Matlab Basics FIN250f: Lecture 3 Spring 2010 Grifths Web Notes.
Time Series Basics Fin250f: Lecture 3.1 Fall 2005 Reading: Taylor, chapter 3.1-3.3.
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Transcript of Time Series Basics Fin250f: Lecture 3.1 Fall 2005 Reading: Taylor, chapter 3.1-3.3.
Time Series Basics
Fin250f: Lecture 3.1
Fall 2005
Reading: Taylor, chapter 3.1-3.3
Outline
Random variablesDistributionsCentral limit theoremTwo variables IndependenceTime series definitions
Random Variables: Discrete
∑ −=
∑=
=∑
==
=
=
=
N
iii
N
iii
N
ii
ii
ii
XExpXVar
xpXE
p
xXp
px
1
2
1
1
))(()(
)(
1
)Pr(
,
Random Variables: Continuous
∫ =
≥
=
=∞=−∞≤=
∞
∞−1)(
0)(
)()(
1)(,0)(
)Pr()(
dxxf
xfdxxdF
xf
FF
xXxF
Random Variables: Continuous
∫ −=−=
=∫ −=
∫=
=∫=
∞
∞−
∞
∞−
∞
∞−
∞
∞−
dxxfxXEm
dxxfxXVar
dxxfxgXgE
dxxxfXE
nn
n )()()(
)()()(
)()())((
)()(
22
μμ
σμ
μ
Random Variables: Continuous
44
224
5.1
2
3
)()(
)(
σσmmXkurtosis
mmXskewness
==
=
Important Distributions
UniformNormalLog normalStudent-tStable
Normal/Gaussian
€
f (x :μ ,σ ) =1
σ 2πexp −
12
x −μσ
⎛ ⎝ ⎜
⎞ ⎠ ⎟2 ⎛
⎝ ⎜
⎞
⎠ ⎟
X ~ N(μ ,σ )
Normal Picture: Sample = 2000
Normal Exponential Expectations
€
X ~ N (μ ,σ 2 )
E(eαX ) = exp(αμ +12α 2σ 2 )
Why Important in Finance?
Central limit theoremMany returns almost normal
Log Normal
( )
∑ +=∑=
−==
==
T
tt
T
ttT
TT
TT
rrW
PPW
PPW
NX
1
'
1
0
0
)1log()log(
log)log()log(
/
),(~)log( σμ
Log Normal
Not symmetric Long right tail
Log Normal Histogram (Sample = 5000)
Chi-square
)(~
),(~
2
1
2
2
nY
XY
NX
n
i
i
χσμ
σμ
∑ ⎟⎠⎞
⎜⎝⎛ −=
=
Student-t
rVW
T
rV
NW
/
)(~
)1,0(~2
=
χ
Student-t Moments
All moments > r do not exist
rmXE m ≥∞=)(
Stable Distribution
Similar shape to normal Infinite varianceSums of stable RV’s are stable
Central Limit Theorem (casual)
),(~
nlargeFor
)Var(Z
tindependen variable,randomany
2
1
i
σμNU
ZU
Z
n
ii
i
∑=
∞<
=
Consequence of CLT and continuous compounding
€
rt+1,2 = rt + rt+1
rt+1,k = rt+ii=0
k−1
∑
If var(rt ) < ∞
rt+1,k ~ N (μ ,σ 2 )
Two Variables
€
F(x, y) = P(X ≤ x,Y ≤ y)
f (x, y) =∂ 2F∂x∂y
f (x | y) =f (x, y)fX (x)
E(Y | x) = yf (y | x)dx−∞
∞
∫
E(Y | X)
More on Two Variables
€
cov(X,Y ) = E[(X − E(X))(Y − E(Y ))]
= E(XY ) − E(X)E(Y )
cor(X,Y ) =cov(X,Y )σ Xσ Y
= ρ
−1 ≤ ρ ≤ 1
More Two Variables
€
cov(a +bX,c + dY ) = bd cov(X,Y )
cor(a +bX,c + dY ) = cor(X,Y )
E(aX +bY ) = aE(X) +bE(Y )
var(aX +bY ) = a2 var(X) +b2 var(Y ) + 2abcov(X,Y )
Independent Random Variables
€
f (x, y) = fX (x) fY (y)
f (y | x) = fY (y)
E(Y | x) = E(Y )
E(YX) = E(X)E(Y )
cov(Y ,X) = cor(Y ,X) = 0
More than Two RV’s
€
F(y1, y2 , y3 K ) = Pr(Y1 ≤ y1,Y2 ≤ y2 ,Y3 ≤ y3,K )
E(a + biYi )i=1
n
∑ = a + biE(Yi )i=1
n
∑
var(a + biYi )i=1
n
∑ = bi
2
i=1
n
∑ var(Yi ) + 2bibj cov(Yi ,Y j )j=i+1
n
∑i=1
n−1
∑
Multivariate Normal
€
Y ~ N (μ ,Ω)
μ i = E(Yi )
Ωi , j = cov(Yi ,Y j )
Independence
€
E(YiY j ) = E(Yi )E(Y j )
f (y1, y2 , y3,K ) = f (y1) f (y2 ) f (y3 )K
f (y1 | y2 , y3 K ) = fY1(y1)
Independent Identically Distributed
All random variables drawn from same distribution
All are independent of each otherCommon assumption IID IID Gaussian
Stochastic Processes
€
Xt
(x1, x2 , x3,K xT )
E(Xt | xt−1, xt−2 ,K ) = E(Xt | t −1)
Time Series Definitions
Strictly stationaryCovariance stationaryUncorrelated White noiseRandom walkMartingale
Strictly Stationary
All distributional features are independent of time
€
F(xt , xt−1,K xt−m )
Covariance Stationary
Variances and covariances independent of time
€
cov(Xt ,Xt− j )
var(Xt )
Uncorrelated
€
cor(Xt ,Xt− j ) = cov(Xt ,Xt− j ) = 0
White Noise
Covariance stationaryUncorrelatedMean zero
Random Walk
€
pt = pt−1 +etet IID
Geometric Random Walk
€
log(pt ) = log(pt−1) +etet IID
Martingale
€
E(Pt+1 | t) = Pt
Autocovariances/correlations
€
ρ j = cor(Xt ,Xt− j ) =cov(Xt ,Xt− j )
σ X2
Uncorrelated : ρ j = 0 j > 0
Outline
Random variablesDistributionsCentral limit theoremTwo variables IndependenceTime series definitions