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
