Download Spatial Chow-Lin models for completing growth rates in cross

References
Spatial Chow-Lin models for completing growth
rates in cross-sections
Wolfgang Polasek
November 16, 2012
References
Overview
• extension of the spatial Chow-Lin procedure for cross-sectional
growth rates (extensive variables)
• compares classical and Bayesian estimation and prediction
methods
• demonstrate the procedure for Spanish regional GDP growth
rates between 2000 and 2004 at a NUTS-3 level
• evaluate the growth rate forecasts by accuracy criteria
References
Introduction to the original Chow-Lin Procedure
• Chow-Lin (1971) developed a method to forecast
(”construct”) quarterly times series observations from yearly
observations
• There is a well defined aggregation matrix C that connects
aggregated and disaggregated variables
• Use appropriate ”indicators” or auxiliary regressors for
predicting the quarterly series
• New: The approach can be extended for constructing
disaggregated observations in the spatial context
• This problem is different from ’kriging’, utilizes more
information because of the aggregation structure
References
The 4 ingredients of any Chow-Lin Procedure
Summary of the data completion (= fine-forecasting) method:
1. First, decide on a forecasting or base model with only
‘intensive’ (or aggregable) regression variables for the
unobserved data at the disaggregated level.
2. Decide on an aggregation matrix C that aggregates the
disaggregated model into a fully observed aggregated model.
3. Estimates the disaggregated parameters using the aggregated
reduced form of the base model.
4. Compute the disaggregated Chow-Lin forecasts based on
known regression indicators in the base model.
References
The Chow-Lin Procedure
We assume a linear relationship for the high frequency
(disaggregate) data yd and the indicators Xd , i.e.
yd = Xd β + with
∼ N [0, σ 2 Ω],
(1)
Chow and Lin (1971) showed that the BLUE for the regression
parameter β̂ and the unobserved high frequency data ŷd is given
by:
β̂ = (Xd0 C 0 (C ΩC 0 )−1 CXd )−1 Xd0 C 0 (C ΩC 0 )−1 ya
ŷd
= Xd β̂ + ΩC 0 (C ΩC 0 )−1 (ya − CXd β̂),
(2)
where ya = Cyd is the observed aggregated dependent variable.
Interestingly, the Chow-Lin formula is the same for time and space,
only the C matrix changes.
References
Chow-Lin fine-forecasting
It is interesting to note that G is a right generalized inverse of C
(i.e. is orthogonal to the aggregation matrix C ), because of
CG = IN and the aggregated Chow-Lin forecasts have the property
C ŷˆd = C ŷd + ˆa ,
or
agg .CL = agg .plain + agg .res.
(3)
References
Why Chow-Lin fine-forecasting?
The Chow-Lin forecasts for the disaggregates are an improvement
over the RF forecasts.
• The reduced form forecasts are corrected by an allocator
(gain-in-mean), which distributes the aggregate residual
across the dis-aggregates.
• The Chow-Lin forecasts have a smaller variance:
• ŷCL = N[ŷRF + g , V̂RF − G ]
References
Statistical Properties of the Chow-Lin Forecasts I/II
• The 1st property: on average the Chow-Lin forecasts and the
plain forecasts are equal (just post-multiply (22) by a vector
of 1’s).
C Ave(ŷˆd ) = C Ave(ŷd ).
• 2nd property: the aggregated Chow-Lin forecasts have a larger
variance than the aggregated plain forecasts:
ŷˆd0 C 0 C ŷˆd > ŷd0 C 0 C ŷd .
References
Statistical Properties of the Chow-Lin Forecasts II/II
• The third property is based on
ŷd = Xd β̂d + Qˆ
a
with the ’reverse projection’ matrix Q = ΩC 0 (C ΩC 0 )−1 and
leads to the following error sum of squares (ESS)
decomposition
ESSCL = ESSplain + ESSgain + noise
ŷd0 ŷd
or
= β̂d0 Xd0 Xd β̂d + ˆ0a Q 0 Qˆ
a + noise.
(4)
The relative decomposition takes the form
1=
β̂d0 Xd0 Xd β̂d
ˆ0a Q 0 Qˆ
a
+
+ rest .
0
0
ŷd ŷd
ŷd ŷd
where the ’rest’ is the remainder of the decomposition that
adds up to 1.
(5)
References
Why to forecast Growth Rates differently?
We need a different method for extensive (or non-aggregable)
∆y2
1
variables: consider 2 disaggregated regions: ∆y
y1 and y2 , which
have to be combined to the growth rate of the aggregated region:
∆y1 +∆y2
y1 +y2 .
The growth rate requires temporal differences between 2 periods,
i.e. ∆y1 = y1t − y1,t−1 and ∆y2 = y2t − y2,t−1 . Since this is not a
simple sum we have to aggregate the nominator and the
denominator separately. This leads to the bivariate disaggregated
Chow-Lin model
∆yd
yd
=
Xd1
1
β1
+
β2
2
Xd2
with
0
1
∼N
, Σ ⊗ In ,
0
2
References
The ’system Chow-Lin’ model
combines these two equations into a regression system, the basis
for CL forecasting:
e = Σ ⊗ In ]
ỹd = X̃d β̃ + ˜ with e
∼ N [0, Σ
Xd1
β1
1
d
with ỹd = ∆y
,
X̃
=
,
β̃
=
,
and
e
=
d
yd
Xd2
β2
2 .
(6)
References
Aggregated and disaggregated reduced form (RF)
The aggregated reduced form
e ỹd = C̃ X̃d β̃ + C̃ ˜ with
C
ee
eΣ
eC
e 0 ].
C
∼ N [0, ΩC = C
ˆ
For the regression parameter βe and the unobserved disaggregated
(high frequency) data ŷd is given by
ˆ
b̃ C̃ 0 )−1 C̃ X̃ )−1 X̃ 0 C̃ 0 (C̃ Σ
b̃ C̃ 0 )−1 y
βe = (X̃d0 C̃ 0 (C̃ Σ
a
d
d
ˆ
ˆ
b̃
b̃
0
0
−1
ˆ = X
ed βe + ΣC̃ (C̃ ΣC̃ ) (ỹa − C̃ X̃d β),
e
ye
d
(7)
(8)
References
Covariance estimation
The unknown covariance matrix is estimated by the OLS estimate
of the system equation:
σ̂11 σ̂12
b
Σ=
(9)
./. σ̂22
with σ̂11 = Var (ˆ
1 ), σ̂22 = Var (ˆ
2 ), and σ̂12 = Cov (ˆ
1 , ˆ2 ). The
estimated residuals are ˆ1 = ∆y − Xd1 β̂1 and ˆ2 = y − Xd2 β̂2 with
the OLS estimates
0
0
β̂1 = (Xd1
C 0 (CC 0 )−1 CXd1 )−1 Xd1
C 0 (CC 0 )−1 ∆ya
β̂2 =
0
0
(Xd2
C 0 (CC 0 )−1 CXd2 )−1 Xd2
C 0 (CC 0 )−1 ya
(10)
(11)
or β̂i = (Xai0 DN−1 Xai )−1 Xai0 DN−1 yai with
CC 0 = DN = diag (n1 , ..., nN ) : N × N, where the ni are the number
of sub-units in each aggregated unit and ya1 = ∆ya and ya2 = ya .
References
The disaggregated forecasts of the growth rates
r (y d ) is the ratio of the Chow-Lin predicted nominator and
denominator
c ./.b
r (y d ) = ∆y
yd .
d
(12)
The system Chow-Lin forecasts are
ỹˆd
= X̃d β̃ˆ + ΩC̃ 0 (C̃ ΩC̃ 0 )−1 (ỹa − C̃ X̃d β̃ˆd ),
(13)
and the gain term G̃ can be simplified by
G̃ = (Σ⊗In )(I2 ⊗C 0 )((I2 ⊗C )(Σ⊗In )(I2 ⊗C 0 ))−1 = (I2 ⊗C 0 (CC 0 )−1 ),
which shows that the classical Chow-Lin forecasts in the SUR
system Chow-Lin can be made independently for both equations
c
∆y
d
ŷd
= Xd1 β̂1 + C 0 (CC 0 )−1 (∆ya − Xa1 β̂1 ),
0
0 −1
= Xd2 β̂2 + C (CC )
(ya − Xa2 β̂2 ).
(14)
(15)
References
The reduced form for disaggregates
Given the cross-sectional SAR model of n regions
∼ N [0, Σ ⊗ In ](16)
ed = diag (X1 , X2 ), βe = β1 and
with Σ to be estimated as in (9), X
β2
f = I2 ⊗ W . The spread matrix R
e = diag (In − ρ1 W , In − ρ2 W )
W
for a chosen weight matrix W : n × n leads to the reduced form
f yed + X
ed βe + ˜d ,
yed = diag (ρ1 , ρ2 )W
˜d
ed βe + R̃ −1 ˜d , .
ỹd = R̃ −1 X
with
R̃ −1 ˜d ∼ N [0, Ω = (R̃ 0 Σ̃−1 R̃)−1 ]
(17)
References
Reduced Form
e R̃ −1 X̃d βe + C̃ R̃ −1 ˜,
ỹa = C̃ ỹd = C
e R̃
C
−1
˜d
with
0
∼ N [0, C̃ ΩC̃ ] or
= X̃a βe + ˜a with
ỹa
−1
e
C R̃ ˜a ∼ N [0, ΩC ]
with
e R̃ −1 X̃d and
ỹa = C̃ ỹd , X̃a = C
e R̃ −1 ˜d .
˜a = C
(18)
References
The Covariance Matrix Ω : 2N × 2N
of the aggregated residuals is
Ω = C̃ R̃ −1 Σ̃R̃ 0
−1
C̃ 0 =
= (I2 ⊗ C )(diag (R1 , R2 )−1 (Σ ⊗ In )diag (R1 , R2 )0−1 (I2 ⊗ C 0 ) =
σ11 C (R10 R1 )−1 C 0 σ12 C (R10 R2 )−1 C 0
Ω11 Ω12
=
=
.
./.
σ22 C (R20 R2 )−1 C 0
./. Ω22
References
The GLS estimate of the aggregates
The GLS estimate of βe is
βeGLS = (X̃a0 Ω−1 X̃a )−1 X̃a Ω−1 ỹa .
(19)
References
The ’plain’ or no-gain forecast
is the point forecast of the reduced form at the observed
low-frequency indicator Xd (the mean of the conditional model 16):
c
∆y
d
ŷd
= ỹˆd =
b̃
e −1 X
ed β
R
GLS
ρ̂
=
R̂1−1 Xd,1 β̂1,GLS
R̂2−1 Xd,2 β̂2,GLS
eρ̂ = diag (R̂1 , R̂2 ) with
and the estimated spread matrix is R
R̂i = In − ρ̂i W for i = 1, 2.
(20)
References
The joint distribution of the aggregates and disaggregates
Note that the aggregated model has always a completely observed
data set. Therefore, we can estimate a β regression response by
GLS or maximum likelihood methods, although aggregate
estimates can become quite unreliable because only fewer
observations are available for estimation on an aggregate level.
We construct the joint distribution of the aggregated (18) and the
disaggregated model (17) is
ỹd
µ̃d
(R̃ 0 Σ̃−1 R̃)−1
(R 0 Σ̃−1 R̃)−1 C̃ 0
,
.(21)
∼N
C̃ (R̃ 0 Σ̃−1 R̃)−1 C̃ (R̃ 0 Σ̃−1 R̃)−1 C̃ 0
C ỹd
µ̃a
The conditional mean ỹˆd for the disaggregated observations, given
the aggregated data ỹa = C̃ ỹd , has to be calculated by the
partitioned inverse rule as well.
References
The CL forecasting formula: ’gain-in-mean’
The Chow-Lin forecasting formula: CL.forecast = plain.RF + gain
ŷd,CL = R −1 X̃d β̂GLS + g̃ ,
(22)
where the g̃ = Qêa is the ’gain-in-mean’ term of the forecast since
it is an improvement over the plain or reduced form RF forecast of
the missing y -value.
Note: The ’gain-in-mean’ term g̃ can be interpreted as an
allocation of the estimated aggregated residual.
References
The covariance matrix Ω
of the reduced form model in (19) of the spatial Chow-Lin system for growth
rates is
ỹˆd
=
X̃d β̃ˆ + g̃ = X̃d β̃ˆ + ΩC̃ 0 (C̃ ΩC̃ 0 )−1 (ỹa − C̃ X̃d β̃ˆGLS ),
(23)
where the gain-in-mean term g̃ is given by the estimated aggregated residual
ˆ
˜a = ỹa − C̃ R̃ −1 X̃d β̃ˆGLS by
−1 Ω11 C 0 Ω12 C 0
C Ω11 C 0 C Ω12 C 0
ˆa1
g̃ = ΩC̃ 0 (C̃ ΩC̃ 0 )−1 ˆ
˜a =
.
0
0
./.
Ω22 C
./.
C Ω22 C
ˆa2
(24)
and the ’gain-in-variance’ matrix G̃ , first used by Goldberger (1962), is defined
by
G̃ = ΩC̃ 0 (C̃ ΩC̃ 0 )−1 C̃ Ω.
(25)
References
Two-step (feasible GLS) estimation
Based on the above system extension of the Chow-Lin method we
suggest the following 2-step estimation of the spatial system to
complete growth rates.
1. Estimate by ML (or LS) the SAR models using the first
differences and levels to get ρ̂1 , ρ̂2 .
2. Compute the LS residuals from the SAR models and estimate
the covariance matrix Σ̂ = Σ(ρ̂1 , ρ̂2 ).
3. Compute the system estimates β̃GLS using the estimated Ω̂
matrix.
4. Compute the vector of system Chow-Lin forecasts as in (23).
5. Compute the vector of growth rate Chow-Lin forecasts
∆ŷd ./.ŷd .
This procedure can be easily implemented along the existing
statistical program packages that allow SAR estimation.
References
The Bayesian Chow-Lin model for completing growth rates
The prior distribution for the parameters of the SAR-CL model
θ = (β̃, Σ−1 , ρ1 , ρ2 ) is proportional to
p(β̃, Σ−1 , ρ1 , ρ2 ) ∝ p(β̃) · p(Σ−1 )
= N [β̃ | β̃∗ , H∗ ] · W[Σ−1 | Σ−1
∗ , n∗ ],
where W stands for the Wishart distribution of dimension n and
where we assume a uniform prior for ρi ∼ U[−1, 1], i = 1, 2.
The joint distribution of θ = (β̃, ρ1 , ρ2 , Σ−1 ) of the Bayesian
SAR-CL model is
e σ 2 Σρ ] · N [β | β∗ , H∗ ] · W[Σ−1 | S∗ , n∗ ].(26)
p(θ | ỹd ) = N [C̃ R̃ −1 X̃ β,
Note: The system estimates βGLS in ( ) using the estimated Ω̂
matrix.
Compute the vector of system Chow-Lin forecasts as in (23).
References
MCMC for the system SAR Chow-Lin Model
The Markov Chain Monte Carlo (MCMC) procedure consists of 4
blocks of sampling, as is shown in the next theorem:
Theorem (MCMC for the system SAR Chow-Lin model)
The MCMC estimation for the Bayesian system SAR model
(6), with the joint distribution defined in (26),
involves the following iteration steps:
h
i
Step 1: Draw β̃d from N β̃ | b̃∗∗ , H̃∗∗ ;
Step 2: Draw ρi by a Metropolis step:
ρi,new = ρi,old + N [0, τi2 ], i = 1, 2;
Step 3: Draw Σ−1 from W[Σ−1 | S∗∗ , n∗∗ ];
Step 4: Repeat until convergence.
References
Finecasting = Completing growth rates by prediction
In Bayesian inference, we obtain the posterior predictive
distribution for yep in the following way, by integrating over the
conditional predictive distribution with the posterior distribution
e ρ, Σ−1 | ỹ )
p(β,
Z Z Z
e ρ, Σ−1 )p(β,
e ρ, Σ−1 | ỹ )d βe dρ dΣ−1
p(e
yp | ỹd ) =
p(ỹp | β,
with ρ = (ρ1 , ρ2 ) and the posterior normal-gamma density
e ρ, Σ−1 | ỹa ) is found numerically by the MCMC sample,
p(β,
yielding a posterior sample of the parameters in θ:
ΘMCMC = {(βej , ρj,1 , ρj,2 , Σ−1
j ),
j = 1, ..., J}.
References
The predictive sample
Next, we compute a numerical predictive sample of the unknown
vector ỹd by drawing from the reduced form (which depends on the
spread matrix R̃ and on the known regressors X̃d ):
(j)
ỹd ∼ N [R̃j−1 X̃d β̃j + g̃j , Ωj − G̃j ],
(27)
with R̃ = diag (R1 , R2 ) and the spread matrices
Rj = In − ρj W , j = 1, ..., J. g̃ is the gain-in-mean vector as in
(24) and G̃ is the gain-in-variance matrix as in (25) for the mean
and covariance matrix of the predictions, which are computed by
g̃j
= Ωj C̃ 0 (C̃ Ωj C̃ 0 )−1 C̃ Ωj
= Ωj C̃ 0 (C̃ Ωj C̃ 0 )−1b̃
e a,j
Ωj
= R̃j−1 (Σj ⊗ In )R̃j−1 ,
G̃j
(28)
using the covariance structure of the reduced form (17) where we
use the aggregated residuals b̃
e a,j = ỹa − ỹˆa,j and the current
−1
aggregate fit yb̃a,j = C̃ R̃j X̃d βej .
References
The disaggregate forecasts of the growth rates
in vector r (y d ) is given by the ratio of the Chow-Lin predicted
nominator and denominator similar to (12). The forecast sample of
the n × 1 vectors of growth rates is
c ./.b
rMCMC (y d ) = {∆y
yd , j = 1, ..., J},
d
(29)
from where we can compute numerically the mean vector
Ave(rMCMC (y d )) and interval predictions (e.g. by quantiles) for all
of the n sub-units.
References
Empirical Application for Spain
• We estimated GDP growth rates using employment,
population, exports and imports
• for 18 Spanish regions (NUTS-2) and break it down to the 52
provinces (NUTS-3)
• via the extensive CL method.
References
Bayesian Difference Estimation Results
Spatial autoregressive model estimates
Dependent Variable =
gdp_d
R-squared
=
0.9543 , Rbar-squared = 0.9402
sigma^2
= 404140.4
Nobs, Nvars
=
18, 5 , min and max rho
=
-1.0,
1.0
log-likelihood
=
-135.73 , # of iterations = 17
total time in secs =
0.1100 , time for lndet = 0.0160
time for x-impacts =
0.0940 ; Pace and Barry, 1999 MC lndet approx used
order for MC appr =
50 , iter for MC appr =
30
***************************************************************
Variable
Coefficient Asymp t-stat
z-probability
c_d
2235.74
2.50
0.012
emp_d
33.71
2.78
0.005
pop_d
0.020
3.19
0.001
exp_d
0.001
1.91
0.057
imp_d
0.0003
0.64
0.520
rho
-0.605
-2.23
0.026
References
Bayesian Level Estimation Results
Spatial autoregressive Model Estimates
Dependent Variable =
gdp
R-squared
=
0.996 , Rbar-squared = 0.9953
sigma^2
= 6489676.4935
Nobs, Nvars
=
18, 5 , min and max rho = -1.0, 1.0
log-likelihood
=
-160.51 ,# of iterations = 11
total time in secs =
0.0940 , time for x-impacts = 0.0780
Pace and Barry, 1999 MC lndet approximation used
order for MC appr =
50 , iter for MC appr =
30
***************************************************************
Variable
Coefficient Asymptot t-stat z-probability
c
11672.664
27060805.9
0.000
emp
48.344
3.8
0.000
pop
-0.004
-0.0
0.375
exp
-0.001
-3.7
0.000
imp
0.001
4.2
0.000
rho
-0.235
-11.5
0.000
References
Bayesian Combined Difference/Level Estimation Results
Bayesian spatial autoregressive : Heteroscedastic model
Dependent Variable =
gdp/_d
R-squared
=
0.996 , Rbar-squared = 0.995
mean of sige draws = 2803257.1
sige, epe/(n-k)
= 6716996.4 , r-value = 4
Nobs, Nvars
=
36, 10 , ndraws,nomit = 15000, 1500
total time in secs =
21.01 , time for sampling = 18.40
Pace and Barry, 1999 MC lndet approximation used
order for MC appr =
50 , iter for MC appr = 30
min and max rho
=
-1.0000,
1.0000
***************************************************************
Posterior Estimates
Variable
Coefficient
Std Dev
p-level
c_d
177.57
209.50
0.185
emp_d
24.16
43.39
0.281
pop_d
0.023
0.02
0.143
exp_d
0.001
0.001
0.325
imp_d
0.001
0.001
0.327
c
340.699
407.626
0.172
emp
44.693
13.152
0.001
pop
-0.003
0.004
0.278
exp
-0.001
0.000
0.033
imp
0.001
0.000
0.001
rho
-0.005
0.020
0.405
References
Chow-Lin Prediction Accuracy: Classical vs. Bayesian
estimates
growth
Classic
rates
simple
spatial
Bayesian
MCMC
spatial
(no gain)
with gain
with gain
gain
no gain
with gain
CORR1
-0.005
0.204
0.217*)
0.101
0.200
0.211*)
MAE2
1.666
0.5146*)
0.5180
0.377695
0.327792
0.308276*)
MAPE3
0.109
0.03475*)
0.03501
0.025228
0.021307
0.020109*)
RMSE4
0.340
0.05115
0.05067*)
.038274
.035276
.033292*)
References
Spanish regional aggregated growth rates: Chow-Lin
forecast comparison
References
Spanish regional disaggregated growth rates: Chow-Lin
forecasts
References
Residuals from aggregated level data: NUTS-2 regions
References
Residuals from aggregated differenced data
References
Conclusions
• We have developed a new spatial Chow-Lin procedure for
growth rates, an example of non-summable random variables.
• Our new approach has shown that it pays to get a good
spatial model if one is interested in good predictions of
missing data in a cross-sectional model
• An important condition for finding a good model is the
existence of good indicators and the modeling skills to find
the appropriate weight matrix to estimate the spatial effects.
References
Contact:
Prof. Dr. Wolfgang Polasek
Phone: ++43-1-59991-155
e-mail: [email protected]
IHS - Institute for Advanced Studies
Stumpergasse 56
A-1060 Wien
References
References
Chow, G. C., Lin, A., 1971. Best linear unbiased interpolation,
distribution, and extrapolation of time series by related series.
The Review of Economics and Statistics 53 (4), 372–375.
References
Assumptions for applying Chow-Lin
Assumption
1. Structural similarity: The aggregated model for yc and the
disaggregated model for y are structurally similar. This implies
that variable relationships that are observed on an aggregated level
are following the same empirical law as on a disaggregated level:
the regression parameters in both models are the same.
Assumption
2. Error similarity: The spatially correlated errors have a similar
error structure on an aggregated level and on a disaggregated level:
The spatial correlations are not significantly different.
Assumption
3. Reliable indicators: The indicators to make the formats on a
disaggregated level have sufficiently large predictive power: The R 2
(or the F test) is significantly different from zero.
References
Thm: MCMC for the extensive Chow-Lin (eCL-SAR)
model
The MCMC estimation for the Bayesian system SAR model (2),
with the joint distribution defined in (26) involves the following
iteration steps:
h
i
Step 1: Draw β̃d from N β̃d | b̃∗∗ , H̃∗∗ ;
Step 2: Draw ρ1 and ρ2 by griddy Gibbs;
Step 3: Draw Σ−1 from W[Σ−1 | S∗∗ , n∗∗ ];
Step 4: Repeat until convergence.
References
Predicting the disaggregate growth rates rd by MCMC
1. Draw
e (j) Y
d
ea(j)
Y
from the joint density
2. Compute the conditional draws given the observed aggregate
values ya
e (j) |ya = Y (j) − g (j)
Y
d
d
d
with the gain-in-mean term
(j)
(j)
gd = Ω(j) C̃ 0 (C̃ Ω(j) C̃ 0 )−1 (Ya − ya ) as in (24) and compute
(j)
g̃d
=
(j)
Ω(j) C̃ 0 (C̃ Ω(j) C̃ 0 )−1 ˜a
(j)
(j)
=
(j)
(j)
Ω11 C 0 Ω12 C 0
(j)
./.
Ω22 C 0
!
(j)
ik (R 0 R )−1 and with Ωik = σ(j)
˜a = ỹa − C̃ R̃j−1 X̃d βed
j j
(j)
(j)
e (j) |ya =
3. Compute the ratio r (j) = Yd1 ./.Yd2 from Y
d
all j.
(j)
C Ω11 C 0 C Ω
./.
CΩ
(30)
(j)
= a1
(j) .
a2
(j)
Yd1
(j)
Yd2
for
References
The Douvet (2010) simulator for a conditional r.v.
from a joint normal distribution. A draw from the conditional
density N [µx|y , Σx|y ] based on the joint density
Σxx Σxy
µx
N [µ, Σ] = N
,
(31)
Σ0xy Σyy
µy
can be obtained in the following way:
Ẋ from N [µ, Σ] ;
1. Draw the bivariate r.v. Ẏ
2. Compute the conditional r.v. Ẋ |y = Ẋ − Σxy Σ−1
yy (Ẏ − y )
given a known y .
For the prediction of the disaggregate observations this translates
to
Ẏ 1. Draw Ẏd from the joint density (31);
a
2. Compute the conditional
Ẏd |ya = Ẏd − (R 0 R)−1 C 0 (C (R 0 R)−1 C 0 )−1 (Ẏa − ya ) given the
observed aggregate values ya .
The conditional system (panel) forecasts are made in the same way.
References
The joint distribution
A simpler way is found by applying the method of Douvet (2010).
We start from the joint distribution as in (??)


µ̃(j) = R̃ −1 X
ed βe(j) (j)
(j)
0
ỹd
d
d
Ω
Ω C̃
(j)
 (32)
,
∼N
(j)
(j)
(j)
C̃ Ω(j) C̃ 0
C̃ Ω
ỹa
µ̃a = C̃ µ̃
d
with Ω given in (19). We propose the following procedure to
predict the disaggregate growth rates rd by MCMC:
References
Model selection by marginal likelihood
The marginal likelihood of model M is computed by the harmonic
mean formula
!−1
nrep
n
X
X
1
m̂(y | M)−1 =
l(Di | M, θj )
(33)
nrep
j=1
i=1
where Di = (∆yi , yi ) is the i-th data observation and with the
likelihood given by the SAR Chow-Lin model. We also use the 1%
trimmed harmonic estimator.
References
MCMC estimation for the SARX model
The MCMC estimation for the SARX model with the joint
distribution p(θ, y ) = N[ρWy + X β, σy2 In ] p(θ) involves the
following iteration steps:
Step 1: Set ρ = 0
h
i
Step 2: Draw β̃d from N β̃d | b̃∗∗ , H̃∗∗ ;
Step 3: Draw Σ−1 from W[Σ−1 | S∗∗ , n∗∗ ];
Step 4: Draw ρi by griddy Gibbs;
Step 5: Repeat until convergence.
References
References