Last Update: December 15, 2020
Multiple regression assumptions consist of independent variables correct specification, independent variables no linear dependence, regression correct functional form, residuals no autocorrelation, residuals homoscedasticity and residuals normality.
This topic is part of Multiple Regression Analysis with Python course. Feel free to take a look at Course Curriculum.
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Residuals homoscedasticity consists of evaluating whether regression residuals or forecasting errors have a constant variance.
This is evaluated through Breusch-Pagan heteroscedasticity test  which consists of using squared original regression residuals data as dependent variable together with original regression independent variables and assessing if independent variables are jointly statistically significant.
1. Formula notation.
1.1. Breusch-Pagan test formula notation.
- Note: number of original regression independent or explanatory variables not fixed and only included for educational purposes.
Where = squared original regression estimated residuals or forecasting errors, = regression constant or intercept, = regression coefficients, = original regression independent or explanatory variables data, = regression residuals or forecasting errors.
1.2. Breusch-Pagan test.
Breusch-Pagan Lagrange multiplier statistic :
- If Breusch-Pagan Lagrange multiplier statistic level of statistical significance then residuals were heteroscedastic with level of statistical confidence.
- If Breusch-Pagan Lagrange multiplier statistic level of statistical significance then residuals were homoscedastic with level of statistical confidence.
2. Python code example.
2.1. Import Python packages .
import numpy as np import pandas as pd import statsmodels.regression.linear_model as rg import statsmodels.tools.tools as ct import statsmodels.stats.diagnostic as dg
2.2. Breusch-Pagan test data.
- Data: S&P 500® index replicating ETF (ticker symbol: SPY) adjusted close prices arithmetic monthly returns, 1 Year U.S. Treasury Bill Yield, 10 Years U.S. Treasury Note Yield, Merrill Lynch U.S. High Yield Corporate Bond Index Yield effective monthly yields, U.S. Consumer Price Index, U.S. Producer Price Index monthly inflations or deflations, West Texas Intermediate Oil prices arithmetic monthly returns, U.S. Industrial Production Index value, U.S. Personal Consumption Expenditures arithmetic monthly changes (1997-2016).
data = pd.read_csv('Data//Breusch-Pagan-Test-Data.txt', index_col='Date', parse_dates=True)
2.3. Breusch-Pagan test calculation and output.
data.loc[:, 'const'] = ct.add_constant(data) ivar = ['const', 't1y', 't10y', 'hyield', 'cpi', 'ppi', 'oil', 'indpro', 'pce'] reg = rg.OLS(data['stocks'], data[ivar], hasconst=bool).fit() res = reg.resid
In: print('== Residuals Homoscedasticity Breusch-Pagan Test ==') print('') print('Breusch-Pagan LM Test Statistic:', np.round(dg.het_breuschpagan(res, exog_het=data[ivar]), 6)) print('Breusch-Pagan LM Test P-Value:', np.round(dg.het_breuschpagan(res, exog_het=data[ivar]), 6))
Out: == Residuals Homoscedasticity Breusch-Pagan Test == Breusch-Pagan LM Test Statistic: 49.943053 Breusch-Pagan LM Test P-Value: 0.0
 Breusch, T. S.; Pagan, A. R. “A Simple Test for Heteroskedasticity and Random Coefficient Variation”. Econometrica. 1979.
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