![]() ![]() The resulting model is known as Regression with Seasonal ARIMA Errors or SARIMAX for short. We apply this property of SARIMA models to model the auto-correlated residuals of the Linear Regression model after it is fitted to the time series data set. Now let’s return to our little conundrum about auto-correlated residuals.Īs mentioned earlier, (S)ARIMA models are perfectly suited for forecasting time series data, and particularly for dealing with auto-correlated data. The complete SARIMA model specification is ARIMA(p,d,q)(P,D,Q)m. Just as with p,d and q, there are well established rules for estimating the values of P, D, Q and m. m=12 months for a time series that exhibits yearly seasonality. The final parameter in SARIMA models is ‘m’ which is the seasonal period. SAR, SMA, D and m: The Seasonal ARIMA or SARIMA model simply extends the above concepts concepts of AR, MA and differencing to the seasonal realm by introducing a Seasonal AR (SAR) term of order P, Seasonal MA (SMA) term of order Q, and a Seasonal Difference of order D. The order of differencing is denoted by the d parameter in the ARIMA(p,d,q) model specification. Differencing is applied by ARIMA models before the AR and the MA terms are brought into play. The difference operation in ARIMA models is denoted by the I letter. ![]() y_i is a linear combination of y_(i-1), y_(i-2),… y_(i-p) as follows:Ī time series with a linear trend, and the de-trended time series after applying a first order of differencing: y_i-y_(i-1) (Image by Author) The SARIMA modelĪ SARIMA model consists of the following 7 components:ĪR: The Auto-Regressive (AR) component is a linear combination of past values of the time series up to some number of lags p. We harness this ability of SARIMA, by modeling the residual errors of linear regression using the SARIMA model. (S)ARIMA models are perfectly suited for dealing with auto-correlated data. Just this one problem of auto-correlation in the residual errors, causes a cascading litany of problems with the Linear Regression model, rendering it practically useless for modeling time series data.īut instead of throwing away the powerful Linear Regression model, one can fix the problem of auto-correlated residuals by bringing in another powerful model, namely ARIMA (or Seasonal ARIMA). This in turn makes the confidence intervals of the regression coefficients, and those of the model’s forecasts, unreliable. Neither can one rely on the standard errors of the regression coefficients. A major problem caused by auto-correlated residual errors is that one cannot use statistical tests of significance such as the F-test or the Student’s t-test to determine whether the regression coefficients are significant. But if the residual errors are auto-correlated, they cannot be independent, causing many problems. We have seen in the section on the Assumptions of Linear Regression that Linear Regression models assume that the residual errors of regression are independent random variables with identical normal distributions. Therefore, if you fit a straight-up linear regression model to the ( y, X) data set, these auto-correlations will leak into the the residual errors of regression ( ε), making the ε auto-correlated! Linear regression models are unable to ‘explain’ such auto-correlations. A given value of y_i in y is influenced by previous values of y i.e. Ε is a vector of size (n x 1), assuming a data set spanning n time steps.īut alas, our elegant Linear Regression model will not work for time series data for a very simple reason: time series data are auto i.e. Ε, the residual errors of regression is the difference between the actual y and the value y(cap) predicted by the model. In the above model specification, β(cap) is an (m x 1) size vector storing the fitted model’s regression coefficients. A linear regression model (Image by Author) ![]()
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