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))
df1053| PAYNSA | |
|---|---|
| DATE | |
| 1939-01-01 | 29296 |
| 1939-02-01 | 29394 |
| 1939-03-01 | 29804 |
| ... | ... |
| 2026-07-01 | 158709 |
| 2026-08-01 | 158905 |
| 2026-09-01 | 159238 |
1053 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 |
| ... | ... |
| 2026-07-01 | 158709 |
| 2026-08-01 | 158905 |
| 2026-09-01 | 159238 |
1053 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 |
| ... | ... | ... |
| 2026-07-01 | 158709 | 1051 |
| 2026-08-01 | 158905 | 1052 |
| 2026-09-01 | 159238 | 1053 |
1053 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: (1053, 1)
Y: (1053,)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: 0.985
Model: OLS Adj. R-squared: 0.985
Method: Least Squares F-statistic: 6.729e+04
Date: Wed, 28 Oct 2026 Prob (F-statistic): 0.00
Time: 23:12:35 Log-Likelihood: -10426.
No. Observations: 1053 AIC: 2.086e+04
Df Residuals: 1051 BIC: 2.087e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 2.694e+04 298.232 90.326 0.000 2.64e+04 2.75e+04
time_step 127.1640 0.490 259.411 0.000 126.202 128.126
==============================================================================
Omnibus: 5.586 Durbin-Watson: 0.048
Prob(Omnibus): 0.061 Jarque-Bera (JB): 5.408
Skew: 0.142 Prob(JB): 0.0669
Kurtosis: 2.794 Cond. No. 1.22e+03
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.22e+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 26937.950744
time_step 127.164044
dtype: float64
------------
y = 127.164x + 26937.951df["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 -1004.655791
2 151.889957
3 45.435705
4 816.609785
Name: residual, dtype: float64df.groupby("month")["residual"].mean()month
1 -1288.506899
2 -1068.205033
3 -657.255440
4 -340.692211
5 153.996018
6 642.366065
7 -186.172978
8 -51.541567
9 374.021662
10 738.417507
11 840.460360
12 870.951488
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 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| 2026-07-01 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 2026-08-01 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 2026-09-01 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
1053 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: 0.011
Method: Least Squares F-statistic: 2.060
Date: Wed, 28 Oct 2026 Prob (F-statistic): 0.0206
Time: 23:12:35 Log-Likelihood: -10415.
No. Observations: 1053 AIC: 2.085e+04
Df Residuals: 1041 BIC: 2.091e+04
Df Model: 11
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Jan -1288.5069 512.373 -2.515 0.012 -2293.908 -283.106
Feb -1068.2050 512.373 -2.085 0.037 -2073.606 -62.804
Mar -657.2554 512.373 -1.283 0.200 -1662.656 348.145
Apr -340.6922 512.373 -0.665 0.506 -1346.093 664.709
May 153.9960 512.373 0.301 0.764 -851.405 1159.397
Jun 642.3661 512.373 1.254 0.210 -363.035 1647.767
Jul -186.1730 512.373 -0.363 0.716 -1191.574 819.228
Aug -51.5416 512.373 -0.101 0.920 -1056.942 953.859
Sep 374.0217 512.373 0.730 0.466 -631.379 1379.423
Oct 738.4175 515.309 1.433 0.152 -272.745 1749.580
Nov 840.4604 515.309 1.631 0.103 -170.702 1851.623
Dec 870.9515 515.309 1.690 0.091 -140.211 1882.114
==============================================================================
Omnibus: 6.239 Durbin-Watson: 0.025
Prob(Omnibus): 0.044 Jarque-Bera (JB): 5.955
Skew: 0.147 Prob(JB): 0.0509
Kurtosis: 2.777 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)