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))
df1037| PAYNSA | |
|---|---|
| DATE | |
| 1939-01-01 | 29296 |
| 1939-02-01 | 29394 |
| 1939-03-01 | 29804 |
| ... | ... |
| 2025-03-01 | 158402 |
| 2025-04-01 | 159238 |
| 2025-05-01 | 159964 |
1037 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 |
| ... | ... |
| 2025-03-01 | 158402 |
| 2025-04-01 | 159238 |
| 2025-05-01 | 159964 |
1037 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 |
| ... | ... | ... |
| 2025-03-01 | 158402 | 1035 |
| 2025-04-01 | 159238 | 1036 |
| 2025-05-01 | 159964 | 1037 |
1037 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: (1037, 1)
Y: (1037,)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: 6.354e+04
Date: Wed, 25 Jun 2025 Prob (F-statistic): 0.00
Time: 01:08:44 Log-Likelihood: -10275.
No. Observations: 1037 AIC: 2.055e+04
Df Residuals: 1035 BIC: 2.056e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 2.689e+04 302.613 88.848 0.000 2.63e+04 2.75e+04
time_step 127.3117 0.505 252.065 0.000 126.321 128.303
==============================================================================
Omnibus: 5.576 Durbin-Watson: 0.048
Prob(Omnibus): 0.062 Jarque-Bera (JB): 5.162
Skew: 0.124 Prob(JB): 0.0757
Kurtosis: 2.760 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 26886.551293
time_step 127.311701
dtype: float64
------------
y = 127.312x + 26886.551df["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 -1000.962502
2 148.291623
3 54.324899
4 808.835533
Name: residual, dtype: float64df.groupby("month")["residual"].mean()month
1 -1284.459231
2 -1065.069782
3 -653.358494
4 -342.049505
5 153.408909
6 639.157602
7 -181.049447
8 -43.733241
9 387.757384
10 728.434055
11 831.354912
12 866.717630
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 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| 2025-03-01 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2025-04-01 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2025-05-01 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1037 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: 1.987
Date: Wed, 25 Jun 2025 Prob (F-statistic): 0.0266
Time: 01:08:44 Log-Likelihood: -10264.
No. Observations: 1037 AIC: 2.055e+04
Df Residuals: 1025 BIC: 2.061e+04
Df Model: 11
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Jan -1284.4592 519.039 -2.475 0.013 -2302.959 -265.959
Feb -1065.0698 519.039 -2.052 0.040 -2083.570 -46.570
Mar -653.3585 519.039 -1.259 0.208 -1671.859 365.142
Apr -342.0495 519.039 -0.659 0.510 -1360.550 676.451
May 153.4089 519.039 0.296 0.768 -865.091 1171.909
Jun 639.1576 522.048 1.224 0.221 -385.247 1663.562
Jul -181.0494 522.048 -0.347 0.729 -1205.454 843.355
Aug -43.7332 522.048 -0.084 0.933 -1068.138 980.671
Sep 387.7574 522.048 0.743 0.458 -636.647 1412.162
Oct 728.4341 522.048 1.395 0.163 -295.971 1752.839
Nov 831.3549 522.048 1.592 0.112 -193.050 1855.760
Dec 866.7176 522.048 1.660 0.097 -157.687 1891.122
==============================================================================
Omnibus: 6.309 Durbin-Watson: 0.025
Prob(Omnibus): 0.043 Jarque-Bera (JB): 5.720
Skew: 0.128 Prob(JB): 0.0573
Kurtosis: 2.741 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)