Stochastic Process, ACF, PACF, White Noise, Estimation
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Outline
1 §2.1: Stationarity
2 §2.2: Autocovariance and Autocorrelation Functions
3 §2.4: White Noise
4 R: Random Walk
5 Homework 1b
Arthur Berg Stationarity, ACF, White Noise, Estimation 2/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stochastic Process
Definition (stochastic process)
A stochastic process is sequence of indexed random variables denoted as Z(ω, t)where ω belongs to a sample space and t belongs to an index set.
From here on out, we will simply write a stochastic process (or time series) as{Zt} (dropping the ω).
NotationFor the time units t1, t2, . . . , tn, denote the n-dimensional distribution function ofZt1 ,Zt2 , . . . ,Ztn as
FZt1 ,...,Ztn(x1, . . . , xn) = P (Zt1 ≤ x1, . . . ,Ztn ≤ xn)
Example
Let Z1, . . . ,Zniid∼ N (0, 1). Then FZt1 ,...,Ztn
(x1, . . . , xn) =∏n
j=1 Φ(xj).
Arthur Berg Stationarity, ACF, White Noise, Estimation 3/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stochastic Process
Definition (stochastic process)
A stochastic process is sequence of indexed random variables denoted as Z(ω, t)where ω belongs to a sample space and t belongs to an index set.
From here on out, we will simply write a stochastic process (or time series) as{Zt} (dropping the ω).
NotationFor the time units t1, t2, . . . , tn, denote the n-dimensional distribution function ofZt1 ,Zt2 , . . . ,Ztn as
FZt1 ,...,Ztn(x1, . . . , xn) = P (Zt1 ≤ x1, . . . ,Ztn ≤ xn)
Example
Let Z1, . . . ,Zniid∼ N (0, 1). Then FZt1 ,...,Ztn
(x1, . . . , xn) =∏n
j=1 Φ(xj).
Arthur Berg Stationarity, ACF, White Noise, Estimation 3/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stochastic Process
Definition (stochastic process)
A stochastic process is sequence of indexed random variables denoted as Z(ω, t)where ω belongs to a sample space and t belongs to an index set.
From here on out, we will simply write a stochastic process (or time series) as{Zt} (dropping the ω).
NotationFor the time units t1, t2, . . . , tn, denote the n-dimensional distribution function ofZt1 ,Zt2 , . . . ,Ztn as
FZt1 ,...,Ztn(x1, . . . , xn) = P (Zt1 ≤ x1, . . . ,Ztn ≤ xn)
Example
Let Z1, . . . ,Zniid∼ N (0, 1). Then FZt1 ,...,Ztn
(x1, . . . , xn) =∏n
j=1 Φ(xj).
Arthur Berg Stationarity, ACF, White Noise, Estimation 3/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stochastic Process
Definition (stochastic process)
A stochastic process is sequence of indexed random variables denoted as Z(ω, t)where ω belongs to a sample space and t belongs to an index set.
From here on out, we will simply write a stochastic process (or time series) as{Zt} (dropping the ω).
NotationFor the time units t1, t2, . . . , tn, denote the n-dimensional distribution function ofZt1 ,Zt2 , . . . ,Ztn as
FZt1 ,...,Ztn(x1, . . . , xn) = P (Zt1 ≤ x1, . . . ,Ztn ≤ xn)
Example
Let Z1, . . . ,Zniid∼ N (0, 1). Then FZt1 ,...,Ztn
(x1, . . . , xn) =∏n
j=1 Φ(xj).
Arthur Berg Stationarity, ACF, White Noise, Estimation 3/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stationarity
Definition (Strongly Stationarity (aka Stricktly, aka Completely))
A time series {xt} is strongly stationary if any collection
{xt1 , xt2 , . . . , xtn}
has the same joint distribution as the time shifted set
{xt1+h, xt2+h, . . . , xtn+h}
Strong stationarity implies the following:
All marginal distributions are equal, i.e. P(Zs ≤ c) = P(Zt ≤ c) for all s, t,and c. This is what is called “first-order stationary”.
The covariance of Zs and Zt (if exists) is shift-independent, i.e.E(Zs + h,Zt + h) = E(Zs,Zt) for any choice of h.
Strong stationarity typically assumes too much. This leads us to the weakerassumption of weak stationarity.
Arthur Berg Stationarity, ACF, White Noise, Estimation 4/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stationarity
Definition (Strongly Stationarity (aka Stricktly, aka Completely))
A time series {xt} is strongly stationary if any collection
{xt1 , xt2 , . . . , xtn}
has the same joint distribution as the time shifted set
{xt1+h, xt2+h, . . . , xtn+h}
Strong stationarity implies the following:
All marginal distributions are equal, i.e. P(Zs ≤ c) = P(Zt ≤ c) for all s, t,and c. This is what is called “first-order stationary”.
The covariance of Zs and Zt (if exists) is shift-independent, i.e.E(Zs + h,Zt + h) = E(Zs,Zt) for any choice of h.
Strong stationarity typically assumes too much. This leads us to the weakerassumption of weak stationarity.
Arthur Berg Stationarity, ACF, White Noise, Estimation 4/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stationarity
Definition (Strongly Stationarity (aka Stricktly, aka Completely))
A time series {xt} is strongly stationary if any collection
{xt1 , xt2 , . . . , xtn}
has the same joint distribution as the time shifted set
{xt1+h, xt2+h, . . . , xtn+h}
Strong stationarity implies the following:
All marginal distributions are equal, i.e. P(Zs ≤ c) = P(Zt ≤ c) for all s, t,and c. This is what is called “first-order stationary”.
The covariance of Zs and Zt (if exists) is shift-independent, i.e.E(Zs + h,Zt + h) = E(Zs,Zt) for any choice of h.
Strong stationarity typically assumes too much. This leads us to the weakerassumption of weak stationarity.
Arthur Berg Stationarity, ACF, White Noise, Estimation 4/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Stationarity
Definition (Strongly Stationarity (aka Stricktly, aka Completely))
A time series {xt} is strongly stationary if any collection
{xt1 , xt2 , . . . , xtn}
has the same joint distribution as the time shifted set
{xt1+h, xt2+h, . . . , xtn+h}
Strong stationarity implies the following:
All marginal distributions are equal, i.e. P(Zs ≤ c) = P(Zt ≤ c) for all s, t,and c. This is what is called “first-order stationary”.
The covariance of Zs and Zt (if exists) is shift-independent, i.e.E(Zs + h,Zt + h) = E(Zs,Zt) for any choice of h.
Strong stationarity typically assumes too much. This leads us to the weakerassumption of weak stationarity.
Arthur Berg Stationarity, ACF, White Noise, Estimation 4/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Mean Function
Definition (Mean Function)
The mean function of a time series {Zt} (if it exists) is given by
µt = E(Zt) =∫ ∞−∞
x ft(x) dx
Example (Mean of an iid MA(q))
Let atiid∼ N (0, σ2) and
Zt = at + θ1at−1 + θ2at−2 + · · ·+ θqat−q
then
µt = E(Zt) = E(at) + θ1 E(at−1) + θ2 E(at−2) + · · ·+ θq E(at−q) ≡ 0
(free of the time variable t)
Arthur Berg Stationarity, ACF, White Noise, Estimation 5/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Mean Function
Definition (Mean Function)
The mean function of a time series {Zt} (if it exists) is given by
µt = E(Zt) =∫ ∞−∞
x ft(x) dx
Example (Mean of an iid MA(q))
Let atiid∼ N (0, σ2) and
Zt = at + θ1at−1 + θ2at−2 + · · ·+ θqat−q
then
µt = E(Zt) = E(at) + θ1 E(at−1) + θ2 E(at−2) + · · ·+ θq E(at−q) ≡ 0
(free of the time variable t)
Arthur Berg Stationarity, ACF, White Noise, Estimation 5/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Mean Function
Definition (Mean Function)
The mean function of a time series {Zt} (if it exists) is given by
µt = E(Zt) =∫ ∞−∞
x ft(x) dx
Example (Mean of an iid MA(q))
Let atiid∼ N (0, σ2) and
Zt = at + θ1at−1 + θ2at−2 + · · ·+ θqat−q
then
µt = E(Zt) = E(at) + θ1 E(at−1) + θ2 E(at−2) + · · ·+ θq E(at−q) ≡ 0
(free of the time variable t)
Arthur Berg Stationarity, ACF, White Noise, Estimation 5/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
White Noise
Definition (White Noise)White noise is a collection of uncorrelated random variables with constant meanand variance.
Notationat ∼WN(0, σ2) — white noise with mean zero and variance σ2
IID WNIf as is independent of at for all s 6= t, then wt ∼ IID(0, σ2)Gaussian White Noise ⇒ IIDSuppose at is normally distributed.uncorrelated + normality⇒ independentThus it follows that at ∼ IID(0, σ2) (a stronger assumption).
Arthur Berg Stationarity, ACF, White Noise, Estimation 6/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
White Noise
Definition (White Noise)White noise is a collection of uncorrelated random variables with constant meanand variance.
Notationat ∼WN(0, σ2) — white noise with mean zero and variance σ2
IID WNIf as is independent of at for all s 6= t, then wt ∼ IID(0, σ2)Gaussian White Noise ⇒ IIDSuppose at is normally distributed.uncorrelated + normality⇒ independentThus it follows that at ∼ IID(0, σ2) (a stronger assumption).
Arthur Berg Stationarity, ACF, White Noise, Estimation 6/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
White Noise
Definition (White Noise)White noise is a collection of uncorrelated random variables with constant meanand variance.
Notationat ∼WN(0, σ2) — white noise with mean zero and variance σ2
IID WNIf as is independent of at for all s 6= t, then wt ∼ IID(0, σ2)Gaussian White Noise ⇒ IIDSuppose at is normally distributed.uncorrelated + normality⇒ independentThus it follows that at ∼ IID(0, σ2) (a stronger assumption).
Arthur Berg Stationarity, ACF, White Noise, Estimation 6/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
White Noise
Definition (White Noise)White noise is a collection of uncorrelated random variables with constant meanand variance.
Notationat ∼WN(0, σ2) — white noise with mean zero and variance σ2
IID WNIf as is independent of at for all s 6= t, then wt ∼ IID(0, σ2)Gaussian White Noise ⇒ IIDSuppose at is normally distributed.uncorrelated + normality⇒ independentThus it follows that at ∼ IID(0, σ2) (a stronger assumption).
Arthur Berg Stationarity, ACF, White Noise, Estimation 6/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Now your turn!
Mean of a Random Walk with DriftSuppose Z0 = 0, and for t > 0, Zt = δ + Zt−1 + at where δ is a constant andat ∼WN(0, σ2). What is the mean function of Zt? (That is compute E [Zt].)
Note thatZt = δ + Zt−1 + at
= 2δ + Zt−2 + at + at−1 (Zt−1 = δ + Zt−2 + at−1)...
= δt +t∑
j=1
aj
Therefore we see
µt = E [Zt] = E [δt] + E
t∑j=1
aj
= δt +t∑
j=1
E [aj] = δt
Arthur Berg Stationarity, ACF, White Noise, Estimation 7/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Now your turn!
Mean of a Random Walk with DriftSuppose Z0 = 0, and for t > 0, Zt = δ + Zt−1 + at where δ is a constant andat ∼WN(0, σ2). What is the mean function of Zt? (That is compute E [Zt].)
Note thatZt = δ + Zt−1 + at
= 2δ + Zt−2 + at + at−1 (Zt−1 = δ + Zt−2 + at−1)...
= δt +t∑
j=1
aj
Therefore we see
µt = E [Zt] = E [δt] + E
t∑j=1
aj
= δt +t∑
j=1
E [aj] = δt
Arthur Berg Stationarity, ACF, White Noise, Estimation 7/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
The Covariance Function
Definition (Covariance Function)
The covariance function of a random sequence {Zt} is
γ(s, t) = cov(Zs,Zt) = E [(Zs − µs) (Zt − µt)]
So in particular, we have
var(Zt) = cov(Zt,Zt) = γ(t, t) = E[(Zt − µt)2]
Also note that γ(s, t) = γ(t, s) since cov(Zs,Zt) = cov(Zt,Zs).
Example (Covariance Function of White Noise)
Let at ∼WN(0, σ2). By the definition of white noise, we have
γ(s, t) = E(as, at) =
{σ2, s = t0, s 6= t
Arthur Berg Stationarity, ACF, White Noise, Estimation 8/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
The Covariance Function
Definition (Covariance Function)
The covariance function of a random sequence {Zt} is
γ(s, t) = cov(Zs,Zt) = E [(Zs − µs) (Zt − µt)]
So in particular, we have
var(Zt) = cov(Zt,Zt) = γ(t, t) = E[(Zt − µt)2]
Also note that γ(s, t) = γ(t, s) since cov(Zs,Zt) = cov(Zt,Zs).
Example (Covariance Function of White Noise)
Let at ∼WN(0, σ2). By the definition of white noise, we have
γ(s, t) = E(as, at) =
{σ2, s = t0, s 6= t
Arthur Berg Stationarity, ACF, White Noise, Estimation 8/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
The Covariance Function
Definition (Covariance Function)
The covariance function of a random sequence {Zt} is
γ(s, t) = cov(Zs,Zt) = E [(Zs − µs) (Zt − µt)]
So in particular, we have
var(Zt) = cov(Zt,Zt) = γ(t, t) = E[(Zt − µt)2]
Also note that γ(s, t) = γ(t, s) since cov(Zs,Zt) = cov(Zt,Zs).
Example (Covariance Function of White Noise)
Let at ∼WN(0, σ2). By the definition of white noise, we have
γ(s, t) = E(as, at) =
{σ2, s = t0, s 6= t
Arthur Berg Stationarity, ACF, White Noise, Estimation 8/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Example (Covariance Function of MA(1))
Let at ∼WN(0, σ2) and Zt = at + θat−1 (where θ is a constant), then
γ(s, t) = cov(at + θat−1, as + θas−1)= cov(at, as) + θ cov(at, as−1)+
+ θ cov(at−1, as) + θ2 cov(at−1, as−1)If s = t, then
γ(s, t) = γ(t, t) = σ2 + θ2σ2 = (θ2 + 1)σ2
If s = t − 1 or s = t + 1, then
γ(s, t) = γ(t, t + 1) = γ(t, t − 1) = θσ2
So all together we have
γ(s, t) =
(θ2 + 1)σ2, ifs = tθσ2, if|s− t| = 10, else
Arthur Berg Stationarity, ACF, White Noise, Estimation 9/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Example (Covariance Function of MA(1))
Let at ∼WN(0, σ2) and Zt = at + θat−1 (where θ is a constant), then
γ(s, t) = cov(at + θat−1, as + θas−1)= cov (at, as) + θ cov (at, as−1)+
+ θ cov (at−1, as) + θ2 cov (at−1, as−1)If s = t, then
γ(s, t) = γ(t, t) = σ2 + θ2σ2 = (θ2 + 1)σ2
If s = t − 1 or s = t + 1, then
γ(s, t) = γ(t, t + 1) = γ(t, t − 1) = θσ2
So all together we have
γ(s, t) =
(θ2 + 1)σ2, ifs = tθσ2, if|s− t| = 10, else
Arthur Berg Stationarity, ACF, White Noise, Estimation 9/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Example (Covariance Function of MA(1))
Let at ∼WN(0, σ2) and Zt = at + θat−1 (where θ is a constant), then
γ(s, t) = cov(at + θat−1, as + θas−1)= cov (at, as) + θ cov (at, as−1)+
+ θ cov (at−1, as) + θ2 cov (at−1, as−1)If s = t, then
γ(s, t) = γ(t, t) = σ2 + θ2σ2 = (θ2 + 1)σ2
If s = t − 1 or s = t + 1, then
γ(s, t) = γ(t, t + 1) = γ(t, t − 1) = θσ2
So all together we have
γ(s, t) =
(θ2 + 1)σ2, ifs = tθσ2, if|s− t| = 10, else
Arthur Berg Stationarity, ACF, White Noise, Estimation 9/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Example (Covariance Function of MA(1))
Let at ∼WN(0, σ2) and Zt = at + θat−1 (where θ is a constant), then
γ(s, t) = cov(at + θat−1, as + θas−1)= cov (at, as) + θ cov (at, as−1)+
+ θ cov (at−1, as) + θ2 cov (at−1, as−1)If s = t, then
γ(s, t) = γ(t, t) = σ2 + θ2σ2 = (θ2 + 1)σ2
If s = t − 1 or s = t + 1, then
γ(s, t) = γ(t, t + 1) = γ(t, t − 1) = θσ2
So all together we have
γ(s, t) =
(θ2 + 1)σ2, ifs = tθσ2, if|s− t| = 10, else
Arthur Berg Stationarity, ACF, White Noise, Estimation 9/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Example (Covariance Function of MA(1))
Let at ∼WN(0, σ2) and Zt = at + θat−1 (where θ is a constant), then
γ(s, t) = cov(at + θat−1, as + θas−1)= cov (at, as) + θ cov (at, as−1)+
+ θ cov (at−1, as) + θ2 cov (at−1, as−1)If s = t, then
γ(s, t) = γ(t, t) = σ2 + θ2σ2 = (θ2 + 1)σ2
If s = t − 1 or s = t + 1, then
γ(s, t) = γ(t, t + 1) = γ(t, t − 1) = θσ2
So all together we have
γ(s, t) =
(θ2 + 1)σ2, ifs = tθσ2, if|s− t| = 10, else
Arthur Berg Stationarity, ACF, White Noise, Estimation 9/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Now your turn!
Covariance Function of a Random Walk with DriftSuppose Z0 = 0, and for t > 0, Zt = δ + Zt−1 + at where δ is a constant andat ∼WN(0, σ2). What is the covariance function of Zt? (That is computecov (Zs,Zt).)
From the representation
Zt = δt +t∑
j=1
aj
We see
γ(s, t) = cov(Zs,Zt) = cov
s∑j=1
aj,t∑
k=1
ak
=
s∑j=1
t∑k=1
cov (aj, ak) = min(s, t)σ2
Arthur Berg Stationarity, ACF, White Noise, Estimation 10/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Now your turn!
Covariance Function of a Random Walk with DriftSuppose Z0 = 0, and for t > 0, Zt = δ + Zt−1 + at where δ is a constant andat ∼WN(0, σ2). What is the covariance function of Zt? (That is computecov (Zs,Zt).)
From the representation
Zt = δt +t∑
j=1
aj
We see
γ(s, t) = cov(Zs,Zt) = cov
s∑j=1
aj,t∑
k=1
ak
=
s∑j=1
t∑k=1
cov (aj, ak) = min(s, t)σ2
Arthur Berg Stationarity, ACF, White Noise, Estimation 10/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
The Autocorrelation Function
Definition (Correlation Function)
The correlation function of a random sequence {Zt} is
ρ(s, t) =γ(s, t)√
γ(s, s)γ(t, t)
Cauchy-Schwartz inequality⇒ |ρ(s, t)| ≤ 1
Arthur Berg Stationarity, ACF, White Noise, Estimation 11/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
The Autocorrelation Function
Definition (Correlation Function)
The correlation function of a random sequence {Zt} is
ρ(s, t) =γ(s, t)√
γ(s, s)γ(t, t)
Cauchy-Schwartz inequality⇒ |ρ(s, t)| ≤ 1
Arthur Berg Stationarity, ACF, White Noise, Estimation 11/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Weak Stationarity
Definition (Weak Stationarity)A weakly stationary time series is a finite variance process that
(i) has a mean function, µ, that is constant (so it doesn’t depend on t);
(ii) has a covariance function, γ(s, t), that dependents on s and t only throughthe difference |s− t|.
From now on when we say stationary, we mean weakly stationary.
For stationary processes, the mean function is µ (no t).
Since γ(s, t) = γ(s + h, t + h) for stationary processes, we haveγ(s, t) = γ(s− t, 0).Therefore we can view the covariance function as a function of only onevariable (h = s− t); this is the autocovariance function γ(h).
Arthur Berg Stationarity, ACF, White Noise, Estimation 12/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Weak Stationarity
Definition (Weak Stationarity)A weakly stationary time series is a finite variance process that
(i) has a mean function, µ, that is constant (so it doesn’t depend on t);
(ii) has a covariance function, γ(s, t), that dependents on s and t only throughthe difference |s− t|.
From now on when we say stationary, we mean weakly stationary.
For stationary processes, the mean function is µ (no t).
Since γ(s, t) = γ(s + h, t + h) for stationary processes, we haveγ(s, t) = γ(s− t, 0).Therefore we can view the covariance function as a function of only onevariable (h = s− t); this is the autocovariance function γ(h).
Arthur Berg Stationarity, ACF, White Noise, Estimation 12/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Weak Stationarity
Definition (Weak Stationarity)A weakly stationary time series is a finite variance process that
(i) has a mean function, µ, that is constant (so it doesn’t depend on t);
(ii) has a covariance function, γ(s, t), that dependents on s and t only throughthe difference |s− t|.
From now on when we say stationary, we mean weakly stationary.
For stationary processes, the mean function is µ (no t).
Since γ(s, t) = γ(s + h, t + h) for stationary processes, we haveγ(s, t) = γ(s− t, 0).Therefore we can view the covariance function as a function of only onevariable (h = s− t); this is the autocovariance function γ(h).
Arthur Berg Stationarity, ACF, White Noise, Estimation 12/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Weak Stationarity
Definition (Weak Stationarity)A weakly stationary time series is a finite variance process that
(i) has a mean function, µ, that is constant (so it doesn’t depend on t);
(ii) has a covariance function, γ(s, t), that dependents on s and t only throughthe difference |s− t|.
From now on when we say stationary, we mean weakly stationary.
For stationary processes, the mean function is µ (no t).
Since γ(s, t) = γ(s + h, t + h) for stationary processes, we haveγ(s, t) = γ(s− t, 0).Therefore we can view the covariance function as a function of only onevariable (h = s− t); this is the autocovariance function γ(h).
Arthur Berg Stationarity, ACF, White Noise, Estimation 12/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Outline
1 §2.1: Stationarity
2 §2.2: Autocovariance and Autocorrelation Functions
3 §2.4: White Noise
4 R: Random Walk
5 Homework 1b
Arthur Berg Stationarity, ACF, White Noise, Estimation 13/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Autocovariance and Autocorrelation Functions
Definition (Autocovariance Function)The autocovariance function of a stationary time series is
γh = γ(h) = E [(Zt+h − µ) (Zt − µ)]
(for any value of t).
Definition (Autocorrelation Function)The autocorrelation function of a stationary time series is
ρh = ρ(h) =γ(h)γ(0)
Arthur Berg Stationarity, ACF, White Noise, Estimation 14/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Autocovariance and Autocorrelation Functions
Definition (Autocovariance Function)The autocovariance function of a stationary time series is
γh = γ(h) = E [(Zt+h − µ) (Zt − µ)]
(for any value of t).
Definition (Autocorrelation Function)The autocorrelation function of a stationary time series is
ρh = ρ(h) =γ(h)γ(0)
Arthur Berg Stationarity, ACF, White Noise, Estimation 14/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Properties of the Autocovariance/Autocorrelation Functionsγ0 = var(Zt); ρ0 = 1|γk| ≤ γ0; |ρk| ≤ 1γk = γ−k; ρk = ρ−k
γk and ρk are positive semidefinite (p.s.d.), i.e.n∑
i=1
n∑j=1
αiαjγ|ti−tj| ≥ 0
andn∑
i=1
n∑j=1
αiαjρ|ti−tj| ≥ 0
for any set of times t1, . . . , tn and any real numbers α1, . . . , αn.Proof.
Define X =∑n
i=1 αiZti , then
0 ≤ var(X) =n∑
i=1
n∑j=1
αiαj cov(Zti ,Ztj) =n∑
i=1
n∑j=1
αiαjγ|ti−tj|
Arthur Berg Stationarity, ACF, White Noise, Estimation 15/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Properties of the Autocovariance/Autocorrelation Functionsγ0 = var(Zt); ρ0 = 1|γk| ≤ γ0; |ρk| ≤ 1γk = γ−k; ρk = ρ−k
γk and ρk are positive semidefinite (p.s.d.), i.e.n∑
i=1
n∑j=1
αiαjγ|ti−tj| ≥ 0
andn∑
i=1
n∑j=1
αiαjρ|ti−tj| ≥ 0
for any set of times t1, . . . , tn and any real numbers α1, . . . , αn.Proof.
Define X =∑n
i=1 αiZti , then
0 ≤ var(X) =n∑
i=1
n∑j=1
αiαj cov(Zti ,Ztj) =n∑
i=1
n∑j=1
αiαjγ|ti−tj|
Arthur Berg Stationarity, ACF, White Noise, Estimation 15/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Properties of the Autocovariance/Autocorrelation Functionsγ0 = var(Zt); ρ0 = 1|γk| ≤ γ0; |ρk| ≤ 1γk = γ−k; ρk = ρ−k
γk and ρk are positive semidefinite (p.s.d.), i.e.n∑
i=1
n∑j=1
αiαjγ|ti−tj| ≥ 0
andn∑
i=1
n∑j=1
αiαjρ|ti−tj| ≥ 0
for any set of times t1, . . . , tn and any real numbers α1, . . . , αn.Proof.
Define X =∑n
i=1 αiZti , then
0 ≤ var(X) =n∑
i=1
n∑j=1
αiαj cov(Zti ,Ztj) =n∑
i=1
n∑j=1
αiαjγ|ti−tj|
Arthur Berg Stationarity, ACF, White Noise, Estimation 15/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Properties of the Autocovariance/Autocorrelation Functionsγ0 = var(Zt); ρ0 = 1|γk| ≤ γ0; |ρk| ≤ 1γk = γ−k; ρk = ρ−k
γk and ρk are positive semidefinite (p.s.d.), i.e.n∑
i=1
n∑j=1
αiαjγ|ti−tj| ≥ 0
andn∑
i=1
n∑j=1
αiαjρ|ti−tj| ≥ 0
for any set of times t1, . . . , tn and any real numbers α1, . . . , αn.Proof.
Define X =∑n
i=1 αiZti , then
0 ≤ var(X) =n∑
i=1
n∑j=1
αiαj cov(Zti ,Ztj) =n∑
i=1
n∑j=1
αiαjγ|ti−tj|
Arthur Berg Stationarity, ACF, White Noise, Estimation 15/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Properties of the Autocovariance/Autocorrelation Functionsγ0 = var(Zt); ρ0 = 1|γk| ≤ γ0; |ρk| ≤ 1γk = γ−k; ρk = ρ−k
γk and ρk are positive semidefinite (p.s.d.), i.e.n∑
i=1
n∑j=1
αiαjγ|ti−tj| ≥ 0
andn∑
i=1
n∑j=1
αiαjρ|ti−tj| ≥ 0
for any set of times t1, . . . , tn and any real numbers α1, . . . , αn.Proof.
Define X =∑n
i=1 αiZti , then
0 ≤ var(X) =n∑
i=1
n∑j=1
αiαj cov(Zti ,Ztj) =n∑
i=1
n∑j=1
αiαjγ|ti−tj|
Arthur Berg Stationarity, ACF, White Noise, Estimation 15/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Outline
1 §2.1: Stationarity
2 §2.2: Autocovariance and Autocorrelation Functions
3 §2.4: White Noise
4 R: Random Walk
5 Homework 1b
Arthur Berg Stationarity, ACF, White Noise, Estimation 16/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
White Noise
White noise, denoted by at ∼WN(0, σ2), is by definition a weakly stationaryprocess with autocovariance function
γk =
{σ2, k = 00, k 6= 0
and autocorrelation function
ρk =
{1, k = 00, k 6= 0
Not all white noise is boring like iid N (0, σ2).For example, stationary GARCH processes can have all the properties of whitenoise.
Arthur Berg Stationarity, ACF, White Noise, Estimation 17/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
White Noise
White noise, denoted by at ∼WN(0, σ2), is by definition a weakly stationaryprocess with autocovariance function
γk =
{σ2, k = 00, k 6= 0
and autocorrelation function
ρk =
{1, k = 00, k 6= 0
Not all white noise is boring like iid N (0, σ2).For example, stationary GARCH processes can have all the properties of whitenoise.
Arthur Berg Stationarity, ACF, White Noise, Estimation 17/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Outline
1 §2.1: Stationarity
2 §2.2: Autocovariance and Autocorrelation Functions
3 §2.4: White Noise
4 R: Random Walk
5 Homework 1b
Arthur Berg Stationarity, ACF, White Noise, Estimation 18/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Random Walk Simulation in R
Simulate the random walk Zt = .2 + Zt−1 + at where atiid∼ N (0, σ2).
Hint: use the representation Zt = .2t +∑t
j=1 aj!
> w = rnorm(200,0,1)> x = cumsum(w)> wd = w +.2> xd = cumsum(wd)> plot.ts(xd, ylim=c(-5,55))> lines(x)> lines(.2*(1:200), lty="dashed")
Arthur Berg Stationarity, ACF, White Noise, Estimation 19/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Outline
1 §2.1: Stationarity
2 §2.2: Autocovariance and Autocorrelation Functions
3 §2.4: White Noise
4 R: Random Walk
5 Homework 1b
Arthur Berg Stationarity, ACF, White Noise, Estimation 20/ 21
§2.1: Stationarity §2.2: Autocovariance and Autocorrelation Functions §2.4: White Noise R: Random Walk Homework 1b
Homework 1b
Read the following sections from the textbook§2.5: Estimation of the Mean, Autocovariances, and Autocorrelations
§2.6: Moving Average and Autoregressive Representations of Time SeriesProcesses
Do the following exercise.Prove that a time series of iid random variables is strongly stationary.
Arthur Berg Stationarity, ACF, White Noise, Estimation 21/ 21
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