Bayesian Econometrics Software

5.2 Ineﬃciency-eﬀects stochastic frontier

Mathematical representation

y_{i} = x_{i}^{'} β + v_{i} \pm u_{i}, v_{i} \sim N (0, \frac{1}{τ}), u_{i} \sim D (𝜃, z_{i})

(5.2)

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
$z_{i}$ is an $L \times 1$ vector that stores the values of the $L$ determinants of ineﬃciency for observation $i$
$τ$ is the precision of the noise component of the error term: $σ_{v}^{2} = \frac{1}{τ}$
ui is the ineﬃciency component of the error term and it can have any non-negative distribution, represented in the equation above by D 𝜃,zi; BayES supports the following distributions for ui:
- exponential: $p (u_{i}) = λ_{i} e^{- λ_{i} u_{i}}$ , with $λ_{i} = e^{z_{i}^{'} δ}$ and $δ$ being an $L \times 1$ vector of parameters to be estimated
- truncated normal: $p (u_{i}) = \frac{ϕ^{1 ∕ 2} exp \{- \frac{ϕ}{2} {(u_{i} - μ_{i})}^{2}\}}{{(2 π)}^{1 ∕ 2} Φ^{1 ∕ 2} (ϕ^{1 ∕ 2} μ_{i})}$ , with $μ_{i} = z_{i}^{'} δ$ and $δ$ being an $L \times 1$ vector of parameters and $ϕ$ a scalar parameter to be estimated

When

u_{i}

enters the speciﬁcation with a plus sign then the model represents a cost frontier, while when

u_{i}

enters with a minus sign the model represents a production frontier. For the eﬃciency scores generated by a stochastic frontier model to be meaningful, the dependent variable in both cases must be in logarithms.

Priors


Parameter	Probability density function	Default hyperparameters

Common to all models
$β$	$p (β) = \frac{\| P_{β} \|^{1 ∕ 2}}{{(2 π)}^{K ∕ 2}} exp \{- \frac{1}{2} {(β - m_{β})}^{'} P_{β} (β - m_{β})\}$	$m_{β} = 0_{K}$ , $P_{β} = 0.001 \cdot I_{K}$
$τ$	$p (τ) = \frac{b_{τ}^{a_{τ}}}{Γ (a_{τ})} τ^{a_{τ} - 1} e^{- τ b_{τ}}$	$a_{τ} = 0.001$ , $b_{τ} = 0.001$
Exponential model
$δ$	$p (δ) = \frac{\| P_{δ} \|^{1 ∕ 2}}{{(2 π)}^{L ∕ 2}} exp \{- \frac{1}{2} {(δ - m_{δ})}^{'} P_{δ} (δ - m_{δ})\}$	$m_{δ} = 0_{L}$ , $P_{δ} = 0.01 \cdot I_{L}$
Truncated normal model
$δ$	$p (δ) = \frac{\| P_{δ} \|^{1 ∕ 2}}{{(2 π)}^{L ∕ 2}} exp \{- \frac{1}{2} {(δ - m_{δ})}^{'} P_{δ} (δ - m_{δ})\}$	$m_{δ} = 0_{L}$ , $P_{δ} = 0.01 \cdot I_{L}$
$ϕ$	$p (ϕ) = \frac{b_{ϕ}^{a_{ϕ}}}{Γ (a_{ϕ})} ϕ^{a_{ϕ} - 1} e^{- ϕ b_{ϕ}}$	$a_{ϕ} = 4$ , $b_{ϕ} = 0.5$

Syntax

[

<model name> =

]

sf( y ~ x1 x2

\dots

|

z1 z2

\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
z1 z2 $\dots$ zL is a list of the $L$ variable names that aﬀect $u_{i}$ (determinants of ineﬃciency), 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 ineﬃciency-eﬀects stochastic frontier 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
Model specification

"udist"	speciﬁcation of the distribution of the ineﬃciency component of the error term; the following options are available, corresponding to the distributions presented at the beginning of this section: "exp" "tnorm" the default value is "exp"
"production"	boolean specifying the type of frontier (production/cost); it could be set to either true (production) or false (cost); the default value is true
Hyperparameters

Common to all models
"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 $0.001 \cdot I_{K}$
"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 $0.001$
Exponential model
"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 $0.01 \cdot I_{L}$
Truncated normal model
"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 $4$
"b_phi"	rate parameter of the prior for $ϕ$ (positive number); the default value is $0.5$
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


Common to all models

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

$τ$	tau	precision parameter of the noise component of the error term, $v_{i}$

$σ_{v}$	sigma_v	standard deviation of the noise component of the error term, $σ_{v} = 1 ∕ τ^{1 ∕ 2}$

Exponential model

$δ$	variable_name	vector of parameters associated with the independent variables in the z list

Truncated normal model

$δ$	variable_name	vector of parameters associated with the independent variables in the z list

$ϕ$	phi	precision parameter of the distribution of the ineﬃciency component of the error term, $u_{i}$

$σ_{u}$	sigma_u	scale parameter of the ineﬃciency component of the error term: $σ_{u} = 1 ∕ ϕ^{1 ∕ 2}$ .

Stored values and post-estimation analysis
If a left-hand-side id value is provided when an ineﬃciency-eﬀects stochastic frontier 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 $τ$ , and, depending on the estimated model, $δ$ , or $δ$ 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 z lists can be distinguished)
tau	vector containing the draws from the posterior of $τ$
u$z1, $\dots$ ,u$zL	vectors containing the draws from the posterior of the parameters associated with variables z1, $\dots$ ,zL (the names of these vectors are the names of the variables that were included in the z list, in the right-hand side of the model, prepended by u$; this is done so that the samples on the parameters associated with a variable that appears in both x and z lists can be distinguished)
phi	vector containing the draws from the posterior of $ϕ$ (available after the estimation of the truncated-normal model)
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
eff_i	$N \times 1$ vector that stores the expected values of the observation-speciﬁc eﬃciency scores, $E (e^{- u_{i}})$ ; the values in this vector are not guaranteed to be in the same order as the order in which the observations appear in the dataset used for estimation; use the store() function to associate the values in eff_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()
mfx()

The ineﬃciency-eﬀects stochastic frontier model uses the store() function to associate the estimates of the eﬃciency scores (eff_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 an ineﬃciency-eﬀects stochastic frontier model is:

store( eff_i, <new variable name>

[

, "model"=<model name>

]

);

The ineﬃciency-eﬀects stochastic frontier model uses the mfx() function to calculate and report the marginal eﬀects of the variables in the z list on the expected value of $u$ and on the expected value of the eﬃciency score ( $= e^{- u}$ ). The two types of marginal eﬀects can be requested by setting the "type" argument of the mfx() function equal to 1 or 2. The generic syntax for a statement involving the mfx() function after estimation of an ineﬃciency-eﬀects stochastic frontier model is:

mfx(

[

"type"=1

]

[

, "point"=<point of calculation>

]

[

, "model"=<model name>

]

);

and:

mfx( "type"=2

[

, "point"=<point of calculation>

]

[

, "model"=<model name>

]

);

for calculation of the marginal eﬀects on $E (u)$ and on $E (e^{- u})$ , respectively. The default value of the "type" option is 1. See the general documentation of the mfx() function (section B.14) for details on the other optional arguments.

Examples

Example 1

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

sf( y ~ constant x2 x3 x4 | constant z2 z3, "production"=false );

Example 2

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

expSF = sf( y ~ constant x2 x3 x4 | constant z2 z3,
"udist"="exp", "production"=false );

tnormSF = sf( y ~ constant x2 x3 x4 | constant z2 z3,
"udist"="tnorm", "production"=false );

store( eff_i, eff_exp, "model" = expSF );
store( eff_i, eff_tnorm, "model" = tnormSF );

mfx( "point"="mean", "model"=expSF, "type"=1 );
mfx( "point"="mean", "model"=tnormSF, "type"=1 );

mfx( "point"="mean", "model"=expSF, "type"=2 );
mfx( "point"="mean", "model"=tnormSF, "type"=2 );

pmp( { expSF, tnormSF } );

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

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