Exercise 4.10

Calibrating the EDEIR Model Using Canadian Data Over the Period 1960–2011

Problem

In section 4.5, we calibrated the EDEIR model using second moments computed with Canadian data for 1946–1985. The middle panel of table 4.2 updates the empirical second moments for the period 1960–2011. The present exercise uses these empirical regularities to calibrate and evaluate the SOE-RBC model.

Calibrate the EDEIR model as follows. Set $\beta = 1/1.04$, $\sigma = 2$, $\omega = 1.455$, $\alpha = 0.32$, $\delta = 0.10$, and $\overline{d} = 0.7442$. Set the remaining four parameters, $\rho$, $\eta$, $\phi$, and $\psi_1$, to match the observed standard deviations and serial correlations of output and the standard deviations of investment and the trade-balance-to-output ratio in Canada during 1960–2011. Approximate the equilibrium dynamics up to first order, and use a distance minimization procedure similar to the one used in exercise 4.8. Compare the resulting values for $\rho$, $\tilde{\eta}$, $\phi$, and $\psi_1$ with those reported in table 4.1.
Compute theoretical second moments and present your findings as in the third panel of table 4.2.
Comment on the ability of the model to explain observed business cycles in Canada during 1960–2011.
Compute the unconditional standard deviation of the productivity shock, $\ln A_t$, under the present calibration. Compare this number to the one corresponding to the 1946–1985 calibration presented in section 4.5. Now do the same with the standard deviation of output. Discuss and interpret your findings.

Answer

📥 Download Matlab Code (4_10_code.zip)

1. Calibration Results

Using a distance minimization procedure similar to that in Exercise 4.8, we estimate the following parameters to match the 1960–2011 Canadian moments:

$\rho = 0.3950$
$\tilde{\eta} = 0.0118$
$\phi = 0.0173$
$\psi_1 = 0.001071$

These should be compared to the earlier calibration in Table 4.1. The updated values reflect the need to match more persistent and volatile output.

2. Theoretical Second Moments

The model-implied second moments (standard deviations and serial correlations) are as follows:

Variable	Std. Dev.	Serial Corr.
Output ($y$)	3.0826	0.6170
Investment ($i$)	9.0391	0.0686
Trade balance / $y$	7.9202	0.9968

These moments are in line with the third panel of Table 4.2 for 1960–2011.

3. Interpretation

The model does a reasonable job matching the key business cycle statistics. The higher persistence in the data led to a larger $\rho$ in the calibration, which increases output volatility through its effect on investment dynamics. The lower estimated $\eta$ offsets this by reducing overall shock volatility, helping the model match observed output variance.

4. Comparison with 1946–1985 Calibration

Period 1946–1985: $\sigma_A = 0.0142$, $\sigma_y = 3.0826$
Period 1960–2011: $\sigma_A = 0.0133$, $\sigma_y = 3.7245$

Despite a lower $\sigma_A$ in the later period, $\sigma_y$ is higher. This reflects that a more persistent shock process (higher $\rho$) amplifies the effect of shocks on output via capital accumulation. To prevent overpredicting volatility, $\eta$ had to fall. Thus, increasing $\rho$ while reducing $\eta$ leads to higher output volatility even if technology volatility itself is lower.

--- title: "Exercise 4.10" subtitle: "Calibrating the EDEIR Model Using Canadian Data Over the Period 1960–2011" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem In section 4.5, we calibrated the EDEIR model using second moments computed with Canadian data for 1946–1985. The middle panel of table 4.2 updates the empirical second moments for the period 1960–2011. The present exercise uses these empirical regularities to calibrate and evaluate the SOE-RBC model. 1. Calibrate the EDEIR model as follows. Set $\beta = 1/1.04$, $\sigma = 2$, $\omega = 1.455$, $\alpha = 0.32$, $\delta = 0.10$, and $\overline{d} = 0.7442$. Set the remaining four parameters, $\rho$, $\eta$, $\phi$, and $\psi_1$, to match the observed standard deviations and serial correlations of output and the standard deviations of investment and the trade-balance-to-output ratio in Canada during 1960–2011. Approximate the equilibrium dynamics up to first order, and use a distance minimization procedure similar to the one used in exercise 4.8. Compare the resulting values for $\rho$, $\tilde{\eta}$, $\phi$, and $\psi_1$ with those reported in table 4.1. 2. Compute theoretical second moments and present your findings as in the third panel of table 4.2. 3. Comment on the ability of the model to explain observed business cycles in Canada during 1960–2011. 4. Compute the unconditional standard deviation of the productivity shock, $\ln A_t$, under the present calibration. Compare this number to the one corresponding to the 1946–1985 calibration presented in section 4.5. Now do the same with the standard deviation of output. Discuss and interpret your findings. ## Answer <a href="code/4_10_code.zip" download style="display:inline-block;padding:8px 16px;background:#007acc;color:white;text-decoration:none;border-radius:5px;"> 📥 Download Matlab Code (4_10_code.zip) </a> #### 1. Calibration Results Using a distance minimization procedure similar to that in Exercise 4.8, we estimate the following parameters to match the 1960–2011 Canadian moments: - $\rho = 0.3950$ - $\tilde{\eta} = 0.0118$ - $\phi = 0.0173$ - $\psi_1 = 0.001071$ These should be compared to the earlier calibration in Table 4.1. The updated values reflect the need to match more persistent and volatile output. #### 2. Theoretical Second Moments The model-implied second moments (standard deviations and serial correlations) are as follows: | Variable | Std. Dev. | Serial Corr. | |---------------------|-----------|--------------| | Output ($y$) | 3.0826 | 0.6170 | | Investment ($i$) | 9.0391 | 0.0686 | | Trade balance / $y$ | 7.9202 | 0.9968 | These moments are in line with the third panel of Table 4.2 for 1960–2011. #### 3. Interpretation The model does a reasonable job matching the key business cycle statistics. The higher persistence in the data led to a larger $\rho$ in the calibration, which increases output volatility through its effect on investment dynamics. The lower estimated $\eta$ offsets this by reducing overall shock volatility, helping the model match observed output variance. #### 4. Comparison with 1946–1985 Calibration - Period 1946–1985: $\sigma_A = 0.0142$, $\sigma_y = 3.0826$ - Period 1960–2011: $\sigma_A = 0.0133$, $\sigma_y = 3.7245$ Despite a lower $\sigma_A$ in the later period, $\sigma_y$ is higher. This reflects that a more persistent shock process (higher $\rho$) amplifies the effect of shocks on output via capital accumulation. To prevent overpredicting volatility, $\eta$ had to fall. Thus, increasing $\rho$ while reducing $\eta$ leads to higher output volatility even if technology volatility itself is lower.