$ARIMA(p,d,q)+X_t$, Simulation over Forecasting period

Question

I have time series data and I used an $ARIMA(p,d,q)+X_t$ as the model to fit the data. The $X_t$ is an indicator random variable that is either 0 (when I don’t see a rare event) or 1 (when I see the rare event). Based on previous observations that I have for $X_t$ , I can develop a model for $X_t$ using Variable Length Markov Chain methodology. This enables me to simulate the $X_t$ over the forecasting period and gives a sequence of zeros and ones. Since this is a rare event, I will not see $X_t=1$ often. I can forecast and obtain the prediction intervals based on the simulated values for $X_t$.

Question:

How can I develop an efficient simulation procedure to take into account the occurrence of 1’s in the simulated $X_t$ over the forecasting period? I need to obtain the mean and the forecasting intervals.

The probability of observing 1 is too small for me to think that the regular Monte Carlo simulation will work well in this case. Maybe I can use “importance sampling”, but I am not sure exactly how.

Thank you.

Answer 1

Firstly we consider a more general case. Let $Y = Y(A, X)$, where $A \sim f_A(\cdot)$ and $X \sim f_X(\cdot)$. Then, assuming the support of $g_x(\cdot)$ dominates the one of $f_X(\cdot)$ and all the integrals below exist, we have: $$ P(Y \le y) = \mathbb{E}_{f_A, f_X}\left[I(Y \le y)\right] = \mathbb{E}_{f_X}\left[\mathbb{E}_{f_A}\left[I(Y \le y) \mid X \right]\right] = \int_{supp(f_X)}{\mathbb{E}_{f_A}\left[I(Y \le y) \mid X = x \right]f_X(x)dx} = \int_{supp(f_X)}{\mathbb{E}_{f_A}\left[I(Y \le y) \mid X = x \right]\frac{f_X(x)}{g_X(x)}g_X(x)dx} = \int_{supp(g_X)}{\mathbb{E}_{f_A}\left[I(Y \le y) \frac{f_X(X)}{g_X(X)} \mid X = x \right]g_X(x)dx} = \mathbb{E}_{g_X}\left[\mathbb{E}_{f_A}\left[I(Y \le y) \frac{f_X(X)}{g_X(X)} \mid X \right]\right] = \mathbb{E}_{f_A, g_X}\left[I(Y \le y) \frac{f_X(X)}{g_X(X)}\right] $$

In your case $$ f_X(x) = \left\{ \begin{array}{cc} p & x = 1 \\ 1 - p & x = 0\\ \end{array} \right. $$ and $g_X(\cdot)$ can be defined like this: $$ g_X(x) = \left\{ \begin{array}{cc} 0.5 & x = 1 \\ 0.5 & x = 0\\ \end{array} \right. $$ Therefore, you can simulate $X$ via distribution $g_X(\cdot)$, but all the observations with $X=1$ will have the weight $\frac{p}{0.5}=2p$ and all the observations with $X=0$ will have the weight $\frac{1-p}{0.5}=2(1-p)$. Simulation of the ARIMA process will not be affected.