Autoregressive Model Explained: Forecasting Made Simple

An autoregressive model predicts future values based on previous observations, assuming a linear relationship between past and future data points. It’s commonly used for time series forecasting, where historical data influences future values.

The model is highly effective in real-time applications, such as weather and sales forecasting, where past data trends are crucial. Python libraries such as statsmodels, NumPy, and Pandas are commonly used to implement and process AR models.

In this blog, you’ll learn about the autoregressive model, its functionality, effectiveness, and real-life applications.

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Autoregressive Model: Ideal Order Selection Explained

An autoregressive model is a time series model that predicts current values using past values and a random error term. The model uses its previous values to predict future values. Mathematically, an AR model of order p(AR(p)) can be represented as:

Where:

• Yt  = current value of the time series at time t.
• ${\varphi }_1, {\varphi }_2,....,{\varphi }_p$

  = AR Parameters (coefficients) that are estimated from the data.

• p = order of the AR model, representing previous time steps.
• ${\epsilon }_t$

  = is the white noise error term at time 𝑡, assumed to be independent.

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Now, the primary task is to determine the appropriate order p of the model. This ensures that the AR coefficients 

can effectively capture the underlying temporal structure of the data. Let’s take a closer look:

1. Autocorrelation Function & Partial Autocorrelation Function

The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are key tools in identifying the structure of a time series. They are also essential for determining the appropriate order of an AR model.

• Autocorrelation Function (ACF): It measures the correlation between a time series and its lagged versions. It helps identify how past values influence the current value at various time lags.
• Autocorrelation Function (PACF): It shows the direct correlation between the time series and its lagged values, after adjusting for the influence of intermediate lags. This makes PACF especially valuable for identifying the exact order p of the AR model.

The point where the PACF cuts off (drops to zero or becomes insignificant) is a strong indicator. It helps determine the ideal number of lags to include in the model.

For example:

• If the PACF cuts off sharply after lag p, the optimal order of the AR model would likely be p.
• If the PACF shows significant correlations at higher lags, a higher order p should be considered.

This analysis helps in selecting the most effective AR model by identifying the key lags that influence the series.

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2. Information Criteria (AIC, BIC, and HQIC)

The AIC, BIC, and HQIC are used to select the optimal order. It’s done by evaluating the goodness-of-fit of the model while penalizing model complexity. These criteria balance the model's ability to fit the data with the number of parameters used.

• Akaike Information Criterion (AIC): AIC evaluates the trade-off between model fit and complexity, with a lower AIC suggesting a better model. AIC is calculated as:

Where: 

• k is the number of parameters in the model.
• $\hat{L}$

  is the likelihood of the model. 

• Bayesian Information Criterion (BIC): BIC also considers model complexity but imposes a heavier penalty for the number of parameters compared to AIC. The formula for BIC is:

Where:

• n is the number of observations.
• k is number of parameters in the model.
• $\hat{L}$

  is maximum likelihood of the model (i.e. at the estimated parameters).

The order p that minimizes AIC or BIC is often chosen as the optimal one.

• Hannan-Quinn Information Criterion (HQIC): The HQIC is another criterion used for model selection. Like AIC and BIC, it penalizes model complexity but at a different rate. HQIC is calculated as:

Where:

• k is the number of parameters in the model.
• n is the number of observations.
• $\hat{L}$

   is the maximum likelihood of the model.

HQIC provides a more balanced penalty between AIC and BIC and is useful when the sample size is large.

Also Read: How Forecasting Works in Tableau? Predicting the Future with Data
 

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3. Cross-Validation

Cross-validation assesses model performance by splitting the dataset into training and test sets. Different values of p are tested, and the one minimizing prediction error is selected, ensuring the model generalizes well to unseen data.

To ensure the model generalizes well and avoids overfitting, the following steps are crucial during the evaluation process:

• Training and Test Sets: The dataset is split into training and test sets to assess the model’s ability to generalize.
• Evaluation of Different p Values: Multiple values of p are tested, and the model's performance is evaluated on the test set.
• Minimizing Prediction Error: The value of p that minimizes prediction error is selected for the final model.
• Avoiding Overfitting: Cross-validation helps prevent overfitting by ensuring the model performs well on unseen data.

In k-Fold Cross-Validation, the data is divided into k subsets, with the model tested on each subset. Leave-One-Out Cross-Validation (LOO-CV) is a specific type where k equals the total number of data points, testing the model on each point.

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Also Read: How to Interpret R Squared in Regression Analysis?

Let’s explore the key assumptions of the AR model, which are essential for accurate and reliable forecasting.

Assumptions of the Autoregressive Model

The autoregressive model relies on key assumptions to ensure reliable forecasts and valid results. These assumptions help capture temporal dependencies, providing unbiased and accurate predictions. Violations of these assumptions can compromise the model’s performance and inference.

Below are the detailed assumptions of the AR model:

1. Stationarity

The AR model assumes that the time series is stationary. This means its statistical properties, such as the mean, variance, and autocorrelation, remain constant over time. Stationarity is one of the most critical assumptions for the AR model to function correctly.

• Mean and Variance: The time series should have a constant mean and variance. If these change over time, the series is non-stationary and needs transformation (e.g., differencing) to make it stationary.
• Autocorrelation: The autocorrelation structure should depend only on the lag, not on the specific period. If the autocorrelation changes over time, the model would struggle to capture consistent patterns.

If the time series is non-stationary, it will lead to unreliable parameter estimates and predictions. Differencing, detrending, or applying other transformation methods are often used to achieve stationarity.

2. Linearity

The AR model assumes a linear relationship between past values and future predictions. This means the current value of the time series is a weighted sum of previous values, with the relationship between them being constant and proportional.

• Linear Relationship: The influence of past values on the current value is assumed to be constant over time, and there is no complex, nonlinear interaction between past and future values.

If the relationship between past values and future predictions is nonlinear, the AR model may not accurately capture the underlying patterns. In such cases, other models (like nonlinear models) might be more appropriate.

3. No Serial Correlation in Residuals

The AR model assumes that the residuals (errors) are uncorrelated. This means that the error term at one time point should not be related to the error term at any other time point. Essentially, the residuals should behave like white noise, with no discernible pattern.

• Uncorrelated Residuals: If residuals are correlated, it indicates that the model hasn't fully captured some aspect of the underlying data structure, and that there’s still some predictable pattern in the residuals.
• Autocorrelation of Residuals: In a properly fitted AR model, the autocorrelation of residuals should be zero at all lags. This can be tested using tools such as the Ljung-Box test or by inspecting the ACF plot of the residuals.

Serial correlation in the residuals indicates that the model fails to capture temporal dependencies, resulting in biased estimates and poor predictions. To address this, consider increasing the order 𝑝 or using an ARMA/ARIMA model.

4. Independence of Error Terms

The AR model assumes that the error terms (residuals) are mutually independent. This means that the error should not influence the error at one time point at any other time point.

• Independent Errors: The errors from each time period should not be correlated with past errors. This ensures that each prediction is based solely on the observed data, without being influenced by previous errors.
• Test for Independence: The independence of error terms can be checked using the Durbin-Watson statistic, which tests for first-order autocorrelation in the residuals. A value close to 2 suggests independence.

If the error terms are not independent, it suggests serial correlation or autocorrelation, where past errors influence future errors. This violates the AR model’s assumptions, leading to biased estimates and inefficient predictions.

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Also Read: Predictive Analytics vs Descriptive Analytics

Now, let’s explore how to overcome AR model limitations across domains and identify where they excel and where they struggle.

Overcoming AR Model Limitations in Various Domains

Autoregressive models are beneficial in domains where past values have a strong influence on future outcomes, such as finance and climate analysis. Their performance may decline when dealing with non-stationary data, seasonal patterns, or sudden structural changes in the time series.

Below are key applications where Autoregressive models are implemented:

1. Stock Market Prediction

AR models predict stock prices by analyzing past closing prices to capture trends and cyclic patterns. The AR model computes the weighted sum of previous stock prices to predict future prices, adjusting the coefficients to minimize forecast error.

2. Electricity Demand Forecasting

AR models predict electricity demand by analyzing historical consumption data to capture trends, seasonal patterns, and fluctuations in demand. The AR model calculates the weighted sum of past demand values to forecast future demand, adjusting the coefficients to minimize forecasting error.

3. Weather Modeling

AR models predict weather patterns by analyzing historical data, including temperature, precipitation, and wind speed. The AR model computes the weighted sum of past weather observations to forecast future conditions, adjusting the coefficients to minimize forecast error.

4. Economic Indicators

AR models predict economic indicators, such as GDP, inflation, or unemployment rates, by analyzing historical data. The AR model calculates the weighted sum of past values of the indicator to forecast future trends, adjusting the coefficients to minimize forecast error.

The AR model is commonly used in data science, AI, machine learning, and data analytics for time series forecasting. Its ability to capture past dependencies makes it valuable for predicting trends in fields such as finance, economics, and operations.

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Also Read: An Intuition Behind Sentiment Analysis: How To Do Sentiment Analysis From Scratch?

Let’s explore how the AR model can be applied to practical stock data for forecasting, using AAPL as an example.

How to Forecast Stock Prices Using the AR Model in Python?

Autoregressive models predict future stock prices by using past price patterns. This example uses AAPL closing prices to show how to train, forecast, and evaluate an AR model using Python.

Step 1: Install Required Libraries

To get started, you'll first need to install the essential Python libraries for data extraction, time series modeling, data visualization, and evaluation. Use the command below to set up your environment.

pip install yfinance statsmodels matplotlib scikit-learn pandas

Explanation: 

• yfinance: For extracting historical data.
• statsmodels: For time series modeling.
• matplotlib: For visualization.
• scikit-learn: For evaluation and machine learning.
• Pandas: For data manipulation.

Step 2: Code - AR Model on Apple Stock (AAPL)

In this step, we’ll implement the Autoregressive model using Python. We’ll begin by fetching historical stock data for AAPL using the yfinance library. Next, we’ll build and evaluate an autoregressive model to forecast its closing prices.

Code Example:

import yfinance as yf
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.ar_model import AutoReg
from sklearn.metrics import mean_squared_error
import numpy as np

# Load Apple stock data
data = yf.download("AAPL", start="2022-01-01", end="2024-12-31")
close_prices = data['Close']

# Keep only the last 300 data points
series = close_prices.tail(300)

# Train-test split
train, test = series[:250], series[250:]

# Fit AR model with lag=5
model = AutoReg(train, lags=5)
model_fit = model.fit()

# Forecast
pred = model_fit.predict(start=len(train), end=len(series)-1, dynamic=False)

# Plot actual vs predicted
plt.figure(figsize=(10, 5))
plt.plot(series.index, series, label='Actual')
plt.plot(test.index, pred, color='red', label='Predicted')
plt.title('AR Model Forecast on Apple Stock')
plt.xlabel('Date')
plt.ylabel('Closing Price (USD)')
plt.legend()
plt.tight_layout()
plt.show()

# Calculate RMSE
rmse = np.sqrt(mean_squared_error(test, pred))
print(f'RMSE: {rmse:.4f}')

Explanation:

• Apple stock prices are downloaded using the Yahoo Finance API.
• Only the 'Close' column is used to represent daily end prices.
• The most recent 300 days of prices are kept for analysis.
• The first 250 values are used to train the Autoregressive model, which uses the past 5 days (lag=5) to forecast the next ones.
• The remaining 50 values are predicted and plotted against actual prices.
• RMSE is calculated to measure the accuracy of a forecast.

Visual Output:

• A line plot with two curves:
  • Blue line: Actual Apple Inc. (AAPL) closing prices over 300 trading days.
  • Red line: Predicted closing prices for the last 50 days using the Autoregressive model.
• The red forecast line closely tracks the blue actual line if the model is well-tuned and the data is stationary.
• The x-axis shows date labels, while the y-axis represents stock prices.
• A legend distinguishes between actual and predicted values.

Text Output: The RMSE (e.g., 2.1394) measures the average error between predicted and actual prices. Lower values indicate better forecast accuracy.

RMSE: 2.1394

Note: RMSE varies based on date range and volatility in prices.

Want to implement AR models efficiently using Python? Consider exploring upGrad's course:  Learn Python Libraries: NumPy, Matplotlib & Pandas. In just 15 hours, you’ll build essential skills in data manipulation, visualization, and analysis.

Also Read: Structured Data vs Semi-Structured Data: Differences, Examples & Challenges

How upGrad Can Help You Excel in Time Series Analysis?

An Autoregressive model predicts future values based on a linear relationship with its past values. To apply this model effectively, you need knowledge of time series analysis, stationarity, autocorrelation, and proficiency in Seaborn and Scikit-learn.

To help you develop these skills, upGrad offers programs that bridge the gap between theory and practical application. Through hands-on projects and tool-based training, you'll gain practical skills in core data technologies relevant to today's analytics field.

Here are a few additional upGrad courses that can help you stand out:

• Basics of Inferential Statistics
• Learn Basic Python Programming
• Data Science in E-commerce
• Generative AI Mastery Certificate for Managerial Excellence
• Analyzing Patterns in Data and Storytelling

Not sure which data science program best aligns with your career goals? Contact upGrad for personalized counseling and valuable insights, or visit your nearest upGrad offline center for more details.

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Reference:
https://www.bis.org/publ/work1250.pdf