Description Usage Arguments Details Value Author(s) References Examples

'nets' is used to fit sparse VARs using the NETS algorithm.

1 2 3 |

`y` |
data, an T x N matrix, each column being a time series. |

`GN` |
Estimate the Autoregressive VAR matrices (default true) |

`CN` |
Estimate the Concentration matrix of the VAR innovations (default true) |

`p` |
VAR order (default 1) |

`lambda` |
shrinkage parameter either a scalar or a vector of size two |

`alpha.init` |
initial value of the alpha parameter |

`rho.init` |
initial value of the rho parameter |

`algorithm` |
lasso optimization algorithm 'shooting' or 'activeshooting' (default) |

`weights` |
lasso weights: 'none' or 'adaptive' (default) |

`iter.in` |
maximum number of in iterations (default 100) |

`iter.out` |
maximum number of out iterations (default 2) |

`verbose` |
extra output messages |

The nets procedure estimates sparse Vector Autoregression (VAR) models by LASSO.

In particular, the routine can be used to estimate: (i) the autoregressive matrices of the VAR model and the concentration matrix of the VAR innovations via the nets algorithm (GN=TRUE and CN=TRUE) (ii) the autoregressive matrices of the VAR model via the LASSO algorithm (GN=TRUE and CN=FALSE), (iii) the concentration matrix of data via the space algorithm of Peng et al. (2009) (GN=FALSE and CN=TRUE). Notice that in this last case the order p of the VAR is automatically set to zero.

In case (i) the penalty parameter lambda can be either set to a single scalar or two a vector of size two.In the former case all the parameters of the model are penalized using the single value of lambda provided while in the latter the first entry is the vector is used to penalize the autoregressive coefficients and the second entry for the contemporaneous correlation coefficients. In case (ii) and (iii) the penalty parameter lambda has to be a single scalar.

Two variants of the LASSO algorithm are implemented: "shooting" and "activeshooting". The second can be significantly faster when the model is sparse.

If the weights variable is set to 'none' the model is estimated using the standard lasso. If the weights variable is set to 'adaptive' the model is estimated using the 'adaptive' lasso.

The iter.in variable sets the maximum number of lasso iterations. The iter.out variable sets the number of times the lasso algorithm is reiterated. This is only relevant for the space and nets algorithms. See Peng et al. (2009) and Barigozzi and Brownlees (2016) for details.

The nets procedure returns an object of class "nets"

The function "print" is used to print a summary of the estimation results of a "nets" object.

The function "predict" is used to predict the future realizations of the VAR usin a "nets" object and a sample of new observation.

An object of class "nets" is a list containing at least the following components:

A.hat: an N x N x P array containing the estimated autoregressive matrices

C.hat: an N x N matrix containing the estimated concentration matrix

alpha.hat: (N^2 * P) x 1 vector of autoregressive parameters stacked in a vector

rho.hat: (N*(N+1)/2) x 1 vector of partial correlations associated with the concentration matrix of the var innovations stacked in a vector

c.hat: N x 1 vector of diagonal entries of the diagonal entries of the concentration matrix of the var innovations

g.adj: Adjancency matrix associated with the Granger network implied by the VAR autoregressive matrices

c.adj: Adjancency matrix associated with the contemporaneous network implied by the VAR innovation concentration matrix

Christian Brownlees

Barigozzi, M. and Brownlees, C. (2016) NETS: Network Estimation for Time Series, Working Paper

Peng, J., Wang, P., Zhou, N. and Zhu, J. (2009), Partial Correlation Estimation by Joint Sparse Regression Model, JASA, 104, 735-746.

Meinshausen, N. and Buhlmann, P. (2006), High Dimensional Graphs and Variable Selection with the Lasso, Annals of Statistics, 34, 1436-1462.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | ```
N <- 5
P <- 3
T <- 1000
A <- array(0,dim=c(N,N,P))
C <- matrix(0,N,N)
A[,,1] <- 0.7 * diag(N)
A[,,2] <- 0.2 * diag(N)
A[1,2,1] <- 0.2
A[4,3,2] <- 0.2
C <- diag(N)
C[1,1] <- 2
C[4,2] <- -0.2
C[2,4] <- -0.2
C[1,3] <- -0.1
C[1,3] <- -0.1
Sig <- solve(C)
L <- t(chol(Sig))
y <- matrix(0,T,N)
eps <- rep(0,N)
for( t in (P+1):T ){
z <- rnorm(N)
for( i in 1:N ){
eps[i] <- sum( L[i,] * z )
}
for( l in 1:P ){
for( i in 1:N ){
y[t,i] <- y[t,i] + sum(A[i,,l] * y[t-l,])
}
}
y[t,] <- y[t,] + eps
}
lambda <- c(1,2)
system.time( mdl <- nets(y,P,lambda=lambda*T,verbose=TRUE) )
mdl
``` |

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