# VectorAutoregressions.jl

Vector autoregressive models for Julia

## Installation

`Pkg.clone("https://github.com/lucabrugnolini/VectorAutoregressions.jl")`

## Introduction

This package is a work in progress for the estimation and identification of Vector Autoregressive (VAR) models.

## Status

- VAR
- VAR(1) form
- Lag-length selection
- AIC
- AICC
- BIC
- HQC

- VAR impulse response function (IRFs)
- Identification
- Reduce form
- Cholesky
- Long-run restrictions
- Sign restrictions
- Heteroskedasticity
- External instruments (ex. high-frequency,narrative)
- Wild bootstrap

- Confidence bands
- Asymptotic
- Bootstrap
- Bootstrap-after-bootstrap

- Identification
- Forecasting
- BVAR
- FAVAR

- Local projection IRFs
- Lag-length selection
- Confidence bands
- Standard
- Bootstrap

- Bayesian Local Projection

## Example

```
## Example: fit a VAR(`p`) to the data and derive structural IRFs with asymptotic and bootstrap conf. bands.
using VectorAutoregressions, Plots
y = readdlm(joinpath(Pkg.dir("VectorAutoregressions"),"test","cholvar_test_data.csv"), ',') #read example file with data
intercept = false #intercept in the estimation
p = 2 #select lag-length
H = 15 # IRFs horizon
nrep = 500 #bootstrap sample
# Fit VAR(2) to data
V = VAR(y,p,intercept)
# Estimate IRFs - Cholesky identification
T,K = size(y)
mIRFa = IRFs_a(V,H,intercept) #asymptotic conf. bandf
mIRFb = IRFs_b(V,H,nrep,intercept) #bootstrap conf. bands
# Plot irf + asy ci
pIRF_asy = plot(layout = grid(K,K));
[plot!(pIRF_asy, [mIRFa.CI.CIl[i,:] mIRFa.IRF[i,:] mIRFa.CI.CIh[i,:]], color = ["red" "red" "red"],
line = [:dash :solid :dash], legend = false, subplot = i) for i in 1:K^2]
gui(pIRF_asy)
# Plot irf + bootstraped ci
pIRF_boot = plot(layout = grid(K,K));
[plot!(pIRF_boot, [mIRFb.CI.CIl[i,:] mIRFb.IRF[i,:] mIRFb.CI.CIh[i,:]], color = ["blue" "blue" "blue"],
line = [:dash :solid :dash], legend = false, subplot = i) for i in 1:K^2]
gui(pIRF_boot)
```

More in details, `y`

is a matrix with data, `p`

is the lag-length of the VAR we fit to the data and `i`

is a Boolean for including an intercept (default is true). `VAR(y,p,intercept)`

returns a fitted VAR(`p`

) model in `V`

with the following structure:

```
struct VAR
Y::Array # dep. variables
X::Array # covariates
β::Array # parameters
ϵ::Array # residuals
Σ::Array # VCV matrix
p::Int64 # lag-length
i::Bool # true or false for including an intercept (default is true)
end
```

You can access to each element writing `V.`

and than the element you are interested in (for example for the covariates `V.X`

).