# Vector Autoregression vs ARIMAX [This Key Difference]

In time series analysis, selecting the right model for forecasting can be challenging.

Two popular models often competing for the spotlight are Vector Autoregression (VAR) and Autoregressive Integrated Moving Average with Exogenous Variables (ARIMAX).

Both models have their unique strengths, but the choice ultimately depends on the structure of your data and the type of problem you’re trying to solve.

The main difference between the two is their ability to handle multiple time series: VAR is built for multivariate time series analysis, while ARIMAX focuses on univariate time series with exogenous variables.

Below, we’ll go more in-depth on the VAR and ARIMAX models, discuss some differences between moving averages and autoregressive formulation and explain some of the tough-to-understand terms used above.

You’re not going to want to miss this one.

## Differences Between Autoregression and Moving Average

Understanding the difference between Autoregression (AR) and Moving Average (MA) is essential when diving into the world of time series analysis.

Let’s break down these concepts in a way that everyone can understand.

Autoregression (AR) is about using the past values, or “lags,” of a time series to predict future values.

Imagine you’re trying to forecast the temperature for tomorrow. If you know that today’s temperature was 75 degrees and yesterday’s was 72 degrees, you could use this information to make a prediction.

In other words, AR models rely on the idea that the past can help predict the future.

Moving Average (MA), however, is focused on the errors, or “error lags,” in the time series. Let’s say you tried to predict the temperature for yesterday and made an error in your forecast.

An MA model would look at your past errors to help better predict today and tomorrow. This way, the model learns from its mistakes and improves its forecasting ability over time – based on the assumption that errors have some trend.

Understanding the difference between these two forecasting ideologies is HUGE when trying to understand the difference between ARIMAX and VAR.

## One Vs. Many

Before we continue diving into the differences between VAR and ARIMAX, we must understand the terms “multivariate” and “univariate.”

In time series analysis, “multivariate” means working with multiple time series simultaneously, while “univariate” means focusing on just one time series.

Now, let’s explore how VAR and ARIMAX are designed for these different situations.

Vector Autoregression (VAR) is designed explicitly for multivariate time series analysis.

This means it can handle multiple time series that might be related to each other.

For example, if you wanted to forecast the prices of several stocks in the market, a VAR model could consider how the prices of these stocks influence each other over time.

This makes VAR a powerful tool for understanding complex relationships between multiple time series.

On the other hand, Autoregressive Integrated Moving Average with Exogenous Variables (ARIMAX) is built for univariate time series analysis, which means it focuses on just one time series.

However, it has an added twist: it can incorporate exogenous variables.

Exogenous variables are simply just external factors that might affect the time series but aren’t part of it.

For instance, if you were forecasting the sales of a particular product, you might want to consider factors like the price, advertising campaigns, or even the weather. These external factors can help improve the accuracy of the ARIMAX model’s forecasts.

## Is VAR better than Arimax?

Asking if Vector Autoregression is better than ARIMAX is the wrong way to think about things.

Deciding between (VAR) and ARIMAX mostly depends on the specific problem you’re working on and the nature of your data.

Each model has advantages; the best choice depends on your unique situation.

Let’s review some factors to consider when choosing between VAR and ARIMAX:

### The number of time series

If you are dealing with interconnected time series, VAR is the better choice because it is designed for multivariate analysis. On the other hand, if you are working with a single time series, ARIMAX would be more appropriate.

### Exogenous variables

If external factors influence your time series, ARIMAX is useful because it allows you to incorporate these exogenous variables. VAR does not have this feature, so if exogenous variables are critical to your analysis, ARIMAX may be the better choice.

### Model complexity

VAR models can become quite complex when dealing with multiple time series, which may require more computational power and time to estimate. If you need a simpler model and have only one time series to analyze, ARIMAX might be more suitable.

### Interpretability

ARIMAX models can be easier to interpret when dealing with exogenous variables, as you can directly see the impact of these external factors on your time series. In contrast, VAR models focus on the relationships between multiple time series, which can be more challenging to understand and explain.

## Are Arimax and VAR the only two Time Series Models?

While ARIMAX and VAR are popular time series models, they are not the only options for time series analysis. There is a wide variety of models to choose from, each with its strengths and weaknesses. Here are a few other common time series models to consider:

### Autoregressive (AR) model

This univariate model uses the time series’s past values, or lags, to make predictions. It is a simpler version of ARIMAX without the integrated moving average or exogenous variables components.

### Moving Average (MA) model

Another univariate model, the MA model, focuses on past errors, or error lags, to improve its forecasting ability.

### Autoregressive Integrated Moving Average (ARIMA) model

Combining the AR and MA models, the ARIMA model also accounts for differencing to make the time series stationary. It is essentially an ARIMAX model without exogenous variables.

### Seasonal Decomposition of Time Series (STL)

This technique breaks down a time series into its trend, seasonal, and residual components. It can help analyze time series with strong seasonality.

### Exponential Smoothing State Space Model (ETS)

This family of models includes simple, double, and triple exponential smoothing, which can be used for forecasting univariate time series with different levels of trend and seasonality.

### Long Short-Term Memory (LSTM) networks

These are a type of recurrent neural network designed explicitly for sequence data, such as time series. They can be helpful for complex problems and large datasets where traditional time series models may struggle. (Ever Heard of ChatGPT?)

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