Length of X in in ARX/ARMAX/ARIMAX

Hi,

I want to an ARX1 model where X is an Indicator function which is 1 during weekends and 0 during weekdays. I have daily data starting on a Monday from January 2 until December 31 (364 obs). I created my X as follows:

X = repmat([0 0 0 0 0 1 1]', 52, 1);

Hence X has also 364 obs. However when I now estimate my model I get an error saying that X must contact at least 365 obs.

I know this has something to do with a X0, however I am not sure what this X0 should be (to be honest I'm not really sure why I need X0). Adding a single 0 or 1 at the beginning of my X does the trick but can someone please tell me whether this should be a 0 or a 1 (even though results are almost the same) and whether my AR1 and X will still be correctly 'aligned' after adding a single observation at the beginning of my X. That is, will all '1's be corresponding to weekends and '0's to weekdays.

Thank you!

Answer 1

[prabhat kumar sharma]

Hello Victor,

When you're working with an ARX (AutoRegressive with eXogenous inputs) model, the requirement for the exogenous input X to have at least 365 observations is likely due to the way ARX models handle lagged variables and align data.Understanding the Requirement

1. ARX Model Structure:

• An ARX model uses past values of the dependent variable (AR part) and current/past values of exogenous variables (X part) to predict the current value of the dependent variable.
• If your ARX model is of order 1 (ARX1), it means it uses one lag of the dependent variable.

2. Data Alignment:

• When you fit an ARX model, the lagged dependent variable and the exogenous variable X need to be aligned correctly.
• For 364 observations of your dependent variable, you'll need 365 observations of X to cover the initial period before the first observation of the dependent variable is used.

Adding an Initial Observation to X

Given that your data starts on a Monday, your X should ideally start with a 0 to maintain the alignment where 0 corresponds to weekdays and 1 to weekends. Here's why:

• Initial Day (X0): Since your data starts on a Monday, the first observation of X should be 0, aligning with a weekday.
• Weekly Pattern: By starting with a 0, your weekly pattern [0 0 0 0 0 1 1] will correctly represent the weekdays and weekends throughout the dataset.

% Initial X with 364 observations
X = repmat([0 0 0 0 0 1 1]', 52, 1);

% Add an initial 0 for the first Monday
X = [0; X];

I hope it helps!