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ECON207 Session 6 Slide 1
Regressions on Time Series Data
1{ , }Tt t tY =x t t tY ε′= +x β
What are the issues?
Generally, cannot consider data to be independent draws
Intertemporal correlations / Serial Correlations / Autocorrelations
ECON207 Session 6 Slide 2
Or even identical:
Singapore IP Index
Other issues: changing variances, seasonalities
25
50
75
100
125
1990 2000 2010T
IP_S
G
ECON207 Session 6 Slide 3
Presence of serial correlation, must use appropriate formulas:
E.g. 1{ }Tt tY = , [ ]tE Y µ= , 2var[ ]tY σ= , but not serially uncorrelated
Want to estimate µ .
1 1
1 1[ ] [ ]T T
t tt t
E Y E Y E YT T
µ= =
= = =
∑ ∑
21
1 2 121 2 3
2
1 2 122 3
1var var
1 var[ ] 2 cov[ , ] 2 cov[ , ] 2cov[ , ]
1 2 cov[ , ] 2 cov[ , ] 2cov[ , ]
T
tt
T T T
t t t t t Tt t t
T T
t t t t Tt t
Y YT
Y Y Y Y Y Y YT
Y Y Y Y Y YT Tσ
=
− −= = =
− −= =
=
= + + + +
= + + + +
∑
∑ ∑ ∑
∑ ∑
Not 2
var[ ]YTσ
= .
ECON207 Session 6 Slide 4
Spurious Regressions:
E.g. SG_IP on simulated (fake) tY where 0 1t t tY Yδ ε−= + + , 0 0.5δ = , ~ (0,1)t Nε
1000 times (same SG_IP, different tY )
Histogram of t-statistics and R-squares
ECON207 Session 6 Slide 5
Dynamic Specifications
0 1t t tY Xβ β ε= + + “Static Model” may be too restrictive?
0 1 2 1 1t t t p t p tY X X Xβ β β β ε− + −= + + + + + “Distributed Lag model”
- for dynamic causal parameters?
- Parameter interpretation?
0 1 1 2 3 1 2...t t t t P t p tY Y X X Xβ β β β β ε− − + −= + + + + + + “Autoregressive DL”
- Interpretation?
0 1 1 2 1t t t tY Y Xβ β β ε− −= + + + For forecasting?
ECON207 Session 6 Slide 6
Structural Change
E.g. 0 1
0 1
for for
t tt
t t
X tY
X tβ β ε τδ δ ε τ
+ + ≤= + + >
- How to detect?
- How to accommodate?
Certain OLS assumptions too strong?
1[ | ,..., ] 0i NE ε =x x for all i Required for unbiasedness
In Time Series data
1[ | ,..., ] 0t TE ε =x x for all t “Strong exogeneity”
Too strong!
ECON207 Session 6 Slide 7
E.g. tε may contain information that can predict future 1 2,t tX X+ +
E.g. tX may be decision variable influenced by 1 2, ,...t tε ε− −
E.g. Regression may contain lagged dependent variable 0 1 1t t tY Yβ β ε−= + + “Autoregression of Order 1”
ECON207 Session 6 Slide 8
More reasonable:
[ | ] 0t tE ε =x “contemporaneous exogeneity”
But then OLS not unbiased
Consistent? Sometimes…
Properties of OLS estimators, and remedies, depends on ‘nature’ of time series
Study OLS estimators for different classes of time series
Key statistic: autocovariance / autocorrelation
ECON207 Session 6 Slide 9
Autocovariances and Autocorrelations
Autocovariance at lag k:
, cov[ , ] [( [ ])( [ ])]k t t t k t t t k t kY Y E Y E Y Y E Yγ − − −= = − − , 1,2,3,...k =
Autocovariance at lag 0 is the variance
0, cov[ , ] [( [ ])( [ ])] var[ ]t t t t t t t tY Y E Y E Y Y E Y Yγ = = − − =
Will generally change with k, possibly also with t
Important special case is when ,k t kγ γ=
1
2
3
4
5
6
1 2 3 4 5 6Y_1
Y
1
2
3
4
5
6
1 2 3 4 5 6Ya_1
Ya
1
2
3
4
5
6
1 2 3 4 5 6Yb_1
Yb
1
2
3
4
5
6
1 2 3 4 5 6Yc_1
Yc
ECON207 Session 6 Slide 10
Sometimes prefer to work with autocorrelations:
0
cov[ , ] cov[ , ]var[ ]var var
t t k t t k kk
tt t k
Y Y Y YYY Y
γργ
− −
−
= = =
(we assume here variance and correlations don’t change over time.)
Measuring autocovariances and autocorrelations
Sample Autocov 1
1ˆ ( )( )T
k t t kt k
Y Y Y YT
γ −= +
= − −∑ where 11 T
ttY YT =
= ∑
Sample Autocorr 0
ˆˆˆk
kγργ
=
0,1,2,3,...k =
1ˆ{ }k kγ ∞= sample autocovariance function
0ˆ{ }k kρ ∞= sample autocorrelation function (correllogram)
ECON207 Session 6 Slide 11
-1.0
-0.5
0.0
0.5
1.0
4 8 12 16 20 24 28Lag
ACF
Series: Y
25
50
75
100
125
1990 2000 2010T
IP_S
G
-1.0
-0.5
0.0
0.5
1.0
0 5 10 15 20 25 30Lag
ACF
Series: IP_SG
ECON207 Session 6 Slide 12
1
1
_ __ __
t tt
t
IP SG IP SGIP SG GIP SG
−
−
−=
-0.2
0.0
0.2
0.4
1990 2000 2010Time
IP_S
G_G
-0.2
0.0
0.2
0.4
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth
year1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
Seasonal plot: IP_SG_G
-1.0
-0.5
0.0
0.5
1.0
12 24Lag
ACF
Series: IP_SG_G
ECON207 Session 6 Slide 13
Important Classes of Time Series
tY covariance stationary if for all t
- [ ]tE Y µ= < ∞
- cov[ , ]t t k kY Y γ− = < ∞ , 0,1,2,...k =
Remarks: Main point is – does not change with t (is constant)
Finiteness important as well
Autocovariance requirement includes variance
Autocovariance may change with k (but does not have to)
Alt names: weak stationarity, second-order stationarity,…
ECON207 Session 6 Slide 14
tY strictly stationary if
Joint pdf of any collection of data points depends only on relative spacing
e.g.,
2 1 3 7 7 10 12 16 3 5 9( , , , ) ( , , , ) ( , , , )t t t tf Y Y Y Y f Y Y Y Y f Y Y Y Y− + + += = for all t
1 10 5 14 9( , ) ( , ) ( , )t tf Y Y f Y Y f Y Y += = for all t
1 2( ) ( ) ( )t t tf Y f Y f Y+ += = etc.
If mean, variance, autocovariances are finite, then strict stationarity implies
covariance stationarity
E.g. A white noise process (zero-mean, constant variance, serially uncorrelated) is
covariance stationary
E.g. An independent white noise process (a white noise process that is iid) is
strictly stationary
ECON207 Session 6 Slide 15
E.g. Stationary Autoregressive Processes
20 1 1 , ~ (0, )
iid
t t t tY Yβ β ε ε σ−= + + ? “AR(1)”
ECON207 Session 6 Slide 16
Some remarks about AR(1)
- Pure stats model with no ‘economics’ underlying it?
- Yes and no…
E.g.
0 1 2 1
0 1
(supply equation)
(demand equation)
(market clearing)
s st t t td dt t td st t
Q P P
Q P
Q Q
α α α ε
δ δ ε−= + + +
= + +
=
Equating dd and ss and solving for tP
0 0 21
1 1 1 1 1 1
s dt t
t tP Pα δ ε εαδ α δ α δ α−
− −= + +
− − −
ECON207 Session 6 Slide 17
An AR(1) can be cov-stationary
- Requires 1| | 1β ≤
0 1 1t t tY Yβ β ε−= + + , 1| | 1β < , 2~ (0, )iid
tε σ , ...,1,2,..., ,...t T=
- Assume 0β , 1β , 2σ not changing over time
We derive properties to show cov-stationary
Write
0
1
0 1 12
0 1 0 1 2 1 0 1 0 1 2 1 1
2 2 10 1 1 1 1 1 1 2 1 1 1
assume 0 as 1
201 1 1 2 1
1
( )...
(1 )
1
t t t
t t t t t t
k k kt t t t k t k
k
kt t t
Y Y
Y Y
Yββ
β β ε
β β β β ε ε β β β β ε β ε
β β β β ε β ε β ε β ε β
β ε β ε β ε ββ
−
− − − −
+− − − − −
→→ →∞
−
− −
= + +
= + + + + = + + + +
=
= + + + + + + + + + +
= + + + + +−
t kε − +
ECON207 Session 6 Slide 18
Then
(a) 20 01 1 1 2
1 1
[ ] [ ] [ ] [ ]1 1t t t tE Y E E Eβ βε β ε β ε
β β− −= + + + + =− −
for all t
(b) 2
2 41 1 1 2 2
1
var[ ] var[ ] var[ ] var[ ]1t t t tY σε β ε β ε
β− −= + + + =−
for all t
(a) means tY fluctuates around a constant
(b) mean size of fluctuations constant
Possibly constant variance Not constant variance
ECON207 Session 6 Slide 19
Autocovariance function 2 2
1 1 1 2 1 1 1 22 2 2
1 1 12
12
1
[( )( )]
[ ] [ ]
, 0,1,2,...1
k t t t t k t k t kk k
t k t kk
E
E E
k
γ ε β ε β ε ε β ε β ε
β ε β ε
β σβ
− − − − − − −
+− − −
= + + + + + +
= + +
= =−
Autocorrelation function 10
, 0,1,2,...kkk kγρ β
γ= = =
Diminishes with k
But for each k ~ does not depend on t
-1.0
-0.5
0.0
0.5
1.0
4 8 12 16 20 24 28Lag
ACF
Series: Y
ECON207 Session 6 Slide 20
Remark Unconditional vs Conditional Moments
ECON207 Session 6 Slide 21
tY Stationary Ergodic
- Stationary - tY and t kY − asymptotically independent as k →∞
E.g. White noise process is stationary ergodic E.g. Stationary AR(1) is stationary ergodic
-1.0
-0.5
0.0
0.5
1.0
4 8 12 16 20 24 28Lag
ACF
Series: Y
ECON207 Session 6 Slide 22
Non-Stationary Series Series can be non-stationary in many many many ways… We focus on two special cases… Difference Stationary Processes Non-stationary, but can be made stationary by taking first differences
- tY non-stationary - 1t t tY Y Y −∆ = − stationary
Alt names: “ tY is integrated of order one”,
“ tY is I(1)”,
“ tY has a unit root”,…
ECON207 Session 6 Slide 23
Example
Random Walk 20 1 , ~(0, )
iid
t t t tY Yδ ε ε σ−= + +
i.e., tY changes by average of 0δ evert period
1 0t t tY Y δ ε−− = +
How does this series behave
1 0 0 1Y Yδ ε= + + 21 0 0 0 1 0[ | ] , var[ | ]E Y Y Y Y Yδ σ= + =
2 0 1 2 0 0 2 12Y Y Yδ ε δ ε ε= + + = + + + 22 0 0 0 2 0[ | ] 2 , var[ | ] 2E Y Y Y Y Yδ σ= + =
3 0 2 3 0 0 3 2 13Y Y Yδ ε δ ε ε ε= + + = + + + + 23 0 0 0 3 0[ | ] 3 , var[ | ] 3E Y Y Y Y Yδ σ= + =
0 1 0 0 1 1t t t t tY Y t Yδ ε δ ε ε ε− −= + + = + + + + + 20 0 0 0[ | ] , var[ | ]t tE Y Y t Y Y Y tδ σ= + =
ECON207 Session 6 Slide 24
When 0 0δ ≠ there is a linear deterministic trend
Even if 0 0δ = variance increases without bound
ECON207 Session 6 Slide 25
A random selection of random walks (without drift)
0
0 1 0 0 1 1
"drift" "stochastic trend" drift parmeter
t t t t tY Y t Y
δ
δ ε δ ε ε ε− −= + + = + + + + +
The RW is the simplest difference stationary series
Many others that include cycles and other features…
ECON207 Session 6 Slide 26
Trend Stationary Processes ( ) (stationary process)tY f t= +
E.g. 0 1t tY tβ β ε= + + “linear deterministic trend
E.g. 20 1 2t tY t tβ β β ε= + + + “linear deterministic quadratic trend”
Many more…
Application to log(SG_IP) (can estimate by OLS)
ECON207 Session 6 Slide 27
E.g. 20 0 1, , | | 1 , ~ (0, )
iid
t t t t t tY t u u uβ β ρ ε ρ ε σ−= + + = + <
E.g. Random Walk with Drift is not trend stationary
0
0 1 0 0 1 1
"drift" "stochastic trend" drift parmeter ~ not stationary
t t t t tY Y t Y
δ
δ ε δ ε ε ε− −= + + = + + + + +
ECON207 Session 6 Slide 28
OLS Estimation of Linear Regression Models with Time Series Data
t t tY ε′= +x β
Reminder: tx may include lags of regressors and lags of dependent variable
E.g. 0 1 2 1 1t t t p t p tY X X Xβ β β β ε− + −= + + + + +
11t t t t pY X X− −′ = x 0 1 2 1pβ β β β +′ = β
E.g. 0 1 1 2 3 1 2t t t t t t p t p tY Y X X Xε β β β β β ε− − + −′= + = + + + + + +x β
11t t t t pY X X− −′ = x 0 1 2 2pβ β β β +′ = β
ECON207 Session 6 Slide 29
Case 1 Time Series are Stationary Ergodic
B1 The stochastic processes in 1{ , }Tt t tY =x are stationary and
ergodic, and related according to t t tY ε′= +x β , where B2 1 1 2 2[ | , , , , ,...] 0t t t t t tE ε ε ε− − − − =x x x . This assumption implies all of the following:
B2a 1 2[ | , ,...] 0t t tE ε ε ε− − = , B2a [ ] 0tE ε = , B2b [ | ] 0t tE ε =x , (i.e., tε is contemporaneously exogenous) B2c [ ] 0t tE ε =x , B2d 1 1 2 2[ | , ,...] 0t t t t t tE ε ε ε− − − − =x x x , B2e 2var[ ] [ ]t t t t tEε ε ′=x x x .
B3 [ ]t tE ′ = xxx x Σ is finite and non-singular. B4 2[ ]t t tE ε ′ =x x S is positive definite. B5 2[( ) ]tk tjE X X is finite for all 1,...,t T= , , 1,...,j k K=
ECON207 Session 6 Slide 30
Remarks: tY is a martingale difference series (mds) if 1 2[ | , ,...] 0t t tE Y Y Y− − = for all t MDS are - zero-mean:
1 2[ ] [ [ | , ,...]] [0] 0t t t tE Y E E Y Y Y E− −= = = - serially uncorrelated:
1 1[ ] [ [ | ,...]] [ [ | ,...]] [ 0] 0t t k t t k t t k t t t kE Y Y E E Y Y Y E Y E Y Y E Y− − − − − −= = = = - 1 2var[ | , ,...]t t tY Y Y− − can depend on 1 2, ,...t tY Y− − Our assumptions imply tε is
- mds (therefore zero-mean, serially uncorrelated) - Can be conditionally heteroskedastic, but unconditionally homoskedastic
since { , }t tY x are stationary
ECON207 Session 6 Slide 31
Implicit in our assumption that tε is serially uncorrelated:
- We are considering dynamically complete models
- Enough lags of tY and regressors included in regression
E.g. (*) 20 1 1, , | | 1, ~ (0, )t t t t t t t iidY X u u uβ β ρ ε ρ ε σ−= + + = + <
Not dynamically complete
(*) ⇒ 21 0 1 1(1 ) ( ) , ~ (0, )t t t t t t iidY Y X Xρ β ρ β ρ ε ε σ− −− = − + − +
⇒ 20 1 1 1 2 1 , ~ (0, )t t t t t t iidY Y X Xα α α α ε ε σ− −= + + + + (**)
(**) is dynamically complete specification
ECON207 Session 6 Slide 32
Consistency
Relies on following extension to LLN
(LLN) If tz is stationary and ergodic with [ ]tE = < ∞z μ , then
11 T
t ptT =→∑ z μ.
Note that serial correlation in tz is allowed.
Asymptotic Normality
Relies on following extension to CLT
(CLT) If tz is a stationary and ergodic mds with [ ]t tE ′ =z z Σ, then
11 ( , )T
t dt NT =
→∑ z 0 Σ .
ECON207 Session 6 Slide 33
Proof of consistency of OLS estimator (B1-B3)
( )( )
1
1 1
11
1 1 1 1
ˆ
1 1
T Tt t t tt t
T T T Tt t t t t t t tt t t t
Y
T Tε ε
−
= =
−−
= = = =
′=
′ ′= + = +
∑ ∑
∑ ∑ ∑ ∑
β x x x
β x x x β x x x
Assumptions (and LLN) gives us 1
11
1 Tt t ptT
−−
= ′ → ∑ xxx x Σ
11 T
t t ptTε
=→∑ x 0
1
1
1 11 1ˆ
pp
T Tt t t t pt tT T
ε
−
−
= =
→→
′= + → ∑ ∑
xx0Σ
β β x x x β
Impt: note that serial non-correlation (though assumed) is not required here.
ECON207 Session 6 Slide 34
Proof of Asymptotic Normality of OLS Estimators (B1-B4) 1 1ˆ( ) ( , )dT N − −− → xx xxβ β 0 Σ SΣ
where 2[ ]t t tE ε ′ =x x S (var matrix of t tεx )
From 1
1 11 1ˆ T T
t t t tt tT Tε
−
= = ′= + ∑ ∑β β x x x
we have 1
11ˆ( ) T
t ttT T
Tε
−
= ′− = ∑β β x x x where
1(1/ ) T
t ttTε ε
== ∑x x
1
11 1
1( , )
1ˆ( ) ( , )d
p
Tt t dt
N
T T NT
ε
−
−− −
=→
→
′− = → ∑
xx
xx xx0 S
Σ
β β x x x 0 Σ SΣ
where we use the fact that t tεx is a stationary ergodic mds with var[ ]t tε =x S
1 11ˆvar[ ]T
− −≈ xx xxβ Σ SΣ
ECON207 Session 6 Slide 35
How to estimate 1 11ˆvar[ ]T
− −≈ xx xxβ Σ SΣ ?
- 1
(1/ ) Tt t pt
T=
′ →∑ xxx x Σ Use 1
11
1 Tt ttT
−−
= ′= ∑xxΣ x x
- If B5 also holds, then 21
1ˆ ˆTt t t ptTε
=′= →∑S x x S (proof omitted)
Therefore, use
1 11ˆ ˆˆ ˆvar[ ] T− −= xx xxβ Σ SΣ
ECON207 Session 6 Slide 36
If not working with dynamically complete models, then may have serial correlation in
tε , t tεx
OLS estimators still consistent, but 1 11ˆ ˆˆ ˆvar[ ] T− −= xx xxβ Σ SΣ not appropriate
Must incorporate serial correlation
- Use “HAC” estimator for the var matrix
- One version:
1 2 1111 1 1
**
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆvar[ ] [1 ] ( )T q Tvt t t t t t v t v t v t v t tT qt v t vε ε ε ε ε− −
− − − −+= = = +
′ ′ ′= + − + ∑ ∑ ∑xx xxβ Σ x x x x x x Σ
ECON207 Session 6 Slide 37
Simulation Example
10.8 0.8 , ~ (0,1)iid
t t t tX X Nε ε−= + +
10.8 , 0.95 , ~ (0,1)iid
t t t t t tY u u u v v N−= + = +
Regress 0 1t t tY X wβ β= + +
Variable est. std. err. t-stat. p. val. OLS (Intercept) 2.830 0.505 5.602 0.000 *** X -0.382 0.131 -2.910 0.0045 *** HC (Intercept) 2.830 0.484 5.842 0.0000 *** X -0.382 0.122 -3129 0.0023 *** HAC (Intercept) 2.830 0.959 2.951 0.0040 *** X -0.382 0.237 -1.609 0.1109
ECON207 Session 6 Slide 38
Suppose (with same data set) we take a dynamically complete approach
Suppose we try 0 1 1 2 3 1t t t t tY Y X X uα α α α− −= + + + +
Residual
ACF
Variable est. std. err. t-stat. p. val. (Intercept) 0.237 0.309 0.767 0.445
tX 0.017 0.094 0.184 0.855
1tY − 0.893 0.051 17.549 0.000 ***
1tX − -0.051 0.094 -0.545 0.587
ECON207 Session 6 Slide 39
Alternative to OLS – Generalized Least Squares
Suppose SLR with zero-mean stat. AR(1) errors
0 1t t tY Xβ β ε= + + , 1t t tuε ρε −= + , | | 1ρ <
Try to get efficient estimates
- Modify regression so that errors are not serially uncorrelated
- Run OLS on modified regression
Given set-up, how? Do quasi-differencing
0
1 0 1 1 1
" "" " " "
(1 ) ( ) ( )tt t
t t t t t t
uY X
Y Y X Xα
ρ β ρ β ρ ε ρε− − −− = − + − + −
0 1t t tY X uα β= + + run OLS on this regression
GLS estimators 11 2
1
( )
( )
Tt tt
Ttt
X X Y
X Xβ =
=
−=
−
∑∑
, 0 1Y Xα β= −
, 0β = 0 / (1 )α ρ−
If suitable assumptions satisfied, get consistent and efficient estimators
ECON207 Session 6 Slide 40
Issues:
- Need to know form of the serial correlation
o Maybe not so bad – run OLS and check residual dynamics, maybe residual
acf may suggest a form
o Modification to original regression relies on form of dynamics (our
modification suitable for AR(1) errors only)
- Need to estimate parameters (in our example, ρ is actually not known)
o Maybe not so bad – run OLS, and use residuals to estimate ρ
o Do quasi-differencing with ρ̂ instead of ρ
ECON207 Session 6 Slide 41
- Run OLS, get 0 1ˆ ˆ
t̂ t tY Xε β β= − −
- Regress t̂ε on 1t̂ε − , get 122
12
ˆ ˆˆ ( 0.938 in our example)
ˆ
Tt tt
Ttt
ε ερ
ε−=
−=
= =∑∑
- Do quasi-differencing 1ˆt t tY Y Yρ −= − , 1ˆt t tX X Xρ −= − ,
- Regress tY on ˆ(1 )ρ− and tX
Estimated regression: ˆ 1.097 0.024
(0.786) (0.091)t tY X= +
Variable est. std. err. t-stat. p. val.
HAC (Intercept) 2.830 0.959 2.951 0.0040 ***
X -0.382 0.237 -1.609 0.1109
GLS ˆ1 ρ− 1.097 0.786 1.396 0.1658
*X 0.024 0.091 0.259 0.7961
ECON207 Session 6 Slide 42
What is the consistency requirement?
- We need contemporaneous exogeneity in modified regression, i.e.,
[ | ] 0t tE u X = i.e., 1 1[ | ] 0t t t tE X Xε ρε ρ− −− − =
- Need tε uncorrelated with tX and 1tX −
- Need 1tε − uncorrelated with tX and 1tX −
- Require at least 1 1[ | , , ] 0t t t tE X X Xε + − =
- In terms of original equation, contemporaneous exogeneity not sufficient
A fourth approach? Since 0 1 1 1 1(1 )t t t t tY Y X X uβ ρ ρ β ρβ− −= − + + − + ,
Choose 0 1, ,β ρ β to minimize 20 1 1 1 1
2( (1 ) )
T
t t t tt
Y Y X Xβ ρ ρ β ρβ− −=
− − − − +∑
ECON207 Session 6 Slide 43
Essentially the same as FGLS
Case No. 2
Regression with Trend Stationary Variables
Suppose
- tY and tX are trending linearly, but are trend stationary
- 0 1t t tY Xβ β ε= + + will always produce significant 1β
- Omitted variable problem
Solution
- Include time trend
0 1 2t t tY t Xβ β β ε= + + +
ECON207 Session 6 Slide 44
E.g. Regression on Seasonal Data
- Suppose tY and tX both seasonal
- Regression of tY on tX will produce strong relationship because both seasonal
- Regression simply catches that fact
- Omitted variable problem
Solution 1
- Include Seasonal Dummies
e.g. For monthly data
1 2 , 3 , 12 , 1t feb t mar t dec t t tY d d d Xβ β β β α ε= + + + + + +
Solution 2 - Use seasonally adjusted data
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan 1983 1 0 0 0 0 0 0 0 0 0 0 0 Feb 1983 0 1 0 0 0 0 0 0 0 0 0 0 Mar 1983 0 0 1 0 0 0 0 0 0 0 0 0 Apr 1983 0 0 0 1 0 0 0 0 0 0 0 0 May 1983 0 0 0 0 1 0 0 0 0 0 0 0 Jun 1983 0 0 0 0 0 1 0 0 0 0 0 0 Jul 1983 0 0 0 0 0 0 1 0 0 0 0 0 Aug 1983 0 0 0 0 0 0 0 1 0 0 0 0 Sep 1983 0 0 0 0 0 0 0 0 1 0 0 0 Oct 1983 0 0 0 0 0 0 0 0 0 1 0 0 Nov 1983 0 0 0 0 0 0 0 0 0 0 1 0 Dec 1983 0 0 0 0 0 0 0 0 0 0 0 1 Jan 1984 1 0 0 0 0 0 0 0 0 0 0 0 Feb 1984 0 1 0 0 0 0 0 0 0 0 0 0 Mar 1984 0 0 1 0 0 0 0 0 0 0 0 0 Apr 1984 0 0 0 1 0 0 0 0 0 0 0 0 May 1984 0 0 0 0 1 0 0 0 0 0 0 0 Jun 1984 0 0 0 0 0 1 0 0 0 0 0 0
ECON207 Session 6 Slide 45
Case No. 3 Regression on Difference-Stationary Data
To highlight issues, we explore two special cases
E.g. A Suppose tY and tX are independent random walks…
1 1
1 2
t t t
t t t
Y YX X
εε
−
−
= +
= +
Regress 0 1t t tY Xβ β ε= + + . What happens?
Simulation experiment on 200 pairs of independent random walks…
ECON207 Session 6 Slide 46
E.g. B Suppose tY and tX are I(1), but tY is I(1) because it is related to tX
1
1 2
0.3 0.9t t t
t t t
Y XX X
εε−
= + +
= +
Regress 0 1t t tY Xβ β ε= + + . What happens?
“Superconsistent”
ECON207 Session 6 Slide 47
Example B is example of cointegration
tY and tX are I(1) variables (variances increase without bound)
Yet they ‘stay together’
1(0.3 0.9 ) ~ stationary "I(0)"t t tY X ε− + =
In Example A, tY and tX are I(1) variables but not cointegrated
For all 0 1,β β :
0 1 1 0 1 1 1 1 2( ) ( ) ( )t t t t t tY X Y Xβ β β β ε β ε− −− − = − − + −
0 1( )t tY Xβ β− + is I(1), i.e., tY and 0 1 tXβ β+ wander far from each other
In regression on I(1) variables,
- estimates are superconsistent in cointegration cases
- tend to be spurious in non-cointegrated cases
ECON207 Session 6 Slide 48
Questions:
1. Difference Stationary or Trend Stationary?
2. If variables are difference stationary, Spurious or Superconsistent?
For Question 1 – do a “unit root test”
- There are many many unit root tests
- We will use the “Phillips-Perron” test for this course
ECON207 Session 6 Slide 49
Phillips-Perron Test
Regress either
a. 1t t tY Y uρ −∆ = + use if tY is known to be zero-mean
b. 1t t tY Y uα ρ −∆ = + +
c. 1t t tY t Y uα δ ρ −∆ = + + + use if tY has a clear trend
Test 0 : 0H ρ = vs : 0AH ρ < (left tailed test)
- If 0ρ = , tY is difference stationary.
Issues: tu may be serially correlated
t-statistic does not have t-distribution
Phillips-Perron test uses the appropriate critical values (programmed into eviews, etc.)
ECON207 Session 6 Slide 50
Example Consider SG (log) consumption and (log) GDP, 1975Q1 to 2011Q2.
Both are difference-stationary
Null Hypothesis: LCONS has a unit root Exogenous: Constant, Linear Trend Bandwidth: 0 (Newey-West automatic) using Bartlett kernel
Adj. t-Stat Prob.* Phillips-Perron test statistic -1.799336 0.7002
Test critical values: 1% level -4.022586 5% level -3.441111 10% level -3.145082 *MacKinnon (1996) one-sided p-values.
Null Hypothesis: LGDP has a unit root Exogenous: Constant, Linear Trend Bandwidth: 3 (Newey-West automatic) using Bartlett kernel
Adj. t-Stat Prob.* Phillips-Perron test statistic -1.842081 0.6791
Test critical values: 1% level -4.022586 5% level -3.441111 10% level -3.145082 *MacKinnon (1996) one-sided p-values.
ECON207 Session 6 Slide 51
Question 2 Assuming difference-stationary variables
Is regression spurious?
Cointegrated case: there is 0 1,β β st 0 1 1( ) ~ stationary "I(0)"t t tY Xβ β ε− + =
Non-cointegrated case: there is no 0 1,β β st 0 1 1( ) ~ stationary "I(0)"t t tY Xβ β ε− + =
Solution: check if residuals are I(1)
Example Consumption function 0 1t t tC Yβ β ε= + +
Using Singapore data from 1975Q1 to 2011Q2 ˆ 0.753 0.843
(0.059) (0.006)tt YC = +
t-stat on tY is 143.623, 2 0.993R =
Spurious?
ECON207 Session 6 Slide 52
- Cannot reject unit root in residuals
- Based on this test, spurious
regression
Cointegration Test - Phillips-Ouliaris Date: 07/09/21 Time: 14:30 Equation: EQ1 Specification: LCONS LGDP C Cointegrating equation deterministics: C Null hypothesis: Series are not cointegrated Long-run variance estimate (Bartlett kernel, Newey-West fixed bandwidth = 5.0000) No d.f. adjustment for variances
Value Prob.*
Phillips-Ouliaris tau-statistic -2.277108 0.3880 Phillips-Ouliaris z-statistic -9.568306 0.3782
*MacKinnon (1996) p-values.