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.

Modeling Questions | Features | Related Functions |
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What is the dimension of my response variable? | The conditional mean and variance models in this toolbox are for modeling univariate, discrete data. Separate models are available for multivariate, discrete data, such as VAR and VEC models. State-space models support univariate or multivariate response variables.
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Is my data 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.
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Does my 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.
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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).
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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.
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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.
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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.
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How do I decide between these models? | 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.
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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.
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What if I want to include predictor variables? | ARIMAX and VARX models are available in this toolbox. State-space models support predictor data.
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What if I want to implement regression, but the classical linear model assumptions do not apply? | Regression models with ARIMA errors are available in this toolbox. Regress robustly using FGLS or HAC estimators. 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.
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How do use the Kalman filter to analyze several unobservable, linear, stochastic time series and several, observable, linear, stochastic functions of them? | Linear state-space models are available in this toolbox. | `ssm` |

- Box-Jenkins Model Selection
- Detect Autocorrelation
- Detect ARCH Effects
- Unit Root Tests
- Time Series Regression I: Linear Models
- Time Series Regression II: Collinearity and Estimator Variance
- Time Series Regression III: Influential Observations
- Time Series Regression IV: Spurious Regression
- Time Series Regression V: Predictor Selection
- Time Series Regression VI: Residual Diagnostics
- Time Series Regression VII: Forecasting
- Time Series Regression VIII: Lagged Variables and Estimator Bias
- Time Series Regression IX: Lag Order Selection
- Time Series Regression X: Generalized Least Squares and HAC Estimators

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