What is the Moving Average Model?

Moving Average Models are a type of time series analysis model usually used in econometrics to forecast trends and understand patterns in time series data. In moving average models the present value of the time series depends on the linear combination of the past white noise error terms of the time series. In time series analysis moving average is denoted by the letter “q” which represents the order of the moving average model, or in simple words we can say the current value of the time series will depend on the past q error terms. Therefore, the moving average model of order q could be represented as:

Here,

• is the value of time series at time t
• c is a constant or the mean of the time series
• are the white noise terms associated with the time series at time t, t-1, t-2, … , t-q.
• are the moving average constants.

For example, if we consider MA(1) model, in this model the present value of the time series will only depend on a single past error term and the time series becomes:

From this observation we can also conclude one of the most important aspects of moving average models that the higher the value of the order of moving average model (q), the model will have longer memory and dependence on the past values.

Interpretation of MA model:

There is a difference in the shock wave that is seen in the MA model and AR model that we can mention which might help us get a better understanding how MA and AR model differ. For a better understanding let’s look at the AR model’s general form as well:

• First that the past noise term affects the MA model’s present value directly as we can see in the above equation of the MA model but in AR model the past noise term() have an indirect influence on the present AR model value() since the AR model equation depends on the previous value of the model() and the previous model value depends on it’s noise term().
• The MA model works a a finite impulse model, which means that the current noise value affects the present value of the model as well as “q” further values, as the moving average models only depend on q terms in the past. Whereas AR models acts as infinite impulse model since the current noise affects infinite values of the model in the future. In AutoRegressive model value affects the term which affects the term and so on.

Concept Related to Moving Average:

Now let’s discuss about some of the concepts that can help us in understanding the moving average model in a better way:

• Stationarity: Stationarity is the principle of time series data that conveys that the statistical properties of the data doesn’t change with time, the mean of the data remains the same or we can also say that the data fluctuates around a certain value, the standard deviation of the time series data nearly remains constant, and there must not be any seasonality in the time series data or there is no periodic behavior in the data. We can check for the stationarity of the dataset visually as well as through Augmented Dickey-Fuller(ADF) Test. We consider stationarity to be one of the most important aspect that the time series data must possess in order to be accepted by the models that are applied to time series data for accurate modelling.
• Differencing: Differencing is one of the most important steps to consider during time series analysis, after taking a peek at the original time series data, if the data is not stationary and contains a lot of trends then differencing must be considered since for accurate time series data analysis the data must be stationary. In regular differencing the current time series data is subtracted by the previous data point. , this method removes trends from the data, making it suitable for modelling.
• White Noise: White noise is the error term which has the mean of zero and a constant standard deviation with no correlation of the data points with each other. White noise acts as a benchmark in the forecasting process through time series modelling, if the forecast error is nor white noise further modifications could be performed on the model, but if it reaches a state such that the forecast errors are white noise then the model would need no further improvements. The value of white noise series are random and unpredictable therefore if any time series data is a white noise then there is no method to model or forecast it.

• ACF Plot: Autocorrelation Function plot or the ACF plot is the plot of correlation between the time series and its lagged version. It shows how similar the time series is with it’s different lagged values. Here the lag term is a fixed time displacement, in the ACF plot the x-axis is the lagged time series and the y-axis is the correlation which ranges from -1 to 1.

Let’s look at a code example which can help us thoroughly understand the moving average model:

Data is often collected with respect to time, whether for scientific or financial purposes. When data is collected in a chronological order, it is referred to as time series data. Analyzing time series data provides insights into how the data behaves over time, including underlying patterns that can help solve problems in various domains. Time series analysis can also aid in forecasting future values based on historical data, leading to better production, profits, policy planning, risk management, and other fields. Therefore, analysis of time series data becomes an important aspect of data science.

In this article, we will discuss Moving Average Models, which are essential for time series analysis and forecasting trends.