VectorAutoregression (JMSL Numerical Library)

java.lang.Object
- com.imsl.stat.VectorAutoregression

All Implemented Interfaces:

Serializable, Cloneable
```
public class VectorAutoregression
extends Object
implements Serializable, Cloneable
```
Performs vector autoregression for a multivariate time series.
This class contains methods for modeling multivariate time series of the form
$Y_t=(y_{1t}, y_{2t},ldots,y_{Kt})^top$
where each
$y_{it}, i=1,ldots,K, t =ldots, -2, -1, 0, 1, 2,ldots$
is a real-valued time series indexed by t. Define the K dimensional mean process
$mu_t=E[Y_t]=(E[y_{1t}],ldots,E[y_{Kt}])^top$
and the cross-covariance matrix
$Sigma_{t,h}=text{Cov}(Y_t,Y_{t+h})$
where for each (t,h), $Sigma_{t,h}$ is a K * K real-valued matrix whose (i,j)^t,h element is the covariance between $y_{it},y_{j(t+h)}$ :
$text{Cov}(y_{it},y_{j(t+h)})= E[(y_{it}-mu_{it})(y_{j(t+h)}-mu_{j(t+h)})].$

The vector autoregressive model, or VAR, has a number of equivalent forms. A general form is
where
is a p-th degree matrix polynomial in the lag or backshift operator, L. Note that p is the specified autoregressive lag and u_t is the error process. The error process is assumed to be a K dimensional white noise series, with and nonsingular covariance matrix, , which is constant over time. The coefficient matrices are K * K matrices containing the coefficients of the model. The matrix X_t represents a matrix of deterministic terms such as trend or seasonal variables, and D represents the associated coefficient matrix of X_t.

See Also:
Example, Serialized Form

Constructor Summary

Constructors
Constructor and Description

VectorAutoregression(TimeSeries ts)
Constructor for the class.

Constructors
Constructor and Description
`VectorAutoregression(TimeSeries ts)` Constructor for the class.

Method Summary

Methods
Modifier and Type	Method and Description
`double[]`	`getARConstants()` Returns the current settings of the constants used in the autoregression model.
`int[]`	`getARModel()` Returns the autoregressive model configuration.
`double[][]`	`getEstimates()` Returns the parameter estimates (coefficients) of the vector autoregression model.
`double[][]`	`getForecasts()` Returns the h-step ahead forecast at times t=nT, nT+1, ..., T, where h=1,2, ..., `maxStepsAhead`.
`boolean`	`isA0Flag()` Returns the state of `A0Flag`.
`void`	`setA0Flag(boolean A0Flag)` Sets the flag to include the leading autoregressive coefficient matrix in the model.
`void`	`setARConstants(double[] arConstants)` Sets the constants for the autoregressive model.
`void`	`setARLag(int arLag)` Sets the autoregressive lag parameter.
`void`	`setARModel(int[] arModel)` Sets the form of the autoregressive terms of the model.
`void`	`setCenter(boolean center)` Sets the flag to center the data.
`void`	`setMaxLag(int maxLag)` Sets the maximum lag.
`void`	`setScale(boolean scale)` Sets the flag to scale the data.
`void`	`setTrend(boolean trend)` Sets the flag to fit a trend parameter in the model.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - VectorAutoregression
```
public VectorAutoregression(TimeSeries ts)
```
    Constructor for the class.
    
    Parameters:
    ts - a TimeSeries object.
- Method Detail
  - getARConstants
```
public double[] getARConstants()
```
    Returns the current settings of the constants used in the autoregression model.
    
    Returns:
    a double array containing the constants used in the autoregression model. If the value is null, the default values are active.
  - getARModel
```
public int[] getARModel()
```
    Returns the autoregressive model configuration.
    
    Returns:
    an int array specifying the autoregressive configuration. If null, then all parameters are active in the current model.
  - getEstimates
```
public double[][] getEstimates()
```
    Returns the parameter estimates (coefficients) of the vector autoregression model.
    
    Returns:
    a double array containing the estimated parameters (coefficients).
  - getForecasts
```
public double[][] getForecasts()
```
    Returns the h-step ahead forecast at times t=nT, nT+1, ..., T, where h=1,2, ..., maxStepsAhead.
    The h-step ahead forecast
    $hat{Y}_{t+h} = E[Y_{t+h}|Y_s, 0 le s le t]$
    uses the conditional expectation of $Y_{t+h}$ given the historical information known at time t. The estimated VAR model approximates the conditional expectation and the forecasts are generated from the estimated VAR model via the following recursion:
    $hat{Y}_{t+h} = sum_{j=1}^{p}hat{A}_jhat{Y}_{t+h-j} + delta$
    where is the vector time series and $delta=(I - hat{A}_1-hat{A}_2-cdots-hat{A}_p)mu.$
    
    Returns:
    a double matrix T-nT by maxStepsAhead * K containing the forecasts.
    Note that the forecast $hat{Y_k}_{t+h}$ is stored in location [t-nT][(h-1)*K + k] of the returned forecast matrix.
  - isA0Flag
```
public boolean isA0Flag()
```
    Returns the state of A0Flag.
    
    Returns:
    a boolean which specifies whether or not the leading coefficient matrix is nontrivial.
  - setA0Flag
```
public void setA0Flag(boolean A0Flag)
```
    Sets the flag to include the leading autoregressive coefficient matrix in the model.
    When true, a nontrivial, lower-triangular leading autoregressive coefficient matrix, A₀, will be estimated in the model. When false, A₀ is the constant identity matrix, which is also the default case.
    
    Parameters:
    A0Flag - boolean indicating whether to fit the leading coefficient matrix.
    Default: A0Flag=false.
  - setARConstants
```
public void setARConstants(double[] arConstants)
```
    Sets the constants for the autoregressive model. See discussion in setARModel(int[]) for details.
    
    Parameters:
    arConstants - a double array specifying the autoregressive model constants.
    The input array arConstants must be of length (A0Flag+arLag)*K*K, such that for indices 1=0,...,(A0Flag + arLag), i=0,...,K-1, and j=0,...,K-1, arConstants[l*K*K +i*K + j] specifies a constant value for parameter $a_{ij,l}$ .
    Default: arConstants[i]=0.
  - setARLag
```
public void setARLag(int arLag)
```
    Sets the autoregressive lag parameter. It must be nonnegative and less than maxLag.
    
    Parameters:
    arLag - an int specifying the desired lag for the autoregressive terms. Default: arLag=1.
  - setARModel
```
public void setARModel(int[] arModel)
```
    Sets the form of the autoregressive terms of the model.
    Without loss of generality, assume that the series mean is 0 and that there are no deterministic terms. Then we can write the vector autoregression model of lag p as
    $A_0Y_t + A_1Y_{t-1} + cdots + A_pY_{t-p} = u_t.$
    Each of the matrices A_j represents K * K unknown coefficients. In practice, many of the (p+ 1) * K * K coefficients are restricted to be 0 or otherwise constant. The leading coefficient matrix A₀ must be lower-triangular and is very often equal to the identity matrix (this is the default behavior). Fully expanded the model looks like K equations in K(p+1) unknowns.
    $Y_{t,k} = sum_{j=1}^{K}a_{kj,0}Y_{t,j} + sum_{l=1}^{p}sum_{j=1}^{K}a_{kj,l}Y_{t-l,k}$
    Consider each coefficient matrix as the vector
    $text{vec}(A_{l}) = (a_{11,l}, a_{21,l},ldots,a_{K1,l}, a_{12,l},a_{22,l},ldots,a_{K2,l},cdots, a_{1K,l},a_{2K,l},ldots, a_{KK,l})^{T}$
    for .
    $text{vec}(A_{l})$
    is just the columns of
    $A_{l}$
    appended to each other. To specify a configuration different from the default, append the vectorized coefficient matrices as
    $text{vec}(A_1, A_2,ldots,A_p)$
    or
    $text{vec}(A_0, A_1,ldots,A_p)$
    when including a nontrivial leading coefficient matrix. Set the value at index l*K*K + j*K + i to 1 or -1 to fit $a_{ij,l}$ or $-a_{ij,l}$ in the model. Set the value to 0 to exclude $a_{ij,l}$ as a parameter. Note that for the leading coefficient matrix,l= 0, the value at j*K + i is ignored unless i is at least j, i.e., only the lower triangle of A₀ can be nontrivial.
    
    Parameters:
    arModel - an int array specifying the autoregressive model parameters.
    For indices l=0,...,(A0Flag+arLag), i=0,...,K-1, and j=0,...,K-1, arModel[l*K*K +i*K + j] = {-1,1} indicates that the coefficient $pm a_{ij,l}$ is a parameter (to be estimated) in the model.
    Default: A₀ is the identity matrix and arModel[i]=1 for all i.
  - setCenter
```
public void setCenter(boolean center)
```
    Sets the flag to center the data. If center=true, column means are subtracted from the data.
    
    Parameters:
    center - a boolean indicating whether or not column means should be subtracted from the data.
    Default: center=false.
  - setMaxLag
```
public void setMaxLag(int maxLag)
```
    Sets the maximum lag.
    
    Parameters:
    maxLag - an int specifying the maximum lag to consider in the first (pure VAR) stage regression.
  - setScale
```
public void setScale(boolean scale)
```
    Sets the flag to scale the data.
    If scale=true, the data values are mean-centered and then divided by the standard deviation.
    
    Parameters:
    scale - a boolean
    Default: scale=false.
  - setTrend
```
public void setTrend(boolean trend)
```
    Sets the flag to fit a trend parameter in the model. Set to true to include a deterministic trend in the model.
    
    Parameters:
    trend - a boolean specifying whether or not to include a deterministic trend.
    Default: trend=false.

Class VectorAutoregression

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

VectorAutoregression

Method Detail

getARConstants

getARModel

getEstimates

getForecasts

isA0Flag

setA0Flag

setARConstants

setARLag

setARModel

setCenter

setMaxLag

setScale

setTrend