A data scientist is analyzing a time series of daily product sales. After fitting a first-order Autoregressive AR(1) model, the estimated autoregressive coefficient (φ₁) is 1.05. What is the most critical implication of this coefficient value for the model and the underlying time series?
The model represents a non-stationary, explosive process, making long-term forecasts unreliable.
The time series exhibits strong negative autocorrelation, where high values are consistently followed by low values.
The variance of the model's error term is non-constant, indicating the presence of heteroskedasticity.
The model is underfitting the data, and a higher-order model, such as AR(2) or AR(3), should be used.
The correct answer is that the model represents a non-stationary, explosive process. For a first-order autoregressive AR(1) model to be stationary, the absolute value of its coefficient (φ₁) must be less than 1. A coefficient with an absolute value of 1 indicates a unit root (a random walk process), which is non-stationary. A coefficient with an absolute value greater than 1, such as 1.05, signifies an explosive process. In an explosive process, the effects of past shocks are magnified over time, causing the variance of the series to increase indefinitely and forecasts to diverge rapidly, making them unreliable.
The suggestion that the model is underfitting is incorrect. The order of an AR model is typically determined by analyzing the Partial Autocorrelation Function (PACF) plot, not by the magnitude of the coefficient.
A positive coefficient like 1.05 indicates positive autocorrelation, not negative.
While heteroskedasticity (non-constant error variance) is a potential issue in time series, the primary and direct implication of a φ₁ > 1 is the explosive non-stationarity of the series itself, not necessarily heteroskedasticity in the residuals.
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What does stationarity mean in a time series context?
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