Chapter 3 - Basic concepts - Part A Flashcards
What are the 5 features of economic time series?
- Trend
- Seasonality
- Aberrant observations
- Time-varying conditional variance (alternating periods with high and low volatility in financial markets)
- Nonlinearity (for example, different dynamic behavior in different business cycle regimes)
Autocorrelation ?
correlation between y_t and y_t-k
1st oder autocorrelation
see formula
In general, the k-th autocorrelation can be estimated by :
see formula
The set of all autocorrelations p_k for k = 1,2,… is called the …
Empirical autocorrelation function [EACF].
What are the properties of white noise time series e_t?
E(e_t) = 0
E[e_t^2] = sigma^2
E[e_s*e_t] = 0
we denote the available history up to time t-1 as :
Y_(t-1) = {y_1, y_2,…, y_t-1}
At time t - 1, y_t we may consider an unknown random variable, what is the distribution ?
with a certain conditional distribution f(y_t|Y_t-1)
Suppose that we know the correct specification of g(.) , and also the values of the parameters Θ in the general time series model:
y_t = g(y_t-1, y_t-2,…, y_t-p; Θ) +e_t
then:
The conditional distribtion of f(y_t|Y_t-1) of y_t is the same as the distribution of e_t.
The basic class of so-called autoregressive [AR] models takes g(.) to be a …….. function of y_t-1, …, y_t-p
Linear.
[AR] models is concerned with the “linear” dependence between y_t and y_t-1, … ,y_t-p.
We can also describe y_t directly in terms of the current and past shocks e_t, e_t-1, e_t-2,….
This is done in a … way.
Linear
A first order autoregressive model (AR(1)) is given by:
yt = φ1yt−1 + εt, t = 1,2,3,…,T.
Derive an MA model from the AR(1) model
yt = φ1yt−1 + εt
= φ21yt−2 + φ1εt−1 + εt
= φ31yt−3 + φ21εt−2 + φ1εt−1 + εt .
= φt1y0 + φt−1ε1 + φt−2ε2 + ··· + φ1εt−1 + εt,
yt = φt1y0 + SUM( φi1εt−i; 0:t-1).
Transitory effect
When |φ1| < 1, φi1 → 0 as i increases. The shock εt−i has a transitory effect on the time series yt.
Explosive effect
When φ1 exceeds 1, the effect of shocks εt−i on yt increases with i. In that case the time series yt is called explosive.
ways of writing AR(1) model
yt = y0 + εt−i,
yt = φ1yt−1 + εt
yt = εt + π1εt−1 + π2εt−2 + π3εt−3 + π4εt−4 + …,
with intercept
yt −μ = φ1(yt−1 −μ)+εt,
Conditional mean of the AR(1) model
Given that εt is a white noise series with E[εt|Yt−1] = E[εt] = 0, the conditional mean of yt is equal to
E[yt|Yt−1] = φ1yt−1.
Unconditional mean of y_t in AR(1) model
Recall that the AR(1) model can be written as
y t = φ1^t y0 + φ1^t−1 ε1 + φ1^(t−2) ε2 + · · · + εt ,
It then follows that
E[yt]=φt1y0 and as t→∞, we find E[yt]=0.
Quick and dirty way to compute the unconditional mean E[y_t] = mu
In the AR(1) model it follows that
E[yt] = E[φ1yt−1 + εt] = φ1E[yt−1].
Setting E[yt] = E[yt−1], it follows immediately that
E[yt] = 0.
In practice, times series often have an (unconditional) mean different from 0. How can we account for this?
include an intercept!
yt =δ+φ1yt−1+εt.
Based on yt =δ+φ1yt−1+εt, compute the unconditional mean E[y_t] :
yt = δ + φ1yt−1 + εt
=δ+φ1(δ+φ1yt−2 +εt−1)+εt =δ+φ1δ+φ21(δ+φ1yt−3 +εt−2)+φ1εt−1 +εt
=…
t−1 t−1
= δ φ i1 + φ t1 y 0 + φ i1 ε t − i . i=0 i=0
Hence, we find that as t → ∞ E[yt] =
δ / 1−φ1
Based on yt =δ+φ1yt−1+εt, compute the unconditional mean E[y_t] , use the quick and dirty approach:
yt =δ+φ1yt−1+εt,
E[yt] = E[δ + φ1yt−1 + εt]
= δ + φ1E[yt−1]
Setting E[yt] = E[yt−1] ≡ μ and solving for μ, we find
μ=δ/ 1−φ1
Note that δ = μ(1 − φ1), so we may also rewrite (9) as
yt −μ = φ1(yt−1 −μ)+εt.
What does stationary mean?
Recall that AR(1):
yt = εt + π1εt−1 + π2εt−2 + π3εt−3 + π4εt−4 + …,
where where πi = φi_1 and εt is a white noise time series.
Stationarity means that the unconditional mean, unconditional variance, and autocorrelations of yt are constant over time.
If |φ1| < 1, πi → 0 when i increases (temporary effects of shocks). The AR(1) model (or the resulting time series yt) is called stationary in this case.
In the AR(1) model, |φ1| < 1 is a … condition for …
a. Necessary and sufficient
b. stationarity
From yt = εt + π1εt−1 + π2εt−2 + π3εt−3 + π4εt−4 + …,
where πi = φi1 and εt is a white noise time series.
We observe that :
- E[ytεt] = σ2
- E[ytεt+j] = 0 for j = 1,2,3,….
The current observation yt is not correlated with future shocks. - E[ytεt−k] for k = 1,2,3,… is NOT equal to 0.
The current observation yt is correlated with past shocks.
The k-th order autocorrelation of a time series yt is defined by
ρk = γk/γ0,
where γk is the k-th order autocovariance of yt, that is,
γk = E[(yt − E[yt])(yt−k − E[yt−k])], k = …,−2,−1,0,1,2,…
Given ρk = γk/γ0, (16)
where γk is the k-th order autocovariance of yt, that is,
γk = E[(yt − E[yt])(yt−k − E[yt−k])], k = …,−2,−1,0,1,2,…
it is clear that :
ρ0 = 1 and that ρ−k = ρk for
all k = 1,2,…
From yt −μ = φ1(yt−1 −μ)+εt, derive E[y_t]:
E[yt] = μ + φ1E[yt−1 − μ] + E[εt]
= (1 − φ1)μ + φ1E[yt−1].
For a stationary AR(1) model with |φ1| < 1, the unconditional mean is constant over time, that is E[yt] = E[yt−1]. Hence, E[yt] = (1 − φ1)^(−1)(1 − φ1)μ = μ.
How to obtain the ACF for the AR(1) model
Use the 3 following facts:
1. For stationary AR(1) models, the unconditional variance is
constant, that is E[(yt − E[yt])2] = E[(yt−1 − E[yt−1])2]. 2. The covariance of μ with a time series is equal to zero. 3. E[y_(t−j)εt]=0 for anyj>0.
see slides.
What is the ACF for AR(1) model
ρk = φ1ρ_k−1 or ρk = φ^k1 for k = 1,2,3,…
What is the “unit root” case?
When φ1 = 1.
yt = εt +εt−1 +εt−2 +···+ε2 +ε1 +y0
Since E[εt] = 0 for all t, E[εt−kεt] = 0 for all k ̸= 0, and assuming that y0 = 0, it follows that the unconditional mean E[yt] = 0 for all t.
From (30), it follows that
γ0,t = E[yt2] = tσ2, (31)
where the additional subscript t indicates that the value of the variance depends on time t.
What do we see when comparing
yt = εt +εt−1 +εt−2 +···+ε2 +ε1 +y0
to the one lagged version:
yt−1 = εt−1 +εt−2 +…ε2 +ε1 +y0,
we find that
γ1,t = E[ytyt−1] = (t − 1)σ2, which, together with γ0,t = E[yt2] = tσ2, results in
ρ1,t = (t − 1)/t.
Similarly, it is not difficult to show that in general ρk,t = (t − k)/t
for any k>0.
⇒ When t becomes large, all (theoretical) autocorrelations ρk,t
become equal to 1.
AR(1) model cannot capture EACF patterns as observed for the motorcycles series. what to do
second order autoregressive model
yt = φ1yt−1 + φ2yt−2 + εt, can generate such “richer” autocorrelation patterns
How is the lag operator L defined ?
- L^k*yt = y_t−k for k = …,−2,−1,0,1,2,…
- L4yt = yt−4
- (1−αL)−1 = 1+αL+α2L2 +α3L3 +…
- (1+L2)(1−L2) = 1−L4
Give the formula of the AR(p) model
yt = φ1yt−1 + φ2yt−2 + ··· + φpyt−p + εt,
- φp(L)yt = εt where φp(L) is the so-called AR-polynomial in L of order p
φp(L) = 1−φ1L−···−φpLp.
The characteristic polynomial is the lag polynomial φp(L) given in (41), but considered as a function of z:
φp(z) = 1 − φ1z − · · · − φ_pz^p.
Its roots are the solutions to φp(z) = 0.
⇒ These roots determine whether the effects of shocks are transitory or permanent.
The characteristic polynomial of the AR(1) model is given by φ1(z) = 1 − φ1z, and its root is z = φ1^−1.
Discuss the solution to this equation.
- When φ1 = 1, this solution equals 1, and in that case the AR(1) polynomial is said to have a unit root (and shocks have permanent effects).
- When φ1 is smaller than 1 in absolute value, the root exceeds 1 (and shocks have transitory effects).
Since higher order AR(p) models may have complex roots, the solution is said to be “outside the unit circle” when |φ1| < 1.
MA(q) model can be written as
yt = εt + π1εt−1 + π2εt−2 + π3εt−3 + π4εt−4 + …, (44) where πi = φ1^I
At some point the coefficient πi becomes 0, that is where we can truncate the equation and obtaining
yt = εt + θ1εt−1 + … + θqεt−q => [MA(q)]
we can rewrite any AR(p) model in MA form …
yt = φ_p(L)^−1εt
What are the lagged versions of an MA(2) model?
yt−k = εt−k + θ1εt−k−1 + θ2εt−k−2
What is the autocorrelation function of a MA(2) model?
see last slide
