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13 changes: 7 additions & 6 deletions lectures/ar1_processes.md
Original file line number Diff line number Diff line change
Expand Up @@ -409,13 +409,12 @@ from numba import njit
from scipy.special import factorial2

@njit
def sample_moments_ar1(k, m=100_000, mu_0=0.0, sigma_0=1.0, seed=1234):
np.random.seed(seed)
def sample_moments_ar1(k, rng, m=100_000, mu_0=0.0, sigma_0=1.0):
sample_sum = 0.0
x = mu_0 + sigma_0 * np.random.randn()
x = mu_0 + sigma_0 * rng.standard_normal()
for t in range(m):
sample_sum += (x - mu_star)**k
x = a * x + b + c * np.random.randn()
x = a * x + b + c * rng.standard_normal()
return sample_sum / m

def true_moments_ar1(k):
Expand All @@ -429,7 +428,8 @@ sample_moments = np.empty(len(k_vals), dtype=float)
true_moments = np.empty(len(k_vals), dtype=float)

for k_idx, k in enumerate(k_vals):
sample_moments[k_idx] = sample_moments_ar1(k)
rng = np.random.default_rng(1234)
sample_moments[k_idx] = sample_moments_ar1(k, rng)
true_moments[k_idx] = true_moments_ar1(k)

fig, ax = plt.subplots()
Expand Down Expand Up @@ -622,8 +622,9 @@ s_next = np.sqrt(a**2 * s**2 + c**2)

```{code-cell} ipython3
n = 2000
rng = np.random.default_rng()
x_draws = ψ.rvs(n)
x_draws_next = a * x_draws + b + c * np.random.randn(n)
x_draws_next = a * x_draws + b + c * rng.standard_normal(n)
kde = KDE(x_draws_next)

x_grid = np.linspace(μ - 1, μ + 1, 100)
Expand Down
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