 # Estimating ARIMA Models

### Introduction

The arimaFit function is a convenient tool for estimating the parameters of any ARIMA model, including:

• ARMA models.
• Purely AR models.
• Purely MA models.

It will compute parameter estimates and standard errors for a time series model with ARMA errors using exact maximum likelihood.

## arimaFit for Matrix Inputs

Below are the inputs to arimaFit when passing the time series as a GAUSS matrix:

y
N x 1 vector, containing the time series.
p
Scalar, the autoregressive order.
d
Optional scalar, the order of differencing. Default = 0.
q
Optional scalar, the moving average order. Default = 0.
ctl
Optional control structure. Contains additional settings for the estimation, such as optimization tolerances, and starting values as well as control over printed output and more.

### Estimation Requirements

• The specified model must include sequential, ordered lags. For example, it will estimate the parameters for the first through fourth lag but will not estimate a model that includes ONLY the first and fourth lag.
• The time series data must be stationary and invertible.

## Estimating ARIMA Models with Matrix Input

### Example AR(2) Model

Let's consider estimating an AR(2) model using the same data from our previous tutorials. Assuming the data series, y_sim, is still in memory we can estimate the AR(2) models with a single line of code:

call arimaFit(y_sim, 2);

The output printed to screen reads

Model:  ARIMA(2,0,0)

Final Results:

Log Likelihood:    702.327968         Number of Residuals: 200
AIC           :  -1400.655937         Error Variance     : 0.897749310
SBC           :  -1394.059302         Standard Error     : 0.947496338

DF: 198      SSE: 177.754363357

Coefficients     Std. Err.   T-Ratio    Approx. Prob.
AR[1,1] 0.54960    0.02204    24.93987   0.00000
AR[2,1]-0.76906    0.03084   -24.93999   0.00000

Constant:   1.77120346
Total Computation Time: 0.83 (seconds)

AR Roots and Moduli:

Real :    0.35731   0.35731
Imag.:    1.08287  -1.08287
Mod. :    1.14030   1.14030

The estimated coefficients, 0.54960 and -0.76906, are reasonable estimates given the coefficients used to simulate the series y_sim (0.5 and -0.8).

### Example ARIMA(2,1,0) Model

Now, for the sake of demonstration, consider estimating an ARIMA(2,1,0) model. Because the order of differencing is now different from zero, both the AR order and the differencing must be specified as inputs to arimaFit. However, because the MA order is still zero, it is not required:

call arimaFit(y_sim, 2, 1);

The output printed to screen reads

Model:  ARIMA(2,1,0)

Final Results:

Log Likelihood:    746.618396         Number of Residuals: 199
AIC           :  -1489.236793         Error Variance     : 1.458774290
SBC           :  -1482.650183         Standard Error     : 1.207797289

DF: 197      SSE: 287.378535224

Coefficients     Std. Err.   T-Ratio    Approx. Prob.
AR[1,1] 0.27909    0.04745    5.88164    0.00000
AR[2,1]-0.74585    0.04729   -15.77057   0.00000

Constant:   0.01674604
Total Computation Time: 0.01 (seconds)

AR Roots and Moduli:

Real :    0.18710   0.18710
Imag.:    1.14270  -1.14270
Mod. :    1.15791   1.15791 

Now, because of the improperly specified model, the coefficient estimates at 0.27909 and -0.74585 are no longer as accurate.

### Example ARIMA(2,0,1) Model

Suppose that we now wish to estimate an ARIMA(2,0,1). In this model, the MA order differs from the default value of zero, but the differencing order is equal to the default value. However, because the optional arguments must be specified in the order p, d, q both the differencing order and MA order must be included as inputs:

call arimaFit(y_sim, 2, 0, 1);

The ARIMA(2,0,1) estimates printed to screen reads

Model:  ARIMA(2,0,1)

Final Results:

Log Likelihood:    701.956107         Number of Residuals: 200
AIC           :  -1397.912214         Error Variance     : 0.898957317
SBC           :  -1388.017262         Standard Error     : 0.948133597

DF: 197      SSE: 177.094591531

Coefficients     Std. Err.   T-Ratio    Approx. Prob.
AR[1,1] 0.51708    0.05971    8.65960    0.00000
AR[2,1]-0.75937    0.04818   -15.76125   0.00000
MA[1,1]-0.07902    0.09151   -0.86349    0.38892

Constant:   1.80435249
Total Computation Time: 0.36 (seconds)

AR Roots and Moduli:

Real :    0.34047   0.34047
Imag.:    1.09589  -1.09589
Mod. :    1.14755   1.14755

MA Roots and Moduli:

Real :  -12.65461
Imag.:    0.00000
Mod. :   12.65461 

The table now includes three estimates, two AR estimates and one MA estimate.

### Conclusion

You have learned the basics of estimating ARIMA models with the arimaFit function in GAUSS. The next tutorial in this series shows how to create and customize a time series plot.

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