
ut EQ ( 4) ?Yt aßtrend fYt-1 ?1?Yt-1
ut EQ ( 5) ?Yt aßtrend fYt-1 ut Use
both the F-test and the Schwarz information
Criteria (SC). Reduce number of lags where
F-test accepts. Choose equation where SC is the
lowest i.e. minimise residual variance and
number of estimated parameters.
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Unit Root Testing
(3) Determine Lag length of ADF test Progress to
date Model T p
log-likelihood Schwarz Criteria EQ(
1) 190 7 OLS -156.91128
1.8450 EQ( 2) 190 6 OLS
-157.12068 1.8196 EQ( 3) 190 5
OLS -160.37203 1.8262 EQ( 4)
190 4 OLS -162.16872 1.8175 EQ(
5) 190 3 OLS -162.17130
1.7899 Tests of model reduction EQ( 1) --gt EQ(
2) F(1,183) 0.40382 0.5259 Accept model
reduction EQ( 1) --gt EQ( 3) F(2,183) 3.3947
0.0357 Reject model reduction EQ( 1) --gt EQ(
4) F(3,183) 3.4710 0.0173 EQ( 1) --gt EQ(
5) F(4,183) 2.6046 0.0374 Some
conflict in results. F-tests suggest equation (2)
is preferred to equation (1) and equation (3)
is not preferred to equation (2). Additionally,
the relative performance of these three equations
is confirmed by information criteria. Therefore
adopt equation (2).
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Unit Root Testing
(B) Conduct Formal Tests EQ( 2) Modelling DY by
OLS (using Lab2.in7) The estimation sample
is 1956 (2) to 2003 (3)
Coefficient Std.Error t-value t-prob
Part.R2 Constant 0.505231
0.3552 1.42 0.157 0.0109 Trend
0.00655304 0.004772 1.37 0.171
0.0101 Y_1 -0.0141798
0.01307 -1.08 0.279 0.0064 DY_1
-0.0119522 0.07297 -0.164
0.870 0.0001 DY_2 0.142437
0.07241 1.97 0.051 0.0206 DY_3
0.185573 0.07332 2.53
0.012 0.0336 AR 1-5 test F(5,179)
0.68451 0.6357 Main issue is serial
correlation assumption for this test. Can we
accept the null hypothesis of no serial
correlation? Yes!
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Unit Root Testing
Apply F-type test - Include time trend in
specification F3 ?Yt a ßtrend fYt-1
?1?Yt-1 ?2?Yt-2 ?3?Yt-3 ut (a) Ho f ß
0 Ha f? 0 or ß? 0 PcGive Output
Test/Exclusion Restrictions. Test for excluding
0 Trend 1 Y_1 F(2,184) 2.29 lt 6.39
5 C.V. (by interpolation). Hence accept joint
null hypothesis of unit root and no time
trend (next test whether drift term is
required). NB Critical Values (C.V.) from
Dickey and Fuller (1981) for F3 Sample Size
(n) 25 50 100 250 500 C.V. at 5 7.24 6.73 6.49 6
.34 6.30
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Unit Root Testing
Apply F-type test - Exclude time trend from
specification F1 ?Yt a fYt-1 ?1?Yt-1
?2?Yt-2 ?3?Yt-3 ut (b) Ho f a
0 Ha f? 0 or a? 0 PcGive Output
Test/Exclusion Restrictions. Test for excluding
0 Constant1 Y_1 F(2,185) 10.27 gt 4.65
5 C.V. Hence reject joint null hypothesis of
unit root and no drift. NB Critical Values
(C.V.) from Dickey and Fuller (1981) for
F1 Sample Size (n) 25 50 100 250 500 C.V. at
5 5.18 4.86 4.71 4.63 4.61
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Unit Root Testing
Apply t-type test (tµ) tµ ?Yt a fYt-1
?1?Yt-1 ?2?Yt-2 ?3?Yt-3 ut (b) Ho f
0 Ha f lt 0 tµ 1.64 gt -2.88 5
C.V. Hence accept null of unit root. N.B.
Critical Values (C.V.) from Fuller (1976) for
tµ Sample Size (n) 25 50 100 250
500 C.V. at 5 -3.00 -2.93 -2.89 -2.88 -2.87
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Unit Root Testing
EQ(2a) Modelling DY by OLS (using Lab2.in7)
The estimation sample is 1956 (2) to 2003 (3)
Coefficient Std.Error
t-value t-prob Part.R2 Constant
0.0535255 0.1343 0.399 0.691
0.0009 Y_1 0.00352407 0.002150
1.64 0.103 0.0143 DY_1
-0.0218516 0.07279 -0.300 0.764
0.0005 DY_2 0.131601 0.07215
1.82 0.070 0.0177 DY_3
0.172115 0.07283 2.36 0.019
0.0293 AR 1-5 test F(5,180) 0.50464
0.7725 tµ 1.64 gt -2.88 (5 C.V.) hence
we can not reject the null of unit root.
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Look at the Series Is there a Trend?
Yes
No
?Xt a fXt-1 ut
?Xt a ßtrend fXt-1 ut
Use F1 to test
Use F3 to test
Ho f a 0 vs Ha f? 0 or a? 0
Ho f ß 0 vs Ha f? 0 or ß? 0
Reject
Accept
Reject
Accept
Pure Random Walk
test f0 using the t-stat. from step 1 using
test f0 using the t-stat. from step 1using
Accept
Reject
Accept
Reject
Unit Root Trend
No Unit Root
Stable Series, use normal test to check the drift
Use F2
Normal Test procedure to determine the presence
of Time trend or Drift
Random Walk Drift
To determine if there is a drift as well
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Problems in Unit Root testing using Dickey-Fuller
tests
(1) Trend stationary or difference
stationary. (2) Low power of unit root
tests (3) Structural breaks in time series.
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Problems in Unit root testing
(1) Trend Stationary Process and Difference
Stationary Process. Graph of GDP could be
approximated by linear trend - Nelson and
Plosser (1982) challenged this assumption trend
was a random walk for many series. - Trend
was not fixed but was moved by random shocks,
and would stay as such until hit by another
shock. This problem can be resolved
partially by careful application of F-type
tests. - e.g. from before there is no evidence
of trend for F3
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Problems in Unit root testing
(2) Low power of unit root tests - Is f 0
in ?Xt a fXt-1 ut Test result is based
on the standard error of f - Measure of how
accurate is our estimated coefficient - with
increasing observations we become more certain.
Power of a tests is ability to reject the null
when it is false. e.g. ability to accept
alternative hypothesis of stationarity. Low
power implies a series may be stationary but
Dickey- Fuller test suggests unit root. -
low power is especially a problem when series is
stationary but close to being unit root.
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Problems in Unit root testing
One solution to low power is to increase the
number of observations by increasing the span of
data. However, there may be differences in
economic structure or policy which should be
modelled differently. (3) Structural breaks in
time series. - Perron (1989) movement of trend
could be explained by single break. -
Nelson-Plosser series are not random walk but
linear trend with single breaks. Alternative
solution to low power is a number of joint ADF
tests. - Take information from a number of
countries. - And pool coefficients. (i.e.
combine information).