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#!/usr/bin/env python
# coding: utf-8
# DO NOT EDIT
# Autogenerated from the notebook recursive_ls.ipynb.
# Edit the notebook and then sync the output with this file.
#
# flake8: noqa
# DO NOT EDIT
# # Recursive least squares
#
# Recursive least squares is an expanding window version of ordinary least
# squares. In addition to availability of regression coefficients computed
# recursively, the recursively computed residuals the construction of
# statistics to investigate parameter instability.
#
# The `RecursiveLS` class allows computation of recursive residuals and
# computes CUSUM and CUSUM of squares statistics. Plotting these statistics
# along with reference lines denoting statistically significant deviations
# from the null hypothesis of stable parameters allows an easy visual
# indication of parameter stability.
#
# Finally, the `RecursiveLS` model allows imposing linear restrictions on
# the parameter vectors, and can be constructed using the formula interface.
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import statsmodels.api as sm
from pandas_datareader.data import DataReader
np.set_printoptions(suppress=True)
# ## Example 1: Copper
#
# We first consider parameter stability in the copper dataset (description
# below).
print(sm.datasets.copper.DESCRLONG)
dta = sm.datasets.copper.load_pandas().data
dta.index = pd.date_range("1951-01-01", "1975-01-01", freq="YS")
endog = dta["WORLDCONSUMPTION"]
# To the regressors in the dataset, we add a column of ones for an
# intercept
exog = sm.add_constant(
dta[["COPPERPRICE", "INCOMEINDEX", "ALUMPRICE", "INVENTORYINDEX"]])
# First, construct and fit the model, and print a summary. Although the
# `RLS` model computes the regression parameters recursively, so there are
# as many estimates as there are datapoints, the summary table only presents
# the regression parameters estimated on the entire sample; except for small
# effects from initialization of the recursions, these estimates are
# equivalent to OLS estimates.
mod = sm.RecursiveLS(endog, exog)
res = mod.fit()
print(res.summary())
# The recursive coefficients are available in the `recursive_coefficients`
# attribute. Alternatively, plots can generated using the
# `plot_recursive_coefficient` method.
print(res.recursive_coefficients.filtered[0])
res.plot_recursive_coefficient(range(mod.k_exog), alpha=None, figsize=(10, 6))
# The CUSUM statistic is available in the `cusum` attribute, but usually
# it is more convenient to visually check for parameter stability using the
# `plot_cusum` method. In the plot below, the CUSUM statistic does not move
# outside of the 5% significance bands, so we fail to reject the null
# hypothesis of stable parameters at the 5% level.
print(res.cusum)
fig = res.plot_cusum()
# Another related statistic is the CUSUM of squares. It is available in
# the `cusum_squares` attribute, but it is similarly more convenient to
# check it visually, using the `plot_cusum_squares` method. In the plot
# below, the CUSUM of squares statistic does not move outside of the 5%
# significance bands, so we fail to reject the null hypothesis of stable
# parameters at the 5% level.
res.plot_cusum_squares()
# ## Example 2: Quantity theory of money
#
# The quantity theory of money suggests that "a given change in the rate
# of change in the quantity of money induces ... an equal change in the rate
# of price inflation" (Lucas, 1980). Following Lucas, we examine the
# relationship between double-sided exponentially weighted moving averages
# of money growth and CPI inflation. Although Lucas found the relationship
# between these variables to be stable, more recently it appears that the
# relationship is unstable; see e.g. Sargent and Surico (2010).
start = "1959-12-01"
end = "2015-01-01"
m2 = DataReader("M2SL", "fred", start=start, end=end)
cpi = DataReader("CPIAUCSL", "fred", start=start, end=end)
def ewma(series, beta, n_window):
nobs = len(series)
scalar = (1 - beta) / (1 + beta)
ma = []
k = np.arange(n_window, 0, -1)
weights = np.r_[beta**k, 1, beta**k[::-1]]
for t in range(n_window, nobs - n_window):
window = series.iloc[t - n_window:t + n_window + 1].values
ma.append(scalar * np.sum(weights * window))
return pd.Series(ma,
name=series.name,
index=series.iloc[n_window:-n_window].index)
m2_ewma = ewma(
np.log(m2["M2SL"].resample("QS").mean()).diff().iloc[1:], 0.95, 10 * 4)
cpi_ewma = ewma(
np.log(cpi["CPIAUCSL"].resample("QS").mean()).diff().iloc[1:], 0.95,
10 * 4)
# After constructing the moving averages using the $\beta = 0.95$ filter
# of Lucas (with a window of 10 years on either side), we plot each of the
# series below. Although they appear to move together prior for part of the
# sample, after 1990 they appear to diverge.
fig, ax = plt.subplots(figsize=(13, 3))
ax.plot(m2_ewma, label="M2 Growth (EWMA)")
ax.plot(cpi_ewma, label="CPI Inflation (EWMA)")
ax.legend()
endog = cpi_ewma
exog = sm.add_constant(m2_ewma)
exog.columns = ["const", "M2"]
mod = sm.RecursiveLS(endog, exog)
res = mod.fit()
print(res.summary())
res.plot_recursive_coefficient(1, alpha=None)
# The CUSUM plot now shows substantial deviation at the 5% level,
# suggesting a rejection of the null hypothesis of parameter stability.
res.plot_cusum()
# Similarly, the CUSUM of squares shows substantial deviation at the 5%
# level, also suggesting a rejection of the null hypothesis of parameter
# stability.
res.plot_cusum_squares()
# ## Example 3: Linear restrictions and formulas
# ## Linear restrictions
#
# It is not hard to implement linear restrictions, using the `constraints`
# parameter in constructing the model.
endog = dta["WORLDCONSUMPTION"]
exog = sm.add_constant(
dta[["COPPERPRICE", "INCOMEINDEX", "ALUMPRICE", "INVENTORYINDEX"]])
mod = sm.RecursiveLS(endog, exog, constraints="COPPERPRICE = ALUMPRICE")
res = mod.fit()
print(res.summary())
# ## Formula
#
# One could fit the same model using the class method `from_formula`.
mod = sm.RecursiveLS.from_formula(
"WORLDCONSUMPTION ~ COPPERPRICE + INCOMEINDEX + ALUMPRICE + INVENTORYINDEX",
dta,
constraints="COPPERPRICE = ALUMPRICE",
)
res = mod.fit()
print(res.summary())
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