Many economic datasets are collected over time. Examples include food prices, exchange rates, inflation, GDP, and stock prices. Unlike cross-sectional observations, time-series observations often exhibit persistence. What happens today is frequently related to what happened yesterday.
Regression analysis assumes that error terms are independent from one observation to the next. When this assumption is violated, the model suffers from autocorrelation.
In this chapter, we learn what autocorrelation is, why it occurs, how to detect it, and how to improve inference using Newey-West standard errors.
Can monthly wheat prices be explained by a long-term trend?
Suppose we observe monthly wheat prices over several years.
Economic theory suggests that prices may gradually increase over time because of inflation, population growth, and rising production costs.
However, unexpected shocks rarely disappear immediately. If prices are unusually high this month, they may remain above average next month. This persistence often creates autocorrelation.
Many economic variables exhibit momentum.
Examples include:
A drought that increases wheat prices today may continue affecting prices for several months. Similarly, a recession that reduces consumer demand today may continue influencing economic activity next quarter.
As a result, regression errors often display patterns over time.
The classical regression model assumes:
[ Cov(u_t,u_s)=0 ]
for all (t s).
This means that errors from different time periods are unrelated.
Autocorrelation occurs when errors are correlated across time:
[ Cov(u_t,u_{t-1}) ]
Positive autocorrelation is most common. In this case, positive errors tend to be followed by positive errors and negative errors tend to be followed by negative errors.
To illustrate autocorrelation, we create a synthetic monthly wheat price series. The price follows a trend, while shocks persist through time.
import numpy as np
import pandas as pd
np.random.seed(4107)
n = 120
time = np.arange(n)
errors = np.zeros(n)
for t in range(1, n):
errors[t] = 0.8 * errors[t - 1] + np.random.normal(0, 5)
price = 200 + 0.5 * time + errors
wheat_data = pd.DataFrame({
"Month": time,
"Price": price
})
wheat_data.head()| Month | Price | |
|---|---|---|
| 0 | 0 | 200.000000 |
| 1 | 1 | 204.705084 |
| 2 | 2 | 197.800607 |
| 3 | 3 | 195.408530 |
| 4 | 4 | 193.581992 |
import matplotlib.pyplot as plt
plt.figure(figsize=(10, 5))
plt.plot(
wheat_data["Month"],
wheat_data["Price"]
)
plt.xlabel("Month")
plt.ylabel("Price")
plt.title("Monthly Wheat Prices")
plt.show()
The graph shows an upward trend. Notice that prices do not fluctuate randomly around the trend. Instead, periods of high prices tend to cluster together and periods of low prices tend to cluster together.
This persistence is typical of time-series data.
We estimate a simple trend model:
[ Price_t = _0 + _1 Time_t + u_t ]
import statsmodels.api as sm
X = sm.add_constant(wheat_data["Month"])
y = wheat_data["Price"]
model = sm.OLS(y, X).fit()
print(model.summary()) OLS Regression Results
==============================================================================
Dep. Variable: Price R-squared: 0.879
Model: OLS Adj. R-squared: 0.878
Method: Least Squares F-statistic: 859.9
Date: Thu, 11 Jun 2026 Prob (F-statistic): 5.08e-56
Time: 06:54:23 Log-Likelihood: -411.15
No. Observations: 120 AIC: 826.3
Df Residuals: 118 BIC: 831.9
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 197.5536 1.362 145.059 0.000 194.857 200.250
Month 0.5801 0.020 29.324 0.000 0.541 0.619
==============================================================================
Omnibus: 1.304 Durbin-Watson: 0.586
Prob(Omnibus): 0.521 Jarque-Bera (JB): 0.839
Skew: 0.144 Prob(JB): 0.657
Kurtosis: 3.291 Cond. No. 137.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.Suppose the estimated slope equals 0.52.
The interpretation is:
On average, wheat prices increase by approximately 0.52 units per month.
The coefficient measures the average trend. However, it does not tell us whether the residuals satisfy the assumptions of the regression model.
The first diagnostic is a plot of residuals over time.
residuals = model.resid
plt.figure(figsize=(10, 5))
plt.plot(
wheat_data["Month"],
residuals
)
plt.axhline(
y=0,
linestyle="--"
)
plt.xlabel("Month")
plt.ylabel("Residual")
plt.title("Residuals Through Time")
plt.show()
If residuals are independent, they should bounce randomly above and below zero. Instead, long runs of positive residuals and long runs of negative residuals appear.
This suggests autocorrelation.
Another useful visualization compares each residual with the previous residual.
plt.figure(figsize=(6, 6))
plt.scatter(
residuals[:-1],
residuals[1:],
alpha=0.7
)
plt.xlabel("Residual t-1")
plt.ylabel("Residual t")
plt.title("Residual Lag Plot")
plt.show()
If residuals are independent, the points should form a random cloud. A clear upward pattern indicates positive autocorrelation.
Large positive residuals tend to follow large positive residuals. Large negative residuals tend to follow large negative residuals.
A commonly used diagnostic is the Durbin-Watson statistic.
from statsmodels.stats.stattools import durbin_watson
dw = durbin_watson(model.resid)
print(dw)0.5856399966901465The Durbin-Watson statistic ranges between 0 and 4.
| Value | Interpretation |
|---|---|
| Around 2 | No autocorrelation |
| Below 2 | Positive autocorrelation |
| Above 2 | Negative autocorrelation |
In many economic applications, values substantially below 2 indicate positive autocorrelation.
Autocorrelation creates several problems. The most important consequence is unreliable standard errors.
This can lead to:
Just as heteroskedasticity affects inference, autocorrelation primarily affects how we evaluate uncertainty.
A common solution is to estimate heteroskedasticity and autocorrelation consistent standard errors. These are often called Newey-West standard errors.
nw_model = model.get_robustcov_results(
cov_type="HAC",
maxlags=4
)
print(nw_model.summary()) OLS Regression Results
==============================================================================
Dep. Variable: Price R-squared: 0.879
Model: OLS Adj. R-squared: 0.878
Method: Least Squares F-statistic: 256.5
Date: Thu, 11 Jun 2026 Prob (F-statistic): 2.25e-31
Time: 06:54:24 Log-Likelihood: -411.15
No. Observations: 120 AIC: 826.3
Df Residuals: 118 BIC: 831.9
Df Model: 1
Covariance Type: HAC
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 197.5536 1.993 99.135 0.000 193.607 201.500
Month 0.5801 0.036 16.016 0.000 0.508 0.652
==============================================================================
Omnibus: 1.304 Durbin-Watson: 0.586
Prob(Omnibus): 0.521 Jarque-Bera (JB): 0.839
Skew: 0.144 Prob(JB): 0.657
Kurtosis: 3.291 Cond. No. 137.
==============================================================================
Notes:
[1] Standard Errors are heteroscedasticity and autocorrelation robust (HAC) using 4 lags and without small sample correctioncomparison = pd.DataFrame({
"OLS Standard Error": model.bse,
"Newey-West Standard Error": nw_model.bse
})
comparison| OLS Standard Error | Newey-West Standard Error | |
|---|---|---|
| const | 1.361885 | 1.992766 |
| Month | 0.019781 | 0.036217 |
The coefficient estimates remain unchanged. Only the estimated uncertainty changes.
Newey-West standard errors generally provide more reliable inference when residuals are correlated over time.
Cross-sectional observations are often independent. Time-series observations are rarely independent.
Economic shocks frequently persist. Because of this persistence, economists routinely examine autocorrelation whenever they work with time-series data.
Ignoring autocorrelation can make a model appear more precise than it really is.
A high (R^2) does not imply independent residuals. A trend model can fit well and still suffer from strong autocorrelation.
Time-series observations should never be treated as randomly ordered observations. The sequence matters.
A model can have a high (R^2) and still suffer from severe autocorrelation.
Autocorrelation can remain important even in very large datasets.
OLS standard errors may underestimate uncertainty when residuals are correlated.
Plots often reveal autocorrelation immediately. Always inspect residual patterns.
Autocorrelation involves relationships among errors across time. In the next chapter, we examine multicollinearity, which occurs when explanatory variables contain similar information.