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🌐 [translation-sync] [theil_2.md] Update np.random → Generator API #216

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🌐 [translation-sync] [theil_2.md] Update np.random → Generator API #216

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c8f8b57
Update translation: lectures/theil_2.md
mmcky Jul 31, 2026
5539012
Update translation: .translate/state/theil_2.md.yml
mmcky Jul 31, 2026
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10 changes: 5 additions & 5 deletions .translate/state/theil_2.md.yml
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@@ -1,6 +1,6 @@
source-sha: 78030a3a27f6527046675bcd8a8d27995ca25af6
synced-at: "2026-07-22"
model: claude-opus-4-8
mode: NEW
source-sha: 7f05c342ae6b5261ee86b25f8ad2751104343526
synced-at: "2026-07-31"
model: claude-sonnet-5
mode: UPDATE
section-count: 8
tool-version: 0.20.0
tool-version: 0.24.0
15 changes: 3 additions & 12 deletions lectures/theil_2.md
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Expand Up @@ -49,8 +49,6 @@ translation:
:depth: 2
```



## 概述


Expand Down Expand Up @@ -103,7 +101,6 @@ mpl.font_manager.fontManager.addfont(FONTPATH) # i18n
mpl.rcParams['font.family'] = ['Source Han Serif SC'] # i18n
```


## 回顾：普通确定性等价

{ref}`配套讲座 <certainty_equiv_theil1>` 详细地建立了 CE 性质。
Expand Down Expand Up @@ -177,7 +174,6 @@ plt.tight_layout()
plt.show()
```


## 模型不确定性与稳健性

### 设定与乘数问题
Expand Down Expand Up @@ -343,7 +339,6 @@ plt.show()

这与普通 CE 形成鲜明对比：在稳健性下，*策略增益和值矩阵都依赖于稳健性参数 $\theta$ 和噪声载荷矩阵 $C$*。


## 稳健性下的值函数

在对稳健性有偏好的情况下，{eq}`eq:multiplier` 的最优值再次是二次的，
Expand Down Expand Up @@ -380,7 +375,6 @@ z_{t+1} = f(z_t,\, w_{t+1}),

随机性的存在降低了值（常数 $p$），但不改变决策规则。


## 风险敏感偏好

在 {cite:t}`Jacobson_73` 和 {cite:t}`Whittle_1990` 的基础上，{cite:t}`hansen2004certainty` 证明了同样的决策规则可以通过**风险敏感偏好**来重新解释。
Expand Down Expand Up @@ -428,7 +422,6 @@ U_t = r(y_t, u_t) + \beta\, \mathcal{R}_t(U_{t+1})

> **关键等价性：** 参数为 $\theta$ 的稳健控制与参数为 $\sigma = -\theta^{-1}$ 的风险敏感控制产生相同的决策规则。


## 应用：永久收入模型

我们现在在一个具体的线性二次型永久收入模型中说明上述所有内容。
Expand Down Expand Up @@ -577,7 +570,7 @@ mystnb:
caption: Standard vs robust consumption paths
name: fig-std-vs-robust-paths
---
np.random.seed(0)
rng = np.random.default_rng(0)
T_sim = 100

def simulate_ar1(φ, ν, shocks, mu0=0.0):
Expand All @@ -587,7 +580,7 @@ def simulate_ar1(φ, ν, shocks, mu0=0.0):
path[t] = φ * path[t-1] + ν * ε
return path

shock_path = np.random.randn(T_sim - 1)
shock_path = rng.standard_normal(T_sim - 1)
mu0_init = 10.0
mu_std_path = simulate_ar1(φ_std, ν_std, shock_path, mu0=mu0_init)
mu_rob_path = simulate_ar1(φ_rob, ν_rob, shock_path, mu0=mu0_init)
Expand Down Expand Up @@ -658,7 +651,6 @@ plt.show()

这是因为对稳健性的偏好通过进入随机贴现因子的 $\mathcal{D}(P)$ 矩阵生成了不同的状态价格。


## 小结

下表浓缩了主要结果：
Expand All @@ -673,7 +665,6 @@ plt.show()

差异在于第二步中形成的是哪种信念。


## 练习

```{exercise-start}
Expand Down Expand Up @@ -778,4 +769,4 @@ for s in σ_trio:
正如 {cite:t}`HST_1999` 所表明的，这就是为什么即使储蓄计划一致，这两种参数化仍然意味着不同的资产价格。

```{solution-end}
```
```
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