Bayesian Econometrics Software

4.1 Basic linear model

Mathematical representation

y_{i} = x_{i}^{'} β + 𝜀_{i}, 𝜀_{i} \sim N (0, \frac{1}{τ})

(4.1)

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
$τ$ is the precision of the error term: $σ_{𝜀}^{2} = \frac{1}{τ}$

Priors


Parameter	Probability density function	Default hyperparameters

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

Syntax

[

<model name> =

]

lm( 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

The optional arguments for the simple linear model are:¹

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 $0.001$
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 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 simple 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 $β$ and $τ$
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 $τ$
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/dataset1.csv");
myData.constant = ones(rows(myData), 1);

lm( y ~ constant x1 x2 x3);

Example 2

myData = import("$BayESHOME/Datasets/dataset1.csv");
myData.constant = ones(rows(myData), 1);

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

diagnostics("model"=myModel);

plot(myModel.tau,
    "title"="draws from the posterior of tau",
    "grid"="on");

¹Optional arguments are always given in option-value pairs (eg. "chains"=3).

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