Main Content

Specify ARIMA Error Model Innovation Distribution

About the Innovation Process

A regression model with ARIMA errors has the following general form:

yt=c+Xtβ+uta(L)A(L)(1L)D(1Ls)ut=b(L)B(L)εt,(1)
where

  • t = 1,...,T.

  • yt is the response series.

  • Xt is row t of X, which is the matrix of concatenated predictor data vectors. That is, Xt is observation t of each predictor series.

  • c is the regression model intercept.

  • β is the regression coefficient.

  • ut is the disturbance series.

  • εt is the innovations series.

  • Ljyt=ytj.

  • a(L)=(1a1L...apLp), which is the degree p, nonseasonal autoregressive polynomial.

  • A(L)=(1A1L...ApsLps), which is the degree ps, seasonal autoregressive polynomial.

  • (1L)D, which is the degree D, nonseasonal integration polynomial.

  • (1Ls), which is the degree s, seasonal integration polynomial.

  • b(L)=(1+b1L+...+bqLq), which is the degree q, nonseasonal moving average polynomial.

  • B(L)=(1+B1L+...+BqsLqs), which is the degree qs, seasonal moving average polynomial.

Suppose that the unconditional disturbance series (ut) is a stationary stochastic processes. Then, you can express the second equation in Equation 1 as

ut=a1(L)A1(L)(1L)D(1Ls)1b(L)B(L)εt=Ψ(L)εt,

where Ψ(L) is an infinite degree lag operator polynomial [2].

The innovation process (εt) is an independent and identically distributed (iid), mean 0 process with a known distribution. Econometrics Toolbox™ generalizes the innovation process to εt = σzt, where zt is a series of iid random variables with mean 0 and variance 1, and σ2 is the constant variance of εt.

regARIMA models contain two properties that describe the distribution of εt:

  • Variance stores σ2.

  • Distribution stores the parametric form of zt.

Innovation Distribution Options

  • The default value of Variance is NaN, meaning that the innovation variance is unknown. You can assign a positive scalar to Variance when you specify the model using the name-value pair argument 'Variance',sigma2 (where sigma2 = σ2), or by modifying an existing model using dot notation. Alternatively, you can estimate Variance using estimate.

  • You can specify the following distributions for zt (using name-value pair arguments or dot notation):

    • Standard Gaussian

    • Standardized Student’s t with degrees of freedom ν > 2. Specifically,

      zt=Tνν2ν,

      where Tν is a Student’s t distribution with degrees of freedom ν > 2.

    The t distribution is useful for modeling innovations that are more extreme than expected under a Gaussian distribution. Such innovation processes have excess kurtosis, a more peaked (or heavier tailed) distribution than a Gaussian. Note that for ν > 4, the kurtosis (fourth central moment) of Tν is the same as the kurtosis of the Standardized Student’s t (zt), i.e., for a t random variable, the kurtosis is scale invariant.

    Tip

    It is good practice to assess the distributional properties of the residuals to determine if a Gaussian innovation distribution (the default distribution) is appropriate for your model.

Specify Innovation Distribution

regARIMA stores the distribution (and degrees of freedom for the t distribution) in the Distribution property. The data type of Distribution is a struct array with potentially two fields: Name and DoF.

  • If the innovations are Gaussian, then the Name field is Gaussian, and there is no DoF field. regARIMA sets Distribution to Gaussian by default.

  • If the innovations are t-distributed, then the Name field is t and the DoF field is NaN by default, or you can specify a scalar that is greater than 2.

To illustrate specifying the distribution, consider this regression model with AR(2) errors:

yt=c+Xtβ+utut=α1ut-1+α2ut-2+εt

Mdl = regARIMA(2,0,0);
Mdl.Distribution
ans = struct with fields:
    Name: "Gaussian"

By default, Distribution property of Mdl is a struct array with the field Name having the value Gaussian.

If you want to specify a t innovation distribution, then you can either specify the model using the name-value pair argument 'Distribution','t', or use dot notation to modify an existing model.

Specify the model using the name-value pair argument.

Mdl = regARIMA('ARLags',1:2,'Distribution','t');
Mdl.Distribution
ans = struct with fields:
    Name: "t"
     DoF: NaN

If you use the name-value pair argument to specify the t innovation distribution, then the default degrees of freedom is NaN.

You can use dot notation to yield the same result.

Mdl = regARIMA(2,0,0);
Mdl.Distribution = 't'
Mdl = 
  regARIMA with properties:

     Description: "ARMA(2,0) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = NaN
       Intercept: NaN
            Beta: [1×0]
               P: 2
               Q: 0
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN

If the innovation distribution is t10, then you can use dot notation to modify the Distribution property of the existing model Mdl. You cannot modify the fields of Distribution using dot notation, e.g., Mdl.Distribution.DoF = 10 is not a value assignment. However, you can display the value of the fields using dot notation.

Mdl.Distribution = struct('Name','t','DoF',10)
Mdl = 
  regARIMA with properties:

     Description: "ARMA(2,0) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = 10
       Intercept: NaN
            Beta: [1×0]
               P: 2
               Q: 0
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN
tDistributionDoF = Mdl.Distribution.DoF
tDistributionDoF = 10

Since the DoF field is not a NaN, it is an equality constraint when you estimate Mdl using estimate.

Alternatively, you can specify the t10 innovation distribution using the name-value pair argument.

Mdl = regARIMA('ARLags',1:2,'Intercept',0,...
    'Distribution',struct('Name','t','DoF',10))
Mdl = 
  regARIMA with properties:

     Description: "ARMA(2,0) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = 10
       Intercept: 0
            Beta: [1×0]
               P: 2
               Q: 0
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN

References

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

[2] Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.

See Also

Apps

Objects

Functions

Related Examples

More About