Exercise 2.7

Predicted Second Moments

Problem

In Chapter 1, we showed that two empirical regularities that characterize emerging economies are the countercyclicality of the trade balance-to-output ratio and the fact that consumption growth appears to be more volatile than output growth. In this chapter, we developed a simple small open endowment economy and provided intuitive arguments suggesting that this economy fails to account for these two stylized facts. However, that model does not allow for closed-form solutions of second moments of output growth, consumption growth, or the trade balance-to-output ratio. The goal of this exercise is to obtain these implied statistics numerically.

To this end, consider the following parameterization of the model developed in the present chapter:

\[ y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \epsilon_t, \]

with $\rho=0.9$, $\bar{y}=1$, and $\epsilon_t$ is distributed normally with mean 0 and standard deviation 0.03. Note that the parameter $\bar{y}$, which earlier in this chapter was implicitly assumed to be zero, represents the deterministic steady state of the output process. Assume further that $r=\frac{1}{\beta}-1=0.1$, $d_{-1}=\bar{y}/2$, and $y_{-1}=\bar{y}$.

Simulate the economy for 100 years.
Discard the first 50 years of artificial data to minimize the dependence of the results on initial conditions.
Compute the growth rates of output and consumption and the trade balance-to-output ratio.
Compute the sample standard deviations of output growth and consumption growth and the correlation between output growth and the trade balance-to-output ratio. Here we denote these three statistics by $\sigma_{gy}$, $\sigma_{gc}$, and $\rho_{gy,tby}$, respectively.
Replicate steps 1 to 4 10,000 times. For each replication, keep record of $\sigma_{gy}$, $\sigma_{gc}$, and $\rho_{gy,tby}$.
Report the average of $\sigma_{gc} / \sigma_{gy}$, and $\rho_{gy,tby}$ over the 10,000 replications.
Discuss your results.

Answer

1-4: Stimulate the economy for 100 years

\[ \sigma_{gy} = 3.2897 \] \[ \sigma_{gc} = 1.7388 \] \[ \frac{\sigma_{gc}}{\sigma_{gy}} = 0.5285 \] \[ \rho_{gy,tby} = 0.2564 \]

5-6: Replicate 10,000 times

\[ \sigma_{gy} = 3.1075 \] \[ \sigma_{gc} = 1.6207 \] \[ \frac{\sigma_{gc}}{\sigma_{gy}} = 0.5220 \] \[ \rho_{gy,tby} = 0.2955 \]

7.

consumption growth is less volatile than output growth and the trade balance is procyclical.

📥 Download Python Code (2_7.py)

--- title: "Exercise 2.7" subtitle: "Predicted Second Moments" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem In Chapter 1, we showed that two empirical regularities that characterize emerging economies are the countercyclicality of the trade balance-to-output ratio and the fact that consumption growth appears to be more volatile than output growth. In this chapter, we developed a simple small open endowment economy and provided intuitive arguments suggesting that this economy fails to account for these two stylized facts. However, that model does not allow for closed-form solutions of second moments of output growth, consumption growth, or the trade balance-to-output ratio. The goal of this exercise is to obtain these implied statistics numerically. To this end, consider the following parameterization of the model developed in the present chapter: $$ y_t - \bar{y} = \rho(y_{t-1} - \bar{y}) + \epsilon_t, $$ with $\rho=0.9$, $\bar{y}=1$, and $\epsilon_t$ is distributed normally with mean 0 and standard deviation 0.03. Note that the parameter $\bar{y}$, which earlier in this chapter was implicitly assumed to be zero, represents the deterministic steady state of the output process. Assume further that $r=\frac{1}{\beta}-1=0.1$, $d_{-1}=\bar{y}/2$, and $y_{-1}=\bar{y}$. 1. Simulate the economy for 100 years. 2. Discard the first 50 years of artificial data to minimize the dependence of the results on initial conditions. 3. Compute the growth rates of output and consumption and the trade balance-to-output ratio. 4. Compute the sample standard deviations of output growth and consumption growth and the correlation between output growth and the trade balance-to-output ratio. Here we denote these three statistics by $\sigma_{gy}$, $\sigma_{gc}$, and $\rho_{gy,tby}$, respectively. 5. Replicate steps 1 to 4 10,000 times. For each replication, keep record of $\sigma_{gy}$, $\sigma_{gc}$, and $\rho_{gy,tby}$. 6. Report the average of $\sigma_{gc} / \sigma_{gy}$, and $\rho_{gy,tby}$ over the 10,000 replications. 7. Discuss your results. ## Answer #### 1-4: Stimulate the economy for 100 years $$ \sigma_{gy} = 3.2897 $$ $$ \sigma_{gc} = 1.7388 $$ $$ \frac{\sigma_{gc}}{\sigma_{gy}} = 0.5285 $$ $$ \rho_{gy,tby} = 0.2564 $$ #### 5-6: Replicate 10,000 times $$ \sigma_{gy} = 3.1075 $$ $$ \sigma_{gc} = 1.6207 $$ $$ \frac{\sigma_{gc}}{\sigma_{gy}} = 0.5220 $$ $$ \rho_{gy,tby} = 0.2955 $$ #### 7. consumption growth is less volatile than output growth and the trade balance is procyclical. <a href="code/2_7.py" download style="display:inline-block;padding:8px 16px;background:#007acc;color:white;text-decoration:none;border-radius:5px;"> 📥 Download Python Code (2_7.py) </a>