Specify ARIMA Error Model Innovation Distribution

About the Innovation Process

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

$$\begin{array}{c}{y}_t=c+X_t\beta +u_t\\ a\left(L\right)A\left(L\right){\left(1-L\right)}^D\left(1-L^s\right)u_t=b\left(L\right)B\left(L\right){\epsilon }_t,\end{array}$$

(1)

where

• t = 1,...,T.

• y_t is the response series.

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

• c is the regression model intercept.

• β is the regression coefficient.

• u_t is the disturbance series.

• ε_t is the innovations series.

• $$L^j{y}_t=y_{t-j}.$$

• $$a\left(L\right)=\left(1-a_{1}L-\mathrm{...}-a_p{L}^p\right),$$ which is the degree p, nonseasonal autoregressive polynomial.

• $$A\left(L\right)=\left(1-A_{1}L-\mathrm{...}-A_{p_s}{L}^{p_s}\right),$$ which is the degree p_s, seasonal autoregressive polynomial.

• $${\left(1-L\right)}^D,$$ which is the degree D, nonseasonal integration polynomial.

• $$\left(1-L^s\right),$$ which is the degree s, seasonal integration polynomial.

• $$b\left(L\right)=\left(1+b_{1}L+\mathrm{...}+b_q{L}^q\right),$$ which is the degree q, nonseasonal moving average polynomial.

• $$B\left(L\right)=\left(1+B_{1}L+\mathrm{...}+B_{q_s}{L}^{q_s}\right),$$ which is the degree q_s, seasonal moving average polynomial.

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

$$u_t=a^{-1}(L)A^{-1}(L){(1-L)}^{-D}{(1-L^s)}^{-1}b(L)B(L){\epsilon }_t=\Psi (L){\epsilon }_t,$$

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

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

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

• Variance stores σ².

• Distribution stores the parametric form of z_t.

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 z_t (using name-value pair arguments or dot notation):

  • Standard Gaussian

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

    $$z_t=T_{\nu }\sqrt{\frac{\nu -2}{\nu }},$$

    where T_ν 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 T_ν is the same as the kurtosis of the Standardized Student’s t (z_t), 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:

$$\begin{array}{rcl}{y}_t& =& c+X_t\beta +u_t\\ u_t& =& {\alpha }_1{u}_{t-1}+{\alpha }_2{u}_{t-2}+{\epsilon }_t\end{array}$$

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 $$t_{10}$$, 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 $$t_{10}$$ 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.