What is the long-run mean of an AR process?

The long-run (unconditional) mean is the value the series converges to over an infinite forecast horizon under stationarity. For AR(1), it equals a/(1-rho). For AR(p), it equals a/(1 - rho_1 - rho_2 - ... - rho_p). This value is only meaningful when the process is stationary.

When should you use AR(2) or AR(3) instead of AR(1)?

Use higher-order AR models when data exhibits complex dynamics. AR(2) captures cyclical behavior (dampened oscillations when rho_2 < 0). AR(3) adds flexibility for seasonal-like patterns. The choice should be guided by information criteria (AIC, BIC), significance of higher-order lag coefficients, and residual diagnostics. AR(1) suffices for many macroeconomic and financial time series.

Time Series Forecasting Calculator (AR/ADL) | AR(p) Fan Charts

Model Parameters

Model Order

Intercept (a)

Estimated intercept from regression

ρ₁ (First Lag)

AR coefficient for first lag

ρ₂ (Second Lag)

AR coefficient for second lag

ρ₃ (Third Lag)

AR coefficient for third lag

σ (Standard Error)

Error standard deviation from regression

y_t (Current Value)

Most recent observed value

y_t-1 (One-Period Lag)

One period before current value

y_t-2 (Two-Period Lag)

Two periods before current value

Forecast Horizon (h)

Number of periods ahead (1–100)

Confidence Level

AR Model Formulas

y_t = a + ρ₁y_t-1 + … + ρ_py_t-p + u_t

a = Intercept | ρ = AR coefficients | σ = Error std. dev. | h = Horizon

Calculator by Ryan O'Connell, CFA

Forecast Results

1-Step Forecast --

h-Step Forecast --

Long-Run Mean --

Half-Life --

Forecast Fan Chart

Chart bands: 50%, 80%, 95% confidence. Dashed line = long-run mean.

Step-by-Step Forecasts

Step	Point Forecast	Forecast SE	Lower CI	Upper CI

Formula Breakdown

AR(1): y_t+h = a·(1 − ρ^h)/(1 − ρ) + ρ^h·y_t

Model Assumptions

Stationarity: all companion-matrix eigenvalues have modulus < 1
Constant error variance (homoskedasticity)
No structural breaks in the forecast period
Errors are independent (no serial correlation in innovations)
Parameters estimated consistently from a sufficiently long sample
Linear dynamics (no threshold or regime-switching effects)
Forecast intervals are normal approximations ignoring parameter-estimation uncertainty

For educational purposes. Not financial advice. Market conventions simplified.

Understanding AR Models and Time Series Forecasting

Autoregressive (AR) models are among the most widely used tools in time series econometrics. An AR(p) model predicts the current value of a variable as a linear function of its p most recent past values, an intercept, and a random error term. These models capture the persistence and mean-reverting behavior commonly observed in macroeconomic and financial time series.

How AR Forecasting Works

For AR(1), the h-step-ahead forecast has a convenient closed form: the forecast is a weighted average of the current value and the long-run mean, with the weight on the current value decaying exponentially at rate ρ^h. For AR(2) and AR(3), forecasts must be computed recursively, using previous forecasts as lagged values once the forecast passes the origin.

Forecast Uncertainty and Fan Charts

Forecast intervals widen as the horizon increases because prediction uncertainty accumulates over time. For stationary models, this uncertainty eventually plateaus at the unconditional variance of the process. A fan chart visualizes this by showing progressively wider confidence bands (50%, 80%, 95%) around the point forecast, with the forecast line converging toward the long-run mean. This approach was popularized by the Bank of England for communicating inflation forecasts.

Stationarity and the Unit Circle

A key requirement for meaningful AR forecasts is stationarity, which ensures the process has a stable long-run mean and bounded variance. For AR(1), stationarity requires |ρ| < 1. For higher-order models, all eigenvalues of the companion matrix must have modulus less than 1. Non-stationary processes (unit roots) produce forecasts that diverge without converging, and require differencing before forecasting.

AR Models vs. ADL Models

While this calculator focuses on pure AR models, Wooldridge Chapter 18 also covers Autoregressive Distributed Lag (ADL) models that include lagged values of additional explanatory variables. ADL models are useful when external factors help predict the series, but they require assumptions about the future values of those regressors. For pure forecasting from historical values alone, AR models are the standard approach.

Important: This calculator uses estimated parameters as if they were known. In practice, parameter-estimation uncertainty adds to forecast variance, especially in small samples. The intervals shown here are conditional on the estimated coefficients.

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Frequently Asked Questions

An autoregressive model of order p, or AR(p), predicts a variable's future value using a linear combination of its p most recent past values plus an intercept and error term. AR(1) uses only the immediately preceding value, AR(2) uses the two most recent values, and AR(3) uses three. The model captures how persistent shocks are over time — a high AR coefficient means shocks decay slowly, while a low coefficient means rapid mean reversion. AR models are foundational in time series econometrics (Wooldridge, Chapter 11 and 18).

Stationarity means the statistical properties of the series (mean, variance, autocorrelation) do not change over time. For AR(1), the condition is simply |ρ| < 1. For higher-order models, compute the companion-matrix eigenvalues: all eigenvalues must have modulus less than 1. Non-stationary processes have forecasts that diverge rather than converge to a long-run mean.

The long-run (unconditional) mean is the value the series converges to over an infinite forecast horizon under stationarity. For AR(1), it equals a/(1−ρ). For AR(p), it equals a/(1 − ρ₁ − ρ₂ − … − ρ_p). This value is only meaningful when the process is stationary. The long-run mean appears as a horizontal dashed line on the fan chart, showing how point forecasts gradually approach it.

Forecast intervals account for the uncertainty that accumulates as we forecast further ahead. For each step h, the interval is ŷ_t+h ± z_α/2 × √Var(e_t+h), where z_α/2 is the critical value (1.96 for 95%). For AR(1), the forecast error variance has a closed form: σ²(1−ρ^2h)/(1−ρ²). For AR(2) and AR(3), the variance is computed using psi-weights from the Wold decomposition. These are normal-approximation intervals that ignore parameter-estimation uncertainty. For stationary models, intervals widen with the horizon then plateau at the unconditional variance.

A fan chart visualizes forecast uncertainty by showing progressively wider confidence bands around the point forecast. The solid line in the center represents the best point forecast at each horizon. The darkest shaded band (50%) shows where the actual value is equally likely to fall above or below. The medium band (80%) covers most likely outcomes. The widest, lightest band (95%) captures nearly all reasonable outcomes. The bands expand over time because forecast uncertainty grows with the horizon. Fan charts were popularized by the Bank of England for inflation forecasts.

Use higher-order AR models when the data exhibits more complex dynamics than a simple first-order process. AR(2) can capture cyclical or oscillatory behavior (when ρ₂ < 0, creating dampened oscillations). AR(3) adds further flexibility for modeling seasonal-like patterns at short lags. The choice should be guided by information criteria (AIC, BIC) from the estimated regression, significance of higher-order lag coefficients, and residual diagnostics. In practice, AR(1) suffices for many macroeconomic and financial time series, but AR(2) is common for quarterly GDP and interest rates.

Disclaimer

This calculator is for educational purposes only and assumes estimated AR parameters are known. Actual forecasting involves additional considerations including parameter uncertainty, model selection, structural breaks, and non-normal errors. Results should not be used as the sole basis for investment or policy decisions.

Time Series Forecasting Calculator

Model Parameters

AR Model Formulas

Forecast Results

Forecast Fan Chart

Step-by-Step Forecasts

Formula Breakdown

Model Assumptions

Understanding AR Models and Time Series Forecasting

How AR Forecasting Works

Forecast Uncertainty and Fan Charts

Stationarity and the Unit Circle

AR Models vs. ADL Models

Frequently Asked Questions

What is an autoregressive AR(p) model?

How do you check stationarity of an AR model?

What is the long-run mean of an AR process?

How are forecast confidence intervals computed?

What is a fan chart and how do you read it?

When should you use AR(2) or AR(3) instead of AR(1)?

Disclaimer

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