econometrics ii - vaasan yliopisto |lipas.uwasa.fi/~sjp/teaching/ecmii/lectures/ecmiic4.pdf · the...

Econometrics II

Seppo Pynnonen

Department of Mathematics and Statistics, University of Vaasa, Finland

Spring 2018

Seppo Pynnonen Econometrics II

Financial Time Series

Part IV


As of Feb 5, 2018Seppo Pynnonen Econometrics II


1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return

Three major ”Stylized Facts”

Basic Time Series Models

Wold Decomposition

Autoregressive Moving Average (ARMA) model

Autocorrelation

Partial Autocorrelation

Estimation of acf

Statistical inference

ARIMA-model

Random Walk

Martingale

The Law of Iterated Expectations

Unit root

Testing for unit root



Asset Returns1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Asset Returns

Simple return:

Rt =Pt + dt − Pt−1

Pt−1, (1)

where Pt is the price of an asset at time point t and dt is thedividend.

Gross return:

1 + Rt =Pt + dtPt−1

. (2)

In the following we assume that dividends are included in Pt .



Asset Returns

Multiperiod gross return:

1 + Rt [k] = PtPt−k

= PtPt−1 ×

Pt−1

Pt−2× · · · × Pt−k+1

Pt−k

= (1 + Rt)(1 + Rt−1) · · · (1 + Rt−k+1)

=∏k−1

j=0 (1 + Rt−j).

(3)

Annualized (p.a): Let k denote the return period measured inyears,

R(p.a) = (1 + Rt [k])1/k − 1 (4)

is the simple annualized return.




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Asset Returns

rt = log(Pt/Pt−1) = log(1 + Rt). (5)

rt [k] =k−1∑j=0

log(1 + Rt−j) =k−1∑j=0

rt−j (6)



Asset Returns

Log-returns are called continuously compounded returns.

In daily or higher frequency rt ≈ Rt .

Thus, does not make big difference which one is used.

Log-returns are preferred in research.

Remark 1

Simple returns are multiplicative, log returns are additive. For a

discussion, see Levy, et al. (2001) Management Science




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Asset Returns

Portfolio of n assets with weights w1, . . . ,wn, w1 + · · ·+ wn = 1.

Then

Rp,t =n∑

i=1

wiRit , (7)

where Rp,t is the portfolio return.

In terms of log-returns

rp,t ≈n∑

i=1

wi rit , (8)

where rp,t is the continuously compounded return of the portfolio.




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Asset Returns

r et = rt − r0t , (9)

where r et is the excess return and r0,t is typically the return of ariskless short-term asset, like three months government bond(loosely ”bank account”).

The riskless return is usually given in annual terms. Thus, it mustbe scaled to match the time period of the asset return rt .

r et ”retrun of a zero-investment porfolio”.



Three major ”Stylized Facts”1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





1. Return distribution is non-normal

- approximately symmetric- fat tails- high peak

2. Almost zero autocorrelation (daily)

3. Autocorrelated squared or absolute value returns




Example 1

Google’s weekly returns from Aug 2004 to Jan 2010

-20-10

010

Google's Weekly Returns [2004-2010]

2005 2006 2007 2008 2009 2010



Basic Time Series Models1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





Definition 1

Time series yt , t = 1, . . . ,T is covariance stationary if

E[yt ] = µ, for all t (10)

cov[yt , yt+k ] = γk , for all t (11)

var[yt ] = γ0 (<∞), for all t (12)

Series that are not stationary are called nonstationary.




Definition 2

Definition 2: Time series ut is a white noise process if

E[ut ] = µ, for all t

cov[ut , us ] = 0, for all t 6= s

var[ut ] = σ2u <∞, for all t.

(13)

We denote ut ∼ WN(µ, σ2u).

Remark 2

Usually it is assumed in (13) that µ = 0.

Remark 3

A WN-process is obviously stationary.




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





Theorem 2 (Wold Decomposition)

Any covariance stationary process yt , t = . . . ,−2,−1, 0, 1, 2, . . .can be written as

yt = µ+ ut + a1ut−1 + . . . = µ+∞∑h=0

ahut−h, (14)

where a0 = 1 and ut ∼ WN(0, σ2u), and∑∞

h=0 a2h <∞.




Definition 3

Lag polynomiala(L) = a0 + a1L + a2L

2 + · · ·, (15)

where L is the lag-operator such that

Lyt = yt−1. (16)

Definition 4

Difference operator∆yt = yt − yt−1. (17)




Thus, in terms of the lag polynomial, equation (14) can be writtenin short

yt = µ+ a(L)ut . (18)

Note that Lkyt = yt−k and ∆yt = (1− L)yt .




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





A covariance stationary process is an ARMA(p,q) process ofautoregressive order p and moving average order q if it can bewritten as

yt = φ0 + φ1yt−1 + · · ·+ φpyt−p

+ut − θ1ut−1 − · · · − θqut−q(19)




In terms of lag-polynomials

φ(L) = 1− φ1L− φ2L2 − · · · − φpLp (20)

θ(L) = 1− θ1L− θ2L2 − · · · − θqLq (21)

the ARMA(p,q) in (19) can be written shortly as

φ(L)yt = φ0 + θ(L)ut (22)

or

yt = µ+θ(L)

φ(L)ut , (23)

where

µ = E[yt ] =φ0

1− φ1 − · · · − φp. (24)




If q = 0 the process is called an AR(p)-process and if p = 0 theprocess is called an MA(q)-process.

Example 3

AR(1)-processyt = φ0 + φ1yt−1 + ut . (25)

An AR(1)-process is stationary if |φ1| < 1.

Below is a sample path for an AR(1)-process with T = 100 observations

for φ0 = 2, φ1 = 0.7, and ut ∼ NID(0, σ2u) with σ2

u = 4 (i.e., standard

deviation σu = 2).




-50

510

y

0 20 40 60 80 100time

Sample path of an AR(1)-priocess




A sample path for an MA(1)-process

yt = µ+ ut − θ1ut−1 (26)

with µ = 0.67 and θ1 = −0.7, and ut ∼ NID(0, 4).

-50

5y m

a1

0 20 40 60 80 100time

Sample path of an MA(1)-priocess




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





Autocorvariance Function

γk = cov[yt , yt−k ] = E[(yt − µ)(yt−k − µ)] (27)

k = 0, 1, 2, . . ..

Variance: γ0 = var[yt ].

Autocorrelation function

ρk =γkγ0. (28)

Autocovariances and autocorrelations are symmetric. That is,γk = γ−k and ρk = ρ−k .




For an AR(p)-process the autocorrelation function is of the form

ρk = φ1ρk−1 + φ2ρk−2 + · · ·+ φpρk−p. (29)

k > 0.

For an MA(q)-process the autocorrelation function is of the form

ρk =−θk + θ1θk−1 + · · ·+ θq−kθq

1 + θ21 + · · ·+ θ2q(30)

for k = 1, 2, . . . , q and ρk = 0 for k > q.




Example 4

For an AR(1) process yt = φ0 + φ1yt−1 + ut the autocorrelation functionis

ρk = φk1 . (31)

For an MA(1)-process yt = µ+ ut − θut−1 the autocorrelation function is

ρk =

−θ

1 + θ2, for k = 1

0, for k > 1

(32)




Typically the autocorrelation function is presented graphically bycorrelogram

0.00.2

0.40.6

0.81.0

Autocorrelation function of AR(1) with phi = 0.7

lag

rho

1 2 3 4 5 6

-1.0-0.5

0.00.5

1.0

Autocorrelation function of AR(1) with phi = -0.7

lag

rho

1 2 3 4 5 6




An AR(p)-process is stationary if the roots of the polynomial

φ(L) = 0 (33)

are outside the unit circle (be greater than 1 in absolute value).

Alternatively, if we consider the characteristic polynomial

mp − φ1mp−1 − · · · − φp = 0, (34)

then an AR(p)-process is stationary if the roots of thecharacteristic polynomial are inside the unit circle (be less than onein absolute value).




An MA-process is always stationary.

We say that and MA(q)-process is invertible if the roots of thecharacteristic polynomial

(35)

θ(L) = 1− θ1L− θ2L2 − · · · − θqLq = 0

lie outside the unit circle.

Invertibility means that an MA-process can be represented asinfinite AR-process.




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





Partial autocorrelation of a time series yt at lag k measures thecorrelation of yt and yt−k after adjusting yt for the effects ofyt−1, . . . , yt−k+1.

Partial autocorrelations are measured by φkk which is the lastcoefficient αk , in regression

yt = φ0k + φ1kyt−1 + · · ·+ φkkyt−k + vt (36)

Thus, denoting

yt = yt − (φ0k + φ1kyt−1 + · · ·+ φk−1,kyt−k+1)

then φkk = corr[yt , yt−k ].

For an AR(p)-process φkk = 0 for k > p.




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





Autocorrelation (and partial autocorrelation) functions areestimated by the empirical counterparts

γk =1

T

T−k∑t=1

(yt − y)(yt−k − y), (37)

where

y =1

T

T∑t=1

yt

is the sample mean.

Similarly

rk = ρk =γkγ0. (38)




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





It can be shown that if ρk = 0, then E[rk ] = 0 and asymptotically

var[rk ] ≈ 1

T. (39)

Similarly, if φkk = 0 then E[φkk

]= 0 and asymptotically

var[ρkk ] ≈ 1

T. (40)

In both cases the asymptotic distribution is normal.

Thus, testingH0 : ρk = 0, (41)

can be tested with the test statistic

z =√Trk , (42)

which is asymptotically N(0, 1) distributed under the nullhypothesis (41).




’Portmanteau’ statistics to test the hypothesis

H0 : ρ1 = ρ2 = · · · = ρm = 0 (43)

is due to Box and Pierce (1970)

Q∗(m) = Tm∑

k=1

r2k , (44)

m = 1, 2, . . ., which is (asymptotically) χ2m-distributed under the

null-hypothesis that all the first autocorrelations up to order m arezero.

Mostly people use Ljung and Box (1978) modification that shouldfollow more closely the χ2

m distribution

Q(m) = T (T + 2)m∑

k=1

1

T − kr2k . (45)




On the basis of autocovariance function one can preliminary inferthe order of an ARMA-proces

Theoretically:

=======================================================

acf pacf

-------------------------------------------------------

AR(p) Tails off Cut off after p

MA(q) Cut off after q Tails off

ARMA(p,q) Tails off Tails off

=======================================================

acf = autocorrelation function

pacf = partial autocorrelation function




Other popular tools for detecting the order of the model areAkaike’s (1974) information criterion (AIC)

AIC(p, q) = log σ2u + 2(p + q)/T (46)

or Schwarz’s (1978) Bayesian information criterion (BIC)2

BIC(p, q) = log(σ2) + (p + q) log(T )/T . (49)

There are several other similar criteria, like Hannan and Quinn(HQ).

2More generally these criteria are of the form

AIC(m) = −2`(θm) + 2m (47)

andBIC(m) = −2`(θm) + log(T )m, (48)

where θm is the MLE of θm, a parameter with m components, `(θm) is thevalue of the log-likelihood at θm.




The best fitting model in terms of the chosen criterion is the onethat minimizes the criterion.

The criteria may end up with different orders of the model!




Example 5

Google weekly (adjusted) closing prices Aug 2004 – Jan 2017.

2006 2010 2014

200

400

600

800

Google prices

Time

Price

ObservedEWMA(0.05)

2006 2010 2014

−10

010

20

Google weekly returns

Time

Retu

rn (%

per

wee

k)

Google retuns

Return

Dens

ity

−10 0 10 20

0.00

0.04

0.08

0.12 Normal

Empirical

5 10 15 20 25 30 35

−0.2

−0.1

0.0

0.1

0.2

Lag

ACF

Google return autocorrelations




Autocorrelations (AC) and parial autocorrelations (PAC)

Google’s weekly returns 2004 - 2017

Included observations: 650

================================================

lag AC PAC Q-AC Q-PAC p(Q-AC) p(Q-PAC)

------------------------------------------------

1 -0.043 -0.043 1.198 1.198 0.274 0.274

2 0.071 0.069 4.478 4.322 0.107 0.115

3 0.016 0.022 4.653 4.647 0.199 0.200

4 0.037 0.034 5.557 5.407 0.235 0.248

5 -0.018 -0.018 5.767 5.614 0.330 0.346

6 0.056 0.049 7.801 7.209 0.253 0.302

7 -0.045 -0.040 9.119 8.255 0.244 0.311

8 0.011 0.000 9.196 8.255 0.326 0.409

9 -0.001 0.004 9.197 8.265 0.419 0.508

10 -0.031 -0.034 9.817 9.012 0.457 0.531

================================================

All autocorrelation and partial autocorrelation estimate virtually to zero

and none of the Q-statistics are significant.




============================

p q AIC BIC

----------------------------

0 0 13107.33* 13119.43*

1 0 13108.98 13127.13

2 0 13110.93 13135.13

0 1 13108.98 13127.13

0 2 13110.94 13135.14

1 1 13110.99 13135.19

2 1 13112.31 13142.56

============================

* = minimum

AIC BIC suggest also white noise.




Later we will find that autocorrelations of the squared returns will behighly significant, suggesting that there is still left time dependency intothe series.

The dependency, however, is nonlinear by nature.




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





Consider the process

ϕ(L)yt = θ(L)ut . (50)

If, say d , of the roots of the polynomial ϕ(L) = 0 are on the unitcircle and the rest outside the circle, then ϕ(L) is a nonstationaryautoregressive operator.




We can write then

φ(L)(1− L)d = φ(L)∆d = ϕ(L)

where φ(L) is a stationary autoregressive operator and

φ(L)∆dyt = θ(L)ut (51)

which is a stationary ARMA.

We say that yt follows and ARIMA(p,d ,q)-process.

A symptom of unit roots is that the autocorrelations do not tendto die out.




Example 6

Example 5: Autocorrelations of Google (log) price series

Included observations: 285

===========================================================

Autocorrelation Partial AC AC PAC Q-Stat Prob

===========================================================

.|******* .|******* 1 0.972 0.972 272.32 0.000

.|******* .|. 2 0.944 -0.020 530.12 0.000

.|******* *|. 3 0.912 -0.098 771.35 0.000

.|****** .|. 4 0.880 -0.007 996.74 0.000

.|****** .|. 5 0.850 0.022 1207.7 0.000

.|****** .|. 6 0.819 -0.023 1404.4 0.000

.|****** .|. 7 0.790 0.005 1588.2 0.000

.|****** .|. 8 0.763 0.013 1760.0 0.000

.|***** .|. 9 0.737 0.010 1920.9 0.000

.|***** .|. 10 0.716 0.072 2073.4 0.000

.|***** .|. 11 0.698 0.040 2218.7 0.000

.|***** *|. 12 0.676 -0.088 2355.7 0.000

===========================================================




Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root





We say that a process is a random walk (RW) if it is of the form

yt = µ+ yt−1 + ut , (52)

where µ is the expected change of the process (drift) series andut ∼ i.i.d(0, σ2u).

More general forms of RW assume that ut is independent process(variances can change) or just that ut is uncorrelated process(autocorrelations are zero).

Earlier random walk was considered as a useful model for shareprices.



Martingale1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Martingale

A stochastic process is called a martingale with respect toinformation It available at time point t if for all t ≤ s

E[ys |It ] = yt . (53)

That is, given information at time point t the best prediction for afuture value ys of the stochastic process is the last observed valueyt .

It is assumed that yt ∈ Is for all t ≤ s.

Martingale is considered as a useful model for the so called fairgame, in which the odds of winning (or loosing) for all participantsare the same.

Martingales constitute the basis for derivative pricing.



Martingale

Remark 4

Remark 4: For short the conditional expectation of the form in(53) is usually denoted as

Et [ys ] ≡ E[ys |It ]. (54)



Martingale1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Martingale

For any It ⊂ Is , where t ≤ s and for any random variable y

Et [Es [y ]] = Et [y ]. (55)

AlsoEs [Et [y ]] = Et [y ]. (56)



Unit root1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Unit root

Definition 5

Times series yt is said to be integrated of order 1, if it is of the form

(1− L)yt = δ + ψ(L)ut , (57)

denoted as yt ∼ I (1), where

ψ(L) = 1 + ψ1L + ψ2L2 + ψ3L

3 + · · · (58)

such that∑∞

j=1 |ψj | <∞, ψ(1) 6= 0, roots of ψ(z) = 0 are outside the

unit circle [or the polynomial (58) is of order zero], and ut is a white

noise series with mean zero and variance σ2u.



Unit root

Remark 5

If a time series process is of the form of the right hand side of(57), i.e.,

xt = δ + ψ(L)ut , (59)

where ψ(L) satisfies the conditions of Def 5, it can be shown thatxt is stationary. In such a case we denote xt ∼ I (0), i.e,integrated of order zero.

Accordingly a stationary process is an I (0) process.



Unit root

Remark 6

The assumption ψ(1) 6= 0 is important. It rules out for example the trendstationary series

yt = α + βt + ψ(L)ut . (60)

Because E[yt ] = α + βt, yt is nonstationary. However,

(1− L)yt = β + ψ(L)ut , (61)

whereψ(L) = (1− L)ψ(L). (62)

Now, although, (1− L)yt is stationary, however,

ψ(1) = (1− 1)ψ(1) = 0,

which does not satisfy the rule in Definition 5, and hence a trend

stationary series is not I (1).



Unit root1 Financial Time Series

Asset Returns

Simple returns

Log-returns

Portfolio returns:

Excess Return



Wold Decomposition


Autocorrelation


Estimation of acf


ARIMA-model

Random Walk

Martingale


Unit root




Unit root

Consider the general model

yt = α + βt + φyt−1 + ut , (63)

where ut is stationary.

If |φ| < 1 then the (63) is trend stationary.

If φ = 1 then yt is unit root process (i.e., I (1)) with trend (anddrift).

Thus, testing whether yt is a unit root process reduces to testingwhether φ = 1.



Unit root

Ordinary OLS approach does not work!

One of the most popular tests is the Augmented Dickey-Fuller(ADF). Other tests are e.g. Phillips-Perron and KPSS-test.

Dickey-Fuller regression

∆yt = µ+ βt + γyt−1 + ut , (64)

where γ = φ− 1.

The null hypothesis is: ”yt ∼ I (1)”, i.e.,

H0 : γ = 0. (65)

This is tested with the usual t-ratio.

t =γ

s.e.(γ). (66)



Unit root

However, under the null hypothesis (65) the distribution is not thestandard t-distribution.

Distributions fractiles are tabulated under various assumptions(whether the trend is present (β 6= 0) and/or the drift (α) ispresent.

In practice also AR-terms are added into the regression to makethe residual as white noise as possible.



Unit root

Elliot, Rosenberg and Stock (1996) Econometrica 64, 813–836,propose a modified version of ADF, where the series is firstde-trended before applying ADF by GLS estimated trend.

In Stata test results are produced at different lags in AR-terms.



Unit root

Example 7

Unit root in Google weekly prices

2006 2008 2010 2012 2014 2016

200

400

600

800

Google (adjusted) closing pricesAug 2004 − Jan 2017

Time

Price



Unit root

=====================

(a) No drift no trend

=====================

df1 <- ur.df(y = log(gw$aclose), lags = 10, select = "AIC") # by default no drift, no trend excluded

Value of test-statistic is: 2.0172

Critical values for test statistics:

1pct 5pct 10pct

tau1 -2.58 -1.95 -1.62

===================

(b) Drift, no trend

===================

df2 <- ur.df(y = log(gw$aclose), type = "drift", lags = 10, select = "AIC") # DF with drift

Value of test-statistic is: -1.4021 3.3159


1pct 5pct 10pct

tau2 -3.43 -2.86 -2.57

phi1 6.43 4.59 3.78

===================

(c) Drift and trend

===================

df3 <- ur.df(y = log(gw$aclose), type = "trend", lags = 10, select = "AIC") # DF with drift and trend

Value of test-statistic is: -2.7589 3.9588 3.8746


1pct 5pct 10pct

tau3 -3.96 -3.41 -3.12

phi2 6.09 4.68 4.03

phi3 8.27 6.25 5.34

The null hypothesis of unit root is not rejected.



Unit root

I(2)? Trend is not needed in ADF here.

======================

(a) No drift, no trend

======================

Value of test-statistic is: -16.9747


1pct 5pct 10pct

tau1 -2.58 -1.95 -1.62

=========================

(b) Drift, no trend

=========================

Value of test-statistic is: -17.1464 146.9996


1pct 5pct 10pct

tau2 -3.43 -2.86 -2.57

phi1 6.43 4.59 3.78

The unit root in log price changes, i.e., returns, is clearly rejected.

The graph below of supports the stationarity of the return (differences of

log price) series.



Unit root

2006 2008 2010 2012 2014 2016

−10

010

20

Google's weekly log returnsAug 2004 − Jan 2017

Time

Retu

rn

Thus, we can conclude that log(close) ∼ I (1).



Unit root

DF-GLS leads to the same conclusion.

====================================

(a) ERS with constant, no trend

====================================

Value of test-statistic is: 1.2058

Critical values of DF-GLS are:

1pct 5pct 10pct

critical values -2.57 -1.94 -1.62

=================================

(b) ERS with trend

================================

Value of test-statistic is: -1.2833

Critical values of DF-GLS are:

1pct 5pct 10pct

critical values -3.48 -2.89 -2.57

The overall conclusion is that Google’s price series is difference

stationary, i.e., I (1), and thus not trend stationary.


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