Bayesian Econometrics Software

4.2 Heteroskedastic linear model

Mathematical representation

\begin{array}{l} y_{i} = x_{i}^{'} β + 𝜀_{i} & 𝜀_{i} \sim N (0, \frac{1}{τ_{i}}) & (4.2) \\ log τ_{i} = w_{i}^{'} δ + v_{i} & v_{i} \sim N (0, \frac{1}{ϕ}) & (4.3) \end{array}

the model is estimated using $N$ observations
$y_{i}$ is the value of the dependent variable for observation $i$
$x_{i}$ is a $K \times 1$ vector that stores the values of the $K$ independent variables for observation $i$
$β$ is a $K \times 1$ vector of parameters
$τ_{i}$ is the precision of the error term for observation $i$ : $σ_{𝜀_{i}}^{2} = \frac{1}{τ_{i}}$
$w_{i}$ is an $L \times 1$ vector that stores the values of the $L$ variables that determine the precision of the error term for observation $i$
$δ$ is an $L \times 1$ vector of parameters
$ϕ$ is the precision of the error term in the equation for $log τ_{i}$ : $σ_{v}^{2} = \frac{1}{ϕ}$

Priors


Parameter	Probability density function	Default hyperparameters

$β$	$p (β) = \frac{\| P_{β} \|^{1 ∕ 2}}{{(2 π)}^{K ∕ 2}} exp \{- \frac{1}{2} {(β - m_{β})}^{'} P_{β} (β - m_{β})\}$	$m_{β} = 0_{K}$ , $P_{β} = 0.001 \cdot I_{K}$
$δ$	$p (δ) = \frac{\| P_{δ} \|^{1 ∕ 2}}{{(2 π)}^{L ∕ 2}} exp \{- \frac{1}{2} {(δ - m_{δ})}^{'} P_{δ} (δ - m_{δ})\}$	$m_{δ} = 0_{L}$ , $P_{δ} = 0.01 \cdot I_{L}$
$ϕ$	$p (ϕ) = \frac{b_{ϕ}^{a_{ϕ}}}{Γ (a_{ϕ})} ϕ^{a_{ϕ} - 1} e^{- ϕ b_{ϕ}}$	$a_{ϕ} = 0.001$ , $b_{ϕ} = 0.001$

Syntax

[

<model name> =

]

lm( y ~ x1 x2 … xK

|

w1 w2

\dots

[

,<options>

]

);

where:

y is the dependent variable name, as it appears in the dataset used for estimation
x1 x2 $\dots$ xK is a list of the $K$ independent variable names, as they appear in the dataset used for estimation; when a constant term is to be included in the model, this must be requested explicitly
w1 w2 $\dots$ wK is a list of the names of the $L$ variables which determine the precision of $𝜀_{i}$ , as they appear in the dataset used for estimation; when a constant term is to be included in the precision equation, this must be requested explicitly

The optional arguments for the heteroskedastic linear model are:²

Gibbs parameters

"chains"	number of chains to run in parallel (positive integer); the default value is 1
"burnin"	number of burn-in draws per chain (positive integer); the default value is 10000
"draws"	number of retained draws per chain (positive integer); the default value is 20000
"thin"	value of the thinning parameter (positive integer); the default value is 1
"seed"	value of the seed for the random-number generator (positive integer); the default value is 42
Hyperparameters

"m_beta"	mean vector of the prior for $β$ ( $K \times 1$ vector); the default value is $0_{K}$
"P_beta"	precision matrix of the prior for $β$ ( $K \times K$ symmetric and positive-deﬁnite matrix); the default value is $0.001 \cdot I_{K}$
"m_delta"	mean vector of the prior for $δ$ ( $L \times 1$ vector); the default value is $0_{L}$
"P_delta"	precision matrix of the prior for $δ$ ( $L \times L$ symmetric and positive-deﬁnite matrix); the default value is $0.01 \cdot I_{L}$
"a_phi"	shape parameter of the prior for $ϕ$ (positive number); the default value is $0.001$
"b_phi"	rate parameter of the prior for $ϕ$ (positive number); the default value is $0.001$
Dataset and log-marginal likelihood

"dataset"	the id value of the dataset that will be used for estimation; the default value is the ﬁrst dataset in memory (in alphabetical order)
"logML_CJ"	boolean indicating whether the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood should be calculated (true $\|$ false); the default value is false

Reported Parameters


$β$	variable_name	vector of parameters associated with the independent variables in the observed equation

$δ$	variable_name	vector of parameters associated with the independent variables in the precision equation

$ϕ$	phi	precision parameter of the error term in the precision equation, $v_{i}$

$σ_{v}$	sigma_v	standard deviation of the error term in the precision equation: $σ_{v} = 1 ∕ ϕ^{1 ∕ 2}$

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a heteroskedastic linear model is created, then the following results are saved in the model item and are accessible via the ‘.’ operator:

Samples	a matrix containing the draws from the posterior of $β$ , $δ$ and $ϕ$
y$x1, $\dots$ ,y$xK	vectors containing the draws from the posterior of the parameters associated with variables x1, $\dots$ ,xK (the names of these vectors are the names of the variables that were included in the right-hand side of the model, prepended by y$, where y is the name of the dependent variable; this is done so that the samples on the parameters associated with a variable that appears in both x and w lists can be distinguished)
logtau$z1, $\dots$ , logtau$zL	vectors containing the draws from the posterior of the parameters associated with variables w1, $\dots$ ,wL (the names of these vectors are the names of the variables that were included in the w list, in the right-hand side of the model, prepended by logtau$; this is done so that the samples on the parameters associated with a variable that appears in both x and w lists can be distinguished)
phi	vector containing the draws from the posterior of $ϕ$
logML	the Lewis & Raftery (1997) approximation of the log-marginal likelihood
logML_CJ	the Chib (1995)/Chib & Jeliazkov (2001) approximation to the log-marginal likelihood; this is available only if the model was estimated with the "logML_CJ"=true option
nchains	the number of chains that were used to estimate the model
nburnin	the number of burn-in draws per chain that were used when estimating the model
ndraws	the total number of retained draws from the posterior ( $=$ chains $\cdot$ draws)
nthin	value of the thinning parameter that was used when estimating the model
nseed	value of the seed for the random-number generator that was used when estimating the model

Additionally, the following functions are available for post-estimation analysis (see section B.14):

diagnostics()
test()
pmp()

Examples

Example 1

myData = import("$BayESHOME/Datasets/dataset3.csv");
myData.constant = ones(rows(myData), 1);

lm( y ~ constant x1 x2 x3 | constant z1 z2);

Example 2

myData = import("$BayESHOME/Datasets/dataset3.csv");
myData.constant = ones(rows(myData), 1);

myModel = lm(y ~ constant x1 x2 x3 | constant z1 z2,
    "m_beta"=ones(4,1), "P_beta" = 0.01*eye(4,4),
    "m_delta"=ones(3,1), "P_delta" = 0.1*eye(3,3),
    "a_phi"=0.01, "b_phi"=0.001,
    "burnin"=10000, "draws"=40000, "thin"=4, "chains"=2,
    "logML_CJ" = true, "dataset"=myData);

diagnostics("model"=myModel);

plotdraws(phi, "model"=myModel);
plotdraws(logtau$z2, "model"=myModel);

²Optional arguments are always given in option-value pairs (eg. "chains"=3).

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