Overview

Teaching: 130 min
Exercises: 130 min

Questions

• Understand the concept of stationarity in time series (constant mean, variance, and autocorrelation over time).

• Recognize the Importance of Stationarity for Modeling: Learn why many time-series forecasting methods assume stationarity.

• Understand how to measure the extent to which current values of a series depend on its past values through autocorrelation and partial autocorrelation functions.

Objectives

• Why do we need to ensure the time series (and related indicators) is stationary for certain modeling approaches?

• Which stationarity test (ADF, KPSS, or others) is most appropriate for our GDP dataset, and what are the interpretation criteria?

• If the time series is non-stationary, what transformations should we consider?

• How does understanding the autocorrelation structure help us in modeling GDP nowcasting?

Stationarity Tests in Time Series

• Time series is said to be stationary if its statistical properties do not change over time (it has constant mean and variance, and covariance is independent of time).
• If the target variable is non-stationary, models might struggle to capture patterns effectively, leading to poor forecasting performance.
• Most machine learning models like Random Forest, XGBoost, LightGBM, LSTMs, and other modern ML algorithms do not require the target variable to be stationary.

How to Check Stationarity

1. Visual
2. Global vs Local Test
3. Statistical Test
   • Augmented Dickey-Fuller (ADF)
   • KPSS (Kwiatkowski, Phillips, Schmidt, and Shin) test

Augmented Dickey-Fuller (ADF)

• The ADF test is used to determine whether a given time series is stationary or not.

• The test uses an autoregressive model across multiple lag values.

  • Null Hypothesis (H0): If accepted, it suggests the time series has some time-dependent structure (non-stationary).
  • Alternate Hypothesis (H1): If the null hypothesis is rejected, the time series is stationary.
• Interpretation of Results:
  • If p-value < 0.05, reject the null hypothesis and infer that the time series is stationary.
  • If p-value > 0.05, fail to reject the null hypothesis (accept that the series is non-stationary).

• ADF Statistic: Value of the test statistic. More negative values indicate stronger evidence against the null hypothesis.

• By setting autolag='AIC', the ADF test chooses the number of lags that yields the lowest AIC.

Install Libraries

pip install statsmodels

Importing Libraries

# Import pandas and numpy
import pandas as pd
import numpy as np

# Plotting libraries
import matplotlib.pyplot as plt
import seaborn as sns

# Import the datetime class from the datetime module
from pandas.tseries.offsets import MonthEnd

# Stationarity and Autocorrelation Tests
from statsmodels.tsa.stattools import adfuller, kpss, acf, pacf

# Statsmodels API
import statsmodels.api as sm

# For loading images
from PIL import Image

Import the data

Data

Import the Data

gdpreal = pd.read_csv('gdpreal.csv')

gdpreal.head()

gdpreal.tail()

# Convert the 'Quarter' column to datetime format and adjust for quarter-end
gdpreal['Quarter'] = pd.PeriodIndex(gdpreal['Quarter'], freq='Q').to_timestamp()
gdpreal['Quarter'] = gdpreal['Quarter'] + MonthEnd(3)

# Convert 'RealGDP' to float (ensure clean data)
gdpreal['RealGDP'] = gdpreal['RealGDP'].str.replace(',', '').astype(float)

gdpreal.head()

Plotting Real GDP Data

plt.figure(figsize=(12, 6))
plt.plot(
    gdpreal['Quarter'],
    gdpreal['RealGDP'],
    marker='o',
    linestyle='-',
    color='blue',
    label='Real GDP (Billion Naira)'
)

# Add titles and labels
plt.title('Nigeria Real GDP (2010 Q1 - 2024 Q3)', fontsize=16)
plt.xlabel('Quarter', fontsize=12)
plt.ylabel('Real GDP (Billion Naira)', fontsize=12)

# Improve y-axis ticks
plt.yticks(fontsize=10)
plt.ticklabel_format(axis='y', style='plain')

# Grid, legend, and layout
plt.grid(True, linestyle='--', alpha=0.6)
plt.xticks(rotation=45)
plt.legend(fontsize=12)
plt.tight_layout()

# Show the plot
plt.show()

Augmented Dickey-Fuller (ADF) Test

def adf_test(series, column_name):
    result = adfuller(series)
    print(f'Augmented Dickey-Fuller Test for {column_name}:')
    print(f'ADF Statistic: {result[0]}')
    print(f'p-value: {result[1]}')
    if result[1] <= 0.05:
        print("The series is stationary.")
    else:
        print("The series is non-stationary.")
    print("-----------------------------")

# Check stationarity of the RealGDP column
adf_test(gdpreal['RealGDP'], 'RealGDP')

KPSS (Kwiatkowski, Phillips, Schmidt, and Shin) Test

• The KPSS test has the null hypothesis opposite to that of the ADF test.

  • Null Hypothesis (H0): The process is trend stationary.
  • Alternate Hypothesis (H1): The time series has some time-dependent structure (non-stationary).

• Regression Parameters:

  • 'c': Includes a constant (or an intercept) in the regression.
  • 'nc': No constant or trend.
  • 'ct': Constant and trend.
  • 'ctt': Constant, linear, and quadratic trend.

def kpss_test(series, column_name):
    result = kpss(series, regression='c')
    print(f'KPSS Test for {column_name}:')
    print(f'KPSS Statistic: {result[0]}')
    print(f'p-value: {result[1]}')
    if result[1] > 0.05:
        print("The series is stationary.")
    else:
        print("The series is non-stationary.")
    print("-----------------------------")

# Check stationarity
kpss_test(gdpreal['RealGDP'], 'RealGDP')

Autocorrelation and Partial Autocorrelation Check

Autocorrelation (ACF)

• Measures the linear relationship between a time series and its lagged values.
• Helps identify MA (Moving Average) order and patterns like seasonality.

Partial Autocorrelation (PACF)

• Measures the direct relationship between a time series and its lagged values, excluding intermediate lags.
• Useful for identifying the AR (Autoregressive) order.

def plot_acf(series, column_name, lags=None):
    plt.figure(figsize=(18, 6))
    sm.graphics.tsa.plot_acf(series, lags=lags, alpha=0.05)
    plt.title(f'Autocorrelation Function (ACF) for "{column_name}"', fontsize=14)
    plt.xlabel("Lags", fontsize=12)
    plt.ylabel("Autocorrelation", fontsize=12)
    plt.grid(True, linestyle='--', alpha=0.7)
    plt.tight_layout()
    plt.show()

plot_acf(gdpreal['RealGDP'], 'RealGDP', lags=40)

Partial Autocorrelation Plot

def plot_pacf(series, column_name, lags=None):
    plt.figure(figsize=(12, 6))
    sm.graphics.tsa.plot_pacf(series, lags=lags, alpha=0.05, method='ywm')
    plt.title(f'Partial Autocorrelation Function (PACF) for "{column_name}"', fontsize=14)
    plt.xlabel("Lags", fontsize=12)
    plt.ylabel("Partial Autocorrelation", fontsize=12)
    plt.grid(True, linestyle='--', alpha=0.7)
    plt.tight_layout()
    plt.show()

plot_pacf(gdpreal['RealGDP'], 'RealGDP')

Interpretation of ACF and PACF

• Significant ACF lags: Indicates strong short-term correlation.
• Significant PACF lags: Indicates direct influence of recent observations.
• After lag 10, ACF becomes insignificant, suggesting mean-reverting behavior.
• PACF suggests an AR(3) structure with significant lags up to 3 and a correction at lag 4.

Key Points

• A stationary time series has statistical properties (mean, variance) that do not change over time.

• Stationarity simplifies modeling and is a common assumption in classical statistical forecasting methods.

• Non-stationary data often exhibit trends, seasonal patterns, or evolving volatility.

• Autocorrelation (ACF): Measures how current values of a series relate to past values at various lags.

• {“Partial Autocorrelation (PACF)”=>”Isolates the direct relationship between a series and its lagged values, removing the influence of intermediate lags."”}

• Statistical tests like the Augmented Dickey-Fuller (ADF), Phillips-Perron, or KPSS help determine if a series is stationary.