Once a model has been specified and estimated its adequacy is usually checked with a range of tests and other statistical procedures. Many of these model checking tools are based on the residuals of the final model. Some of them are applied to the residuals of individual equations and others are based on the full residual vectors. Examples of specification checking tools are visual inspection of the plots of the residuals and their autocorrelations. In addition, autocorrelations of squared residuals may be considered to check for possible autoregressive conditional heteroskedasticity (ARCH). Although it may be quite insightful to inspect the autocorrelations visually, formal statistical tests for remaining residual autocorrelation should also be applied. Such tests are often based on LM (Lagrange multiplier) or Portmanteau statistics. Moreover, normality tests of the Lomnicki – Jarque-Bera type may be applied to the residuals (see, e. g. Lutkepohl, 1991; Doornik and Hendry, 1997).
If model defects are detected at the checking stage this is usually regarded as an indication of the model being a poor representation of the DGP and efforts are made to find a better representation by adding other variables or lags to the model, by including nonlinear terms or changing the functional form, by modifying the sampling period or getting other data.