Stochastic Process Part 1

Stochastic Process Part 1
Econ 475 B3 – Economic Forecasting
Italo Morais Santos
Department of Economics
University of Illinois at Urbana-Champaign
October 18, 2021
Introduction
I Diebold Textbook. Chapter 6 (Page 141).
I Recall our road map:
1. Estimate sample ACF and PACF to characterize it’s dynamics.
2. Match it to some known stochastic process.
3. Estimate the model of the cycle component.
4. Perform forecasts.
I In the next classes: step 2.
Introduction
I Throughout this section assume away the trend and the season component.
I No loss of generality. You could have detrended and SA’ed data and dealing with the
cycle part alone.
yt = Ct + εt
I From the last class we learned we can estimate the sample ACF and PACF if yt
is
covariance stationary.
I Let’s study possible models of yt
.
White Noise
I White Noise
yt = εt
εt ∼ WN(0, σ2
)
I Ct = 0.
I Where εt
is serially uncorrelated.
I Independent White Noise εt ∼iid WN(0, σ2
).
I Gaussian White Noise εt ∼iid N(0, σ2
).
White Noise
I Simple and boring.
I Important because white noise residuals is one of our modeling goals.
I How to characterize it is important to know when we are ”done”.
White Noise
White Noise
I Unconditional Mean and Autocovariance
E(yt) = 0
γ(τ ) = (
σ
2
if τ = 0
0 if τ ≥ 1
I It’s ACF:
ρ(τ ) = (
1 if τ = 0
0 if τ ≥ 1
White Noise
I It’s partial ACF:
yt = β0 + β1yt−1 + … + p(τ )yt−τ + εt
I The β are all zero, such that yt = εt
. That is
p(τ ) = (
1 if τ = 0
0 if τ ≥ 1
I Same as ACF. Past information does not matter for white noise dynamics.
White Noise
White Noise
I We know the populational shape of the white noise ACF and PACF.
I We can calculate the sample equivalents and compare.
I Formally we have tests of serial correlation:
I Durbin-Watson. Tests H0 : p(1) = 0.
I Box-Pierce and Ljung-Box. Tests H0 : ρ(1) = ρ(2) = … = ρ(m) = 0.
I Next topic.
White Noise
I In order to construct forecasts of stochastic process we use the conditional
moments.
I Recall we talked about information sets. For the univariate time series analysis:
Ωt = {yt
, yt−1, …}
I Conditional Mean is
E(yt
|Ωt−1) orEt−1(yt)
I Same for conditional variance
Var(yt
|Ωt−1) orVart−1(yt)
White Noise
I Point Forecasts
yT+h,T = E(yT+h|ΩT )
I Density Forecasts is the conditional density.
p(yT+h,T ) = p(yT+h|ΩT )
I In Gaussian case:
yT+h,T ∼ N(ET (yT+h), VarT (yT+h))
I Intervals obtained by integrating the density.
I More on this later.
White Noise
I Unconditional white noise moments:
Et−1(yt) = E(εt
|εt−1, …)
Et−1(yt) = 0
I And
Vart−1(yt) = Var(εt
|εt−1, …)
Vart−1(yt) = σ
2
I For white noise conditional and unconditional moments are the same.
”Unforecastable”.
Moving Average
I MA(1): Moving Average of order 1 (textbook page 237).
yt = θεt−1 + εt
εt ∼ WN(0, σ2
)
I The cycle is Ct = θεt−1. Disturbances (shocks) in previous period have a lasting
effect. Persistency.
Moving Average
Moving Average
I Unconditional Mean
E(yt) = E(θεt−1 + εt)
E(yt) = 0
I Unconditional Variance
Var(yt) = Var(θεt−1 + εt)
Var(yt) = σ
2
(1 + θ
2
)
I Covariance-stationary for any value of θ.
Moving Average
I Autocovariance with displacement τ = 1:
Cov(ytyt+1) = E(ytyt+1) − E(yt)E(yt+1)
Cov(ytyt+1) = E[(θεt−1 + εt)(θεt + εt+1)]
Cov(ytyt+1) = θσ2
I Autocovariance with displacement τ = 2:
Cov(ytyt+2) = E[(θεt−1 + εt)(θεt+1 + εt+2)]
Cov(ytyt+2) = 0
I Same for τ > 2.
Moving Average
I It’s ACF:
ρ(τ ) =



1 for τ = 0
θ
1+θ
2
for τ = 1
0 Otherwise
Moving Average
I Persistency exists but only lasts one period. Sharp cut-off.
I Conditional Mean and Variance
Et−1(yt) = θεt−1
Vart−1(yt) = σ
2
I Can help us with one step ahead forecasts as Et(yt+1) = θεt
. Two steps ahead or
more is ”unforecastable” Et(yt+h) = 0 for h ≥ 2.
Moving Average
I Can be extended for higher orders. MA(2):
yt = θ2εt−2 + θ1εt−1 + εt
I Here the cycle is Ct = θ2εt−2 + θ1εt−1. The last two shocks persist over time.
I Easy to show the unconditional mean and variance:
E(yt) = 0
Var(yt) = σ
2
(1 + θ
2
1 + θ
2
2
)
Moving Average
I Autocovariances. One displacement
γ(1) = E(ytyt+1) − E(yt)E(yt+1)
γ(1) = E[(θ2εt−2 + θ1εt−1 + εt)(θ2εt−1 + θ1εt + εt+1)]
γ(1) = θ1θ2σ
2 + θ1σ
2
I Two displacements:
γ(2) = E(ytyt+2) − E(yt)E(yt+2)
γ(2) = E[(θ2εt−2 + θ1εt−1 + εt)(θ2εt + θ1εt+1 + εt+2)]
γ(2) = θ2σ
2
I Easy to verify the others are zero.
Moving Average
I The ACF:
ρ(τ ) =



1 for τ = 0
θ1θ2+θ1
1+θ
2
1+θ
2
2
for τ = 1
θ2
1+θ
2
1+θ
2
2
for τ = 2
0 Otherwise
I That is, sharp cut-off at τ = 2.
I Conditional mean:
Et−1(yt) = θ2εt−2 + θ1εt−1
Et−2(yt) = θ2εt−2
Et−τ (yt) = 0 for τ ≥ 3
Moving Average
I Generalizing to MA(q):
yt = θqεt−q + … + θ1εt−1 + εt
I Same properties as in the MA(1) and MA(2) but at the q-th order:
1. Always covariance stationary.
2. Autocorrelation and covariances are non-zero for τ ≤ q and zero otherwise.
I The MA(q) model can help us to account for serial correlation of any finite order.
I For each extra order, we need an extra parameter.
I In vast majority of cases, we would burn our degrees of freedom really fast if we
try this !
Unemployment Rate
Moving Average
I Next class: Autoregressive models