Durbin-Watson (DW) test
In regression analysis, Durbin-Watson (DW) is useful for checking the first-order autocorrelation (serial correlation). It analyzes the residuals for independence over time points (autocorrelation). The autocorrelation varies from -1 (negative autocorrelation) to 1 (positive autocorrelation).
Durbin-Watson test analyzes the following hypotheses,
Null hypothesis (H0): Residuals from the regression are not autocorrelated (autocorrelation coefficient, ρ = 0)
Alternative hypothesis (Ha): Residuals from the regression are autocorrelated (autocorrelation coefficient, ρ > 0)
Learn more about hypothesis testing and interpretation
Durbin-Watson test statistics (d) always ranges between 0 and 4. If the value is near 2, it indicates evidence of non-autocorrelation. If the value is towards 0, it indicates evidence of positive autocorrelation. If the value is towards 4, it indicates evidence of negative autocorrelation
Perform Durbin-Watson test in Python
We will use the
statsmodels package to perform Durbin-Watson test
Suppose, there is a hypothetical time-series dataset of stock prices recorded over 12 months.
import pandas as pd df = pd.read_csv("https://reneshbedre.github.io/assets/posts/reg/stock_price.csv") df.head(2) # output months stock_price 0 1 122 1 2 129
Fit the regression model
To perform the Durbin-Watson test, we first need to get regression residuals. Fit the regression
months as independent variables and
stock_price as the dependent variable,
import statsmodels.api as sm X = df['months'] # independent variable y = df['stock_price'] # dependent variable # to get intercept X = sm.add_constant(X) # fit the regression model reg = sm.OLS(y, X).fit() reg.summary() # output OLS Regression Results ============================================================================== Dep. Variable: stock_price R-squared: 0.892 Model: OLS Adj. R-squared: 0.881 Method: Least Squares F-statistic: 82.42 Date: Fri, 17 Jun 2022 Prob (F-statistic): 3.83e-06 Time: 18:05:58 Log-Likelihood: -40.579 No. Observations: 12 AIC: 85.16 Df Residuals: 10 BIC: 86.13 Df Model: 1 Covariance Type: nonrobust ============================================================================== coef std err t P>|t| [0.025 0.975] ------------------------------------------------------------------------------ const 114.6061 4.799 23.881 0.000 103.913 125.299 months 5.9196 0.652 9.078 0.000 4.467 7.372 ============================================================================== Omnibus: 10.544 Durbin-Watson: 2.585 Prob(Omnibus): 0.005 Jarque-Bera (JB): 5.798 Skew: -1.582 Prob(JB): 0.0551 Kurtosis: 4.259 Cond. No. 15.9 ==============================================================================
Learn more about regression analysis
Calculate Durbin-Watson test in Python
We will use
durbin_watson() function from
from statsmodels.stats.stattools import durbin_watson as dwtest import numpy as np dwtest(resids=np.array(reg.resid)) # output 2.5848268 alternative hypothesis: true autocorrelation is not 0
As the Durbin-Watson statistics (d) is close to 2, we fail to reject the null hypothesis. Hence, we conclude that the residuals are not autocorrelated.
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- Salamon SJ, Hansen HJ, Abbott D. How real are observed trends in small correlated datasets?. Royal Society open science. 2019 Mar 20;6(3):181089.
- Turner SL, Forbes AB, Karahalios A, Taljaard M, McKenzie JE. Evaluation of statistical methods used in the analysis of interrupted time series studies: a simulation study. BMC medical research methodology. 2021 Dec;21(1):1-8.
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