$$
\star\beta
$$

$$
\beta^\star
$$

$$
\beta\top\sigma^\star
$$

$$
\beta^\top\sigma^\star
$$

Click the Refresh button to see progress of the chains
starting worker pid=5360 on localhost:11587 at 22:04:15.786
starting worker pid=1032 on localhost:11587 at 22:04:15.964
starting worker pid=16968 on localhost:11587 at 22:04:16.142
starting worker pid=16676 on localhost:11587 at 22:04:16.325
starting worker pid=17588 on localhost:11587 at 22:04:16.499
starting worker pid=12300 on localhost:11587 at 22:04:16.676

SAMPLING FOR MODEL 'macro.reg' NOW (CHAIN 1).
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SAMPLING FOR MODEL 'macro.reg' NOW (CHAIN 3).
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SAMPLING FOR MODEL 'macro.reg' NOW (CHAIN 4).
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SAMPLING FOR MODEL 'macro.reg' NOW (CHAIN 5).
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SAMPLING FOR MODEL 'macro.reg' NOW (CHAIN 6).
Chain 6:

a_Regs_Times_Real_Equip_Investment_per_Firm",
+                                     "mean_beta_Regs_Times_Real_IP_Investment_per_Firm","hetero_beta_Regs_Times_Real_IP_Investment_per_Firm",
+                                     "mean_beta_Regs_Times_Real_Structs_Investment_per_Firm","hetero_beta_Regs_Times_Real_Structs_Investment_per_Firm",
+                                     "mean_beta_Regs_Times_Real_Fixed_Investment_per_Firm","hetero_beta_Regs_Times_Real_Fixed_Investment_per_Firm",
+                                     "mean_beta_Regs_Times_Real_Interest_Rate_Times_Real_GDP_per_Firm","hetero_beta_Regs_Times_Real_Interest_Rate_Times_Real_GDP_per_Firm",
+                                     "mean_beta_Lagged_Regs","hetero_beta_Lagged_Regs",
+                                     "mean_beta_Lagged_Regs_Times_Dynamic_Variable","hetero_beta_Lagged_Regs_Times_Dynamic_Variable",
+                                     "mean_beta_Lagged_Regs_Times_Real_Equip_Investment_per_Firm","hetero_beta_Lagged_Regs_Times_Real_Equip_Investment_per_Firm",
+                                     "mean_beta_Lagged_Regs_Times_Real_IP_Investment_per_Firm","hetero_beta_Lagged_Regs_Times_Real_IP_Investment_per_Firm",
+                                     "mean_beta_Lagged_Regs_Times_Real_Structs_Investment_per_Firm","hetero_beta_Lagged_Regs_Times_Real_Structs_Investment_per_Firm",
+                                     "mean_beta_Lagged_Regs_Times_Real_Fixed_Investment_per_Firm","hetero_beta_Lagged_Regs_Times_Real_Fixed_Investment_per_Firm",
+                                     "mean_beta_Lagged_Regs_Times_Real_Interest_Rate_Times_Real_GDP_per_Firm","hetero_beta_Lagged_Regs_Times_Real_Interest_Rate_Times_Real_GDP_per_Firm")
>
> # specify the number of Posterior Draws required
> Draws.Required <- 1000
>
>
> ##########################################################################################
> # Regress Real.Equip.Investment.per.Firm on Real.Interest.Rate and Real.GDP.per.Firm
> ##########################################################################################
>
> # Specify which investment variable is the outcome variable
> Investment.Data[["Real_Investment_per_Firm"]] <- Real_Equip_Investment_per_Firm
>
> # run model
> Equip.Investment.Results <- Estimate.Model.Investment.Equation(Investment.Data,Industry.Data.Without.NA,
+                                                                Investment.Variable.Name="Real.Equip.Investment.per.Firm",
+                                                                Draws.Required,Base.Year+1,Terminal.Year-1)
DIAGNOSTIC(S) FROM PARSER:
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.
Info: assignment operator <- deprecated in the Stan language; use = instead.

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or a minimal example that shows this warning you will be contributing
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Error in context_eval(join(src), private$context, serialize) :
  0,248,Failure,-3,

Let $log$ denote the logarithm to the base 2, then informational gain is measured in bits.
Shannon entropy [@S48] states that for a discrete random variable $J$ with probability distribution $p(j)$, where $j$ stands for the different outcomes the random variable $J$ can take, the average number of bits required to optimally encode independent draws from the distribution of $J$ can be calculated as
  
$$
  H_J = - \sum_j p(j) \cdot log \left(p(j)\right).
$$

Strictly speaking, Shannon's formula  is a measure for uncertainty, which increases with the number of bits needed to optimally encode a sequence of realizations of $J$.
In order to measure the information flow between two processes, Shannon entropy is combined with the concept of the Kullback-Leibler distance [@KL51] and by assuming that the underlying processes evolve over time according to a Markov process [@schreiber2000].
Let $I$ and $J$ denote two discrete random variables with marginal probability distributions $p(i)$ and $p(j)$ and joint probability distribution $p(i,j)$, whose dynamical structures correspond to stationary Markov processes of order $k$ (process $I$) and $l$ (process $J$).
The Markov property implies that the probability to observe $I$ at time $t+1$ in state $i$ conditional on the $k$ previous observations is $p(i_{t+1}|i_t,...,i_{t-k+1})=p(i_{t+1}|i_t,...,i_{t-k})$.
The average number of bits needed to encode the observation in $t+1$ if the previous $k$ values are known is given by
  
$$
  h_I(k)=- \sum_i p\left(i_{t+1}, i_t^{(k)}\right) \cdot log \left(p\left(i_{t+1}|i_t^{(k)}\right)\right),
$$

$$
\begin{array}{l}
{x_{1,t}} = {x_{1,t - 1}} + \sigma_1{u_{1,t}}\\
{x_{2,t}} = \phi {x_{2,t - 1}} + \sigma_2{u_{2,t}}\\
{y_t} = {x_{1,t}} + {x_{2,t}}
\end{array}
$$

$$
\begin{array}{l}
{\phi_{z,t}} = {\phi_{z,t - 1}}\\
{x_{2,z,t}} = {x_{2,z,t - 1}} \phi_{z,t}  + 0.8{u_{2,t}}\\
\end{array}
$$

$$r_t = c + \theta r_{t-1} + \epsilon_t$$

$$\sigma^2_t = \kappa + \alpha\sigma^2_{t-1} + \phi\epsilon^2_{t-1} + \psi[\epsilon_{t-1}<0]\epsilon^2_{t-1}$$

$$z_t = \epsilon_t/\sigma_t$$ i.i.d. distributed $$t(\nu)$$

$$
\begin{aligned}
y_{cst} &= Z^{1}_{cst} \beta^{1}_{cs} + X^{1}_{cst} \beta_{1} +\epsilon^{1}_{cst} \\
\beta^{1}_{cs} &= Z^{2}_{cs} \beta^{2}_{c} + X^{2}_{cs} \beta_{2} +\epsilon^{2}_{cs} \\
\beta^{2}_{c} &= X^{3}_{c} \beta_{3} +\epsilon^{3}_{c}.
\end{aligned}
$$