We’ve explored using a regression for time series forecasting, but what if there are seasonal or cyclical patterns in the data?
Let’s explore an example of how to use regression to identify cyclical patterns and perform seasonality analysis with time series data.
For a time series dataset that exemplifies cyclical patterns, let’s consider this dataset of U.S. employment over time, from the Federal Reserve Economic Data (FRED).
Fetching the data, going back as far as possible:
from pandas_datareader import get_data_fred
from datetime import datetime
DATASET_NAME = "PAYNSA"
df = get_data_fred(DATASET_NAME, start=datetime(1900,1,1))
print(len(df))
df1040| PAYNSA | |
|---|---|
| DATE | |
| 1939-01-01 | 29296 |
| 1939-02-01 | 29394 |
| 1939-03-01 | 29804 |
| ... | ... |
| 2025-06-01 | 160256 |
| 2025-07-01 | 159210 |
| 2025-08-01 | 159410 |
1040 rows × 1 columns
Here is some more information about the “PAYNSA” dataset:
“All Employees: Total Nonfarm, commonly known as Total Nonfarm Payroll, is a measure of the number of U.S. workers in the economy that excludes proprietors, private household employees, unpaid volunteers, farm employees, and the unincorporated self-employed.”
“Generally, the U.S. labor force and levels of employment and unemployment are subject to fluctuations due to seasonal changes in weather, major holidays, and the opening and closing of schools.”
“The Bureau of Labor Statistics (BLS) adjusts the data to offset the seasonal effects to show non-seasonal changes: for example, women’s participation in the labor force; or a general decline in the number of employees, a possible indication of a downturn in the economy.
To closely examine seasonal and non-seasonal changes, the BLS releases two monthly statistical measures: the seasonally adjusted All Employees: Total Nonfarm (PAYEMS) and All Employees: Total Nonfarm (PAYNSA), which is not seasonally adjusted.”
This “PYYNSA” data is expressed in “Thousands of Persons”, and is “Not Seasonally Adjusted”.
The dataset frequency is “Monthly”.
Wrangling the data, including renaming columns and converting the date index to be datetime-aware, may make it easier for us to work with this data:
from pandas import to_datetime
df.rename(columns={DATASET_NAME: "employment"}, inplace=True)
df.index.name = "date"
df.index = to_datetime(df.index)
df| employment | |
|---|---|
| date | |
| 1939-01-01 | 29296 |
| 1939-02-01 | 29394 |
| 1939-03-01 | 29804 |
| ... | ... |
| 2025-06-01 | 160256 |
| 2025-07-01 | 159210 |
| 2025-08-01 | 159410 |
1040 rows × 1 columns
Visualizing the data:
import plotly.express as px
px.line(df, y="employment", height=450,
title="US Employment by month (non-seasonally adjusted)",
labels={"employment": "Employment (in thousands of persons)"},
)Cyclical Patterns
Exploring cyclical patterns in the data:
px.line(df[(df.index.year >= 1970) & (df.index.year <= 1980)], y="employment",
title="US Employment by month (selected years)", height=450,
labels={"Employment": "Employment (in thousands)"},
)Hover over the dataviz to see which month(s) typically have higher employment, and which month(s) typically have lower employment.
Trend Analysis
Exploring trends:
import plotly.express as px
px.scatter(df, y="employment", height=450,
title="US Employment by month (vs Trend)",
labels={"employment": "Employment (in thousands)"},
trendline="ols", trendline_color_override="red"
)Looks like evidence of a possible linear relationship. Let’s perform a more formal regression analysis.
Because we need numeric features to perform a regression, we convert the dates to a linear time step of integers (after sorting the data first for good measure):
df.sort_values(by="date", ascending=True, inplace=True)
df["time_step"] = range(1, len(df) + 1)
df| employment | time_step | |
|---|---|---|
| date | ||
| 1939-01-01 | 29296 | 1 |
| 1939-02-01 | 29394 | 2 |
| 1939-03-01 | 29804 | 3 |
| ... | ... | ... |
| 2025-06-01 | 160256 | 1038 |
| 2025-07-01 | 159210 | 1039 |
| 2025-08-01 | 159410 | 1040 |
1040 rows × 2 columns
We will use the numeric time step as our input variable (x), to predict the employment (y).
X/Y Split
Identifying dependent and independent variables:
x = df[["time_step"]]
y = df["employment"]
print("X:", x.shape)
print("Y:", y.shape)X: (1040, 1)
Y: (1040,)Adding Constants
We are going to use statsmodels, so we add a column of constant ones representing the intercept:
import statsmodels.api as sm
# adding in a column of constants, as per the OLS docs
x = sm.add_constant(x)
x.head()| const | time_step | |
|---|---|---|
| date | ||
| 1939-01-01 | 1.0 | 1 |
| 1939-02-01 | 1.0 | 2 |
| 1939-03-01 | 1.0 | 3 |
| 1939-04-01 | 1.0 | 4 |
| 1939-05-01 | 1.0 | 5 |
Train/Test Split
Splitting into training vs testing datasets:
#from sklearn.model_selection import train_test_split
#
#x_train, x_test, y_train, y_test = train_test_split(x, y, random_state=99)
#print("TRAIN:", x_train.shape, y_train.shape)
#print("TEST:", x_test.shape, y_test.shape)Splitting data sequentially where earlier data is used in training and recent data is use for testing:
#print(len(df))
#
#training_size = round(len(df) * .8)
#print(training_size)
#
#x_train = x.iloc[:training_size] # slice all before
#y_train = y.iloc[:training_size] # slice all before
#
#x_test = x.iloc[training_size:] # slice all after
#y_test = y.iloc[training_size:] # slice all after
#print("TRAIN:", x_train.shape)
#print("TEST:", x_test.shape)For this example, we will not split the data. To help illustrate a story about predictions over the entire time period.
Training a linear regression model on the training data:
import statsmodels.api as sm
model = sm.OLS(y, x, missing="drop")
print(type(model))
results = model.fit()
print(type(results))<class 'statsmodels.regression.linear_model.OLS'>
<class 'statsmodels.regression.linear_model.RegressionResultsWrapper'>Examining training results:
print(results.summary()) OLS Regression Results
==============================================================================
Dep. Variable: employment R-squared: 0.984
Model: OLS Adj. R-squared: 0.984
Method: Least Squares F-statistic: 6.428e+04
Date: Mon, 10 Nov 2025 Prob (F-statistic): 0.00
Time: 01:07:36 Log-Likelihood: -10303.
No. Observations: 1040 AIC: 2.061e+04
Df Residuals: 1038 BIC: 2.062e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 2.688e+04 301.748 89.094 0.000 2.63e+04 2.75e+04
time_step 127.3191 0.502 253.534 0.000 126.334 128.305
==============================================================================
Omnibus: 5.341 Durbin-Watson: 0.048
Prob(Omnibus): 0.069 Jarque-Bera (JB): 4.988
Skew: 0.124 Prob(JB): 0.0826
Kurtosis: 2.767 Cond. No. 1.20e+03
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.2e+03. This might indicate that there are
strong multicollinearity or other numerical problems.print(results.params)
print("------------")
print(f"y = {results.params['time_step'].round(3)}x + {results.params['const'].round(3)}")const 26883.985876
time_step 127.319106
dtype: float64
------------
y = 127.319x + 26883.986df["prediction"] = results.fittedvalues
df["residual"] = results.resid#from pandas import DataFrame
#
## get all rows from the original dataset that wound up in the training set:
#training_set = df.loc[x_train.index].copy()
#print(len(training_set))
#
## create a dataset for the predictions and the residuals:
#training_preds = DataFrame({
# "prediction": results.fittedvalues,
# "residual": results.resid
#})
## merge the training set with the results:
#training_set = training_set.merge(training_preds,
# how="inner", left_index=True, right_index=True
#)
#
## calculate error for each datapoint:
#training_setRegression Trends
Plotting trend line:
px.line(df, y=["employment", "prediction"], height=350,
title="US Employment (monthly) vs linear trend",
labels={"value":""}
)Regression Residuals
Removing the trend, plotting just the residuals:
px.line(df, y="residual",
title="US Employment (monthly) vs linear trend residuals", height=350
)There seem to be some periodic movements in the residuals.
Observe there may be some cyclical patterns in the residuals, by calculating periodic means:
df["year"] = df.index.year
df["quarter"] = df.index.quarter
df["month"] = df.index.monthHere we are grouping the data by quarter and calculating the average residual. This shows us for each quarter, on average, whether predictions are above or below trend:
df.groupby("quarter")["residual"].mean()quarter
1 -1002.233195
2 150.932091
3 53.244581
4 807.542623
Name: residual, dtype: float64df.groupby("month")["residual"].mean()month
1 -1285.722517
2 -1066.340474
3 -654.636592
4 -343.461446
5 151.725195
6 644.532525
7 -179.740604
8 -43.174653
9 386.479286
10 727.148551
11 830.062003
12 865.417315
Name: residual, dtype: float64Let’s perform a regression using months as the features and the trend residuals as the target. This can help us understand the degree to which employment will be over or under trend for a given month.
# https://pandas.pydata.org/docs/reference/api/pandas.get_dummies.html
# "one hot encode" the monthly values:
from pandas import get_dummies as one_hot_encode
x_monthly = one_hot_encode(df["month"])
x_monthly.columns=["Jan", "Feb", "Mar", "Apr",
"May", "Jun", "Jul", "Aug",
"Sep", "Oct", "Nov", "Dec"]
x_monthly = x_monthly.astype(int)
x_monthly| Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| date | ||||||||||||
| 1939-01-01 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1939-02-01 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1939-03-01 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| 2025-06-01 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2025-07-01 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 2025-08-01 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
1040 rows × 12 columns
y_monthly = df["residual"]
ols_monthly = sm.OLS(y_monthly, x_monthly)
print(type(ols_monthly))
results_monthly = ols_monthly.fit()
print(type(results_monthly))
print(results_monthly.summary())<class 'statsmodels.regression.linear_model.OLS'>
<class 'statsmodels.regression.linear_model.RegressionResultsWrapper'>
OLS Regression Results
==============================================================================
Dep. Variable: residual R-squared: 0.021
Model: OLS Adj. R-squared: 0.010
Method: Least Squares F-statistic: 1.997
Date: Mon, 10 Nov 2025 Prob (F-statistic): 0.0257
Time: 01:07:36 Log-Likelihood: -10292.
No. Observations: 1040 AIC: 2.061e+04
Df Residuals: 1028 BIC: 2.067e+04
Df Model: 11
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Jan -1285.7225 518.285 -2.481 0.013 -2302.739 -268.706
Feb -1066.3405 518.285 -2.057 0.040 -2083.357 -49.324
Mar -654.6366 518.285 -1.263 0.207 -1671.653 362.380
Apr -343.4614 518.285 -0.663 0.508 -1360.478 673.555
May 151.7252 518.285 0.293 0.770 -865.291 1168.742
Jun 644.5325 518.285 1.244 0.214 -372.484 1661.549
Jul -179.7406 518.285 -0.347 0.729 -1196.757 837.276
Aug -43.1747 518.285 -0.083 0.934 -1060.191 973.842
Sep 386.4793 521.289 0.741 0.459 -636.433 1409.392
Oct 727.1486 521.289 1.395 0.163 -295.764 1750.061
Nov 830.0620 521.289 1.592 0.112 -192.850 1852.974
Dec 865.4173 521.289 1.660 0.097 -157.495 1888.330
==============================================================================
Omnibus: 6.053 Durbin-Watson: 0.025
Prob(Omnibus): 0.048 Jarque-Bera (JB): 5.541
Skew: 0.127 Prob(JB): 0.0626
Kurtosis: 2.749 Cond. No. 1.01
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.The coefficients tell us how each month contributes towards the regression residuals, in other words, for each month, to what degree does the model predict we will be above or below trend?
Monthly Predictions of Residuals
df["prediction_monthly"] = results_monthly.fittedvalues
df["residual_monthly"] = results_monthly.residDecomposition of the original data into trend, seasonal component, and residuals:
px.line(df, y=["employment", "prediction"], title="Employment vs trend", height=350)px.line(df, y="prediction_monthly", title="Employment seasonal component", height=350)px.line(df, y="residual_monthly", title="Employment de-trended residual", height=350)