Bayesian Econometrics Software

4.3 Random-eﬀects linear model

Mathematical representation

y_{i t} = α_{i} + x_{i t}^{'} β + 𝜀_{i t}, 𝜀_{i t} \sim N (0, \frac{1}{τ}), α_{i} \sim N (0, \frac{1}{ω})

(4.4)

the model is estimated using observations from $N$ groups, each group observed for $T_{i}$ periods (balanced or unbalanced panels); the total number of observations is $\sum_{i = 1}^{N} T_{i}$
$y_{i t}$ is the value of the dependent variable for group $i$ , observed in period $t$
$x_{i t}$ is a $K \times 1$ vector that stores the values of the $K$ independent variables for group $i$ , observed in period $t$
$β$ is a $K \times 1$ vector of parameters
$τ$ is the precision of the observation-speciﬁc error term: $σ_{𝜀}^{2} = \frac{1}{τ}$
$α_{i}$ is the group-speciﬁc error term for group $i$
$ω$ is the precision of the group-speciﬁc error term: $σ_{α}^{2} = \frac{1}{ω}$

The mean of the distribution of the

α_{i}

s is restricted to zero and, therefore, these are simply group-speciﬁc errors terms. However, including a constant term in the set of independent variables is valid and leads to a speciﬁcation equivalent to one where the group eﬀects are draws from a normal distribution with mean equal to the parameter associated with the constant term and precision

ω

Priors


Parameter	Probability density function	Default hyperparameters

$β$	$p (β) = \frac{\| P \|^{1 ∕ 2}}{{(2 π)}^{K ∕ 2}} exp \{- \frac{1}{2} {(β - m)}^{'} P (β - m)\}$	$m = 0_{K}$ , $P = 0.001 \cdot I_{K}$
$τ$	$p (τ) = \frac{b_{τ}^{a_{τ}}}{Γ (a_{τ})} τ^{a_{τ} - 1} e^{- τ b_{τ}}$	$a_{τ} = 0.001$ , $b_{τ} = 0.001$
$ω$	$p (ω) = \frac{b_{ω}^{a_{ω}}}{Γ (a_{ω})} ω^{a_{ω} - 1} e^{- ω b_{ω}}$	$a_{ω} = 0.01$ , $b_{ω} = 0.001$

Syntax

[

<model name> =

]

lm_re( y ~ x1 x2 … xK

[

, <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

Before using the lm_re() function the dataset used for estimation must be declared as a panel dataset using the set_pd() function (see section B.13).

BayES automatically drops from the sample used for estimation groups which are observed only once. This is because for these groups the group eﬀect (

α_{i}

) cannot be distinguished from the error term (

𝜀_{i t}

The optional arguments for the random-eﬀects linear model are:³

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"	mean vector of the prior for $β$ ( $K \times 1$ vector); the default value is $0_{K}$
"P"	precision matrix of the prior for $β$ ( $K \times K$ symmetric and positive-deﬁnite matrix); the default value is $0.001 \cdot I_{K}$
"a_tau"	shape parameter of the prior for $τ$ (positive number); the default value is $0.001$
"b_tau"	rate parameter of the prior for $τ$ (positive number); the default value is $0.001$
"a_omega"	shape parameter of the prior for $ω$ (positive number); the default value is $0.01$
"b_omega"	rate parameter of the prior for $ω$ (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

$τ$	tau	precision parameter of the observation-speciﬁc error term, $𝜀_{i t}$

$ω$	omega	precision parameter of the group-speciﬁc error term, $α_{i}$

$σ_{𝜀}$	sigma_e	standard deviation of the observation-speciﬁc error term: $σ_{𝜀} = 1 ∕ τ^{1 ∕ 2}$

$σ_{α}$	sigma_alpha	standard deviation of the group-speciﬁc error term: $σ_{α} = 1 ∕ ω^{1 ∕ 2}$

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a random-eﬀects 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 $β$ , $τ$ and $ω$
x1, $\dots$ ,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)
tau	vector containing the draws from the posterior of $τ$
omega	vector containing the draws from the posterior of $ω$
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
alpha_i	$N \times 1$ vector that stores the group-speciﬁc errors; the values in this vector are not guaranteed to be in the same order as the order in which the groups appear in the dataset used for estimation; use the store() function to associate the values in alpha_i with the observations in the dataset
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()
store()

The random-eﬀects linear model uses the store() function to associate the group eﬀects (alpha_i) with speciﬁc observations and store their values in the dataset used for estimation. The generic syntax for a statement involving the store() function after estimation of a random-eﬀects linear model is:

store( alpha_i, <new variable name>,

[

"model"=<model name>

]

);

Examples

Example 1

myData = import("$BayESHOME/Datasets/dataset2.csv", ",");
myData.constant = ones(rows(myData), 1);
set_pd( year, id, "dataset" = myData);

lm_re( y ~ constant x1 x2 x3 );

Example 2

myData = import("$BayESHOME/Datasets/dataset2.csv", ",");
myData.constant = ones(rows(myData), 1);
set_pd( year, id, "dataset" = myData);

myModel = lm_re(y ~ constant x1 x2 x3,
    "m"=ones(4,1), "P" = 0.1*eye(4,4),
    "a_tau"=0.01, "b_tau"=0.01,
    "a_omega"=0.1, "b_omega"=0.01,
    "burnin"=10000, "draws"=40000, "thin"=4, "chains"=2,
    "logML_CJ" = true, "dataset"=myData);

diagnostics("model"=myModel);

store( alpha_i, re, "model" = myModel );

test( myModel.omega > 8 );

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

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