update ch11

markdown/Ch11. Further Issues in Using OLS with Time Series Data.md

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+---
+jupyter:
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+    formats: notebooks//ipynb,markdown//md,scripts//py
+    text_representation:
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+    language: python
+    name: python3
+---
+
+# 11. Further Issues in Using OLS with Time Series Data
+
+```python
+%pip install matplotlib numpy pandas statsmodels wooldridge scipy -q
+```
+
+```python
+import matplotlib.pyplot as plt
+import numpy as np
+import pandas as pd
+import statsmodels.formula.api as smf
+import wooldridge as wool
+from scipy import stats
+```
+
+## 11.1 Asymptotics with Time Seires
+
+### Example 11.4: Efficient Markets Hypothesis
+
+```python
+nyse = wool.data("nyse")
+nyse["ret"] = nyse["return"]
+
+# add all lags up to order 3:
+nyse["ret_lag1"] = nyse["ret"].shift(1)
+nyse["ret_lag2"] = nyse["ret"].shift(2)
+nyse["ret_lag3"] = nyse["ret"].shift(3)
+
+# linear regression of model with lags:
+reg1 = smf.ols(formula="ret ~ ret_lag1", data=nyse)
+reg2 = smf.ols(formula="ret ~ ret_lag1 + ret_lag2", data=nyse)
+reg3 = smf.ols(formula="ret ~ ret_lag1 + ret_lag2 + ret_lag3", data=nyse)
+results1 = reg1.fit()
+results2 = reg2.fit()
+results3 = reg3.fit()
+
+# print regression tables:
+table1 = pd.DataFrame(
+    {
+        "b": round(results1.params, 4),
+        "se": round(results1.bse, 4),
+        "t": round(results1.tvalues, 4),
+        "pval": round(results1.pvalues, 4),
+    },
+)
+print(f"table1: \n{table1}\n")
+```
+
+```python
+table2 = pd.DataFrame(
+    {
+        "b": round(results2.params, 4),
+        "se": round(results2.bse, 4),
+        "t": round(results2.tvalues, 4),
+        "pval": round(results2.pvalues, 4),
+    },
+)
+print(f"table2: \n{table2}\n")
+```
+
+```python
+table3 = pd.DataFrame(
+    {
+        "b": round(results3.params, 4),
+        "se": round(results3.bse, 4),
+        "t": round(results3.tvalues, 4),
+        "pval": round(results3.pvalues, 4),
+    },
+)
+print(f"table3: \n{table3}\n")
+```
+
+## 11.2 The Nature of Highly Persistent Time Series
+
+```python
+# set the random seed:
+np.random.seed(1234567)
+
+# initialize plot:
+x_range = np.linspace(0, 50, num=51)
+plt.ylim([-18, 18])
+plt.xlim([0, 50])
+
+# loop over draws:
+for r in range(30):
+    # i.i.d. standard normal shock:
+    e = stats.norm.rvs(0, 1, size=51)
+
+    # set first entry to 0 (gives y_0 = 0):
+    e[0] = 0
+
+    # random walk as cumulative sum of shocks:
+    y = np.cumsum(e)
+
+    # add line to graph:
+    plt.plot(x_range, y, color="lightgrey", linestyle="-")
+
+plt.axhline(linewidth=2, linestyle="--", color="black")
+plt.ylabel("y")
+plt.xlabel("time")
+```
+
+```python
+# set the random seed:
+np.random.seed(1234567)
+
+# initialize plot:
+x_range = np.linspace(0, 50, num=51)
+plt.ylim([0, 100])
+plt.xlim([0, 50])
+
+# loop over draws:
+for r in range(30):
+    # i.i.d. standard normal shock:
+    e = stats.norm.rvs(0, 1, size=51)
+
+    # set first entry to 0 (gives y_0 = 0):
+    e[0] = 0
+
+    # random walk as cumulative sum of shocks plus drift:
+    y = np.cumsum(e) + 2 * x_range
+
+    # add line to graph:
+    plt.plot(x_range, y, color="lightgrey", linestyle="-")
+
+plt.plot(x_range, 2 * x_range, linewidth=2, linestyle="--", color="black")
+plt.ylabel("y")
+plt.xlabel("time")
+```
+
+## 11.3 Differences of Highly Persistent Time Series
+
+```python
+# set the random seed:
+np.random.seed(1234567)
+
+# initialize plot:
+x_range = np.linspace(1, 50, num=50)
+plt.ylim([-1, 5])
+plt.xlim([0, 50])
+
+# loop over draws:
+for r in range(30):
+    # i.i.d. standard normal shock and cumulative sum of shocks:
+    e = stats.norm.rvs(0, 1, size=51)
+    e[0] = 0
+    y = np.cumsum(2 + e)
+
+    # first difference:
+    Dy = y[1:51] - y[0:50]
+
+    # add line to graph:
+    plt.plot(x_range, Dy, color="lightgrey", linestyle="-")
+
+plt.axhline(y=2, linewidth=2, linestyle="--", color="black")
+plt.ylabel("y")
+plt.xlabel("time")
+```
+
+## 11.4 Regression with First Differences
+
+### Example 11.6: Fertility Equation
+
+```python
+fertil3 = wool.data("fertil3")
+T = len(fertil3)
+
+# define time series (years only) beginning in 1913:
+fertil3.index = pd.date_range(start="1913", periods=T, freq="YE").year
+
+# compute first differences:
+fertil3["gfr_diff1"] = fertil3["gfr"].diff()
+fertil3["pe_diff1"] = fertil3["pe"].diff()
+print(f"fertil3.head(): \n{fertil3.head()}\n")
+```
+
+```python
+# linear regression of model with first differences:
+reg1 = smf.ols(formula="gfr_diff1 ~ pe_diff1", data=fertil3)
+results1 = reg1.fit()
+
+# print regression table:
+table1 = pd.DataFrame(
+    {
+        "b": round(results1.params, 4),
+        "se": round(results1.bse, 4),
+        "t": round(results1.tvalues, 4),
+        "pval": round(results1.pvalues, 4),
+    },
+)
+print(f"table1: \n{table1}\n")
+```
+
+```python
+# linear regression of model with lagged differences:
+fertil3["pe_diff1_lag1"] = fertil3["pe_diff1"].shift(1)
+fertil3["pe_diff1_lag2"] = fertil3["pe_diff1"].shift(2)
+
+reg2 = smf.ols(
+    formula="gfr_diff1 ~ pe_diff1 + pe_diff1_lag1 + pe_diff1_lag2",
+    data=fertil3,
+)
+results2 = reg2.fit()
+
+# print regression table:
+table2 = pd.DataFrame(
+    {
+        "b": round(results2.params, 4),
+        "se": round(results2.bse, 4),
+        "t": round(results2.tvalues, 4),
+        "pval": round(results2.pvalues, 4),
+    },
+)
+print(f"table2: \n{table2}\n")
+```