### 4.3 Random-eﬀects linear model

Mathematical representation

 ${y}_{it}={\alpha }_{i}+{\mathbf{x}}_{it}^{\prime }\beta +{𝜀}_{it},\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{𝜀}_{it}\sim \mathrm{N}\left(0,\frac{1}{\tau }\right),\phantom{\rule{1em}{0ex}}{\alpha }_{i}\sim \mathrm{N}\left(0,\frac{1}{\omega }\right)$ (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 ${\sum }_{i=1}^{N}{T}_{i}$
• ${y}_{it}$ is the value of the dependent variable for group $i$, observed in period $t$
• ${\mathbf{x}}_{it}$ is a $K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector that stores the values of the $K$ independent variables for group $i$, observed in period $t$
• $\beta$ is a $K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector of parameters
• $\tau$ is the precision of the observation-speciﬁc error term: ${\sigma }_{𝜀}^{2}=\frac{1}{\tau }$
• ${\alpha }_{i}$ is the group-speciﬁc error term for group $i$
• $\omega$ is the precision of the group-speciﬁc error term: ${\sigma }_{\alpha }^{2}=\frac{1}{\omega }$

 The mean of the distribution of the ${\alpha }_{i}$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 $\omega$.

Priors

 Parameter Probability density function Default hyperparameters $\beta$ $p\left(\beta \right)=\frac{|\mathbf{P}{|}^{1∕2}}{{\left(2\pi \right)}^{K∕2}}exp\left\{-\frac{1}{2}{\left(\beta -\mathbf{m}\right)}^{\prime }\mathbf{P}\left(\beta -\mathbf{m}\right)\right\}$ $\mathbf{m}={\mathbf{0}}_{K}$, $\mathbf{P}=0.001\cdot {\mathbf{I}}_{K}$ $\tau$ $p\left(\tau \right)=\frac{{b}_{\tau }^{{a}_{\tau }}}{\Gamma \left({a}_{\tau }\right)}{\tau }^{{a}_{\tau }-1}{e}^{-\tau {b}_{\tau }}$ ${a}_{\tau }=0.001$, ${b}_{\tau }=0.001$ $\omega$ $p\left(\omega \right)=\frac{{b}_{\omega }^{{a}_{\omega }}}{\Gamma \left({a}_{\omega }\right)}{\omega }^{{a}_{\omega }-1}{e}^{-\omega {b}_{\omega }}$ ${a}_{\omega }=0.01$, ${b}_{\omega }=0.001$

Syntax

$\left[$<model name> = $\right]$ lm_re( y ~ x1 x2  xK $\left[$, <options> $\right]$ );

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 (${\alpha }_{i}$) cannot be distinguished from the error term (${𝜀}_{it}$).

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 $\beta$ ($K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ vector); the default value is ${\mathbf{0}}_{K}$ "P" precision matrix of the prior for $\beta$ ($K\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}K$ symmetric and positive-deﬁnite matrix); the default value is $0.001\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}{\mathbf{I}}_{K}$ "a_tau" shape parameter of the prior for $\tau$ (positive number); the default value is $0.001$ "b_tau" rate parameter of the prior for $\tau$ (positive number); the default value is $0.001$ "a_omega" shape parameter of the prior for $\omega$ (positive number); the default value is $0.01$ "b_omega" rate parameter of the prior for $\omega$ (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

 $\beta$ variable_name vector of parameters associated with the independent variables $\tau$ tau precision parameter of the observation-speciﬁc error term, ${𝜀}_{it}$ $\omega$ omega precision parameter of the group-speciﬁc error term, ${\alpha }_{i}$ ${\sigma }_{𝜀}$ sigma_e standard deviation of the observation-speciﬁc error term: ${\sigma }_{𝜀}=1∕{\tau }^{1∕2}$ ${\sigma }_{\alpha }$ sigma_alpha standard deviation of the group-speciﬁc error term: ${\sigma }_{\alpha }=1∕{\omega }^{1∕2}$

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 $\beta$, $\tau$ and $\omega$ 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 $\tau$ omega vector containing the draws from the posterior of $\omega$ 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 $N\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}1$ 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>, $\left[$"model"=<model name>$\right]$ );

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).