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## Goodness of Fit

After specifying a model and estimating its parameters, it is good practice to perform goodness-of-fit checks to diagnose the adequacy of your fitted model. When assessing model adequacy, areas of primary concern are:

• Violations of model assumptions, potentially resulting in bias and inaccurate standard errors

• Poor predictive performance

• Missing explanatory variables

Goodness-of-fit checks can help you identify areas of model inadequacy. They can also suggest ways to improve your model. For example, if you conduct a test for residual autocorrelation and get a significant result, you might be able to improve your model fit by adding additional autoregressive or moving average terms.

Some strategies for evaluating goodness of fit are:

• Compare your model against an augmented alternative. Make comparisons, for example, by conducting a likelihood ratio test. Testing your model against a more elaborate alternative model is a way to assess evidence of inadequacy. Give careful thought when choosing an alternative model.

• Making residual diagnostic plots is an informal—but useful—way to assess violation of model assumptions. You can plot residuals to check for normality, residual autocorrelation, residual heteroscedasticity, and missing predictors. Formal tests for autocorrelation and heteroscedasticity can also help quantify possible model violations.

• Predictive performance checks. Divide your data into two parts: a training set and a validation set. Fit your model using only the training data, and then forecast the fitted model over the validation period. By comparing model forecasts against the true, holdout observations, you can assess the predictive performance of your model. Prediction mean square error (PMSE) can be calculated as a numerical summary of the predictive performance. When choosing among competing models, you can look at their respective PMSE values to compare predictive performance.

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