Section 6 Ssc 389
16
Time Series and Dynamic Models Section 6 Dynamic Regression Model Carlos M. Carvalho The University of Texas at Austin 1
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Transcript of Section 6 Ssc 389
Time Series and Dynamic Models
Section 6Dynamic Regression Model
Carlos M. CarvalhoThe University of Texas at Austin
1
Normal/Gamma
Quick review...
Let φ ∼ Ga(
n02 ,
d02
)and (X |φ) ∼ N(m,Cφ−1)
hence (φ|X ) ∼ Ga(
n12 ,
d12
)where
n1 = n0 + 1 and d1 = d0 +(X −m)2
C
Also,p(X ) ∝ [n0 + (X −m)2/R]−
n+12
so that X ∼ Tn0(m,R), where R = C(
d0n0
)
1
Dynamic Regression Model
The Model:
yt = βtXt + εt
βt = βt−1 + wt
I εt ∼ N(0, σ2) and wt ∼ Tnt−1(0,Wt)
I Wt = Ct−1
(1−δ
δ
)I δ is the discount factor ∈ (0, 1)
Posterior at t − 1 :
(βt−1|Dt−1) ∼ Tnt−1(mt−1,Ct−1) (σ−2|Dt−1) ∼ Ga
(nt−1
2,dt−1
2
)
2
Dynamic Regression Model
Priors at t :
(βt |Dt−1) ∼ Tnt−1(mt−1,Rt) (yt |Dt−1) ∼ Tnt−1(ft ,Qt)
whereRt = Ct−1/δ, ft = mt−1Xt , Qt = X 2
t Rt + St−1 and St−1 = dt−1
nt−1
Posteriors at t :
(βt |Dt) ∼ Tnt (mt ,Ct) (σ−2|Dt) ∼ Ga
(nt
2,dt
2
)where nt = nt−1 + 1, dt = dt−1 + St−1e
2t /Qt ,
mt = mt−1 + Atet Ct = RtSt/Qt
et = (yt − ft), At = XtRt/Qt and St = dtnt
3
Example: APPL vs. MKTδ = 0.90
Posterior Mean (beta_t|D_t)
Time
m
0 100 200 300 400
0.0
0.5
1.0
1.5
2.0
2.5
3.0
4
Example: APPL vs. MKTδ = 0.90
Posterior summary (sigma|D_t)
Time
d/n
0 100 200 300 400
510
1520
5
Example: APPL vs. MKTδ = 0.95
Posterior Mean (beta_t|D_t)
Time
m
0 100 200 300 400
0.5
1.0
1.5
2.0
2.5
6
Example: APPL vs. MKTδ = 0.95
Posterior summary (sigma|D_t)
Time
d/n
0 100 200 300 400
510
1520
7
Example: APPL vs. MKTδ = 0.99
Posterior Mean (beta_t|D_t)
Time
m
0 100 200 300 400
1.0
1.5
2.0
2.5
8
Example: APPL vs. MKTδ = 0.99
Posterior Mean (beta_t|D_t)
Time
m
0 100 200 300 400
1.0
1.5
2.0
2.5
9
Example: APPL vs. MKTWe can compute the “model likelihood” for different values of δ...
p(δi |Dt) ∝ p(y1:T |δi ) =T∏
t=1
p(yt |Dt−1, δi )
0.95 0.96 0.97 0.98 0.99 1.00
-1474
-1473
-1472
-1471
-1470
max: delta = 0.9872
DEL
LikOUT
10
Now let’s try the MCMC
The Model:
yt = βtXt + εt
βt = βt−1 + wt
I εt ∼ N(0, σ2) and wt ∼ N(0, ω2)
I σ−2 ∼ Ga(aσ2 ,
bσ2 )
I ω−2 ∼ Ga(aω2 ,
bω2 )
11
Example: APPL vs. MKT
Time
mBeta
0 100 200 300 400
0.8
1.0
1.2
1.4
1.6
12
Example: APPL vs. MKT
Time
mBeta
0 100 200 300 400
0.5
1.0
1.5
2.0
2.5
13
Example: APPL vs. MKT
Time
Sigma2
0 200 400 600 800 1000
2224
2628
3032
Sigma2
PARS[, 1]
Density
22 24 26 28 30 32
0.00
0.05
0.10
0.15
0.20
Time
Omega2
0 200 400 600 800 1000
0.005
0.015
0.025
Omega2
PARS[, 2]
Density
0.005 0.010 0.015 0.020 0.025
020
4060
80120
14
Homework
1. Replicate the dynamic regression example from section 6.I First make sure you can implement the closed-form solution
using discount factorsI evaluate the the model likelihood for different values of δ
2. Simulate, for different values of r , data from the AR(1) plusnoise model with α = 0 and β = 1. r = W /V the ratiobetween the variance in the evolution equation and thevariance in the observation equation. For the different choicesof r plot the time series generated and the smoothed posteriormean from the distribution of θt |DT .
3. Prove the result in slide 6 in Section 5.
15
