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) Dependent variable: Y_ar2
Time Span: Unknown Valid cases: 200
SSE: 177.754 Degrees of freedom: 198
Log Likelihood: 702.328 RMSE: 0.943
AIC: 702.328 SEE: 0.947
SBC: -1394.059 Durbin-Watson: 1.909
R-squared: 0.630 Rbar-squared: 0.626
================================================================================
Coefficient Estimate Std. Err. T-Ratio Prob |>| t
================================================================================
AR[1,1] 0.549 0.045 12.101 0.000
AR[2,1] -0.770 0.045 -17.015 0.000
Constant 1.773 0.951 1.864 0.064
================================================================================
Total Computation Time: 0.04 (seconds)
MA Roots and Moduli:
---------------------------------------------
Real : 0.35674 0.35674
Imag. : 1.08266 -1.08266
Mod. : 1.13992 1.13992
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) Dependent variable: Y_ar2
Time Span: Unknown Valid cases: 199
SSE: 287.379 Degrees of freedom: 197
Log Likelihood: 746.618 RMSE: 1.202
AIC: 746.618 SEE: 1.208
SBC: -1482.650 Durbin-Watson: 2.595
R-squared: 0.565 Rbar-squared: 0.561
================================================================================
Coefficient Estimate Std. Err. T-Ratio Prob |>| t
================================================================================
AR[1,1] 0.279 0.047 5.882 0.000
AR[2,1] -0.746 0.047 -15.771 0.000
Constant 0.017 1.214 0.014 0.989
================================================================================
Total Computation Time: 0.01 (seconds)
MA 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) Dependent variable: Y_ar2
Time Span: Unknown Valid cases: 200
SSE: 177.095 Degrees of freedom: 197
Log Likelihood: 701.956 RMSE: 0.941
AIC: 701.956 SEE: 0.948
SBC: -1388.017 Durbin-Watson: 2.002
R-squared: 0.631 Rbar-squared: 0.625
================================================================================
Coefficient Estimate Std. Err. T-Ratio Prob |>| t
================================================================================
AR[1,1] 0.517 0.060 8.665 0.000
AR[2,1] -0.759 0.048 -15.764 0.000
MA[1,1] -0.079 0.091 -0.865 0.388
Constant 1.804 0.952 1.896 0.059
================================================================================
Total Computation Time: 0.08 (seconds)
MA Roots and Moduli:
---------------------------------------------
Real : 0.34047 0.34047
Imag. : 1.09589 -1.09589
Mod. : 1.14755 1.14755
MA Roots and Moduli:
------------------------------
Real : -12.65466
Imag. : 0.00000
Mod. : 12.65466
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.
For your convenience, the entire code is available here.