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3.Analysis of asset price dynamics
3.1Introduction
Price – continuous function yet sampled discretely (usually - equal
spacing).
Stochastic process – has a random component, hence cannot be
exactly predicted.
Sequence of random variables => time series
Basic notions from Statistics & Probability Theory:
- distribution function
- mean (expectation)
- variance / standard deviation
- (auto) correlation
- stationary process (mean & variance do not change)
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Statistical concepts 1
• Consider a random variable (or variate) X. Probability density
function f(x) defines the probability to find X between a and b:
Pr(a ≤ X ≤ b) =
b
 f ( x)dx
a
The probability density must satisfy the normalization condition
X max
 f ( x)dx = 1
X min
• Cumulative distribution bfunction:
Pr(X≤b) =  f ( x)dx

Obviously, Pr(X > b) = 1 – Pr(X ≤ b)
Statistical concepts 2
• Two characteristics are used to describe the most probable values
of random variables: (1) mean (or expectation), and (2) median.
Mean of X is the average of all possible values of X that are weighed
with the probability density f(x):
m = E[X] = ∫ x f(x) dx
Median of X is the value M for which
Pr(X > M) = Pr(X < M) = 0.5
• Variance, Var, and the standard deviation, σ, are the conventional
estimates of the deviations from the mean values of X
Var[X] ≡ σ2 = ∫(x – m)2 f(x) dx
Statistical concepts 3
Higher-order moments of the probability distributions are defined as
mn = E[Xn] = ∫ xn f(x) dx
According to this definition, mean is the first moment (m ≡ m1), and
variance can be expressed via the first two moments, σ2 = m2 – m2.
Two other important parameters, skewness S and kurtosis K, are related to
the third and fourth moments, respectively:
S = E[(x – m)3] / σ3 , K = E[(x – m)4] / σ4
Both S and K are dimensionless. Zero skewness implies that f(x) is
symmetrical around its mean value. The positive and negative values of
skewness indicate long positive tails and long negative tails, respectively.
Kurtosis characterizes the distribution peakedness. Kurtosis of the normal
distribution equals three. The excess kurtosis, Ke = K – 3, is often used as
a measure of deviation from the normal distribution.
Statistical concepts 4
• Joint distribution of two random variables
X and Y
b c
Pr(X ≤ b, Y ≤ c) =   h(x, y) dx dy
- -
h(x, y) is the joint density that satisfies the normalization
condition
 
  h(x, y) dx dy =1
- -
Two random variables are independent if their joint density
function is the product of the univariate density functions:
h(x, y) = f(x) g(y).
• Covariance between two variates provides a measure of their
simultaneous change. Consider two variates X and Y that have the
means mX and mY, respectively. Their covariance equals
Cov(x, y) = σXY = E[(x – mX)(y – mY)] = E[xy] – mX mY
Statistical concepts 5
Positive (negative) covariance between two variates implies that these variates
tend to change simultaneously in the same (opposite) direction.
•
•
•
•
Another popular measure of simultaneous change is correlation coefficient:
Corr(x, y) = Cov(x, y)/(σX σY); -1 ≤ Corr(x, y) ≤ 1
Autocovariance: γ(k, t) = E[y(t) – m)(y(t – k) – m)]
Autocorrelation function (ACF): ρ(k) = γ(k)/γ(0); ρ(0) = 1; |ρ(k)| < 1
Ljung-Box test
H0 hypothesis: ρ(1) = ρ(2) = … ρ(k) = 0; p-value.
In the general case with N variates X1, . . ., XN (where N > 2), correlations
among variates are described with the covariance matrix, which has the
following elements
Cov(xi, xj) = σij = E[(xi – mi)(xj – mj)]
Statistical concepts 6
• Uniform distribution has a constant value within the given interval [a,
b] and equals zero outside this interval
fU = 0, x < a and x > b
fU = 1/(b – a), a ≤ x ≤ b
mU = 0.5(a+b), σ2U = (b – a)2/12, SU = 0, KeU = –6/5
• Normal (Gaussian) distribution has the form
fN(x) = exp[–(x – m)2/2σ2]
It is often denoted N(m, σ). Skewness and excess kurtosis of the
normal distribution equal zero. The transform z = (x – m)/σ converts the
normal distribution into the standard normal distribution
fSN(x) = exp[–z2/2]
Statistical concepts 7
Estimation for a given data sample
Sample mean:
m=
1
𝑁
𝑁
𝑖=1 𝑥𝑖
Sample variance:
σ2
=
1
𝑁−1
𝑁
𝑖=1(𝑥𝑖 -
m)2
Sample standard error:
SE =
σ
𝑁
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3.Analysis of asset price dynamics
3.1Introduction (continued)
Time series analysis:
- ARMA model
- linear regression
- trends (deterministic vs stochastic)
- vector autoregressions /simultaneous equations
- cointegration
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3.Analysis of asset price dynamics
3.2 Autoregressive model AR(p)
Univariate time series y(t) observed at moments t = 0, 1, …, n;
y(tk) ≡ y(k) ≡ yk
y(t) = a1y(t-1) + a2y(t-2) + …+ apy(t-p) + ε(t), t > p (lag)
Random process ε(t) (noise, shock, innovation)
White noise:
E[ε(t)] = 0;
Lag operator: Lp = y(t-p);
E[ε2(t)] = 2;
E[ε(t) ε(s)] = 0, if t  s.
Ap(L) = 1 – a1L – a2L2 - … - apLp
AR(p): Ap(L)y(t) = ε(t),
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3.Analysis of asset price dynamics
3.2 Autoregressive model AR(p) (continued 1)
AR(1): y(t) = a1y(t-1) + ε(t),
t
y(t) =  a1i ε(t-i)
i 0
Mean-reverting process: shocks decay and process returns to its mean.
“Old” noise converges with time to zero when | a1| < 1
If a1= 1, AR(1) is the random walk (RW):
y(t) = y(t-1) + ε(t) => y(t) =
ε(t-i)
RW is not mean-reverting.
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3. Analysis of asset price dynamics
3.2 Autoregressive model AR(p) (continued 2)
The 1st difference of RW: x(t) = y(t) – y(t-1) = ε(t) => mean-reverting
Processes that must be differenced d times in order to exclude nontransitory noise shocks are named integrated of order d: I(d).
Unit roots exist for AR(p) when shocks are not transitory. Then
modulus of solutions to the characterisitc equation
1 – a1z – a2z2 - … - ap zp = 0
must be lower than 1 (inside unit circle):
y(t) = 0.5y(t-1) – 0.2y(t-2) => 1- 0.5z + 0.2z2 = 0;
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3. Analysis of asset price dynamics
3.2 Autoregressive model AR(p) (continued 3)
AR(p) with non-zero mean:
If E[y(t)] = m, RW: y(t) = c + a1y(t-1) + ε(t), c = m(1- a1)
AR(p): Ap(L)y(t) = c + ε(t), c = m(1- a1 - ... - ap)
Autocorrelation coefficients:
y(t) is covariance-stationary (or weakly stationary) if γ(k, t) = γ(k).
AR(1): ρ(1) = a1 , ρ(k) = a1ρ(k-1)
AR(2): ρ(1) = a1/(1 – a2), ρ(k) = a1ρ(k-1) + a2ρ(k-2), k ≥ 2
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3. Analysis of asset price dynamics
3.3 Moving average model MA(q)
y(t) = ε(t) + b1ε(t-1) + b2ε(t-2) + ... + bqε(t-q) = Bq(L) ε(t)
Bq(L) = 1 + b1L + b2L2 + … + bqLq
MA(1): y(t) = ε(t) + b1ε(t-1), ε(0) = 0;
MA(1) incorporates past like AR(): y(t)(1-b1L + b1L2-b1L3+ ...) = ε(t)
MA(1): ρ(1) = b1/( b12 + 1) , ρ(k>1) = 0
MA(q) is invertible if it can be transformed into AR().
In this case, all solutions to 1 + b1z + b2z2 + … + bq zq = 0 must be
outside unit circle. Hence MA(1) is invertible when |b1| < 1.
MA(q) with non-zero mean m: y(t) = c + Bp(L)ε(t), c = m
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3. Analysis of asset price dynamics
3.4 The ARMA(p, q) model
y(t) = a1y(t-1) + a2y(t-2) + …+ apy(t-p) +
ε(t) + b1ε(t-1) + b2ε(t-2) + ... + bqε(t-q)
Strict stationarity when higher moments do not depend on time.
Any MA(q) is covariance-stationary. AR(p) is covariance-stationary only if
the roots of its polynomial are outside the unit circle.
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3. Analysis of asset price dynamics
3.5 Linear regression
Empirical TS: yi = a + bxi + εi., i = 1, 2, .., N.
a – intercept; b – slope.
Estimator: y = A + Bx;
N
Residual: ei = yi - A - Bxi; RSS =
e
i 1
N
2
i
  ( y i  A  Bx i ) 2
i 1
N
MSE => OLS = min(RSS) => A = ym - Bxm ; B =
N
xm =
x /N
i 1
i
N
; ym =
y /N
i 1
i
Xi = xi – xm
N
X Y /X
i 1
i i
i 1
2
i
Yi = yi – ym
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3. Analysis of asset price dynamics
3.5 Linear regression (continued)
Assumptions:
1) E[εi] = 0; otherwise intercept is biased.
2) Var(εi) = σ2 = const ;
3) E[ε(t) ε(s)] = 0, if t  s.
4) Independent variable is deterministic.
Goodness of fit (coefficient of determination; R2)
N
R2 = 1 -
N
 e / Y
i 1
2
i
i 1
2
i
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3. Analysis of asset price dynamics
3.5 Linear regression (continued)
2.5
y(t)
2
1.5
y = 0.0958x + 0.3695
2
R = 0.9743
1
0.5
t
0
0
10
20
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3.Analysis of asset price dynamics
3.6 Multiple regression
yi = a + b1x1,i + b2x2,i +... + bKxK,i + εi
Additional assumption: no perfect collinearity, i.e. no Xi is a
linear combination of other Xi.
Overspecification => no bias in estimates of bi but overstates σ2
Underspecification => yields biased bi and understates σ2
N
Adjusted R2 = 1 -
e
i 1
2
i
N
/( N  K ) /  Yi 2 /( N  1)
i 1
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3. Analysis of asset price dynamics
3.7 Trends
Trends => non-stationary time series
Deterministic trend vs stochastic trend
AR(1): y(t) – m – ct = a1[y(t – 1) – m – c(t – 1)] + ε(t)
t
z(t) = y(t) – m – ct = a1
tz(0)
+  a 1 ε(t)
t -i
i 1
If |a1| < 1, shocks are transitory.
If a1=1, random walk with drift: y(t) = c + y(t – 1) + ε(t)
For m=0, deterministic trend: y(t) = at + ε(t)
stochastic trend: y(t) = a + y(t – 1) + ε(t)
May look similar for some time.
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3. Analysis of asset price dynamics
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y(t)
6
5
y(t)= y(t-1) + 0.1 + ε(t)
4
3
y(t) = 0.1t + ε(t)
2
1
t
0
0
10
20
30
40
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3. Analysis of asset price dynamics
3.8 Multivariate time series
A multivariate time series y(t) = (y1(t), y2(t),..., yn(t))' is a vector of n processes
Multivariate moving average models are rarely used. Therefore we focus on the
vector autoregressive model (VAR).
Bivariate VAR(1) process:
y1(t) = a10 + a11y1(t - 1) + a12y2(t - 1) + ε1(t)
y2(t) = a20 + a21y1(t - 1) + a22y2(t - 1) + ε2(t)
Matrix form:
y(t) = a0 + Ay(t - 1) + ε(t)
 a 11 a 12 

y(t) = (y1(t), y2(t))', a0 = (a10, a20)', ε(t) = (ε1(t), ε2(t))', A = 
 a 21 a 22 
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3. Analysis of asset price dynamics
3.9 Multivariate time series (continued)
Simultaneous dynamic models
y1(t) = a11y1(t - 1) + a12y2(t) + ε1(t)
y2(t) = a21y1(t) + a22y2(t - 1) + ε2(t)
can be transformed to VAR:
 y1 (t) 


 y 2 (t) 
 a 11 a 12 a 22   y1 (t - 1) 
-1
 a a a   y (t - 1)  + (1 - a12 a21)
 11 21 22   2

= (1 - a12 a21)-1
1 a 12    1 (t) 

 


a
1
 21    2 (t) 
Two covariance stationary processes are x(t) and y(t) are jointly
covariance-stationary if Cov(x(t), y(t – s)) depends on lag s only.
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