com.imsl.stat

Class VectorAutoregression

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 $$A(L)(Y_t - \mu_t)=u_t + DX_t$$ where $$A(L)=A_0 + A_1L + A_2L^2 + \ldots + A_pL^p$$ 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 $E[u_t]=0$ and nonsingular covariance matrix, $\Sigma_u$, which is constant over time. The coefficient matrices $A_j, j = 0,\ldots,p$ 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.

Method Summary

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

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.

isA0Flag

public boolean isA0Flag()

Returns the state of A0Flag.

Returns:

a boolean which specifies whether or not the leading coefficient matrix is nontrivial.

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.

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.

setARConstants

public void setARConstants(double[] arConstants)

Sets the constants for the autoregressive model. See also the discussion in VectorAutoregression.setARModel(int[]).

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.

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.

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 $l=0,\ldots,p$. $$\text{vec}(A_{l})$$ 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.

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: $$\hat{Y}_{t+h} = \sum_{j=1}^{p}\hat{A}_j\hat{Y}_{t+h-j} + \delta$$ where $Y =(Y_1,...,Y_K)^T$ 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.