Brock-Mirman Model — SMM Estimation Exercise

Summary

Evans (2024) Ch. 19 uses the Brock and Mirman (1972) stochastic optimal growth model as a canonical SMM exercise. The model has a known analytical policy function but latent productivity shocks, making it a clean test case for simulation-based estimation. Four parameters $(\alpha ,\rho ,\mu ,\sigma )$ are identified from six empirical moments computed on quarterly macroeconomic time series.

Overview

The Brock-Mirman (1972) model is a workhorse of quantitative macroeconomics: a representative agent maximises expected discounted utility from consumption, subject to a stochastic production technology. The model is well-suited to SMM because:

• It is fully simulable given parameters $(\alpha ,\beta ,\rho ,\mu ,\sigma )$
• It has a closed-form policy function (savings rule), making simulation efficient
• The productivity shock $z_t$ is latent — it drives the data but is not directly observed — a canonical motivation for SMM over GMM

See Method of Simulated Moments for the general MSM theory and SMM Python Implementation for the Python workflow this exercise follows.

Model Equations

The Brock-Mirman (1972) stochastic growth model is characterised by six equations:

Definition: Brock-Mirman (1972) Structural System

Euler equation (consumption):

$$(c_t{)}^{-1}-\beta E[r_{t+1}(c_{t+1}{)}^{-1}]=0\text{\hspace{1em}(BM.1)}$$

Budget constraint:

$$c_t+k_{t+1}-w_t-r_t{k}_t=0\text{\hspace{1em}(BM.2)}$$

Wage (labour FOC):

$$w_t-(1-\alpha )e^{z_t}(k_t{)}^{\alpha }=0\text{\hspace{1em}(BM.3)}$$

Return on capital (capital FOC):

$$r_t-\alpha e^{z_t}(k_t{)}^{\alpha -1}=0\text{\hspace{1em}(BM.4)}$$

TFP process (AR(1)):

$$z_t=\rho z_{t-1}+(1-\rho )\mu +{\epsilon }_t,{\epsilon }_t\sim N(0,{\sigma }^2)\text{\hspace{1em}(BM.5)}$$

Output:

$$y_t=e^{z_t}(k_t{)}^{\alpha }\text{\hspace{1em}(BM.6)}$$

Variables: $c_t$ = consumption; $k_{t+1}$ = savings/investment (= next period’s capital, full depreciation assumed); $w_t$ = wage; $r_t$ = return on capital; $z_t$ = log TFP (latent); $y_t$ = GDP.

Parameters and constraints:

$$\alpha ,\beta \in (0,1),\mu ,\sigma >0,\rho \in (-1,1)$$

Analytical Policy Function

Theorem: Brock-Mirman Policy Function

The household’s optimal savings decision has the closed-form solution:

$$k_{t+1}=\alpha \beta e^{z_t}(k_t{)}^{\alpha }\text{\hspace{1em}(BM.7)}$$

This eliminates the need to numerically solve the Euler equation at each guess of $\theta$ during SMM estimation.

Simulation Algorithm

Given parameters $\theta =(\alpha ,\rho ,\mu ,\sigma )$ with $\beta =0.99$ fixed:

1. Initialise: Set $z_0=\mu$, $k_1=\overline{k}$ (sample mean of $k_t$ from data). These are fixed across all simulations.
2. Draw uniform shocks once: Sample $u_{s,t}\sim U(0,1)$ for $s=1,\dots ,S$ and $t=1,\dots ,T$. Never re-draw inside the optimizer. See Practical Issues in Simulation Estimation for the common random numbers principle.
3. Generate ${\epsilon }_{s,t}$: Convert uniform draws via inverse normal CDF: $${\epsilon }_{s,t}=\sigma \cdot {\Phi }^{-1}(u_{s,t})$$
4. Simulate $z_{s,t}$ using (BM.5) recursively: $$z_{s,t}=\rho z_{s,t-1}+(1-\rho )\mu +{\epsilon }_{s,t}$$
5. Simulate $k_{s,t+1}$ using the policy function (BM.7).
6. Simulate $w_{s,t}$, $r_{s,t}$ from (BM.3)–(BM.4).
7. Simulate $c_{s,t}$ from the budget constraint (BM.2).
8. Simulate $y_{s,t}$ from (BM.6).

SMM Estimation Setup

Exercise 19.1: Estimating the Brock-Mirman Model by SMM

Data: NewMacroSeries.txt — 100 quarterly observations of $(c_t,k_t,w_t,r_t,y_t)$ for $t=1,\dots ,100$.

Calibrated: $\beta =0.99$

Parameters to estimate: $\hat{\theta }=(\hat{\alpha },\hat{\rho },\hat{\mu },\hat{\sigma })$

Estimation bounds:

$$\alpha \in [0.01,0.99],\rho \in [-0.99,0.99],\mu \in [5,14],\sigma \in [0.01,1.1]$$

Six moments to match ($R=6>K=4$, so the model is overidentified):

Moment	Description
$\text{mean}(c_t)$	Average consumption
$\text{mean}(k_t)$	Average capital stock
$\text{mean}(c_t/y_t)$	Average consumption share
$\text{var}(y_t)$	Variance of output
$\text{corr}(c_t,c_{t-1})$	First-order autocorrelation of consumption
$\text{corr}(c_t,k_t)$	Cross-correlation of consumption and capital

Simulation design: $S=1,000$ simulations, $T=100$ periods each.

Part 1: Estimate with identity weighting matrix $W=I_6$. Report $\hat{\theta }$, moment errors, criterion function value, and standard errors ${\hat{\Sigma }}_{SMM}$.

Part 2: Re-estimate with the two-step optimal weighting matrix ${\hat{W}}_{2\text{step}}={\hat{\Omega }}_2^{-1}$. Report the same quantities and compare standard errors to Part 1.

See SMM Weighting Matrix and Inference for the two-step procedure and the Jacobian-based ${\hat{\Sigma }}_{SMM}$ formula.

Moment Selection Rationale

The six moments are chosen to identify the four parameters through distinct channels:

• $\text{mean}(c_t)$ and $\text{mean}(k_t)$ are driven primarily by $\mu$ (long-run TFP level) and $\alpha$ (capital share determining steady-state ratios)
• $\text{mean}(c_t/y_t)$ pins down the consumption-to-output ratio, which is a function of $\alpha \beta$ in the Brock-Mirman solution
• $\text{var}(y_t)$ and $\text{corr}(c_t,c_{t-1})$ are sensitive to the TFP volatility $\sigma$ and persistence $\rho$
• $\text{corr}(c_t,k_t)$ reflects the capital-consumption co-movement induced by the AR(1) TFP process

The overidentification ($R=6>K=4$) allows for a specification test: moments not used in estimation can be verified against the estimated model — a key diagnostic recommended in SMM Weighting Matrix and Inference.

Why SMM and Not GMM?

The productivity shock $z_t$ in (BM.5) is latent: it is not directly observed in the data. Although the budget constraint and FOCs pin down the model structure, computing theoretical moments like $E[y_t^2]$ or $\text{corr}(c_t,k_t)$ requires integrating over the distribution of the entire TFP path — a high-dimensional integral with no closed form.

SMM resolves this by simulating the TFP path $\{z_{s,t}\}$ directly, making any moment computable via Monte Carlo averaging. See Method of Simulated Moments for the general consistency result showing SMM works for any $R\ge 1$ simulations.

Connections

• SMM Python Implementation — Python workflow (trunc_norm_draws, err_vec, criterion, two-step W, Jac_err) directly applicable to this exercise with BM-specific simulation replacing trunc_norm_draws
• SMM Weighting Matrix and Inference — identity W, two-step W, and ${\hat{\Sigma }}_{SMM}$ procedures used in both parts of the exercise
• Method of Simulated Moments — consistency and $(1+1/R)$ variance inflation theorem
• Practical Issues in Simulation Estimation — common random numbers (fix $u_{s,t}$ before optimizer), step-size for Jacobian

See Also

• Simulation-Based Estimation - Overview — situates SMM within the family of simulation-based methods
• Indirect Inference — alternative that uses AR(1) or VAR regression coefficients as moments (natural choice for BM model)
• Efficient Method of Moments — simulation-based approach that matches auxiliary-model coefficients rather than raw moments; applicable to the BM model via VAR auxiliary regression
• SMM Estimator for Copulas — another application of the same SMM framework

Sources

• Computational Methods for Economists — Ch. 19 — Evans (2024), Sections 19.4.2, 19.7
• Brock, W.A. and L.J. Mirman (1972), “Optimal Economic Growth and Uncertainty: The Discounted Case,” Journal of Economic Theory 4(3), 479–513
• Smith, A.A. Jr. (2020), “Indirect Inference,” New Palgrave Dictionary of Economics, Palgrave MacMillan