Learn Autoregressive (AR) Models

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Autoregressive models are a foundational class of time series models that forecast future values based on past observations. In an autoregressive model of order $p$ , denoted as AR(p), the current value of the series is expressed as a linear combination of its previous $p$ values plus a random error term. The general form of an AR(p) model is:

X_t = c + φ₁X_{t-1} + φ₂X_{t-2} + ... + φₚX_{t-p} + ε_t

where:

$X_t$ is the value at time $t$ ;
$c$ is a constant;
$φ₁, φ₂, ..., φₚ$ are parameters (coefficients) to be estimated;
$ε_t$ is white noise (a random error term with zero mean and constant variance).

A key concept in AR models is the use of lagged values. Lags refer to prior time points in the series; for example, $X_{t-1}$ is the first lag, $X_{t-2}$ is the second lag, and so on. The order $p$ determines how many lagged values are used to predict the current value.

For AR models to be valid, several assumptions should be met:

The time series should be stationary, meaning its statistical properties like mean and variance do not change over time;
The error terms should be uncorrelated and normally distributed;
The model parameters should ensure the process does not explode (i.e., the series remains stable over time).


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import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.ar_model import AutoReg

# Generate a synthetic AR(1) time series
np.random.seed(42)
n = 100
phi = 0.7
c = 2
errors = np.random.normal(0, 1, n)
series = [0]
for t in range(1, n):
    series.append(c + phi * series[-1] + errors[t])
series = pd.Series(series)

# Fit an AR(1) model
model = AutoReg(series, lags=1)
model_fit = model.fit()

# Predict the next 10 values
pred_start = len(series)
pred_end = pred_start + 9
predictions = model_fit.predict(start=pred_start, end=pred_end)

# Plot the original series and predictions
plt.figure(figsize=(10, 5))
plt.plot(series, label="Original Series")
plt.plot(range(pred_start, pred_end + 1), predictions, label="AR(1) Predictions", marker='o')
plt.legend()
plt.title("AR(1) Model Fit and Forecast")
plt.xlabel("Time")
plt.ylabel("Value")
plt.show()

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Section 1. Chapter 5

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Section 1. Chapter 5