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LOS d: Discuss the structure of an autoregressive (AR) model of order p, and calculate one- and two-period-ahead forecasts given the estimated coefficients.

An analyst wants to model quarterly sales data using an autoregressive model. She has found that an AR(1) model with a seasonal lag has significant slope coefficients. She also finds that when a second and third seasonal lag are added to the model, all slope coefficients are significant too. Based on this, the best model to use would most likely be an:

A)	AR(1) model with no seasonal lags.

B)	AR(1) model with 3 seasonal lags.

ARCH(1).

She has found that all the slope coefficients are significant in the model x_t = b₀ + b₁x_t–1 + b₂x_t–4 + e_t. She then finds that all the slope coefficients are significant in the model x_t = b₀ + b₁x_t–1 + b₂x_t–2 + b₃x_t–3 + b₄x_t–4 + e_t. Thus, the final model should be used rather than any other model that uses a subset of the regressors.

The model x_t = b₀ + b₁ x_t-1 + b₂ x_t-2+ b₃ x_t-3 + b₄ x_t-4 + ε_t is:

A)	an autoregressive model, AR(4).

B)	an autoregressive conditional heteroskedastic model, ARCH.

C)	a moving average model, MA(4).

This is an autoregressive model (i.e., lagged dependent variable as independent variables) of order p=4 (that is, 4 lags).

A)	an autoregressive model, AR(2).

B)	an autoregressive conditional heteroskedastic model, ARCH.

C)	a moving average model, MA(2).

This is an autoregressive model (i.e., lagged dependent variable as independent variables) of order p = 2 (that is, 2 lags).

Consider the estimated model x_t = -6.0 + 1.1 x_t-1 + 0.3 x_t-2 + ε_t that is estimated over 50 periods. The value of the time series for the 49^th observation is 20 and the value of the time series for the 50^th observation is 22. What is the forecast for the 51^st observation?

Forecasted x₅₁ = -6.0 + 1.1 (22) + 0.3 (20) = 24.2.

Consider the estimated model x_t = ?6.0 + 1.1 x_{t ? 1} + 0.3 x_{t ? 2} + ε_t that is estimated over 50 periods. The value of the time series for the 49^th observation is 20 and the value of the time series for the 50^th observation is 22. What is the forecast for the 52^nd observation?

Using the chain-rule of forecasting,
Forecasted x₅₁ = ?6.0 + 1.1(22) + 0.3(20) = 24.2.
Forecasted x₅₂ = ?6.0 + 1.1(24.2) + 0.3(22) = 27.22.

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