4.2 Time Series with Trends

Definition of Trend¶

The first primary cause of non-stationarity we will discuss is the presence of a trend. At its most basic level, a trend is simply a long-term increase or decrease in values.

Recall from the first chapter that time series can exhibit highly convincing spurious trends. We will cover some statistical methods to determine if a trend is real in later chapters, but none of these are fool-proof. As a data scientist, you must always combine your prior domain knowledge to assist in determining if it is reasonable to assume the existence of a trend.

Trends and Detrending¶

S&P 500 Trend¶

The following plot of the S&P 500 values appears to exhibit a stable overall trend.

You can experiment with the values from Figure 1 by downloading the data using the following cell:

import pandas as pd
import matplotlib.pyplot as plt

df = pd.read_csv("https://fred.stlouisfed.org/graph/fredgraph.csv?bgcolor=%23ebf3fb&chart_type=line&drp=0&fo=open%20sans&graph_bgcolor=%23ffffff&height=450&mode=fred&recession_bars=on&txtcolor=%23444444&ts=12&tts=12&width=1320&nt=0&thu=0&trc=0&show_legend=yes&show_axis_titles=yes&show_tooltip=yes&id=SP500&scale=left&cosd=2016-01-25&coed=2026-01-23&line_color=%230073e6&link_values=false&line_style=solid&mark_type=none&mw=3&lw=3&ost=-99999&oet=99999&mma=0&fml=a&fq=Daily%2C%20Close&fam=avg&fgst=lin&fgsnd=2020-02-01&line_index=1&transformation=lin&vintage_date=2026-01-25&revision_date=2026-01-25&nd=2016-01-25",
                 index_col=0,
                 parse_dates=True,
                 )
# statsmodels is very particular regarding missing values, we will fill in missing values with the previous value.
df.ffill(inplace=True)
# examine a quick plot
plt.plot(df)

Trend Stationary Process¶

Given the apparent upward trend, it is reasonable to explore the possibility that the S&P 500 is trend stationary, i.e. the time series takes the form

where $x_t$ is the observed time series, $\mu_t$ is a trend, and $y_t$ is an underlying stationary time series. Eq. (1) suggests recovering $y_t$ via the process of detrending

What form should our estimate $\hat{\mu}_t$ take? Let’s start with the most basic model, ordinary least squares (OLS)

where $\beta_0$ is the intercept and $\beta_1$ is the slope.

Detrending S&P 500¶

We will fit an OLS regression of the form in Eq. (3) using the following code:

import statsmodels.api as sm
import numpy as np

y = sp_500_df['SP500']
X = np.arange(len(sp_500_df))
# By default, statsmodels does not include an intercept term.
# We will add an intercept manually using add_constant.
X = sm.add_constant(X)

model = sm.OLS(y, X)
results = model.fit()

print(results.summary())

You should now see something like the following:

                            OLS Regression Results
==============================================================================
Dep. Variable:                  SP500   R-squared:                       0.907
Model:                            OLS   Adj. R-squared:                  0.907
Method:                 Least Squares   F-statistic:                 2.543e+04
Date:                Sun, 25 Jan 2026   Prob (F-statistic):               0.00
Time:                        18:33:02   Log-Likelihood:                -19337.
No. Observations:                2610   AIC:                         3.868e+04
Df Residuals:                    2608   BIC:                         3.869e+04
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const       1644.6932     15.634    105.199      0.000    1614.037    1675.350
x1             1.6550      0.010    159.469      0.000       1.635       1.675
==============================================================================
Omnibus:                       10.728   Durbin-Watson:                   0.011
Prob(Omnibus):                  0.005   Jarque-Bera (JB):               10.076
Skew:                          -0.117   Prob(JB):                      0.00649
Kurtosis:                       2.806   Cond. No.                     3.01e+03
==============================================================================

Note that the $R^2$ is quite high, and both the intercept and slope are statistically significant.

Based on the results of our OLS, we will detrend the S&P 500 results using the formula

where $t$ consists of $[0,1,\ldots, n-1]$ for $n$ observations. We can apply this formula using the code

detrend_component = results.params["const"] + results.params["x1"] * np.arange(len(sp_500_df))
sp_500_df_detrended = sp_500_df["SP500"] - pd.Series(detrend_component, index=sp_500_df.index)