[topic 8-random parameters] 1/83 topics in microeconometrics william greene department of economics...
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Topics in Microeconometrics
William Greene
Department of Economics
Stern School of Business
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8. Random Parameters and Hierarchical Linear Models
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Heterogeneous Dynamic Model
i,t i i i,t 1 i it i,t
ii
i
logY logY x
Long run effect of interest is 1
Average (over countries) effect: or 1
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“Fixed Effects” Approach
i,t i i i,t 1 i it i,t
ii
i
i i i
N Nii 1 i 1 i
i
Ni 1 i
Ni 1
logY logY x
, = 1 1
ˆ ˆ(1) Separate regressions; , ,ˆ
ˆ1 1ˆ ˆ(2) Average estimates = orˆN N1
ˆ(1/ N)ˆ Function of averages: =1 (1/ N)
i
i
N
ii=1
ˆ
In each case, each term i has variance O(1/T)
Each average has variance O(1/N) (1/N)O(1/T)
Expect consistency of estimates of long run effects.
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A Mixed/Fixed Approach
Ni,t i i 1 i i,t i,t 1 i i,t i,t
i,t
i i i
logy d logy x
d = country specific dummy variable.
Treat and as random, is a 'fixed effect.'
This model can be fit consistently by OLS and
efficiently by GLS.
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A Mixed Fixed Model Estimator
Ni,t i i 1 i i,t i,t 1 i i,t i,t
i i
Ni,t i i 1 i i,t i,t 1 i i,t i i,t i,t
2 2 2i i,t i,t w i,t
logy d logy x
w
logy d logy x (wx )
Heteroscedastic : Var[wx ]= x
Use two step least squares.
(1) Linear regressi
i,t
i,t-1 i,t
2i,t
2 2w
on of logy on dummy variables, dummy
variables times logy and x .
(2) Regress squares of OLS residuals on x and 1 to
estimate and .
(3) Return to (1) but now use weighted least squares.
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Baltagi and Griffin’s Gasoline Data
World Gasoline Demand Data, 18 OECD Countries, 19 yearsVariables in the file are
COUNTRY = name of country YEAR = year, 1960-1978LGASPCAR = log of consumption per carLINCOMEP = log of per capita incomeLRPMG = log of real price of gasoline LCARPCAP = log of per capita number of cars
See Baltagi (2001, p. 24) for analysis of these data. The article on which the analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An Application of Pooling and Testing Procedures," European Economic Review, 22, 1983, pp. 117-137. The data were downloaded from the website for Baltagi's text.
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Baltagi and Griffin’s Gasoline Market
COUNTRY = name of country YEAR = year, 1960-1978LGASPCAR = log of consumption per car yLINCOMEP = log of per capita income zLRPMG = log of real price of gasoline x1LCARPCAP = log of per capita number of cars x2
yit = 1i + 2i zit + 3i x1it + 4i x2it + it.
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FIXED EFFECTS
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Parameter Heterogeneity
it i it
it i it
i i
i i
i i
X i i
i i
y c
y
u ,
E[u | ] 0 --> Random effects
E[u | ] 0 --> Fixed effects
E E[u | ] 0.
Var[u | ] not yet defined -
it
it
Unobserved Effects Random C
x β
on
x β
X
X
st
X
a
X
nts
so far, constant.
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Parameter Heterogeneity
it it
i
i i
X i i
i i
y
E[ | ] zero or nonzero - to be defined
E [E[ | ]] =
Var[ | ] to be defined, constant or variable
it i
i
Generalize to
x β
β β u
u X
u X
Random Sl
0
u X
opes
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Fixed Effects (Hildreth, Houck, Hsiao, Swamy)
it it
i i
i
i
i i i
i i i i
X i i X i i
i
y , each observation
, T observations
Assume (temporarily) T > K.
E[ | ] =g( ) (conditional mean)
P[ | ] =( -E[ ]) (projection)
E [E[ | ]] = E [P[ | ]] =
Var[ |
it i
i i i
i
x β
y Xβ ε
β β u
u X X
u X X X θ
u X u X 0
u Xi] constant but nonzeroΓ
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OLS and GLS Are Inconsistent
i i
i
i i i i
i i
i i i i i i
, T observations
, T observations
E[ | ] E[ | ] E[ | ]
i i i
i
i i
i
i
y Xβ ε
β β u
y Xβ Xu ε
y Xβ w
w X X u X ε X 0
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Estimating the Fixed Effects Model
Ni 1
Estimator: Equation by equation OLS or (F)GLS
1 ˆEstimate ? is consistent inN
1 1 1 1
2 2 2 2
N N N N
i
y X 0 ... 0 β ε
y 0 X ... 0 β ε
... ... ... ... ... ... ...
y 0 0 ... X β ε
β β N for E[ ].iβ
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Random Effects and Random Parameters
it it
i i
i
i
i i
i i
Random Parameters Model
y , each observation
, T observations
Assume (temporarily) T > K.
E[ | ] =
Var[ | ] constant but nonzero
it i
i i i
i
THE
x β
y Xβ ε
β β u
u X 0
u X Γ
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Estimating the Random Parameters Model
i i
i
i i i i
i i
i i i i i i
2 2i i i i ,i ,i
2,i
, T observations
, T observations
E[ | ] E[ | ] E[ | ]
Var[ | ] Should vary by i?
,
i i i
i
i i
i
i
y Xβ ε
β β u
y Xβ Xu ε
y Xβ w
w X X u X ε X 0
w X XΓX I <==
Objects of estimation: β, Γ
Second le ivel estimation: β
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Estimating the Random Parameters Model by OLS
i i
i
i i i i
i i
N 1 Ni 1 i i i 1 i i
N 1 Ni 1 i i i 1 i i
N 1 Ni 1 i i i 1 i i
, T observations
, T observations
= [ ] [ ]
= [ ] [ ]
[ ]= [ ] [ (
i i i
i
i i
i
y Xβ ε
β β u
y Xβ Xu ε
y Xβ w
b XX Xy
β XX Xw
Var b| X XX X XΓ
2 N 1i i 1 i i
2 N 1 N 1 N N 1i 1 i i i 1 i i i 1 i i i i 1 i i
) ][ ]
= [ ] [ ] [ ( ) ( )][ ]
= the usual + the variation due to the random parameters
Robust estimator
E
X I X XX
XX XX XX Γ XX XX
N 1 N N 1i 1 i i i 1 i i i i i 1 i iˆ ˆst.Var[ ] [ ] [ ][ ]b XX Xw w X XX
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Estimating the Random Parameters Model by GLS
i i
i
i i i i
2i i i i i i ,i
N 1 Ni 1 i i i 1 i i
2,i
, T observations
, T observations
, Var[ | ] = =( )
ˆ [ ] [ ]
ˆFor FGLS, we need and .ˆ
i i i
i
i i
i i
-1 -1i i
y Xβ ε
β β u
y Xβ Xu ε
y Xβ w w X Ω XΓX I
β XΩ X XΩ y
Γ
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Estimating the RPM
i
1
1
2 1,i
T 22 t 1 it,i
i
i
( ) , = +
= ( )
Var[ | ]= + ( )
(y ) is unbiasedˆ
T K
(but not consistent because T is fixed).
i i i i i i i i i
i i i i i
i i i i
it i
b β X X X w w Xu ε
β u X X Xε
b X Γ X X
x b
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An Estimator for Γ
2 1,i
X X
2 1X ,i
2 1X ,i
i
E[ | ]
Var[ | ]= + ( )
Var[ ] Var E[ | ] E Var[ | ]
= 0 E [ + ( ) ]
+E [ ( ) ]
1Estimate Var[ ] with
N
i i
i i i i
i i i i i
i i
i i
i
b X β
b X Γ X X
b b X b X
Γ X X
Γ X X
b
N1
2 1 N 2 1X ,i i 1 ,i
N N 2 1i 1 i 1 ,i
( )( )
1EstimateE [ ( ) ] with ( )ˆ
N1 1ˆ= ( )( ) ( )ˆN N
i i
i i i i
i i i i
b b b b '
X X X X
Γ b b b b ' - X X
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A Positive Definite Estimator for Γ
N N 2 1i 1 i 1 ,i
i
1 1ˆ= ( )( ) - ( )ˆN N
May not be positive definite. What to do?
(1) The second term converges (in theory) to 0 in T. Drop it.
(2) Various Bayesian "shrinkage" estimators,
(3)
i i i iΓ b b b b ' X X
An ML estimator
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Estimating βi
Ni 1
N 2 1 1 2 1i 1 ,i ,i
2 1 1 1,i
ˆ
{ [ ( ) ]} [ ( ) ]
Best linear unbiased predictor based on GLS is
ˆ ˆ ˆ + ( - ) ( )
{ [ ( ) ] }
GLS i i,OLS
i i i i i
i i GLS i i,OLS i,OLS i GLS i,OLS
-1i i i
β Wb
W Γ X X Γ X X
β Aβ I A b b A β b
A Γ X X Γ
ˆ ˆVar[ | all data]= Var[ ]
ˆVar[ ] Var[ ] [ ( - )]
( - )Var[ ] Var[ ]
-1
i i GLS i
GLS i,OLS i ii i
ii,OLS i i,OLS
β A β A
β b W AA I A
I AW b b
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OLS and FGLS Estimates+----------------------------------------------------+| Overall OLS results for pooled sample. || Residuals Sum of squares = 14.90436 || Standard error of e = .2099898 || Fit R-squared = .8549355 |+----------------------------------------------------++---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |+---------+--------------+----------------+--------+---------+ Constant 2.39132562 .11693429 20.450 .0000 LINCOMEP .88996166 .03580581 24.855 .0000 LRPMG -.89179791 .03031474 -29.418 .0000 LCARPCAP -.76337275 .01860830 -41.023 .0000+------------------------------------------------+| Random Coefficients Model || Residual standard deviation = .3498 || R squared = .5976 || Chi-squared for homogeneity test = 22202.43 || Degrees of freedom = 68 || Probability value for chi-squared= .000000 |+------------------------------------------------++---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |+---------+--------------+----------------+--------+---------+ CONSTANT 2.40548802 .55014979 4.372 .0000 LINCOMEP .39314902 .11729448 3.352 .0008 LRPMG -.24988767 .04372201 -5.715 .0000 LCARPCAP -.44820927 .05416460 -8.275 .0000
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Best Linear Unbiased Country Specific Estimates
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Estimated Price Elasticities
Est imated Country Specific Price E last icit ies
CN TRY
-.40
-.30
-.20
-.10
.00
-.50
2 4 6 8 10 12 14 16 180
EL
AS
T
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Estimated Γ
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Two Step Estimation (Saxonhouse)
it i it
i
i
it it
A Fixed Effects Model
y
Secondary Model
Two approaches
(1) Reduced form is a linear model with time constant z
y
(2) Two step
(a) FEM at step 1
it
i
it i
x β
zδ
x β zδ
i i i i i
2 1i i
i
i
(b) a (a ) v
1 Var[v ] ( )
T
Use weighted least squares regression of a on
i
ii D i i
i
zδ
x XM X x
z
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A Hierarchical Model
it i it
i i i
i
it i it
Fixed Effects Model
y
Secondary Model
u <======== (No u in Saxonhouse)
Two approaches
(1) Reduced form is an REM with time constant z
y u
(2) Two step
(a) FE
it
i
it i
x β
zδ
x β zδ
i i i i i i
2 2 1i i u i
i
M at step 1
(b) a (a ) u v
1 Var[u v ] ( )
T
i
ii D i i
zδ
x XM X x
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Analysis of Fannie Mae
• Fannie Mae• The Funding Advantage• The Pass Through
Passmore, W., Sherlund, S., Burgess, G., “The Effect of Housing Government-Sponsored Enterprises on Mortgage Rates,” 2005, Real Estate Economics
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Two Step Analysis of Fannie-Mae
0 1 2 3i,s,t s,t s,t s,t i,s,t s,t i,s,t
4 5s,t i,s,t s,t i,s,t s,t i,s,t i,s,t
Fannie Mae's GSE Funding Advantage and Pass Through
RM ( LTV) Small Fees
New MtgCo J
i,s, t individual,state,month
1,036,252 observations in 370 state,months.
RM mortgage
LTV= 3 dummy variables for loan to value
Small = dummy variable for small loan
Fees = dummy variable for whether fees paid up front
New = dummy varia
ble for new home
MtgCo = dummy variable for mortgage company
J = dummy variable for whether this is a JUMBO loan
THIS IS THE COEFFICIENT OF INTEREST.
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Average of 370 First Step RegressionsSymbol Variable Mean S.D. Coeff S.E.
RM Rate % 7.23 0.79
J Jumbo 0.06 0.23 0.16 0.05
LTV1 75%-80% 0.36 0.48 0.04 0.04
LTV2 81%-90% 0.15 0.35 0.17 0.05
LTV3 >90% 0.22 0.41 0.15 0.04
New New Home
0.17 0.38 0.05 0.04
Small < $100,000
0.27 0.44 0.14 0.04
Fees Fees paid 0.62 0.52 0.06 0.03
MtgCo Mtg. Co. 0.67 0.47 0.12 0.05 R2 = 0.77
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Second Step Uses 370 Estimates of st
s,t 0
1 s,t
2 s,t
3 s,t
4 s,t
5 s,t
6
GSE Funding Advantage - estimated separately
Risk free cost of credit
Corporate debt spreads - estimated 4 different ways
Prepayment spread
Maturity mismatch risk
A
s,t
7 s,t
8 s,t
9 s,t
10-13 s,t
14-16 s,t
ggregate Demand
Long term interest rate
Market Capacity
Time trend
4 dummy variables for CA, NJ , MD, VA
3 dummy variables for calendar quarters
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Estimates of β1
Second step based on 370 observations. Corrected for
"heteroscedasticity, autocorrelation, and monthly clustering."
Four estimates based on different estimates of corporate
credit spread:
0.07 (0.11) 0
11 1
21 11 2 3 4
1 1 1 1 1 1 1 1 31 1
41 1
.31 (0.11) 017 (0.10) 0.10 (0.11)
Reconcile the 4 estimates with a minimum distance estimator
ˆ( - )
ˆ( - )ˆˆ ˆ ˆ ˆMinimize [( - ),( - ),( - ),( - )]'ˆ( - )
ˆ( - )
-1Ω
Estimated mortgage rate reduction: About 7 basis points. .07%.
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RANDOM EFFECTS - CONTINUOUS
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Continuous Parameter Variation (The Random Parameters Model)
it it
i i
i
i i
i i
i i i
i
y , each observation
, T observations
E[ | ] =
Var[ | ] constant but nonzero
f( | )= g( , ), a density that does
not involve
it i
i i i
i
x β
y Xβ ε
β β u
u X 0
u X Γ
u X u Γ
X
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OLS and GLS Are Consistent
i i
i
i i i i
i i
i i i i i i
2i i i i
, T observations
, T observations
E[ | ] E[ | ] E[ | ]
Var[ | ]
(Discussed earlier - two step GLS)
i i i
i
i i
i
i
y Xβ ε
β β u
y Xβ Xu ε
y Xβ w
w X X u X ε X 0
w X I XΓX
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ML Estimation of the RPM
i
i
i,t i,t
T
i1 iT itt 1
it
Sample data generation
y
Individual heterogeneity
Conditional log likelihood
log f(y ,...,y | , , ) log f(y | , , )
Unconditional log likelihood
logL( , ) log f(y |
i,t i
i i
i i it i
x β
β =β u
X β x β
β,Γ
iT
t 1
i
, , )g( | , )d
(1) Using simulated ML or quadrature, maximize to
estimate , .
(2) Using data and estimated structural parameters,
compute E[ | data , , ]
iit i i iβ
i
x β β β Γ β
β,Γ
β β,Γ
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RP Gasoline Market
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Parameter Covariance matrix
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RP vs. Gen1
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Modeling Parameter Heterogeneity
i,t i,t
i,k i k
Conditional Linear Regression
y
Individual heterogeneity in the means of the parameters
+
E[ | , ]
Heterogeneity in the variances of the parameters
Var[u | data] exp(
i,t i
i i i
i i i
i
x β
β =β Δz u
u X z
h )
Estimation by maximum simulated likelihoodkδ
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Hierarchical Linear Model
COUNTRY = name of country YEAR = year, 1960-1978LGASPCAR = log of consumption per car yLINCOMEP = log of per capita income zLRPMG = log of real price of gasoline x1LCARPCAP = log of per capita number of cars x2
yit = 1i + 2i x1it + 3i x2it + it. 1i=1+1 zi + u1i
2i=2+2 zi + u2i
3i=3+3 zi + u3i
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Estimated HLM
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RP vs. HLM
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Hierarchical Bayesian Estimation
2i,t i,t i,t
i
Sample data generation: y , ~ N[0, ]
Individual heterogeneity: , u ~N[ ]
What information exists about 'the model?'
p(log )= uni
i,t i
i i
x β
β =β u 0,Γ
Prior densities for structural parameters :0
0 0
form density with (large) parameter A
p( ) = N[ , ], e.g., and (large) v
p( ) = Inverse Wishart[...]
p( )= N[ , ]
p( ) = as above.
0
i
β β Σ 0 I
Γ
Priors for parameters of interest :
β β Γ
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Estimation of Hierarchical Bayes Models
(1) Analyze 'posteriors' for hyperparameters ,
(2) Analyze posterior for group level parameters,
Estimators are Means and Variances of posterior
distributions.
Algorithm: Generally,
i
β,Γ
β
Gibbs sampling from posteriors
with resort to laws of large numbers
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A Hierarchical Linear Model
German Health Care DataHsat = β1 + β2AGEit + γi EDUCit + β4 MARRIEDit + εit
γi = α1 + α2FEMALEi + ui
Sample ; all $Setpanel ; Group = id ; Pds = ti $Regress ; For [ti = 7] ; Lhs = newhsat ; Rhs = one,age,educ,married ; RPM = female ; Fcn = educ(n) ; pts = 25 ; halton ; panel ; Parameters$Sample ; 1 – 887 $Create ; betaeduc = beta_i $Dstat ; rhs = betaeduc $Histogram ; Rhs = betaeduc $
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OLS ResultsOLS Starting values for random parameters model...Ordinary least squares regression ............LHS=NEWHSAT Mean = 6.69641 Standard deviation = 2.26003 Number of observs. = 6209Model size Parameters = 4 Degrees of freedom = 6205Residuals Sum of squares = 29671.89461 Standard error of e = 2.18676Fit R-squared = .06424 Adjusted R-squared = .06378Model test F[ 3, 6205] (prob) = 142.0(.0000)--------+--------------------------------------------------------- | Standard Prob. Mean NEWHSAT| Coefficient Error z z>|Z| of X--------+---------------------------------------------------------Constant| 7.02769*** .22099 31.80 .0000 AGE| -.04882*** .00307 -15.90 .0000 44.3352 MARRIED| .29664*** .07701 3.85 .0001 .84539 EDUC| .14464*** .01331 10.87 .0000 10.9409--------+---------------------------------------------------------
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Maximum Simulated LikelihoodNormal exit: 27 iterations. Status=0. F= 12584.28------------------------------------------------------------------Random Coefficients LinearRg ModelDependent variable NEWHSATLog likelihood function -12583.74717Estimation based on N = 6209, K = 7Unbalanced panel has 887 individualsLINEAR regression modelSimulation based on 25 Halton draws--------+--------------------------------------------------------- | Standard Prob. Mean NEWHSAT| Coefficient Error z z>|Z| of X--------+--------------------------------------------------------- |Nonrandom parametersConstant| 7.34576*** .15415 47.65 .0000 AGE| -.05878*** .00206 -28.56 .0000 44.3352 MARRIED| .23427*** .05034 4.65 .0000 .84539 |Means for random parameters EDUC| .16580*** .00951 17.43 .0000 10.9409 |Scale parameters for dists. of random parameters EDUC| 1.86831*** .00179 1044.68 .0000 |Heterogeneity in the means of random parameterscEDU_FEM| -.03493*** .00379 -9.21 .0000 |Variance parameter given is sigmaStd.Dev.| 1.58877*** .00954 166.45 .0000--------+---------------------------------------------------------
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Simulating Conditional Means for Individual Parameters
,,1 1
,
1 1
1
ˆ ˆ( )1 1ˆ ˆ( )ˆ ˆ
ˆ ( | , )ˆ ˆ( )1 1
ˆ ˆ
1 ˆˆ =
i
i
TR it i r iti rr t
i i iTR it i r it
r t
R
ir irr
y
RE
y
R
WeightR
Lw xLw
y XLw x
Posterior estimates of E[parameters(i) | Data(i)]
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“Individual Coefficients”--> Sample ; 1 - 887 $--> create ; betaeduc = beta_i $--> dstat ; rhs = betaeduc $Descriptive StatisticsAll results based on nonmissing observations.==============================================================================Variable Mean Std.Dev. Minimum Maximum Cases Missing==============================================================================All observations in current sample--------+---------------------------------------------------------------------BETAEDUC| .161184 .132334 -.268006 .506677 887 0
Fre
quency
BETAEDUC
-.268 -.157 -.047 .064 .175 .285 .396 .507
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A Hierarchical Linear Model
• A hedonic model of house values• Beron, K., Murdoch, J., Thayer, M.,
“Hierarchical Linear Models with Application to Air Pollution in the South Coast Air Basin,” American Journal of Agricultural Economics, 81, 5, 1999.
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Three Level HLM
ijk
M m mijk jk ijk ijkm 1
mijk
y log of home sale price i, neighborhood j, community k.
y x (linear regression model)
x sq.ft, #baths, lot size, central heat, AC, pool, good view,
age, distance to b
m
qm
Qm q qjk j jk jkq 1
qjk
Sq s qmj j js 1
qmj
each
Random coefficients
N w
N %population poor, race mix, avg age, avg. travel to work,
FBI crime index, school avg. CA achievement test score
E v
E air qu
ality measure, visibility
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Mixed Model Estimation
• WinBUGS: • MCMC • User specifies the model – constructs the Gibbs Sampler/Metropolis Hastings
• MLWin:• Linear and some nonlinear – logit, Poisson, etc.• Uses MCMC for MLE (noninformative priors)
• SAS: Proc Mixed. • Classical• Uses primarily a kind of GLS/GMM (method of moments algorithm for loglinear models)
• Stata: Classical• Several loglinear models – GLAMM. Mixing done by quadrature.• Maximum simulated likelihood for multinomial choice (Arne Hole, user provided)
• LIMDEP/NLOGIT• Classical• Mixing done by Monte Carlo integration – maximum simulated likelihood• Numerous linear, nonlinear, loglinear models
• Ken Train’s Gauss Code• Monte Carlo integration• Mixed Logit (mixed multinomial logit) model only (but free!)
• Biogeme• Multinomial choice models• Many experimental models (developer’s hobby)
Programs differ on the models fitted, the algorithms, the paradigm, and the extensions provided to the simplest RPM, i = +wi.
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GEN 2.1 – RANDOM EFFECTS - DISCRETE
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Heterogeneous Production Model
i,t i i i,t i i,t i,tHealth HEXP EDUC
i country, t=year
Health = health care outcome, e.g., life expectancy
HEXP = health care expenditure
EDUC = education
Parameter heterogeneity:
Discrete? Aids domin
ated vs. QOL dominated
Continuous? Cross cultural heterogeneity
World Health Organization, "The 2000 World Health Report"
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Parameter Heterogeneity• Fixed and Random Effects Models
• Latent common time invariant “effects”• Heterogeneity in level parameter – constant term –
in the model• General Parameter Heterogeneity in Models
• Discrete: There is more than one time of individual in the population – parameters differ across types. Produces a Latent Class Model
• Continuous; Parameters vary randomly across individuals: Produces a Random Parameters Model or a Mixed Model. (Synonyms)
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Parameter Heterogeneity
i,t it
it i,t
(1) Regression model
y ε
(2) Conditional probability model
f(y | x , )
(3) Heterogeneity - how are parameters distributed across
individuals?
(a) Discrete - the populatio
i,t i
i
x β
β
n contains a mixture of J
types of individuals.
(b) Continuous. Parameters are part of the stochastic
structure of the population.
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Discrete Parameter Variation
q ,q
The Latent Class Model
(1) Population is a (finite) mixture of Q types of individuals.
q = 1,...,Q. Q 'classes' differentiated by ( , )
(a) Analyst does not know class memberships. ('latent
β
Q1 Q q=1 q
i,t it i,t q ,q
.')
(b) 'Mixing probabilities' (from the point of view of the
analyst) are ,..., , with 1
(2) Conditional density is
P(y | class q) f(y | x , , )β
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Example: Mixture of Normals
q q
2
it q it qit
q j qq
Q normal populations each with a mean and standard deviation
For each individual in each class at each period,
y y1 1 1f(y | class q) exp = .
22
Panel data,
T 2
it qTi1 iT t 1
T 2
N Q it qTq t 1i 1 q 1
T observations on each individual i,
y1 1f(y ,...,y | class q) exp
22
Log Likelihood
y1 1logL log exp
22
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An Extended Latent Class Model
it itx ,x βit it q
q
(1) There are Q classes, unobservable to the analyst
(2) Class specific model: f(y | ,class q) g(y , )
(3) Conditional class probabilities
Common multinomial logit form for prior class
δ
Q
qq=1
qq QJ
qj 1
q
probabilities
to constrain all probabilities to (0,1) and ensure 1;
multinomial logit form for class probabilities;
exp( ) P(class=q| ) , = 0
exp( )
Note, = log( q Q/ ).
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Log Likelihood for an LC Model
i
x x β
X ,β x βi
i
i,t i,t it i,t q
i
T
i1 i2 i,T q it i,t qt 1
i
Conditional density for each observation is
P(y | ,class q) f(y | , )
Joint conditional density for T observations is
f(y ,y ,...,y | ) f(y | , )
(T may be 1. This is not
iX x βi
i
TQ
i1 i2 i,T q it i,t qq 1 t 1
only a 'panel data' model.)
Maximize this for each class if the classes are known.
They aren't. Unconditional density for individual i is
f(y ,y ,...,y | ) f(y | , )
Log Likelihoo
1β β x βiTN Q
Q 1 Q q it i,t qi 1 q 1 t 1
d
LogL( ,..., ,δ ,...,δ ) log f(y | , )
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Unmixing a Mixed SampleN[1,1] and N[5,1]
Sample ; 1 – 1000$Calc ; Ran(123457)$Create ; lc1=rnn(1,1) ; lc2=rnn(5,1)$ Create ; class=rnu(0,1)$Create ; if(class<.3)ylc=lc1 ; (else)ylc=lc2$ Kernel ; rhs=ylc $Regress ; lhs=ylc;rhs=one;lcm;pts=2;pds=1$
YLC
.045
.090
.135
.180
.224
.000
-2 0 2 4 6 8 10-4
Kernel dens ity estimate for YLC
De
ns
ity
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Mixture of Normals+---------------------------------------------+| Latent Class / Panel LinearRg Model || Dependent variable YLC || Number of observations 1000 || Log likelihood function -1960.443 || Info. Criterion: AIC = 3.93089 || LINEAR regression model || Model fit with 2 latent classes. |+---------------------------------------------++--------+--------------+----------------+--------+--------+----------+|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|+--------+--------------+----------------+--------+--------+----------++--------+Model parameters for latent class 1 ||Constant| 4.97029*** .04511814 110.162 .0000 ||Sigma | 1.00214*** .03317650 30.206 .0000 |+--------+Model parameters for latent class 2 ||Constant| 1.05522*** .07347646 14.361 .0000 ||Sigma | .95746*** .05456724 17.546 .0000 |+--------+Estimated prior probabilities for class membership ||Class1Pr| .70003*** .01659777 42.176 .0000 ||Class2Pr| .29997*** .01659777 18.073 .0000 |+--------+------------------------------------------------------------+| Note: ***, **, * = Significance at 1%, 5%, 10% level. |+---------------------------------------------------------------------+
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Estimating Which Class
i
i
q
i
T
i1 i2 i,T it i,t q ,qt 1
i1 i2 i
Prob[class=q]=
for T observations is
P(y ,y ,...,y | class q) f(y | x , , )
membership is
P(y ,y ,...,y
Prior probability
Joint conditional density
β
Joint density for data and class
i
i
i i
i
i
T
,T q it i,t q ,qt 1
i1 i2 i,T i1 i2 i,Ti1 i2 i,T
i1 i2 i,T i1
,class q) f(y | x , , )
P(y ,y ,...,y ,class q) P(y ,y ,...,y ,class q)P(class q| y ,y ,...,y )
P(y ,y ,...,y ) P(y ,y
β
Posterior probability for class, given the data
i
i
i i
J
i2 i,Tj 1
T
q it i,t q ,qt 1i i1 i2 i,T TQ
q it i,t q ,qq 1 t 1
,...,y ,class q)
Use Bayes Theorem to compute the
f(y | x , , )w(q| data) P(class q| y ,y ,...,y )
f(y | x , , )
posterior probability
β
β
Best guess = the class with the largest posterior probability.
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Posterior for Normal Mixture
i
i
T it qq t 1
q q
iTQ it q
qq 1 t 1q q
iq qQ
iq qq 1
y ˆ1ˆ
ˆ ˆˆ ˆw(q| data) w(q| i)
y ˆ1ˆ
ˆ ˆ
c ˆ =
c ˆ
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Estimated Posterior Probabilities
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More Difficult When the Populations are Close Together
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The Technique Still Works----------------------------------------------------------------------Latent Class / Panel LinearRg ModelDependent variable YLCSample is 1 pds and 1000 individualsLINEAR regression modelModel fit with 2 latent classes.--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Model parameters for latent class 1Constant| 2.93611*** .15813 18.568 .0000 Sigma| 1.00326*** .07370 13.613 .0000 |Model parameters for latent class 2Constant| .90156*** .28767 3.134 .0017 Sigma| .86951*** .10808 8.045 .0000 |Estimated prior probabilities for class membershipClass1Pr| .73447*** .09076 8.092 .0000Class2Pr| .26553*** .09076 2.926 .0034--------+-------------------------------------------------------------
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Estimating E[βi |Xi,yi, β1…, βQ]
q
Q
i qq=1
ˆ(1) Use from the class with the largest estimated probability
(2) Probabilistic
ˆ ˆ = Posterior Prob[class=q|data]i
β
β β
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How Many Classes?
β(1) Q is not a 'parameter' - can't 'estimate' Q with and
(2) Can't 'test' down or 'up' to Q by comparing
log likelihoods. Degrees of freedom for Q+1
vs. Q classes is not well define
1
2
3
d.
(3) Use AKAIKE IC; AIC = -2 logL + 2#Parameters.
For our mixture of normals problem,
AIC 10827.88
AIC 9954.268
AIC 9958.756
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Latent Class Regression
it qit
,q ,q
q itq
Assume normally distributed disturbances
y1f(y | class q)
.
it
it
x β
Mixture of normals sets x β = μ
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Baltagi and Griffin’s Gasoline Data
World Gasoline Demand Data, 18 OECD Countries, 19 yearsVariables in the file are
COUNTRY = name of country YEAR = year, 1960-1978LGASPCAR = log of consumption per carLINCOMEP = log of per capita incomeLRPMG = log of real price of gasoline LCARPCAP = log of per capita number of cars
See Baltagi (2001, p. 24) for analysis of these data. The article on which the analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An Application of Pooling and Testing Procedures," European Economic Review, 22, 1983, pp. 117-137. The data were downloaded from the website for Baltagi's text.
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3 Class Linear Gasoline Model
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Estimated Parameters LCM vs. Gen1 RPM
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An Extended Latent Class Model
it
Class probabilities relate to observable variables (usually
demographic factors such as age and sex).
(1) There are Q classes, unobservable to the analyst
(2) Class specific model: f(y | ,class q) g(itx
it q
qiq qQ
qq 1
y , )
(3) Conditional class probabilities given some information, )
Common multinomial logit form for prior class probabilities
exp( ) P(class=q| , ) , =
exp( )
it
i
ii
i
,x β
z
zδz δ δ 0
zδ
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LC Poisson Regression for Doctor Visits
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Heckman and Singer’s RE Model• Random Effects Model• Random Constants with Discrete Distribution
it it q
q
(1) There are Q classes, unobservable to the analyst
(2) Class specific model: f(y | ,class q) g(y , )
(3) Conditional class probabilities
Common multinomial logit form for prior clas
it itx ,x ,β
Q
qq=1
qq QJ
qj 1
s probabilities
to constrain all probabilities to (0,1) and ensure 1;
multinomial logit form for class probabilities;
exp( ) P(class=q| ) , = 0
exp( )δ
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3 Class Heckman-Singer Form
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The EM Algorithm
i
i,q
i,q
TN Q
c i,q i,t i,ti 1 q 1 t 1
Latent Class is a ' ' model
d 1 if individual i is a member of class q
If d were observed, the complete data log likelihood would be
logL log d f(y | data ,class q)
missing data
(Only one of the Q terms would be nonzero.)
Expectation - Maximization algorithm has two steps
(1) Expectation Step: Form the 'Expected log likelihood'
given the data and a prior guess of the parameters.
(2) Maximize the expected log likelihood to obtain a new
guess for the model parameters.
(E.g., http://crow.ee.washington.edu/people/bulyko/papers/em.pdf)
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Implementing EM for LC Models
, β β β β
β
β
0 0 0 0 0 0 0 0q 1 2 Q q 1 2 Q
q
q
q
Given initial guesses , ,..., , ,...,
E.g., use 1/Q for each and the MLE of from a one class
model. (Must perturb each one slightly, as if all are equal
and all are
0β δ
β
β
0
q
q iq it
the same, the model will satisfy the FOC.)
ˆ ˆˆ(1) Compute F(q|i) = posterior class probabilities, using ,
Reestimate each using a weighted log likelihood
ˆ Maximize wrt F log f(y |
itx β
δ
β
iN T
qi=1 t=1
q
Nq i=1
, )
(2) Reestimate by reestimating
ˆ =(1/N) F(q|i) using old and new ˆ ˆ
Now, return to step 1.
Iterate until convergence.