Creating Univariate Conditional Mean Models

This topic shows how to represent various autoregressive integrated moving average (ARIMA) models, which are univariate conditional mean models, as an arima model object, and it shows how to interpret the property values of a specified object.

Default ARIMA Model

The default ARIMA(p,D,q) model in Econometrics Toolbox™ is the nonseasonal model of the form

$${\Delta }^D{y}_t=c+{\varphi }_1{\Delta }^D{y}_{t-1}+\dots +{\varphi }_p{\Delta }^D{y}_{t-p}+{\theta }_1{\epsilon }_{t-1}+\dots +{\theta }_q{\epsilon }_{t-q}+{\epsilon }_t.$$

You can write this equation in condensed form using lag operator notation:

$$\varphi (L){(1-L)}^D{y}_t=c+\theta (L){\epsilon }_t$$

In either equation, the default innovation distribution is Gaussian with mean zero and constant variance.

At the command line, you can specify a model of this form using the shorthand syntax arima(p,D,q). For the input arguments p, D, and q, enter the number of nonseasonal AR terms (p), the order of nonseasonal integration (D), and the number of nonseasonal MA terms (q), respectively.

When you use this shorthand syntax, arima creates an arima model with these default property values.

Property Name	Property Data Type
AR	Cell vector of NaNs
Beta	Empty vector [] of regression coefficients corresponding to exogenous covariates
Constant	NaN
D	Degree of nonseasonal integration, D
Distribution	"Gaussian"
MA	Cell vector of NaNs
P	Number of AR terms plus degree of integration, p + D
Q	Number of MA terms, q
SAR	Cell vector of NaNs
SMA	Cell vector of NaNs
Variance	NaN

To assign nondefault values to any properties, you can modify the created model object using dot notation.

Notice that the inputs D and q are the values arima assigns to properties D and Q. However, the input argument p is not necessarily the value arima assigns to the model property P. P stores the number of presample observations needed to initialize the AR component of the model. For nonseasonal models, the required number of presample observations is p + D.

To illustrate, consider specifying the ARIMA(2,1,1) model

$$(1-{\varphi }_{1}L-{\varphi }_2{L}^2){(1-L)}^1{y}_t=c+(1+{\theta }_{1}L){\epsilon }_t,$$

where the innovation process is Gaussian with (unknown) constant variance.

Mdl = arima(2,1,1)

Mdl = 

  arima with properties:

     Description: "ARIMA(2,1,1) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 3
               D: 1
               Q: 1
        Constant: NaN
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {NaN} at lag [1]
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN

Notice that the model property P does not have value 2 (the AR degree). With the integration, a total of p + D (here, 2 + 1 = 3) presample observations are needed to initialize the AR component of the model.

The created model, Mdl, has NaNs for all parameters. A NaN value signals that a parameter needs to be estimated or otherwise specified by the user. All parameters must be specified to forecast or simulate the model.

To estimate parameters, input the model object (along with data) to estimate. This returns a new fitted arima model object. The fitted model object has parameter estimates for each input NaN value.

Calling arima without any input arguments returns an ARIMA(0,0,0) model specification with default property values:

DefaultMdl = arima

DefaultMdl = 

  arima with properties:

     Description: "ARIMA(0,0,0) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 0
               D: 0
               Q: 0
        Constant: NaN
              AR: {}
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN

Specify Nonseasonal Models Using Name-Value Arguments

The best way to specify models to arima is using name-value arguments. You do not need, nor are you able, to specify a value for every model object property. arima assigns default values to any properties you do not (or cannot) specify.

In condensed, lag operator notation, nonseasonal ARIMA(p,D,q) models are of the form

$$\varphi (L){(1-L)}^D{y}_t=c+\theta (L){\epsilon }_t.$$

(1)

You can extend this model to an ARIMAX(p,D,q) model with the linear inclusion of exogenous variables. This model has the form

$$\varphi (L)y_t=c^{\ast }+x_t^{\prime }\beta +{\theta }^{\ast }(L){\epsilon }_t,$$

(2)

where c^* = c/(1–L)^D and θ^*(L) = θ(L)/(1–L)^D.

Tip

If you specify a nonzero D, then Econometrics Toolbox differences the response series y_t before the predictors enter the model. You should preprocess the exogenous covariates x_t by testing for stationarity and differencing if any are unit root nonstationary. If any nonstationary exogenous covariate enters the model, then the false negative rate for significance tests of β can increase.

For the distribution of the innovations, ε_t, there are two choices:

• Independent and identically distributed (iid) Gaussian or Student’s t with a constant variance, $${\sigma }_{\epsilon }^2$$.

• Dependent Gaussian or Student’s t with a conditional variance process, $${\sigma }_t^2$$. Specify the conditional variance model using a garch, egarch, or gjr model.

The arima default for the innovations is an iid Gaussian process with constant (scalar) variance.

In order to estimate, forecast, or simulate a model, you must specify the parametric form of the model (e.g., which lags correspond to nonzero coefficients, the innovation distribution) and any known parameter values. You can set any unknown parameters equal to NaN, and then input the model to estimate (along with data) to get estimated parameter values.

arima (and estimate) returns a model corresponding to the model specification. You can modify models to change or update the specification. Input models (with no NaN values) to forecast or simulate for forecasting and simulation, respectively. Here are some example specifications using name-value arguments.

Model	Specification
$$y_t=c+{\varphi }_1{y}_{t-1}+{\epsilon }_t$$ $${\epsilon }_t={\sigma }_{\epsilon }{z}_t$$ zt Gaussian	arima('AR',NaN) or arima(1,0,0)
$$y_t={\epsilon }_t+{\theta }_1{\epsilon }_{t-1}+{\theta }_2{\epsilon }_{t-2}$$ $${\epsilon }_t={\sigma }_{\epsilon }{z}_t$$ zt Student’s t with unknown degrees of freedom	arima('Constant',0,'MA',{NaN,NaN},... 'Distribution','t')
$$(1-0.8L)(1-L)y_t=0.2+(1+0.6L){\epsilon }_t$$ $${\epsilon }_t=0.1z_t$$ zt Student’s t with eight degrees of freedom	arima('Constant',0.2,'AR',0.8,'MA',0.6,'D',1,... 'Variance',0.1^2,'Distribution',struct('Name','t','DoF',8))
$$(1+0.5L){(1-L)}^1\Delta y_t=x_t^{\prime }\left[\begin{array}{c}-5\\ 2\end{array}\right]+{\epsilon }_t$$ $${\epsilon }_t~N(0,1)$$	arima('Constant',0,'AR',-0.5,'D',1,'Beta',[-5 2])

You can specify the following name-value arguments to create nonseasonal arima models.

Name-Value Arguments for Nonseasonal ARIMA Models

Name	Corresponding Model Term(s) in Equation 1	When to Specify
AR	Nonseasonal AR coefficients, $${\varphi }_1,\dots ,{\varphi }_p$$	To set equality constraints for the AR coefficients. For example, to specify the AR coefficients in the model $$y_t=0.8y_{t-1}-0.2y_{t-2}+{\epsilon }_t,$$ specify 'AR',{0.8,-0.2}. You only need to specify the nonzero elements of AR. If the nonzero coefficients are at nonconsecutive lags, specify the corresponding lags using ARLags. Any coefficients you specify must correspond to a stable AR operator polynomial.
ARLags	Lags corresponding to nonzero, nonseasonal AR coefficients	ARLags is not a model property. Use this argument as a shortcut for specifying AR when the nonzero AR coefficients correspond to nonconsecutive lags. For example, to specify nonzero AR coefficients at lags 1 and 12, e.g., $$y_t={\varphi }_1{y}_{t-1}+{\varphi }_{12}{y}_{t-12}+{\epsilon }_t,$$specify 'ARLags',[1,12]. Use AR and ARLags together to specify known nonzero AR coefficients at nonconsecutive lags. For example, if in the given AR(12) model $${\varphi }_1=0.6$$ and $${\varphi }_{12}=-0.3,$$ specify 'AR',{0.6,-0.3},'ARLags',[1,12].
Beta	Values of the coefficients of the exogenous covariates	Use this argument to specify the values of the coefficients of the exogenous variables. For example, use 'Beta',[0.5 7 -2] to specify $$\beta ={\left[\begin{array}{ccc}0.5& 7& -2\end{array}\right]}^{\prime }.$$ By default, Beta is an empty vector.
Constant	Constant term, c	To set equality constraints for c. For example, for a model with no constant term, specify 'Constant',0. By default, Constant has value NaN.
D	Degree of nonseasonal differencing, D	To specify a degree of nonseasonal differencing greater than zero. For example, to specify one degree of differencing, specify 'D',1. By default, D has value 0 (meaning no nonseasonal integration).
Distribution	Distribution of the innovation process	Use this argument to specify a Student’s t innovation distribution. By default, the innovation distribution is Gaussian. For example, to specify a t distribution with unknown degrees of freedom, specify 'Distribution','t'. To specify a t innovation distribution with known degrees of freedom, assign Distribution a data structure with fields Name and DoF. For example, for a t distribution with nine degrees of freedom, specify 'Distribution',struct('Name','t','DoF',9).
MA	Nonseasonal MA coefficients, $${\theta }_1,\dots ,{\theta }_q$$	To set equality constraints for the MA coefficients. For example, to specify the MA coefficients in the model $$y_t={\epsilon }_t+0.5{\epsilon }_{t-1}+0.2{\epsilon }_{t-2},$$ specify 'MA',{0.5,0.2}. You only need to specify the nonzero elements of MA. If the nonzero coefficients are at nonconsecutive lags, specify the corresponding lags using MALags. Any coefficients you specify must correspond to an invertible MA polynomial.
MALags	Lags corresponding to nonzero, nonseasonal MA coefficients	MALags is not a model property. Use this argument as a shortcut for specifying MA when the nonzero MA coefficients correspond to nonconsecutive lags. For example, to specify nonzero MA coefficients at lags 1 and 4, e.g., $$y_t={\epsilon }_t+{\theta }_1{\epsilon }_{t-1}+{\theta }_4{\epsilon }_{t-4},$$ specify 'MALags',[1,4]. Use MA and MALags together to specify known nonzero MA coefficients at nonconsecutive lags. For example, if in the given MA(4) model $${\theta }_1=0.5$$ and $${\theta }_4=0.2,$$ specify 'MA',{0.4,0.2},'MALags',[1,4].
Variance	Scalar variance of the innovation process, $${\sigma }_{\epsilon }^2$$ Conditional variance process, $${\sigma }_t^2$$	To set equality constraints for $${\sigma }_{\epsilon }^2$$. For example, for a model with known variance 0.1, specify 'Variance',0.1. By default, Variance has value NaN. To specify a conditional variance model, $${\sigma }_t^2$$. Set 'Variance' equal to a conditional variance model object, e.g., a garch model object.

Note

You cannot assign values to the properties P and Q. For nonseasonal models,

• arima sets P equal to p + D

• arima sets Q equal to q

Specify Multiplicative Models Using Name-Value Arguments

For a time series with periodicity s, define the degree p_s seasonal AR operator polynomial, $$\Phi (L)=(1-{\Phi }_1{L}^{p_1}-\dots -{\Phi }_{p_s}{L}^{p_s})$$, and the degree q_s seasonal MA operator polynomial, $$\Theta (L)=(1+{\Theta }_1{L}^{q_1}+\dots +{\Theta }_{q_s}{L}^{q_s})$$. Similarly, define the degree p nonseasonal AR operator polynomial, $$\varphi (L)=(1-{\varphi }_{1}L-\dots -{\varphi }_p{L}^p)$$, and the degree q nonseasonal MA operator polynomial,

$$\theta (L)=(1+{\theta }_{1}L+\dots +{\theta }_q{L}^q).$$

(3)

A multiplicative ARIMA model with degree D nonseasonal integration and degree s seasonality is given by

$$\varphi (L)\Phi (L){(1-L)}^D(1-L^s)y_t=c+\theta (L)\Theta (L){\epsilon }_t.$$

(4)

The innovation series can be an independent or dependent Gaussian or Student’s t process. The arima default for the innovation distribution is an iid Gaussian process with constant (scalar) variance.

In addition to the arguments for specifying nonseasonal models (described in Name-Value Arguments for Nonseasonal ARIMA Models), you can specify these name-value arguments to create a multiplicative arima model. You can extend an ARIMAX model similarly to include seasonal effects.

Name-Value Arguments for Seasonal ARIMA Models

Argument	Corresponding Model Term(s) in Equation 4	When to Specify
SAR	Seasonal AR coefficients, $${\Phi }_1,\dots ,{\Phi }_{p_s}$$	To set equality constraints for the seasonal AR coefficients. When specifying AR coefficients, use the sign opposite to what appears in Equation 4 (that is, use the sign of the coefficient as it would appear on the right side of the equation). Use SARLags to specify the lags of the nonzero seasonal AR coefficients. Specify the lags associated with the seasonal polynomials in the periodicity of the observed data (e.g., 4, 8,... for quarterly data, or 12, 24,... for monthly data), and not as multiples of the seasonality (e.g., 1, 2,...). For example, to specify the model $$(1-0.8L)(1-0.2L^{12})y_t={\epsilon }_t,$$ specify 'AR',0.8,'SAR',0.2,'SARLags',12. Any coefficient values you enter must correspond to a stable seasonal AR polynomial.
SARLags	Lags corresponding to nonzero seasonal AR coefficients, in the periodicity of the observed series	SARLags is not a model property. Use this argument when specifying SAR to indicate the lags of the nonzero seasonal AR coefficients. For example, to specify the model $$(1-\varphi L)(1-{\Phi }_{12}{L}^{12})y_t={\epsilon }_t,$$ specify 'ARLags',1,'SARLags',12.
SMA	Seasonal MA coefficients, $${\Theta }_1,\dots ,{\Theta }_{q_s}$$	To set equality constraints for the seasonal MA coefficients. Use SMALags to specify the lags of the nonzero seasonal MA coefficients. Specify the lags associated with the seasonal polynomials in the periodicity of the observed data (e.g., 4, 8,... for quarterly data, or 12, 24,... for monthly data), and not as multiples of the seasonality (e.g., 1, 2,...). For example, to specify the model $$y_t=(1+0.6L)(1+0.2L^{12}){\epsilon }_t,$$ specify 'MA',0.6,'SMA',0.2,'SMALags',12. Any coefficient values you enter must correspond to an invertible seasonal MA polynomial.
SMALags	Lags corresponding to the nonzero seasonal MA coefficients, in the periodicity of the observed series	SMALags is not a model property. Use this argument when specifying SMA to indicate the lags of the nonzero seasonal MA coefficients. For example, to specify the model $$y_t=(1+{\theta }_{1}L)(1+{\Theta }_4{L}^4){\epsilon }_t,$$ specify 'MALags',1,'SMALags',4.
Seasonality	Seasonal periodicity, s	To specify the degree of seasonal integration s in the seasonal differencing polynomial Δs = 1 – Ls. For example, to specify the periodicity for seasonal integration of monthly data, specify 'Seasonality',12. If you specify nonzero Seasonality, then the degree of the whole seasonal differencing polynomial is one. By default, Seasonality has value 0 (meaning periodicity and no seasonal integration).

Note

You cannot assign values to the properties P and Q. For multiplicative ARIMA models,

• arima sets P equal to p + D + p_s + s

• arima sets Q equal to q + q_s

Specify Conditional Mean Model Using Econometric Modeler App

You can specify the lag structure and innovation distribution of seasonal and nonseasonal conditional mean models using the Econometric Modeler app. The app treats all coefficients as unknown and estimable, including the degrees of freedom parameter for a t innovation distribution.

At the command line, open the Econometric Modeler app.

econometricModeler

Alternatively, open the app from the apps gallery (see Econometric Modeler).

In the app, you can see all supported models by selecting a time series variable for the response in the Time Series pane. Then, on the Econometric Modeler tab, in the Models section, click the arrow to display the models gallery.

The ARMA/ARIMA Models section contains supported conditional mean models.

For conditional mean model estimation, SARIMA and SARIMAX are the most flexible models. You can create any conditional mean model that excludes exogenous predictors by clicking SARIMA, or you can create any conditional mean model that includes at least one exogenous predictor by clicking SARIMAX.

After you select a model, the app displays the Type Model Parameters dialog box, where Type is the model type. This figure shows the SARIMAX Model Parameters dialog box.

Adjustable parameters in the dialog box depend on Type. In general, adjustable parameters include:

• A model constant and linear regression coefficients corresponding to predictor variables

• Time series component parameters, which include seasonal and nonseasonal lags and degrees of integration

• The innovation distribution

As you adjust parameter values, the equation in the Model Equation section changes to match your specifications. Adjustable parameters correspond to input and name-value arguments described in the previous sections and in the arima reference page.

For more details on specifying models using the app, see Fitting Models to Data and Specifying Univariate Lag Operator Polynomials Interactively.

What Are Conditional Mean Models?

Unconditional vs. Conditional Mean

For a univariate random variable y_t, the unconditional mean is simply the expected value, $$E\left(y_t\right).$$ In contrast, the conditional mean of y_t is the expected value of y_t given a conditioning set of variables, Ω_t. A conditional mean model specifies a functional form for $$E\left(y_t|{\Omega }_t\right).$$.

Static vs. Dynamic Conditional Mean Models

For a static conditional mean model, the conditioning set of variables is measured contemporaneously with the dependent variable y_t. An example of a static conditional mean model is the ordinary linear regression model. Given $$x_t,$$ a row vector of exogenous covariates measured at time t, and β, a column vector of coefficients, the conditional mean of y_t is expressed as the linear combination

$$E(y_t|x_t)=x_t\beta$$

(that is, the conditioning set is $${\Omega }_t=x_t$$).

In time series econometrics, there is often interest in the dynamic behavior of a variable over time. A dynamic conditional mean model specifies the expected value of y_t as a function of historical information. Let H_t–1 denote the history of the process available at time t. A dynamic conditional mean model specifies the evolution of the conditional mean, $$E\left(y_t|H_{t-1}\right).$$ Examples of historical information are:

• Past observations, y₁, y₂,...,y_t–1

• Vectors of past exogenous variables, $$x_1,x_2,\dots ,x_{t-1}$$

• Past innovations, $${\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_{t-1}$$

Conditional Mean Models for Stationary Processes

By definition, a covariance stationary stochastic process has an unconditional mean that is constant with respect to time. That is, if y_t is a stationary stochastic process, then $$E(y_t)=\mu$$ for all times t.

The constant mean assumption of stationarity does not preclude the possibility of a dynamic conditional expectation process. The serial autocorrelation between lagged observations exhibited by many time series suggests the expected value of y_t depends on historical information. By Wold’s decomposition [2], you can write the conditional mean of any stationary process y_t as

$$E(y_t|H_{t-1})=\mu +{\displaystyle \sum _{i=1}^{\infty }{\psi }_i{\epsilon }_{t-i},}$$

(5)

where $$\left\{{\epsilon }_{t-i}\right\}$$ are past observations of an uncorrelated innovation process with mean zero, and the coefficients $${\psi }_i$$ are absolutely summable. $$E(y_t)=\mu$$ is the constant unconditional mean of the stationary process.

Any model of the general linear form given by Equation 5 is a valid specification for the dynamic behavior of a stationary stochastic process. Special cases of stationary stochastic processes are the autoregressive (AR) model, moving average (MA) model, and the autoregressive moving average (ARMA) model.

References

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

[2] Wold, Herman. "A Study in the Analysis of Stationary Time Series." Journal of the Institute of Actuaries 70 (March 1939): 113–115. https://doi.org/10.1017/S0020268100011574.