%pip install matplotlib numpy pandas statsmodels wooldridge scipy -qNote: you may need to restart the kernel to use updated packages.
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 statsnyse = 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")table1:
b se t pval
Intercept 0.1796 0.0807 2.2248 0.0264
ret_lag1 0.0589 0.0380 1.5490 0.1218
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")table2:
b se t pval
Intercept 0.1857 0.0812 2.2889 0.0224
ret_lag1 0.0603 0.0382 1.5799 0.1146
ret_lag2 -0.0381 0.0381 -0.9982 0.3185
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")table3:
b se t pval
Intercept 0.1794 0.0816 2.1990 0.0282
ret_lag1 0.0614 0.0382 1.6056 0.1088
ret_lag2 -0.0403 0.0383 -1.0519 0.2932
ret_lag3 0.0307 0.0382 0.8038 0.4218
11.2 The Nature of Highly Persistent Time Series¶
# 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")
# 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¶
# 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")
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")fertil3.head():
gfr pe year t tsq pe_1 pe_2 pe_3 pe_4 pill ... \
1913 124.699997 0.00 1913 1 1 NaN NaN NaN NaN 0 ...
1914 126.599998 0.00 1914 2 4 0.0 NaN NaN NaN 0 ...
1915 125.000000 0.00 1915 3 9 0.0 0.0 NaN NaN 0 ...
1916 123.400002 0.00 1916 4 16 0.0 0.0 0.0 NaN 0 ...
1917 121.000000 19.27 1917 5 25 0.0 0.0 0.0 0.0 0 ...
cpe_3 cpe_4 gfr_1 cgfr_1 cgfr_2 cgfr_3 cgfr_4 \
1913 NaN NaN NaN NaN NaN NaN NaN
1914 NaN NaN 124.699997 NaN NaN NaN NaN
1915 NaN NaN 126.599998 1.900002 NaN NaN NaN
1916 NaN NaN 125.000000 -1.599998 1.900002 NaN NaN
1917 0.0 NaN 123.400002 -1.599998 -1.599998 1.900002 NaN
gfr_2 gfr_diff1 pe_diff1
1913 NaN NaN NaN
1914 NaN 1.900002 0.00
1915 124.699997 -1.599998 0.00
1916 126.599998 -1.599998 0.00
1917 125.000000 -2.400002 19.27
[5 rows x 26 columns]
# 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")table1:
b se t pval
Intercept -0.7848 0.5020 -1.5632 0.1226
pe_diff1 -0.0427 0.0284 -1.5045 0.1370
# 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")table2:
b se t pval
Intercept -0.9637 0.4678 -2.0602 0.0434
pe_diff1 -0.0362 0.0268 -1.3522 0.1810
pe_diff1_lag1 -0.0140 0.0276 -0.5070 0.6139
pe_diff1_lag2 0.1100 0.0269 4.0919 0.0001