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This note applies the classic Hansen (1985, JME) on the Vietnamese economy. Note that the economy is closed. There is no government or monetary authority. All variables are measured in real value. While it is highly stylized and maybe not suitable for a small open economy such as Vietnam, it is a useful example on how to assess the model and its applicability.

In preparation of the data, I used only the data of consumption, investment, and hours worked from the GSO. The domestic output is then constructed by adding consumption and investment together. There is no population growth, so all variables such as consumption, investment, and output are measured in per worker terms.

Model

A representative household maximizes consumption $c_t$, hours worked $h_t$, and investment $i_t$:

\[\max_{c_t, h_t, k_{t+1}} E_0 \sum_{t=0}^\infty \beta^t (\ln c_t - \gamma h_t) a_t\]

subject to

\[\begin{aligned} c_t + i_t = w_t h_t + r_t k_t, \\ k_{t+1} = i_t + (1-\delta) k_t, \\ k_0 > 0 \end{aligned}\]

The output is given by

\[y_t = \theta_t k_t^\alpha h_t^{1-\alpha}\]

Factor prices:

\[\begin{aligned} w_t = (1-\alpha) \theta_t k_t^{\alpha} h_t^{-\alpha} = (1-\alpha) \frac{y_t}{h_t}, \\ r_t = \alpha \theta_t k_t^{\alpha-1} h_t^{1-\alpha} = \alpha \frac{y_t}{k_t}. \end{aligned}\]

Furthermore, $a_t,\theta_t$ are structural shocks on demand and supply.

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Equilibrium condition

\[\begin{aligned} y_t = \theta_t k_t^\alpha h_t^{1-\alpha}, \\ \ln \theta_t = (1-\rho_\theta)\ln \bar{\theta} + \rho_\theta \ln \theta_{t-1} + \epsilon_t, \quad \epsilon_t \sim N(0,\sigma^2_\theta), \\ y_t = c_t + i_t, \\ k_{t+1} = (1-\delta)k_t + i_t, \\ \gamma c_t h_t = (1-\alpha) y_t, \\ \frac{a_t}{c_t} = \beta E_t \left[ \frac{a_{t+1}}{c_{t+1}} \left( \alpha \frac{y_{t+1}}{k_{t+1}} + 1 - \delta \right)\right], \\ \ln a_t = \rho_a \ln a_{t-1} + \xi_t, \quad \xi_t \sim N(0,\sigma^2_a) \end{aligned}\]

Note that we do not need $w_t$ here, and $r_{t+1}$ is substituted by a function of $y_{t+1},k_{t+1}$.

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Steady state

\[\begin{align} c_{ss} = \frac{1-\alpha}{\gamma} \bar{\theta} \left[ \frac{\bar{\theta}\alpha}{(1/\beta)-1+\delta} \right]^{\alpha/(1-\alpha)}, \\ k_{ss} = \frac{\alpha c_{ss}}{(1/\beta)-1+\delta -\alpha\delta}, \\ h_{ss} = k_{ss} \left[ \frac{(1/\beta)-1+\delta}{\bar{\theta}\alpha} \right]^{1/(1-\alpha)}, \\ y_{ss} = \bar{\theta} k_{ss}^\alpha h_{ss}^{1-\alpha}, \\ i_{ss} = y_{ss} - c_{ss}, \\ w_{ss} = (1-\alpha) \frac{y_{ss}}{h_{ss}}, \\ r_{ss} = \alpha \frac{y_{ss}}{k_{ss}} - 1. \end{align}\]

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Log-linearization

Define $\hat{x} = \ln x_t - \ln x_{ss}$, we have:

\[\hat{y}_t = \hat{\theta}_t + \alpha \hat{k}_t + (1-\alpha)\hat{h}_t\] \[\hat{\theta}_{t} = \rho_{\theta} \hat{\theta}_{t-1} + \epsilon_t\] \[((1/\beta)-1+\delta)\hat{y}_t = ((1/\beta)-1+\delta-\alpha\delta) \hat{c}_t + \alpha\delta \hat{i}_t\] \[\hat{k}_{t+1} = (1-\delta)\hat{k}_t + \delta \hat{i}_t\] \[\hat{c}_t + \hat{h}_t = \hat{y}_t\] \[(1/\beta) \hat{a}_{t} - (1/ \beta) \hat{c}_t = -(1/\beta) E_t \hat{c}_{t+1} + ((1/\beta)-1+\delta)(E_t\hat{y}_{t+1}-\hat{k}_{t+1}) + (1/\beta)E_t \hat{a}_{t+1}\] \[\hat{a}_t = \rho_{a} \hat{a}_{t-1} + \xi_t.\]

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State-space representation

The contemporary equations:

\[\begin{bmatrix} 1 & 0 & \alpha - 1 \\ 1/\beta - 1 + \delta & -\alpha\delta & 0 \\ 1 & 0 & -1 \end{bmatrix} \begin{bmatrix} \hat{y}_t \\ \hat{i}_t \\ \hat{h}_t \end{bmatrix} = \begin{bmatrix} \alpha & 0 \\ 0 & 1/\beta - 1 + \delta - \alpha\delta \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \hat{k}_t \\ \hat{c}_t \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \hat{\theta}_t \\ \hat{a}_t \end{bmatrix}\]

and

\[\begin{bmatrix} E_t \hat{\theta}_{t+1} \\ E_t \hat{a}_{t+1} \end{bmatrix} = \begin{bmatrix} \rho_\theta & 0 \\ 0 & \rho_a \end{bmatrix} \begin{bmatrix} \hat{\theta}_t \\ \hat{a}_t \end{bmatrix}\]

The dynamic equations:

\[\begin{align} &\begin{bmatrix} 1 & 0 \\ 1/\beta-1+\delta & 1/\beta \end{bmatrix} \begin{bmatrix} \hat{k}_{t+1} \\ E_t \hat{c}_{t+1} \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ -(1/\beta-1+\delta) & 0 & 0 \end{bmatrix} \begin{bmatrix} E_t \hat{y}_{t+1} \\ E_t \hat{i}_{t+1} \\ E_t \hat{h}_{t+1} \end{bmatrix} = \\ &\begin{bmatrix} 1-\delta & 0 \\ 0 & 1/\beta \end{bmatrix} \begin{bmatrix} \hat{k}_t \\ \hat{c}_t \end{bmatrix} + \begin{bmatrix} 0 & \delta & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \hat{y}_t \\ \hat{i}_t \\ \hat{h}_t \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & (-1/\beta)(1-\rho_a) \end{bmatrix} \begin{bmatrix} \hat{\theta}_t \\ \hat{a}_t \end{bmatrix} \end{align}\]

Estimation

The estimation procedure follows Chapter 10 of Novales, A., Fernández, E., Ruiz, J. (2022). Empirical Methods: Frequentist Estimation. In: Economic Growth. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg..

You can obtain the code here.

First, we estimate the trend of the data for output, consumption, and hours worked.

Then, we can run the maximum likelihood estimation for structural parameters.

Iteration	Func-count	f(x)	Step-size	Optimality
0	8	951478	–	1.07e+06
1	16	71127.3	9.34793e-07	8.26e+04
2	24	57735.6	1	6.71e+04
3	32	23347.6	1	2.72e+04
4	40	12516.7	1	1.47e+04
5	48	5992.47	1	7.14e+03
6	56	2964.62	1	3.61e+03
7	64	1419.30	1	1.79e+03
8	72	665.733	1	884
9	80	298.288	1	427
10	88	125.398	1	198
11	96	48.5919	1	83.3

Local minimum found.

The estimated parameters are:

Parameter	Coefficient	Std. Error	t-statistic	p-value
Productivity shock mean ($\bar{\theta}$)	0.38993	0.050089	7.7848	6.8834e-15
Productivity shock persistence ($\rho$)	0.95931	0.025398	37.772	0
Productivity shock std ($\sigma_e$)	0.099209	0.013974	7.0997	1.2506e-12
Preference shock persistence ($\rho_a$)	0.93936	0.01472	63.815	0
Preference shock std ($\sigma_a$)	0.25173	0.0066917	37.619	0
Output elasticity of capital ($\alpha$)	0.26933	0.013686	19.679	0
Utility function parameter ($\gamma$)	0.013093	0.0012716	10.296	0

Calibrated parameters

Note that here, we did not estimate discount factor and the depreciation rate, so these parameters are calibrated.

Parameter	Value
Discount factor ($\beta$)	0.99269
Depreciation rate ($\delta$)	0.014677

To test the goodness of fit, we apply a Kalman filtration on output using consumption and hours. We will use the residuals from the Kalman filtration to simulate the model’s dynamics using the estimated parameters.

Goodness-of-Fit for Output per Worker.

Correlation: 0.6771.

R-squared: 0.4584.

One can also add back the trend components, which would results in:

Now, an IRFs for supply and demand shocks can be generated and analyzed.