Forecasting is the process of predicting future events based on past and present data. Time-series forecasting is a type of forecasting that predicts future events based on time-stamped data points. There are many different types of time-series forecasting models, each with its own strengths and weaknesses. In this blog post, we will discuss the most common time-series forecasting machine learning models such as the following, and provide examples of how they can be used to predict future events.

- Autoregressive (AR) model
- Moving average (MA) model
- Autoregressive moving average (ARMA) model
- Autoregressive integrated moving average (ARIMA) model
- Seasonal autoregressive integrated moving average (SARIMA) model
- Vector autoregressive (VAR) model
- Vector error correction (VECM) model

Table of Contents

## Autoregressive (AR) model

Autoregressive (AR) models are defined as regression models in which the dependent or response variable is a linear function of past values of the dependent/response variable. The order of an autoregressive model is denoted as ‘p’, which represents the number of lags used to predict the current value. For example, if p=0, then it means that we are predicting the current time-step (t) based on the previous time-step (t-0). If p=n, then we are predicting time-step (t) based on n past time-steps.

The general form of an autoregressive model can be represented as:

\(Y_t = c + \phi_p Y_{t-p} + \varepsilon_t\)## Moving average (MA) model

The moving average model can be defined as a regression model in which the dependent/response variable is a linear function of past values of the error term or white noise. The order of a moving average model is denoted by ‘q’, which represents the number of lags used to predict the current value. For example, if q=0, then it means that we are predicting time-step (t) only based on the current time-step (t). If q=n, then we are predicting time-step (t) based on n past time-steps.

The general form of a moving average model can be represented as:

\(Y_t = \mu + \theta_q \varepsilon_{t-q} + \varepsilon_t\)## Autoregressive moving average (ARMA) model

The autoregressive moving average (ARMA) model is a combination of the autoregressive and moving average models. The ARMA model is defined as a regression model in which the dependent/response variable is a linear function of past values of both the dependent/response variable and the error term. The order of an ARMA model is represented by ‘p’ for the autoregressive part and ‘q’ for the moving average part. For example, if p=0 and q=0, then it means that we are predicting time-step (t) based on time-step (t) only. If p=n and q=m, then we are predicting time-step (t) based on n past time-steps of the dependent/response variable and m past time-steps of the error term.

The general form of an ARMA model can be represented as:

\(Y_t = c + \phi_p Y_{t-p} + \theta_q \varepsilon_{t-q} + \varepsilon_t\)## Autoregressive integrated moving average (ARIMA) model

The autoregressive integrated moving average (ARIMA) model is a generalization of the ARMA model. The ARIMA model is defined as a regression model in which the dependent/response variable is a linear function of past values of both the dependent/response variable and the error term, where the error term has been differentiated ‘d’ times. The order of an ARIMA model is represented by ‘p’ for the autoregressive part, ‘q’ for the moving average part, and ‘d’ for the differencing part. For example, if p=0, q=0, and d=0, then it means that we are predicting time-step (t) based on time-step (t) only. If p=n, q=m, and d=k, then we are predicting time-step (t) based on n past time-steps of the dependent/response variable, m past time-steps of the error term, and k past time-steps of the differenced error term.

The general form of an ARIMA model can be represented as:

\(Y_t = c + \phi_p Y_{t-p} + \theta_q \varepsilon_{t-q} + \delta^d Y_t\)The interpretation of autoregressive integrated moving average (ARIMA) models is similar to that of autoregressive moving average (ARMA) models. The main difference between ARIMA and ARMA models is that ARIMA models can be used to model time-series data that is non-stationary, whereas ARMA models can only be used to model time-series data that is stationary.

## Seasonal autoregressive integrated moving average (SARIMA) model

SARIMA is a type of time-series forecasting model that takes into account both seasonality and autocorrelation. SARIMA models are based on a combination of differencing, autoregression, and moving average processes. These models can be used to forecast short-term or long-term trends in data. SARIMA models are generally considered to be more accurate than other types of time-series forecasting models, such as ARIMA models. SARIMA models are also relatively easy to interpret and use.

The SARIMA model can be used to forecast demand for a product or service over the course of a year. The model takes into account sales data from previous years as well as seasonality (e.g., holiday sales). SARIMA can also be used to forecast other time-series data, such as stock prices or weather patterns. The SARIMA model is a generalization of the ARIMA model (which only accounts for autocorrelation), and it can be used to forecast data with seasonality.

## Vector autoregressive (VAR) model

The vector autoregressive (VAR) model can be defined as a multivariate time-series model that captures the linear interdependence among multiple time-series variables. In addition to the dependent variables, the VAR model also includes one or more lagged values of each dependent variable as independent variables. Example: A VAR(p) for a bivariate time-series \({y_t,x_t}\) would be

\(y_t = c + A_iy_{t-i} + A_jx_{t-j} + u_t\)where p is the lag order, c is a vector of constants, A_i and A_j are matrices of coefficients, and u_t is white noise.

The above equation can be rewritten in vector form as

\(Y_t = c + A_0Y_{t-0} + A_pY_{t-p} + u_t\)where Y_t is a vector consisting of the time-series variables y_t and x_t, c is a vector of constants, A_0 is a matrix of coefficients, and u_t is white noise.

The VAR model can be estimated using ordinary least squares (OLS). Once the model has been estimated, it can be used to make forecasts. For example, if we wanted to forecast the value of y at time t+h, we would use the following equation:

\(y_{t+h} = c + A_iy_{t+h-i} + A_jx_{t+h-j}\)where h is the forecasting horizon.

The VAR model is a generalization of the univariate autoregressive (AR) model and the multivariate linear regression model. It can be used to study the dynamics of a single time-series variable or multiple time-series variables. One advantage of the VAR model is that it can be easily estimated using OLS. Another advantage is that it can be used to make forecasts. One disadvantage of the VAR model is that it can be difficult to interpret. For example, it can be hard to determine which time-series variables are influencing each other.

## Vector Error Correction Model (VECM)

The Vector Error Correction Model (VECM) is a type of time-series model that is often used for forecasting economic data. VECM is an extension of the traditional vector autoregression (VAR) model. VECM was first introduced by Engle and Granger (1987). VECM models are similar to VAR models, but they incorporate a “correction” term that accounts for the fact that errors in one period are likely to be corrected in subsequent periods. This makes VECM models more accurate than VAR models when forecasting data with significant serial correlation. VECM models are also relatively easy to interpret since the coefficients can be directly interpreted as the impact of one variable on another. VECM has become a popular tool in econometrics and finance due to its flexibility and ease of use.

The VECM has several advantages over the VAR model. First, it is easier to interpret the results of a VECM than a VAR. Second, the VECM can be used to test for co-integration among time series variables, while the VAR cannot. Finally, the VECM can be applied to non-stationary time series data, while the VAR cannot.

The disadvantages of the VECM include its complexity and the fact that it requires a large amount of data to produce reliable results. In addition, the VECM is sensitive to outliers and changes in the time-series data.

Time-series forecasting models are used to predict future values of time-series data. In this blog post, we discussed different types of time-series forecasting models such as AR, MA, ARMA, ARIMA, the SARIMA model, the vector autoregressive (VAR) model, and the vector error correction model (VECM). All of these models have their own advantages and disadvantages. Choose the time-series forecasting model that is best suited for your data and your forecasting needs.

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