Exam 7: Multivariate Models

What is the appropriate Box-Pierce test statistic?

A recursive forecasting framework is one where

The acf is clearly declining very slowly in this case, which is consistent with their being an autoregressive part to the appropriate model. The pacf is clearly significant for lags one and two, but the question is does it them become insignificant for lags 2 and 4, indicating an AR(2) process, or does it remain significant, which would be more consistent with a mixed ARMA process? Well, given the huge size of the sample that gave rise to this acf and pacf, even a pacf value of 0.001 would still be statistically significant. Thus an ARMA process is the most likely candidate, although note that it would not be possible to tell from the acf and pacf which model from the ARMA family was more appropriate. The DGP for the data that generated this plot was y_t = 0.9 y_(t-1) - 0.3 u_(t-1) + u_t. -If a series, y, is described as "mean-reverting", which model from the following list is likely to produce the best long-term forecasts for that series y?

Consider a series that follows an MA(1) with zero mean and a moving average coefficient of 0.4. What is the value of the autocorrelation function at lag 1?

The acf is clearly declining very slowly in this case, which is consistent with their being an autoregressive part to the appropriate model. The pacf is clearly significant for lags one and two, but the question is does it them become insignificant for lags 2 and 4, indicating an AR(2) process, or does it remain significant, which would be more consistent with a mixed ARMA process? Well, given the huge size of the sample that gave rise to this acf and pacf, even a pacf value of 0.001 would still be statistically significant. Thus an ARMA process is the most likely candidate, although note that it would not be possible to tell from the acf and pacf which model from the ARMA family was more appropriate. The DGP for the data that generated this plot was y_t = 0.9 y_(t-1) - 0.3 u_(t-1) + u_t. -Suppose you had to guess at the most likely value of a one hundred step-ahead forecast for the AR(2) model given in question 14 - what would your forecast be?

Consider the following MA(3) process., y_t = 0.1 + 0.4u_t-1 + 0.2u_t-2 - 0.1u_t-3 + u_t What is the optimal forecast for y_t, 3 steps into the future (i.e. for time t+2 if all information until time t-1 is available), if you have the following data? U_t-1 = 0.3; u_t-2 = -0.6; u_t-3 = -0.3

Which of these is an appropriate way to determine the order of an ARMA model required to capture the dynamic features of a given data?

Use the following to answer questions A researcher is interested in forecasting the house price index in Country Z. The observed price index values from 1996 to 2000 are 101, 103 104, 107 and 111. The researcher uses two different forecasting models, A and B. The forecasts for the price index using Model A are 100.5, 102.4, 103.2, 106 and 111 whilst the forecast using Model B are 100.8, 102.2, 104, 104.2 and 112.1. -Based on the MAE and MSE forecast evaluation metrics, which of these statements are true?

Which of the following statements are true? (I) An MA(q) can be expressed as an AR(infinity) if it is invertible (ii) An AR(p) can be written as an MA(infinity) if it is stationary (iii) The (unconditional) mean of an ARMA process will depend only on the intercept and on the AR coefficients and not on the MA coefficients (iv) A random walk series will have zero pacf except at lag 1

Consider the following single exponential smoothing model: S_t = $\alpha$ X_t + (1- $\alpha$ ) S_t-1 You are given the following data: =0)1, X_t=0.5,S_t-1=0.2 If we believe that the true DGP can be approximated by the exponential smoothing model, what would be an appropriate 2-step ahead forecast for X? (i.e. a forecast of X_t+2 made at time t)