[FIX] Fix String Warnings (#194)

* update strings

* update a few missing cases

Author: HumphreyYang

lectures/BCG_complete_mkts.md

@@ -1090,8 +1090,8 @@ epsgrid = np.linspace(-1,1,1000)
 
 
 fig, ax = plt.subplots(1,2,figsize=(14,6))
-ax[0].plot(epsgrid, mdl1.w11(epsgrid), color='black', label='Agent 1\'s endowment')
-ax[0].plot(epsgrid, mdl1.w21(epsgrid), color='blue', label='Agent 2\'s endowment')
+ax[0].plot(epsgrid, mdl1.w11(epsgrid), color='black', label=r'Agent 1\'s endowment')
+ax[0].plot(epsgrid, mdl1.w21(epsgrid), color='blue', label=r'Agent 2\'s endowment')
 ax[0].plot(epsgrid, mdl1.Y(epsgrid,1), color='red', label=r'Production with $k=1$')
 ax[0].set_xlim([-1,1])
 ax[0].set_ylim([0,7])
@@ -1100,8 +1100,8 @@ ax[0].set_title(r'Model with $\chi_1 = 0$, $\chi_2 = 0.9$')
 ax[0].legend()
 ax[0].grid()
 
-ax[1].plot(epsgrid, mdl2.w11(epsgrid), color='black', label='Agent 1\'s endowment')
-ax[1].plot(epsgrid, mdl2.w21(epsgrid), color='blue', label='Agent 2\'s endowment')
+ax[1].plot(epsgrid, mdl2.w11(epsgrid), color='black', label=r'Agent 1\'s endowment')
+ax[1].plot(epsgrid, mdl2.w21(epsgrid), color='blue', label=r'Agent 2\'s endowment')
 ax[1].plot(epsgrid, mdl2.Y(epsgrid,1), color='red', label=r'Production with $k=1$')
 ax[1].set_xlim([-1,1])
 ax[1].set_ylim([0,7])

lectures/arma.md

@@ -241,7 +241,7 @@ for i, ϕ in enumerate((0.8, -0.8)):
     times = list(range(16))
     acov = [ϕ**k / (1 - ϕ**2) for k in times]
     ax.plot(times, acov, 'bo-', alpha=0.6,
-            label=f'autocovariance, $\phi = {ϕ:.2}$')
+            label=fr'autocovariance, $\phi = {ϕ:.2}$')
     ax.legend(loc='upper right')
     ax.set(xlabel='time', xlim=(0, 15))
     ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)
@@ -479,7 +479,7 @@ for i, ϕ in enumerate((0.8, -0.8)):
     ax = axes[i]
     sd = ar1_sd(ϕ, ωs)
     ax.plot(ωs, sd, 'b-', alpha=0.6, lw=2,
-            label='spectral density, $\phi = {ϕ:.2}$')
+            label=fr'spectral density, $\phi = {ϕ:.2}$')
     ax.legend(loc='upper center')
     ax.set(xlabel='frequency', xlim=(0, np.pi))
 plt.show()
@@ -525,21 +525,21 @@ plt.subplots_adjust(hspace=0.25)
 
 # Autocovariance when ϕ = -0.8
 ax = axes[0]
-ax.plot(times, y1, 'bo-', alpha=0.6, label='$\gamma(k)$')
+ax.plot(times, y1, 'bo-', alpha=0.6, label=r'$\gamma(k)$')
 ax.legend(loc='upper right')
 ax.set(xlim=(0, 15), yticks=(-2, 0, 2))
 ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)
 
 # Cycles at frequency π
 ax = axes[1]
-ax.plot(times, y2, 'bo-', alpha=0.6, label='$\cos(\pi k)$')
+ax.plot(times, y2, 'bo-', alpha=0.6, label=r'$\cos(\pi k)$')
 ax.legend(loc='upper right')
 ax.set(xlim=(0, 15), yticks=(-1, 0, 1))
 ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)
 
 # Product
 ax = axes[2]
-ax.stem(times, y3, label='$\gamma(k) \cos(\pi k)$')
+ax.stem(times, y3, label=r'$\gamma(k) \cos(\pi k)$')
 ax.legend(loc='upper right')
 ax.set(xlim=(0, 15), ylim=(-3, 3), yticks=(-1, 0, 1, 2, 3))
 ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)
@@ -565,21 +565,21 @@ plt.subplots_adjust(hspace=0.25)
 
 # Autocovariance when phi = -0.8
 ax = axes[0]
-ax.plot(times, y1, 'bo-', alpha=0.6, label='$\gamma(k)$')
+ax.plot(times, y1, 'bo-', alpha=0.6, label=r'$\gamma(k)$')
 ax.legend(loc='upper right')
 ax.set(xlim=(0, 15), yticks=(-2, 0, 2))
 ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)
 
 # Cycles at frequency π
 ax = axes[1]
-ax.plot(times, y2, 'bo-', alpha=0.6, label='$\cos(\pi k/3)$')
+ax.plot(times, y2, 'bo-', alpha=0.6, label=r'$\cos(\pi k/3)$')
 ax.legend(loc='upper right')
 ax.set(xlim=(0, 15), yticks=(-1, 0, 1))
 ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)
 
 # Product
 ax = axes[2]
-ax.stem(times, y3, label='$\gamma(k) \cos(\pi k/3)$')
+ax.stem(times, y3, label=r'$\gamma(k) \cos(\pi k/3)$')
 ax.legend(loc='upper right')
 ax.set(xlim=(0, 15), ylim=(-3, 3), yticks=(-1, 0, 1, 2, 3))
 ax.hlines(0, 0, 15, linestyle='--', alpha=0.5)

lectures/asset_pricing_lph.md

@@ -384,7 +384,7 @@ plt.title('mean standard deviation frontier')
 plt.xlabel(r"$\sigma(R^i)$")
 plt.ylabel(r"$E (R^i) $")
 plt.text(.053, 1.015, "(.05,1.015)")
-ax.plot(.05, 1.015, 'o', label="$(\sigma(R^j), E R^j)$")
+ax.plot(.05, 1.015, 'o', label=r"$(\sigma(R^j), E R^j)$")
 # Add a legend and show the plot
 ax.legend()
 plt.show()

lectures/black_litterman.md

@@ -319,8 +319,8 @@ d_m = r_m / σ_m
 x = np.arange(N) + 1
 fig, ax = plt.subplots(figsize=(8, 5))
 ax.set_title(r'Difference between $\hat{\mu}$ (estimate) and $\mu_{BL}$ (market implied)')
-ax.plot(x, μ_est, 'o', c='k', label='$\hat{\mu}$')
-ax.plot(x, μ_m, 'o', c='r', label='$\mu_{BL}$')
+ax.plot(x, μ_est, 'o', c='k', label=r'$\hat{\mu}$')
+ax.plot(x, μ_m, 'o', c='r', label=r'$\mu_{BL}$')
 ax.vlines(x, μ_m, μ_est, lw=1)
 ax.axhline(0, c='k', ls='--')
 ax.set_xlabel('Assets')

lectures/calvo_machine_learn.md

@@ -1230,7 +1230,7 @@ Let's plot the regression line $\mu_t = .0645 + 1.5995 \theta_t$  and the points
 
 ```{code-cell} ipython3
 plt.scatter(θs, μs, label=r'$\mu_t$')
-plt.plot(θs, results1.predict(X1_θ), 'grey', label='$\hat \mu_t$', linestyle='--')
+plt.plot(θs, results1.predict(X1_θ), 'grey', label=r'$\hat \mu_t$', linestyle='--')
 plt.xlabel(r'$\theta_t$')
 plt.ylabel(r'$\mu_t$')
 plt.legend()
@@ -1271,7 +1271,7 @@ Let's plot $\theta_t$ for $t =0, 1, \ldots, T$ along the line.
 
 ```{code-cell} ipython3
 plt.scatter(θ_t, θ_t1, label=r'$\theta_{t+1}$')
-plt.plot(θ_t, results2.predict(X2_θ), color='grey', label='$\hat θ_{t+1}$', linestyle='--')
+plt.plot(θ_t, results2.predict(X2_θ), color='grey', label=r'$\hat θ_{t+1}$', linestyle='--')
 plt.xlabel(r'$\theta_t$')
 plt.ylabel(r'$\theta_{t+1}$')
 plt.legend()
@@ -1321,7 +1321,7 @@ X3_grid = np.column_stack((np.ones(len(θ_grid)), θ_grid, θ_grid**2))
 
 plt.scatter(θs, v_t)
 plt.plot(θ_grid, results3.predict(X3_grid), color='grey', 
-         label='$\hat v_t$', linestyle='--')
+         label=r'$\hat v_t$', linestyle='--')
 plt.axhline(V_CR, color='C1', alpha=0.5)
 
 plt.text(max(θ_grid) - max(θ_grid)*0.025, V_CR, '$V^{CR}$', color='C1', 

lectures/coase.md

@@ -552,7 +552,7 @@ for i, s in enumerate(pc.grid):
     ell_star[i] = e
 
 fig, ax = plt.subplots()
-ax.plot(pc.grid, ell_star, label="$\ell^*$")
+ax.plot(pc.grid, ell_star, label=r"$\ell^*$")
 ax.legend(fontsize=14)
 plt.show()
 ```

lectures/discrete_dp.md

@@ -935,7 +935,7 @@ for beta in discount_factors:
     res0 = ddp0.solve()
     k_path_ind = res0.mc.simulate(init=k_init_ind, ts_length=sample_size)
     k_path = grid[k_path_ind]
-    ax.plot(k_path, 'o-', lw=2, alpha=0.75, label=f'$\\beta = {beta}$')
+    ax.plot(k_path, 'o-', lw=2, alpha=0.75, label=fr'$\beta = {beta}$')
 
 ax.legend(loc='lower right')
 plt.show()

lectures/markov_jump_lq.md

@@ -444,8 +444,8 @@ u1_star = - ex1_a.Fs[0, 0, 1] - ex1_a.Fs[0, 0, 0] * k_grid
 u2_star = - ex1_a.Fs[1, 0, 1] - ex1_a.Fs[1, 0, 0] * k_grid
 
 fig, ax = plt.subplots()
-ax.plot(k_grid, k_grid + u1_star, label="$\overline{s}_1$ (high)")
-ax.plot(k_grid, k_grid + u2_star, label="$\overline{s}_2$ (low)")
+ax.plot(k_grid, k_grid + u1_star, label=r"$\overline{s}_1$ (high)")
+ax.plot(k_grid, k_grid + u2_star, label=r"$\overline{s}_2$ (low)")
 
 # The optimal k*
 ax.scatter([0.5, 0.5], [0.5, 0.5], marker="*")
@@ -546,10 +546,10 @@ for i, λ in enumerate(λ_vals):
 ```{code-cell} python3
 for i, state_var in enumerate(state_vec1):
     fig, ax = plt.subplots()
-    ax.plot(λ_vals, F1[:, i], label="$\overline{s}_1$", color="b")
-    ax.plot(λ_vals, F2[:, i], label="$\overline{s}_2$", color="r")
+    ax.plot(λ_vals, F1[:, i], label=r"$\overline{s}_1$", color="b")
+    ax.plot(λ_vals, F2[:, i], label=r"$\overline{s}_2$", color="r")
 
-    ax.set_xlabel("$\lambda$")
+    ax.set_xlabel(r"$\lambda$")
     ax.set_ylabel("$F_{s_t}$")
     ax.set_title(f"Coefficient on {state_var}")
     ax.legend()
@@ -617,8 +617,8 @@ for i, state_var in enumerate(state_vec1):
     ax.plot_surface(λ_grid, δ_grid, F1_grid[:, :, i], color="b")
     # low adjustment cost, red surface
     ax.plot_surface(λ_grid, δ_grid, F2_grid[:, :, i], color="r")
-    ax.set_xlabel("$\lambda$")
-    ax.set_ylabel("$\delta$")
+    ax.set_xlabel(r"$\lambda$")
+    ax.set_ylabel(r"$\delta$")
     ax.set_zlabel("$F_{s_t}$")
     ax.set_title(f"coefficient on {state_var}")
     plt.show()
@@ -656,11 +656,11 @@ def run(construct_func, vals_dict, state_vec):
     for i, state_var in enumerate(state_vec):
         fig = plt.figure()
         ax = fig.add_subplot(111)
-        ax.plot(λ_vals, F1[:, i], label="$\overline{s}_1$", color="b")
-        ax.plot(λ_vals, F2[:, i], label="$\overline{s}_2$", color="r")
+        ax.plot(λ_vals, F1[:, i], label=r"$\overline{s}_1$", color="b")
+        ax.plot(λ_vals, F2[:, i], label=r"$\overline{s}_2$", color="r")
 
-        ax.set_xlabel("$\lambda$")
-        ax.set_ylabel("$F(\overline{s}_t)$")
+        ax.set_xlabel(r"$\lambda$")
+        ax.set_ylabel(r"$F(\overline{s}_t)$")
         ax.set_title(f"coefficient on {state_var}")
         ax.legend()
         plt.show()
@@ -674,17 +674,17 @@ def run(construct_func, vals_dict, state_vec):
             F = [F1, F2][i]
             c = ["b", "r"][i]
             ax.plot([0, 1], [k_star[i], k_star[i]], "--",
-                    color=c, label="$k^*(\overline{s}_"+str(i+1)+")$")
+                    color=c, label=r"$k^*(\overline{s}_"+str(i+1)+")$")
             ax.plot(λ_vals, - F[:, 1] / F[:, 0], color=c,
-                    label="$k^{target}(\overline{s}_"+str(i+1)+")$")
+                    label=r"$k^{target}(\overline{s}_"+str(i+1)+")$")
 
         # Plot a vertical line at λ=0.5
         ax.plot([0.5, 0.5], [min(k_star), max(k_star)], "-.")
 
-        ax.set_xlabel("$\lambda$")
+        ax.set_xlabel(r"$\lambda$")
         ax.set_ylabel("$k$")
         ax.set_title("Optimal k levels and k targets")
-        ax.text(0.5, min(k_star)+(max(k_star)-min(k_star))/20, "$\lambda=0.5$")
+        ax.text(0.5, min(k_star)+(max(k_star)-min(k_star))/20, r"$\lambda=0.5$")
         ax.legend(bbox_to_anchor=(1., 1.))
         plt.show()
 
@@ -714,9 +714,9 @@ def run(construct_func, vals_dict, state_vec):
         ax = fig.add_subplot(111, projection='3d')
         ax.plot_surface(λ_grid, δ_grid, F1_grid[:, :, i], color="b")
         ax.plot_surface(λ_grid, δ_grid, F2_grid[:, :, i], color="r")
-        ax.set_xlabel("$\lambda$")
-        ax.set_ylabel("$\delta$")
-        ax.set_zlabel("$F(\overline{s}_t)$")
+        ax.set_xlabel(r"$\lambda$")
+        ax.set_ylabel(r"$\delta$")
+        ax.set_zlabel(r"$F(\overline{s}_t)$")
         ax.set_title(f"coefficient on {state_var}")
         plt.show()
 ```

lectures/opt_tax_recur.md

@@ -1242,7 +1242,7 @@ for ax, title, plot in zip(axes, titles, [tax_policy, interest_rate]):
     ax.set(title=title, xlim=(min(gov_debt), max(gov_debt)))
     ax.grid()
 
-axes[0].legend(('Time $t=0$', 'Time $t \geq 1$'))
+axes[0].legend(('Time $t=0$', r'Time $t \geq 1$'))
 axes[1].set_xlabel('Initial Government Debt')
 
 fig.tight_layout()

lectures/rob_markov_perf.md

@@ -469,7 +469,7 @@ The function's code is as follows
 def nnash_robust(A, C, B1, B2, R1, R2, Q1, Q2, S1, S2, W1, W2, M1, M2,
                  θ1, θ2, beta=1.0, tol=1e-8, max_iter=1000):
 
-    """
+    r"""
     Compute the limit of a Nash linear quadratic dynamic game with
     robustness concern.