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This example shows how to choose lags for an ARIMA model by comparing the AIC values of estimated models using the **Econometric Modeler** app. The example also shows how to compare the predictive performance of several models that have the best in-sample fits at the command line. The data set `Data_Airline.mat`

contains monthly counts of airline passengers.

At the command line, load the `Data_Airline.mat`

data set.

`load Data_Airline`

To compare predictive performance later, reserve the last two years of data as a holdout sample.

fHorizon = 24; HoldoutTable = DataTable((end - fHorizon + 1):end,:); DataTable((end - fHorizon + 1):end,:) = [];

At the command line, open the **Econometric Modeler** app.

econometricModeler

Alternatively, open the app from the apps gallery (see **Econometric
Modeler**).

Import `DataTable`

into the app:

On the

**Econometric Modeler**tab, in the**Import**section, click .In the

**Import Data**dialog box, in the**Import?**column, select the check box for the`DataTable`

variable.Click

**Import**.

The variable `PSSG`

appears in the **Time Series** pane, its value appears in the **Preview** pane, and its time series plot appears in the **Time Series Plot(PSSG)** figure window.

The series exhibits a seasonal trend, serial correlation, and possible exponential growth. For an interactive analysis of serial correlation, see Detect Serial Correlation Using Econometric Modeler App.

Address the exponential trend by applying the log transform to `PSSG`

.

In the

**Time Series**pane, select`PSSG`

.On the

**Econometric Modeler**tab, in the**Transforms**section, click**Log**.

The transformed variable `PSSGLog`

appears in the **Time Series** pane, its value appears in the **Preview** pane, and its time series plot appears in the **Time Series Plot(PSSGLog)** figure window.

The exponential growth appears to be removed from the series.

Box, Jenkins, and Reinsel suggest a SARIMA(0,1,1)×(0,1,1)_{12} model without a constant for `PSSGLog`

[1] (for more details, see Estimate Multiplicative ARIMA Model Using Econometric Modeler App). However, consider all combinations of monthly SARIMA models that include up to two seasonal and nonseasonal MA lags. Specifically, iterate the following steps for each of the nine models of the form SARIMA(0,1,*q*)×(0,1,*q*_{12})_{12}, where *q* ∈ {`0`

,`1`

,`2`

} and *q*_{12} ∈ {`0`

,`1`

,`2`

}.

For the first iteration:

Let

*q*=*q*_{12}= 0.With

`PSSGLog`

selected in the**Time Series**pane, click the**Econometric Modeler**tab. In the**Models**section, click the arrow to display the models gallery.In the models gallery, in the

**ARMA/ARIMA Models**section, click**SARIMA**.In the

**SARIMA Model Parameters**dialog box, on the**Lag Order**tab:**Nonseasonal**sectionSet

**Degrees of Integration**to`1`

.Set

**Moving Average Order**to`0`

.Clear the

**Include Constant Term**check box.

**Seasonal**sectionSet

**Period**to`12`

to indicate monthly data.Set

**Moving Average Order**to`0`

.Select the

**Include Seasonal Difference**check box.

Click

**Estimate**.

Rename the new model variable.

In the

**Models**pane, click the new model variable twice to select its name.Enter

`SARIMA01`

. For example, whenx01`q`

`q`

_{12}=`q`

= 0, rename the variable to`q`

_{12}`SARIMA010x010`

.

In the

**Model Summary(SARIMA01**document, in thex01`q`

)`q`

_{12}**Goodness of Fit**table, note the AIC value. For example, for the model variable`SARIMA010x010`

, the AIC is in this figure.For the next iteration, choose values of

*q*and*q*_{12}. For example,*q*= 0 and*q*_{12}= 1 for the second iteration.In the

**Models**pane, right-click`SARIMA01`

. In the context menu, selectx01`q`

`q`

_{12}**Modify**to open the**SARIMA Model Parameters**dialog box with the current settings for the selected model.In the

**SARIMA Model Parameters**dialog box:In the

**Nonseasonal**section, set**Moving Average Order**to.`q`

In the

**Seasonal**section, set**Moving Average Order**to.`q`

_{12}Click

**Estimate**.

After you complete the steps, the **Models** pane contains nine estimated models named `SARIMA010x010`

through `SARIMA012x012`

.

The resulting AIC values are in this table.

Model | Variable Name | AIC |
---|---|---|

SARIMA(0,1,0)×(0,1,0)_{12} | `SARIMA010x010` | -410.3520 |

SARIMA(0,1,0)×(0,1,1)_{12} | `SARIMA010x011` | -443.0009 |

SARIMA(0,1,0)×(0,1,2)_{12} | `SARIMA010x012` | -441.0010 |

SARIMA(0,1,1)×(0,1,0)_{12} | `SARIMA011x010` | -422.8680 |

SARIMA(0,1,1)×(0,1,1)_{12} | `SARIMA011x011` | -452.0039 |

SARIMA(0,1,1)×(0,1,2)_{12} | `SARIMA011x012` | -450.0605 |

SARIMA(0,1,2)×(0,1,0)_{12} | `SARIMA012x010` | -420.9760 |

SARIMA(0,1,2)×(0,1,1)_{12} | `SARIMA012x011` | -450.0087 |

SARIMA(0,1,2)×(0,1,2)_{12} | `SARIMA012x012` | -448.0650 |

The three models yielding the lowest three AIC values are SARIMA(0,1,1)×(0,1,1)_{12}, SARIMA(0,1,1)×(0,1,2)_{12}, and SARIMA(0,1,2)×(0,1,1)_{12}. These models have the best parsimonious in-sample fit.

Export the models with the best in-sample fits.

On the

**Econometric Modeler**tab, in the**Export**section, click .In the

**Export Variables**dialog box, in the**Models**column, click the**Select**check box for the`SARIMA011x011`

,`SARIMA011x012`

, and`SARIMA012x011`

. Clear the check box for any other selected models.Click

**Export**.

The `arima`

model objects `SARIMA011x011`

, `SARIMA011x012`

, and `SARIMA012x011`

appear in the MATLAB^{®} Workspace.

At the command line, estimate two-year-ahead forecasts for each model.

f5 = forecast(SARIMA_PSSGLog5,fHorizon); f6 = forecast(SARIMA_PSSGLog6,fHorizon); f8 = forecast(SARIMA_PSSGLog8,fHorizon);

`f5`

, `f6`

, and `f8`

are 24-by-1 vectors containing the forecasts.

Estimate the prediction mean square error (PMSE) for each of the forecast vectors.

logPSSGHO = log(HoldoutTable.Variables); pmse5 = mean((logPSSGHO - f5).^2); pmse6 = mean((logPSSGHO - f6).^2); pmse8 = mean((logPSSGHO - f8).^2);

Identify the model yielding the lowest PMSE.

[~,bestIdx] = min([pmse5 pmse6 pmse8],[],2)

The SARIMA(0,1,1)×(0,1,1)_{12} model performs the best in-sample and out-of-sample.

[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. *Time Series Analysis: Forecasting and Control*. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.