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## Econometric Modeling

### Model Selection

A probabilistic time series model is necessary for a wide variety of analysis goals, including regression inference, forecasting, and Monte Carlo simulation. When selecting a model, aim to find the most parsimonious model that adequately describes your data. A simple model is easier to estimate, forecast, and interpret.

• Specification tests help you identify one or more model families that could plausibly describe the data generating process.

• Model comparisons help you compare the fit of competing models, with penalties for complexity.

• Goodness-of-fit checks help you assess the in-sample adequacy of your model, verify that all model assumptions hold, and evaluate out-of-sample forecast performance.

Model selection is an iterative process. When goodness-of-fit checks suggest model assumptions are not satisfied—or the predictive performance of the model is not satisfactory—consider making model adjustments. Additional specification tests, model comparisons, and goodness-of-fit checks help guide this process.

### Econometrics Toolbox Features

Modeling QuestionsFeaturesRelated Functions
What is the dimension of my response variable?
• The conditional mean and variance models, regression models with ARIMA errors, and Bayesian linear regression models in this toolbox are for modeling univariate, discrete-time data.

• Separate models are available for multivariate, discrete-time data, such as VAR and VEC models.

• State-space models support univariate or multivariate response variables.

Is my time series stationary?
• Stationarity tests are available. If your data is not stationary, consider transforming your data. Stationarity is the foundation of many time series models.

• Or, consider using a nonstationary ARIMA model if there is evidence of a unit root in your data.

Does my time series have a unit root?
• Unit root tests are available. Evidence in favor of a unit root suggests your data is difference stationary.

• You can difference a series with a unit root until it is stationary, or model it using a nonstationary ARIMA model.

How can I handle seasonal effects?
• You can deseasonalize (seasonally adjust) your data. Use seasonal filters or regression models to estimate the seasonal component.

• Seasonal ARIMA models use seasonal differencing to remove seasonal effects. You can also include seasonal lags to model seasonal autocorrelation (both additively and multiplicatively).

Is my data autocorrelated?
• Sample autocorrelation and partial autocorrelation functions help identify autocorrelation.

• Conduct a Ljung-Box Q-test to test autocorrelations at several lags jointly.

• If autocorrelation is present, consider using a conditional mean model.

• For regression models with autocorrelated errors, consider using FGLS or HAC estimators. If the error model structure is an ARIMA model, consider using a regression model with ARIMA errors.

What if my data is heteroscedastic (exhibits volatility clustering)?
• Looking for autocorrelation in the squared residual series is one way to detect conditional heteroscedasticity.

• Engle’s ARCH test evaluates evidence against the null of independent innovations in favor of an ARCH model alternative.

• To model conditional heteroscedasticity, consider using a conditional variance model.

• For regression models that exhibit heteroscedastic errors, consider using FGLS or HAC estimators.

Is there an alternative to a Gaussian innovation distribution for leptokurtic data?
• You can use a Student’s t distribution to model fatter tails than a Gaussian distribution (excess kurtosis).

• You can specify a t innovation distribution for all conditional mean and variance models, and ARIMA error models in Econometrics Toolbox™.

• You can estimate the degrees of freedom of the t distribution along with other model parameters.

How do I decide between several model fits?
• You can compare nested models using misspecification tests, such as the likelihood ratio test, Wald’s test, or Lagrange multiplier test.

• Information criteria, such as AIC or BIC, compare model fit with a penalty for complexity.

Do I have two or more time series that are cointegrated?
• The Johansen and Engle-Granger cointegration tests assess evidence of cointegration.

• Consider using the VEC model for modeling multivariate, cointegrated series.

• Also consider cointegration when regressing time series. If present, it can introduce spurious regression effects.

What if I want to include predictor variables?
• ARIMAX, VARX, regression models with ARIMA errors, and Bayesian linear regression models are available in this toolbox.

• State-space models support predictor data.

What if I want to implement regression, but the classical linear model assumptions might not apply?
• Regression models with ARIMA errors are available in this toolbox.

• Regress robustly using FGLS or HAC estimators.

• Use Bayesian linear regression.

• For a series of examples on time series regression techniques that illustrate common principles and tasks in time series regression modeling, see Econometrics Toolbox Examples.

• For more regression options, see Statistics and Machine Learning Toolbox™ documentation.

What if observations of a dynamic process include measurement error?

Standard, linear state-space modeling is available in this toolbox.