Bayesian Econometrics Software

4.4 Random-coeﬃcients linear model

Mathematical representation

y_{i t} = z_{i t}^{'} γ_{i} + x_{i t}^{'} β + 𝜀_{i t}, 𝜀_{i t} \sim N (0, \frac{1}{τ}), γ_{i} \sim N (\bar{γ}, Ω^{- 1})

(4.5)

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 $\sum_{i = 1}^{N} T_{i}$ and $T_{i}$ could be equal to one for all $i$ (cross-sectional data)
$y_{i t}$ is the value of the dependent variable for group $i$ , observed in period $t$
$z_{i t}$ is a $K \times 1$ vector that stores the values of the $K$ independent variables which are associated with group-speciﬁc coeﬃcients, for group $i$ , observed in period $t$
$x_{i t}$ is an $L \times 1$ vector that stores the values of the $L$ independent variables which are associated with coeﬃcients common to all groups, for group $i$ , observed in period $t$ ( $L$ could be zero)
$γ_{i}$ is a $K \times 1$ vector of parameters associated with group $i$
$\bar{γ}$ is a $K \times 1$ vector of parameters that represents the mean of the $γ_{i}$ s
$Ω$ is a $K \times K$ precision matrix for the distribution of the $γ_{i}$ s
$β$ is an $L \times 1$ vector of parameters
$τ$ is the precision of the error term: $σ_{𝜀}^{2} = \frac{1}{τ}$

Priors


Parameter	Probability density function	Default hyperparameters

$\bar{γ}$	$p (\bar{γ}) = \frac{\| P_{γ} \|^{1 ∕ 2}}{{(2 π)}^{K ∕ 2}} exp \{- \frac{1}{2} {(\bar{γ} - m_{γ})}^{'} P_{γ} (\bar{γ} - m_{γ})\}$	$m_{γ} = 0_{K}$ , $P_{γ} = 0.001 \cdot I_{K}$
$Ω$	$p (Ω) = \frac{\| Ω \|^{\frac{n - K - 1}{2}} \| V^{- 1} \|^{n ∕ 2}}{2^{n K ∕ 2} Γ_{K} (\frac{n}{2})} exp \{- \frac{1}{2} tr (V^{- 1} Ω)\}$	$n = K^{2}$ , $V = \frac{100}{K} \cdot I_{K}$
$β$	$p (β) = \frac{\| P_{β} \|^{1 ∕ 2}}{{(2 π)}^{L ∕ 2}} exp \{- \frac{1}{2} {(β - m_{β})}^{'} P_{β} (β - m_{β})\}$	$m_{β} = 0_{L}$ , $P_{β} = 0.001 \cdot I_{L}$
$τ$	$p (τ) = \frac{b_{τ}^{a_{τ}}}{Γ (a_{τ})} τ^{a_{τ} - 1} e^{- τ b_{τ}}$	$a_{τ} = 0.001$ , $b_{τ} = 0.001$

Syntax

[

<model name> =

]

lm_rc( y ~ z1 z2 … zK

[

|

x1 x2

\dots

]

[

, <options>

]

);

where:

y is the dependent variable name, as it appears in the dataset used for estimation
z1 z2 $\dots$ zK is a list of the names, as they appear in the dataset used for estimation, of the independent variables which are associated with group-speciﬁc coeﬃcients; when a constant term is to be included in the set of group-speciﬁc coeﬃcients this must be requested explicitly
x1 x2 $\dots$ xL is is a list of the names, as they appear in the dataset used for estimation, of the independent variables which are associated with coeﬃcients common to all groups; when a constant term is to be included in the set of common coeﬃcients, this must be requested explicitly

An independent variable could be included in either the x or the z variable list, depending on whether the parameter associated with this variable is common to all groups or not. However, including a variable in both lists would lead to exact multicollinearity and, in this case, BayES will issue an error.

Before using the lm_rc() function the dataset used for estimation must be declared as a panel dataset using the set_pd() function (see section B.13). In the case of cross-sectional data, the dataset still needs to be declared as a panel, but the group-id variable could be constructed as a list of unique integers using, for example, the range() function.

For groups observed only once, a group-speciﬁc parameter associated with a constant term cannot be distinguished from the error term (

𝜀_{i t}

). Thus, a warning is produced when a constant term is included in the z list and the dataset contains at least one group which is observed only once.

The optional arguments for the random-coeﬃcients linear model are:⁴

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_gamma"	mean vector of the prior for $\bar{γ}$ ( $K \times 1$ vector); the default value is $0_{K}$
"P_gamma"	precision matrix of the prior for $\bar{γ}$ ( $K \times K$ symmetric and positive-deﬁnite matrix); the default value is $0.001 \cdot I_{K}$
"V"	scale matrix of the prior for $Ω$ ( $K \times K$ symmetric and positive-deﬁnite matrix); the default value is $\frac{100}{K} \cdot I_{K}$
"n"	degrees-of-freedom parameter of the prior for $Ω$ (real number greater than or equal to $K$ ); the default value is $K^{2}$
"m_beta"	mean vector of the prior for $β$ ( $L \times 1$ vector); the default value is $0_{L}$
"P_beta"	precision matrix of the prior for $β$ ( $L \times L$ symmetric and positive-deﬁnite matrix); the default value is $0.001 \cdot I_{L}$
"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 $0.001$
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


$\bar{γ}$	variable_name	vector of parameters associated with the independent variables in the z list; these are the means of the group-speciﬁc parameters

$β$	variable_name	vector of parameters associated with the independent variables in the x list

$τ$	tau	precision parameter of the error term, $𝜀_{i}$

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

Stored values and post-estimation analysis
If a left-hand-side id value is provided when a random-coeﬃcients 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 $\bar{γ}$ , $β$ , $τ$ and the unique elements of $Ω$
z1, $\dots$ ,zK	vectors containing the draws from the posterior of the mean of the group-speciﬁc coeﬃcients ( $\bar{γ}$ s) associated with variables z1, $\dots$ ,zK (the names of these vectors are the names of the variables that were included in the right-hand side of the model)
x1, $\dots$ ,xL	vectors containing the draws from the posterior of the parameters associated with variables x1, $\dots$ ,xL (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_i_j	vectors containing the draws from the posterior of the unique elements of $Ω$ ; because $Ω$ is symmetric, only $\frac{(K - 1) K}{2} + K$ of its elements are stored (instead of all $K^{2}$ elements); i and j index the row and column of $Ω$ , respectively, at which the corresponding element is located
Omega	$K \times K$ matrix that stores the posterior mean 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
gamma_i	$N \times K$ matrix that stores the group-speciﬁc coeﬃcients for the variables in the z list; the values in this matrix 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 gamma_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-coeﬃcients linear model uses the store() function to associate the group-speciﬁc parameters (gamma_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-coeﬃcients linear model is:

store( gamma_i, <new variable name prefix>,

[

"model"=<model name>

]

);

This statement will generate $K$ additional variables in the dataset used for estimation of the random-coeﬃcients model, with names constructed by prepending the preﬁx provided as the second argument to store() to the names of the variables which are associated with group-speciﬁc coeﬃcients.

Examples

Example 1

Example 2

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

// only the constant term and the coefficient associated with x1 are
// group-specific; the coefficients on x2 and x3 are common to all groups
lm_rc( y ~ constant x1

|

x2 x3);

Example 3

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

// all rhs variables are associated with group-specific coefficients
model1 = lm_rc(y ~ constant x1 x2 x3,
"burnin"=10000, "draws"=40000, "thin"=4, "chains"=2,
"logML_CJ" = true, "dataset"=myData);
store( gamma_i, rc1_, "model" = model1 );

// only the constant term and the coefficient associated with x1 are
// group-specific; the coefficients on x2 and x3 are common to all groups
model2 = lm_rc( y ~ constant x1

|

x2 x3,
"burnin"=10000, "draws"=40000, "thin"=4, "chains"=2,
"logML_CJ" = true, "dataset"=myData);
store( gamma_i, rc2_, "model" = model2 );

pmp( { model1, model2 } );

⁴Optional arguments are always given in option-value pairs (eg. "chains"=3).

[next] [prev] [prev-tail] [front] [up]