7.3 Instrumental variable
This inferential approach is used when there are endogeneity issues, that is, when the stochastic error is not independent of the regressors. This, in turn, generates bias in posterior mean estimates when we use an inferential approach that does not account for this issue. Endogeneity can be caused by reverse causality, omitting relevant correlated variables, or measurement error in the regressors.42
Let’s specify the dependent variable as a linear function of one endogenous regressor and some exogenous regressors. That is,
y_{i} = \boldsymbol{x}_{ei}^{\top} \boldsymbol{\beta}_1 + \beta_s x_{si} + \mu_{i}
where
x_{si} = \boldsymbol{x}_{ei}^{\top} \boldsymbol{\gamma}_1 + \boldsymbol{z}_i^{\top} \boldsymbol{\gamma}_2 + v_{i},
x_s is the variable that generates the endogeneity issues (\mathbb{E}[\mu \mid x_{s}] \neq 0), \boldsymbol{x}_e are K_1 exogenous regressors (\mathbb{E}[\mu \mid \boldsymbol{x}_{e}] = \boldsymbol{0}), and \boldsymbol{z} are K_2 instruments. The instruments are regressors that drive x_s (\mathbb{E}[x_{s} \boldsymbol{z}] \neq \boldsymbol{0}), but do not have a direct effect on y (\mathbb{E}[y \boldsymbol{z} \mid x_s] = \boldsymbol{0}). The equation for y is called the structural equation, and it is the equation that the researcher is ultimately interested in.
Assuming
(u_{i},v_i)^{\top} \stackrel{i.i.d.}{\thicksim} N(0,\boldsymbol{\Sigma}),
where \boldsymbol{\Sigma}=[\sigma_{lm}], l,m=1,2, the likelihood function is
p(\boldsymbol{\beta},\boldsymbol{\gamma},\boldsymbol{\Sigma} \mid \boldsymbol{y},\boldsymbol{X},\boldsymbol{Z}) = \frac{1}{(2\pi)^\frac{N}{2}|\boldsymbol{\Sigma}|^\frac{N}{2}} \exp\left\{-\frac{1}{2} \sum_{i=1}^N (y_i-\boldsymbol{x}_i^{\top} \boldsymbol{\beta}, x_{si} -\boldsymbol{w}_i^{\top} \boldsymbol{\gamma}) \boldsymbol{\Sigma}^{-1} \begin{pmatrix} y_i - \boldsymbol{x}_i^{\top} \boldsymbol{\beta} \\ x_{si} - \boldsymbol{w}_i^{\top} \boldsymbol{\gamma} \end{pmatrix} \right\},
where
\boldsymbol{\beta}=\begin{bmatrix} \boldsymbol{\beta}_1^{\top} & \beta_s \end{bmatrix}^{\top}, \quad \boldsymbol{\gamma}=\begin{bmatrix} \boldsymbol{\gamma}_1^{\top} & \boldsymbol{\gamma}_2^{\top} \end{bmatrix}^{\top}, \quad \boldsymbol{x}_i=\begin{bmatrix} \boldsymbol{x}_{ei}^{\top} & x_{si} \end{bmatrix}^{\top}, \quad \boldsymbol{w}_i=\begin{bmatrix} \boldsymbol{x}_{ei}^{\top} & \boldsymbol{z}_{i}^{\top} \end{bmatrix}^{\top}.
We obtain the standard conditional posterior densities by specifying the following independent priors:
\boldsymbol{\gamma} \sim N(\boldsymbol{\gamma}_0, \boldsymbol{G}_0), \quad \boldsymbol{\beta} \sim N(\boldsymbol{\beta}_0, \boldsymbol{B}_0), \quad \boldsymbol{\Sigma}^{-1} \sim W(\alpha_0, \boldsymbol{\Psi}_0).
In particular, the conditional distributions are:
\boldsymbol{\beta} \mid \boldsymbol{\gamma}, \boldsymbol{\Sigma}, \boldsymbol{y}, \boldsymbol{X}, \boldsymbol{Z} \sim N(\boldsymbol{\beta}_n, \boldsymbol{B}_n),
\boldsymbol{\gamma} \mid \boldsymbol{\beta}, \boldsymbol{\Sigma}, \boldsymbol{y}, \boldsymbol{X}, \boldsymbol{Z} \sim N(\boldsymbol{\gamma}_n, \boldsymbol{G}_n),
\boldsymbol{\Sigma}^{-1} \mid \boldsymbol{\beta}, \boldsymbol{\gamma}, \boldsymbol{y}, \boldsymbol{X}, \boldsymbol{Z} \sim W(\alpha_n, \boldsymbol{\Psi}_n),
where
\boldsymbol{\beta}_n = \boldsymbol{B}_n \left( \boldsymbol{B}_0^{-1} \boldsymbol{\beta}_0 + \omega_1^{-1} \sum_{i=1}^{N} \left[ \boldsymbol{x}_i \left( y_i - \frac{\sigma_{12}(x_{si} - \boldsymbol{w}_i^{\top} \boldsymbol{\gamma})}{\sigma_{22}} \right) \right] \right),
\boldsymbol{B}_n = \left( \omega_1^{-1} \sum_{i=1}^{N} \boldsymbol{x}_i \boldsymbol{x}_i^{\top} + \boldsymbol{B}_0^{-1} \right)^{-1}, \quad \omega_1 = \sigma_{11} - \frac{\sigma_{12}^2}{\sigma_{22}},
\boldsymbol{G}_n = \left( \omega_2^{-1} \sum_{i=1}^{N} \boldsymbol{w}_i \boldsymbol{w}_i^{\top} + \boldsymbol{G}_0^{-1} \right)^{-1}, \quad \boldsymbol{\gamma}_n = \boldsymbol{G}_n \left( \boldsymbol{G}_0^{-1} \boldsymbol{\gamma}_0 + \omega_2^{-1} \sum_{i=1}^{N} \left[ \boldsymbol{w}_i \left( x_{si} - \frac{\sigma_{12} (y_i - \boldsymbol{x}_i^{\top} \boldsymbol{\beta})}{\sigma_{11}} \right) \right] \right),
\omega_2 = \sigma_{22} - \frac{\sigma_{12}^2}{\sigma_{11}}, \quad \boldsymbol{\Psi}_n = \left[ \boldsymbol{\Psi}_0^{-1} + \sum_{i=1}^N \begin{pmatrix} y_i - \boldsymbol{x}_i^{\top} \boldsymbol{\beta} \\ x_{si} - \boldsymbol{w}_i^{\top} \boldsymbol{\gamma} \end{pmatrix} (y_i - \boldsymbol{x}_i^{\top} \boldsymbol{\beta}, x_{si} - \boldsymbol{w}_i^{\top} \boldsymbol{\gamma}) \right]^{-1},
\alpha_n = \alpha_0 + N, \quad \sigma_{lj} \text{ are the elements of } \boldsymbol{\Sigma}.
We also use a Gibbs sampling algorithm in this model since we have standard conditional posterior distributions.
Example: Simulation exercise
Let’s simulate the simple process y_i=\beta_1+\beta_2x_{si}+\mu_i and x_{si}=\gamma_1+\gamma_2z_i+v_i where [\mu_i \ v_i]^{\top} \sim N(\boldsymbol{0},\boldsymbol{\Sigma}), \boldsymbol{\Sigma}=[\sigma_{lj}] such that \sigma_{12} \neq 0, i=1,2,\dots,100.
Observe that \mu\mid v\sim N\left(\frac{\sigma_{12}}{\sigma_{22}}v,\sigma_{11}-\frac{\sigma_{21}^2}{\sigma_{22}}\right), this implies that \mathbb{E}[\mu\mid x_s]=\mathbb{E}[\mu\mid v]=\frac{\sigma_{12}}{\sigma_{22}}v\neq 0 given \sigma_{12}\neq 0 and \mathbb{E}[\mu\mid z]=0. Let’s set all location parameters equal to 1, and \sigma_{11}=\sigma_{22}=1, \sigma_{12}=0.8, and z\sim N(0,1).
We know from the large sampling properties of the posterior mean that this converges to the maximum likelihood estimator (see Section 1.1, and Lehmann and Casella (2003), Van der Vaart (2000)), which in this setting is
\hat{\beta}_2=\frac{\widehat{\text{Cov}}(x_s,y)}{\widehat{\text{Var}}(x_s)}
which converges in probability to
\beta_2+\frac{\sigma_{12}}{\sigma_{22}\text{Var}(x_s)}=\beta_2+\frac{\sigma_{12}}{\sigma_{22}(\gamma_2^2\text{Var}(z)+\sigma_{22})}=1.4,
that is, the asymptotic bias when using the posterior mean of a linear regression without taking into account endogeneity is 0.4 in this example.
We assess the sampling performance of the Bayesian estimators by simulating this setting 100 times. The following code demonstrates how to do this using a linear model that does not account for the endogeneity issue (see Section 6.1), as well as how to implement the instrumental variable model. In this setup, we use \boldsymbol{B}_0 = 1000 \mathbf{I}_2, \boldsymbol{\beta}_0 = \mathbf{0}_2, and the parameters of the inverse gamma distribution are set to 0.0005. For the instrumental variable model, we additionally set \boldsymbol{\gamma}_0 = \mathbf{0}_2, \boldsymbol{G}_0 = 1000 \mathbf{I}_2, \alpha_0 = 3, and \boldsymbol{\Psi}_0 = 3 \mathbf{I}_2.
rm(list = ls()); set.seed(010101)
N <- 100; k <- 2
B <- rep(1, k); G <- rep(1, 2); s12 <- 0.8
SIGMA <- matrix(c(1, s12, s12, 1), 2, 2)
z <- rnorm(N); Z <- cbind(1, z); w <- matrix(1,N,1); S <- 100
U <- replicate(S, MASS::mvrnorm(n = N, mu = rep(0, 2), SIGMA))
x <- G[1] + G[2]*z + U[,2,]; y <- B[1] + B[2]*x + U[,1,]
# Hyperparameters
d0 <- 0.001/2; a0 <- 0.001/2
b0 <- rep(0, k); c0 <- 1000; B0 <- c0*diag(k)
B0i <- solve(B0); g0 <- rep(0, 2)
G0 <- 1000*diag(2); G0i <- solve(G0)
nu <- 3; Psi0 <- nu*diag(2)
# MCMC parameters
mcmc <- 5000; burnin <- 1000
tot <- mcmc + burnin; thin <- 1
# Gibbs sampling
Gibbs <- function(x, y){
Data <- list(y = y, x = x, w = w, z = Z)
Mcmc <- list(R = mcmc, keep = thin, nprint = 0)
Prior <- list(md = g0, Ad = G0i, mbg = b0, Abg = B0i, nu = nu, V = Psi0)
RestIV <- bayesm::rivGibbs(Data = Data, Mcmc = Mcmc, Prior = Prior)
PostBIV <- mean(RestIV[["betadraw"]])
ResLM <- MCMCpack::MCMCregress(y ~ x + w - 1, b0 = b0, B0 = B0i, c0 = a0, d0 = d0)
PostB <- mean(ResLM[,1]); Res <- c(PostB,PostBIV)
return(Res)
}
PosteriorMeans <- sapply(1:S, function(s) {Gibbs(x = x[,s], y = y[,s])})##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
##
##
## Starting Gibbs Sampler for Linear IV Model
##
## nobs= 100 ; 2 instruments; 1 included exog vars
## Note: the numbers above include intercepts if in z or w
##
## Prior Parms:
## mean of delta
## [1] 0 0
## Adelta
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## mean of beta/gamma
## [1] 0 0
## Abeta/gamma
## [,1] [,2]
## [1,] 0.001 0.000
## [2,] 0.000 0.001
## Sigma Prior Parms
## nu= 3 V=
## [,1] [,2]
## [1,] 3 0
## [2,] 0 3
##
## MCMC parms:
## R= 5000 keep= 1 nprint= 0
## ## [1] 1.408272 1.035428Model <- c(replicate(S, "Ordinary"), replicate(S, "Instrumental"))
postmeans <- c(t(PosteriorMeans))
df <- data.frame(postmeans, Model, stringsAsFactors = FALSE)
library(ggplot2); library(latex2exp)
histExo <- ggplot(df, aes(x = postmeans, fill = Model)) + geom_histogram(bins = 40, position = "identity", color = "black", alpha = 0.5) + labs(title = "Overlayed Histograms", x = "Value", y = "Count") + scale_fill_manual(values = c("blue", "red")) + geom_vline(aes(xintercept = mean(postmeans[1:S])), color = "black", linewidth = 1, linetype = "dashed") + geom_vline(aes(xintercept = mean(postmeans[101:200])), color = "black", linewidth = 1, linetype = "dashed") + geom_vline(aes(xintercept = B[2]), color = "green", linewidth = 1, linetype = "dashed") + xlab(TeX("$E[\\beta_2]$")) + ylab("Frequency") + ggtitle("Histogram: Posterior means simulating 100 samples")
histExo 
The Figure displays the histograms of the posterior means of \beta_2 using the ordinary model, which does not account for endogeneity, and the instrumental variable model. On one hand, the mean of the posterior means for the ordinary model is 1.41 (black dashed line in the red histogram), implying a bias of 0.41, which is very close to the population bias of 0.40. On the other hand, the mean of the posterior means for the instrumental variable model is 1.04 (black dashed line in the blue histogram), which is close to the population value of \beta_2 = 1 (green dashed line).
We also observe that the histogram of the posterior means for the ordinary model is less dispersed. That is, this estimator is more efficient, which is a well-known result in the Frequentist inferential approach when comparing ordinary least squares and two-stage least squares (see Jeffrey M. Wooldridge (2010)).
Two very important aspects in the instrumental variables literature are the weakness and exogeneity of the instruments. The former refers to how strong the relationship is between the instruments and the endogenous regressors, while the latter refers to the independence of the instruments from the stochastic error in the structural equation. In Exercise 6, we ask you to use the previous code as a baseline to study these two aspects. Observe the link between the weakness and exogeneity of the instrument, and the exclusion restrictions (\mathbb{E}[x_s \boldsymbol{z}] \neq \boldsymbol{0} and \mathbb{E}[y \boldsymbol{z} \mid x_s] = \boldsymbol{0}). This is the point of departure of Conley, Hansen, and Rossi (2012), who propose assessing the plausibility of the exclusion restrictions by defining plausible exogeneity as having prior information that the effect of the instrument in the structural equation is near zero, but perhaps not exactly zero.
The following Algorithm can be used to estimate the instrumental variable model using our GUI. We ask in Exercise 8 to replicate the example of the effect of institutions on per capita GDP using our GUI.
Algorithm: Instrumental Variable Model
Select Multivariate Models on the top panel
Select Variable instrumental (two equations) model using the left radio button
Upload the dataset, selecting first if there is a header in the file, and the kind of separator in the csv file of the dataset (comma, semicolon, or tab). Then, use the Browse button under the Choose File legend
Select MCMC iterations, burn-in, and thinning parameters using the Range sliders
Write down the formula of the structural equation in the Main Equation box. This formula must be written using the syntax of the formula command of R software. This equation includes the intercept by default, do not include it in the equation
Write down the formula of the endogenous regressor in the Instrumental Equation box. This formula must be written using the syntax of the formula command of R software. This equation includes the intercept by default, do not include it in the equation
Set the hyperparameters: mean vectors, covariance matrices, degrees of freedom, and the scale matrix. This step is not necessary as by default our GUI uses non-informative priors
Click the Go! button
Analyze results
Download posterior chains and diagnostic plots using the Download Posterior Chains and Download Posterior Graphs buttons
References
Conley, T., C. Hansen, and P. Rossi. 2012. “Plausibly Exogenous.” The Review of Economics and Statistics 94 (1): 260–72.
Lehmann, E. L., and George Casella. 2003. Theory of Point Estimation. Second Edition. Springer.
Van der Vaart, Aad W. 2000. Asymptotic Statistics. Vol. 3. Cambridge university press.
Wooldridge, Jeffrey M. 2010. Econometric Analysis of Cross Section and Panel Data. MIT press.
———. 2016. Introductory Econometrics: A Modern Approach. 6th ed. Boston, MA: Cengage Learning.