#StackBounty: #python #arima #statsmodels What is the exact defenition of ARMA model in statsmodels and how to predict the next value u…

Bounty: 50

I’ve fitted a time series (Y) on the ARMA(2,1) model using statsmodels in python. let’s leave alone that the selected order is not the best for my time series (Y).

The summary of the fitted model can be seen below. The AR and MA coefficients are shown as ar.L1.Y, ar.L2.Y, and ma.L1.Y. The constant value is indicated as const. The standard deviation of the white noise is also given by S.D. of innovations.

Therefore, we may say that our ARMA model is defined as follows:

$Y_{t}= C+ A_{1}Y_{t-1}+A_{2}Y_{t-2}+B_{1}e_{t-1}+e_{t}$

However, I am not certain about it, since I could not find the ARMA model definition in statsmodels package. I know there is no unique definition for the ARMA model through different software packages. For instance, the MA coefficients are defined by a negative sign in some packages and textbooks.

Questions:

1- Is my interpretation of ARMA coefficients and parameters with respect to the summary correct? (Paragraph 2)

2- What is statsmodels definition for ARMA’s equation?

3- Lets say we have fitted the model using $(Y_{0},…, Y_{10})$. How can we predict $Y_{11}$?

Well, my guess is to do as follows. But not sure, especially about the value of $e_{10}$.

$Y_{11}= C+ A_{1}Y_{10}+A_{2}Y_{9}+B_{1}e_{10}+e_{11}$, in which $e_{11}$ is a generated random number based on $N(0,sigma)$. However, what is $e_{10}$ and how can I calculate it?

                                   ARMA Model Results                                  
=======================================================================================
Dep. Variable:                           Y       No. Observations:                  359
Model:                              ARMA(2, 1)   Log Likelihood                -127.666
Method:                                css-mle   S.D. of innovations              0.344
Date:                         Thu, 12 Mar 2020   AIC                            265.331
Time:                                 09:35:15   BIC                            284.748
Sample:                             01-01-2017   HQIC                           273.052
                                  - 01-04-2017                                         
================================================================================================
                              coef       std err      z        P>|z|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        4.4386      0.911      4.873      0.000       2.653       6.224
ar.L1.Y                      1.0940      1.398      0.783      0.434      -1.645       3.833
ar.L2.Y                     -0.1096      1.373     -0.080      0.936      -2.801       2.582
ma.L1.Y                     -0.1028      1.398     -0.074      0.941      -2.843       2.637
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1            1.0179           +0.0000j            1.0179            0.0000
AR.2            8.9667           +0.0000j            8.9667            0.0000
MA.1            9.7242           +0.0000j            9.7242            0.0000
-----------------------------------------------------------------------------


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