7.6 Exercises
Show that \(\mathbb{E}[u_1\text{PAER}] = \frac{\alpha_1}{1 - \beta_1\alpha_1} \sigma_1^2\), assuming that \(\mathbb{E}[u_1 u_2] = 0\), where \(\text{Var}(u_1) = \sigma_1^2\), in the example of the effect of institutions on per capita GDP.
Show that \(\beta_1=\pi_1/\gamma_1\), in the example of the effect of institutions on per capita GDP.
The effect of institutions on per capita gross domestic product continues I
Use the rmultireg command from the bayesm package to perform inference in the example of the effect of institutions on per capita GDP.
Demand and supply simulation
Given the structural demand-supply model:
\[ \begin{aligned} q_i^d &= \beta_1 + \beta_2 p_i + \beta_3 y_i + \beta_4 pc_i + \beta_5 ps_i + u_{i1}, \\ q_i^s &= \alpha_1 + \alpha_2 p_i + \alpha_3 er_i + u_{i2}, \end{aligned} \] where \(q^d\) is demand, \(q^s\) is supply, \(p\), \(y\), \(pc\), \(ps\), and \(er\) are price, income, complementary price, substitute price, and exchange rate, respectively. Complementary and substitute prices refer to the prices of complementary and substitute goods for \(q\). Assume that
\[ \boldsymbol{\beta} = \begin{bmatrix} 5 \\ -0.5 \\ 0.8 \\ -0.4 \\ 0.7 \end{bmatrix}, \quad \boldsymbol{\alpha} = \begin{bmatrix} -2 \\ 0.5 \\ -0.4 \end{bmatrix}, \] \(u_1 \sim N(0, 0.5^2)\), and \(u_2 \sim N(0, 0.5^2)\). Additionally, assume that \(y \sim N(10, 1)\), \(pc \sim N(5, 1)\), \(ps \sim N(5, 1)\), and \(er \sim N(15, 1)\).- Find the reduced-form model by using the condition that in equilibrium, demand and supply are equal, i.e., \(q^d = q^s\). This condition defines the observable quantity, \(q\).
- Simulate \(p\) and \(q\) from the reduced-form equations.
- Perform inference for the reduced-form model using the rmultireg command from the bayesm package.
- Use the posterior draws of the reduced-form parameters to perform inference for the structural parameters. Any issues? Hint: Are all structural parameters exactly identified?
- Find the reduced-form model by using the condition that in equilibrium, demand and supply are equal, i.e., \(q^d = q^s\). This condition defines the observable quantity, \(q\).
Utility demand continues
- Run the Utility demand application using our GUI and the information in the dataset Utilities.csv. Hint: This file should be modified to agree with the structure that our GUI requires (see the dataset 5Institutions.csv in the folder DataApp of our GitHub repository - https://github.com/besmarter/BSTApp - for a template).
- Program from scratch the Gibbs sampler algorithm in this application.
- Run the Utility demand application using our GUI and the information in the dataset Utilities.csv. Hint: This file should be modified to agree with the structure that our GUI requires (see the dataset 5Institutions.csv in the folder DataApp of our GitHub repository - https://github.com/besmarter/BSTApp - for a template).
Simulation exercise of instrumental variables continues I
- Use the setting of the simulation exercise with instrumental variables to analyze the impact of a weak instrument. For instance, set \(\gamma_2 = 0.2\) and compare the performance of the posterior means of the ordinary and instrumental variable models.
- Perform a simulation to analyze how the degree of exogeneity of the instrument affects the performance of the posterior mean in the instrumental variable model.
- Use the setting of the simulation exercise with instrumental variables to analyze the impact of a weak instrument. For instance, set \(\gamma_2 = 0.2\) and compare the performance of the posterior means of the ordinary and instrumental variable models.
Simulation exercise of instrumental variables continues II
Program from scratch the Gibbs sampling algorithm of the instrumental model for the simulation exercise of the instrumental variables.
The effect of institutions on per capita gross domestic product continues II
Estimate the structural Equation (7.1) using the instrumental variable model where the instrument of PAER is \(\log(\textit{Mort})\). Compare the effect of property rights on per capita GDP of this model with the effect estimated in the example of the effect of institutions on per capita gross domestic product. Use the file 6Institutions.csv to do this exercise in our GUI, and set
\[ \boldsymbol{B}_0=100\boldsymbol{I}_5, \quad \boldsymbol{\beta}_0=\boldsymbol{0}_5, \quad \boldsymbol{\gamma}_0=\boldsymbol{0}_2, \quad \boldsymbol{G}_0=100\boldsymbol{I}_2, \quad \alpha_0=3, \quad \boldsymbol{\Psi}_0=3\boldsymbol{I}_2. \] The MCMC iterations, burn-in, and thinning parameters are 50000, 1000, and 5, respectively.Multivariate probit with different regressors
Let’s do a simulation exercise where
\[ \begin{aligned} y_{i1}^* &= 0.5 - 1.2x_{i11} + 0.7x_{i12} + 0.8x_{i3} + \mu_{i1}, \\ y_{i2}^* &= 1.5 - 0.8x_{i21} + 0.5x_{i22} + \mu_{i2}, \end{aligned} \] with
\[ \boldsymbol{\Sigma}= \begin{bmatrix} 1 & 0.5 \\ 0.5 & 1 \end{bmatrix}, \] where all regressors follow a standard normal distribution, and \(N=5000\). Use
\[ \boldsymbol{\beta}_0=\boldsymbol{0}, \quad \boldsymbol{B}_0=1000\boldsymbol{B}, \quad \alpha_0=4, \quad \boldsymbol{\Psi}_0=4\boldsymbol{I}_2. \] Set the number of iterations to 2000 and a thinning parameter equal to 5.- Perform inference using the setting of Section 7.4, that is, assuming that \(x_{i3}\) could have an effect on \(y_{i2}\).
- Program a Gibbs sampling algorithm taking into account that there are different regressors in each binary decision, that is, \(x_{i3}\) does not have an effect on \(y_{i2}\).
- Perform inference using the setting of Section 7.4, that is, assuming that \(x_{i3}\) could have an effect on \(y_{i2}\).