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Econometrics II
Seppo Pynnonen
Department of Mathematics and Statistics, University of Vaasa, Finland
Spring 2018
Seppo Pynnonen Econometrics II
Financial Time Series
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
Seppo Pynnonen Econometrics II
Financial Time Series
Asset Returns1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Asset Returns1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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 .
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Asset Returns1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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)
Seppo Pynnonen Econometrics II
Financial Time Series
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
Seppo Pynnonen Econometrics II
Financial Time Series
Asset Returns1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Asset Returns1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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”.
Seppo Pynnonen Econometrics II
Financial Time Series
Three major ”Stylized Facts”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
Seppo Pynnonen Econometrics II
Financial Time Series
Three major ”Stylized Facts”
1. Return distribution is non-normal
- approximately symmetric- fat tails- high peak
2. Almost zero autocorrelation (daily)
3. Autocorrelated squared or absolute value returns
Seppo Pynnonen Econometrics II
Financial Time Series
Three major ”Stylized Facts”
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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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 <∞.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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)
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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 .
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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)
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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)
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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).
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
-50
510
y
0 20 40 60 80 100time
Sample path of an AR(1)-priocess
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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 .
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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)
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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).
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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)
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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).
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
’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)
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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!
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
============================
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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
===========================================================
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models1 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
Seppo Pynnonen Econometrics II
Financial Time Series
Basic Time Series Models
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Martingale1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
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)
Seppo Pynnonen Econometrics II
Financial Time Series
Martingale1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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)
Seppo Pynnonen Econometrics II
Financial Time Series
Unit root1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
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).
Seppo Pynnonen Econometrics II
Financial Time Series
Unit root1 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
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
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)
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
Financial Time Series
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
Seppo Pynnonen Econometrics II
Financial Time Series
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
Critical values for test statistics:
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
Critical values for test statistics:
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.
Seppo Pynnonen Econometrics II
Financial Time Series
Unit root
I(2)? Trend is not needed in ADF here.
======================
(a) No drift, no trend
======================
Value of test-statistic is: -16.9747
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62
=========================
(b) Drift, no trend
=========================
Value of test-statistic is: -17.1464 146.9996
Critical values for test statistics:
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.
Seppo Pynnonen Econometrics II
Financial Time 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).
Seppo Pynnonen Econometrics II
Financial Time Series
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.
Seppo Pynnonen Econometrics II
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