5. ARMA Time Series Flashcards
What are the conditions for weak stationarity in a time series?
E(Xt) = µ (constant mean), Var(Xt) = γ(0) (constant variance), Cov(Xt, Xs) = γ(h) (depends only on lag h, not on time t or s).
What is the formula for autocorrelation in a time series?
ρ(h) = γ(h) / γ(0), where γ(h) is the autocovariance at lag h and γ(0) is the variance.
Write the AR(1) model equation and its stationarity condition.
AR(1): Yt = ϕ * Yt-1 + εt; Stationarity condition: |ϕ| < 1.
What is the difference operator, and what does it do?
The difference operator is ∇Xt = Xt - Xt-1, and it transforms a non-stationary series into a stationary one by removing trends.
State the formula for the variance of an MA(1) model.
Var(Yt) = σ²ε * (1 + θ²), where θ is the MA(1) parameter and σ²ε is the variance of the noise term.
How do you interpret the autocorrelation function (ACF) of an AR(1) process?
The ACF of an AR(1) process decays exponentially as the lag increases, with ρ(h) = (ϕ^h), where ϕ is the autoregressive coefficient.
What is the purpose of the Dickey-Fuller test in time series analysis?
The Dickey-Fuller test is used to test for stationarity by checking if a time series has a unit root (indicating non-stationarity).
Explain the stationarity condition for a general AR(p) model.
The stationarity condition for an AR(p) model requires the roots of the characteristic equation Φ(z) = 1 - Σ(ϕj * zj) to lie outside the unit circle in the complex plane.
What is the significance of the moving average (MA) coefficients in the autocovariance function?
MA coefficients determine the autocovariance up to the lag equal to the MA order (q), beyond which the autocovariance is zero.
Why is differencing used in ARIMA models, and how is it applied?
Differencing is used to make a non-stationary series stationary by removing trends or seasonality, applied as ∇Xt = Xt - Xt-1 or higher-order differences like ∇²Xt = Xt - 2Xt-1 + Xt-2.
How can the autocorrelation and partial autocorrelation functions (PACF) help identify the order of an AR(p) model?
The ACF of an AR(p) model tails off gradually, while the PACF cuts off sharply after lag p, indicating the model order.
Describe how the Box-Jenkins methodology is applied in building ARIMA models.
The Box-Jenkins methodology involves model identification (using ACF and PACF), parameter estimation (e.g., maximum likelihood), and model diagnostics (checking residuals for white noise and model fit).
Explain the conditions under which the MA(1) model exhibits invertibility and its importance.
An MA(1) model is invertible if |θ| < 1. Invertibility ensures a unique representation of the model in terms of past observations and errors.
How does the maximum likelihood estimation method work in ARIMA models?
Maximum likelihood estimation finds parameter values that maximize the likelihood of observing the given data, assuming the errors follow a Gaussian white noise process.
Discuss the impact of overfitting in ARMA/ARIMA models and how it can be avoided.
Overfitting occurs when too many parameters are included, leading to poor generalization. It can be avoided using criteria like AIC or BIC for model selection and cross-validation for forecasting accuracy.
