Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation for Julia.
The package is registered on METADATA, so to install
Pkg.add("CovarianceMatrices")
This package provides types and methods useful to obtain consistent estimates of the long run covariance matrix of a random process.
Three classes of estimators are considered:
- HAC - heteroskedasticity and autocorrelation consistent (Andrews, 1996; Newey and West, 1994)
- HC - hetheroskedasticity (White, 1982)
- CRVE - cluster robust (Arellano, 1986)
The typical application of these estimators is to conduct robust inference about parameters of a model. This is accomplished by extending methods defined in StatsBase.jl and GLM.jl.
Available kernel types are:
TruncatedKernel
BartlettKernel
ParzenKernel
TukeyHanningKernel
QuadraticSpectralKernel
For example, ParzenKernel(NeweyWest)
return an instance of TruncatedKernel
parametrized by NeweyWest
, the type that corresponds to the optimal bandwidth calculated following Newey and West (1994). Similarly, ParzenKernel(Andrews)
corresponds to the optimal bandwidth obtained in Andrews (1991). If the bandwidth is known, it can be directly passed, i.e. TruncatedKernel(2)
.
The examples below clarify the API, that is however relatively easy to use.
In the regression context, the function vcov
does all the work:
vcov(::DataFrameRegressionModel, ::HAC; prewhite = true)
Consider the following artificial data (a regression with autoregressive error component):
using CovarianceMatrices
using DataFrames
Random.seed!(1)
n = 500
x = randn(n,5)
u = Array{Float64}(2*n)
u[1] = rand()
for j in 2:2*n
u[j] = 0.78*u[j-1] + randn()
end
u = u[n+1:2*n]
y = 0.1 + x*[0.2, 0.3, 0.0, 0.0, 0.5] + u
df = DataFrame()
df[:y] = y
for j in enumerate([:x1, :x2, :x3, :x4, :x5])
df[j[2]] = x[:,j[1]]
end
Using the data in df
, the coefficient of the regression can be estimated using GLM
lm1 = glm(y~x1+x2+x3+x4+x5, df, Normal(), IdentityLink())
To get a consistent estimate of the long run variance of the estimated coefficients using a Quadratic Spectral kernel with automatic bandwidth selection à la Andrews
vcov(lm1, QuadraticSpectralKernel(Andrews), prewhite = false)
If one wants to estimate the long-time variance using the same kernel, but with a bandwidth selected à la Newey-West
vcov(lm1, QuadraticSpectralKernel(NeweyWest), prewhite = false)
The standard errors can be obtained by the stderror
method
stderror(::DataFrameRegressionModel, ::HAC; prewhite::Bool)
Sometime is useful to access the bandwidth selected by the automatic procedures. This can be done using the optimalbw
method
optimalbw(NeweyWest, QuadraticSpectralKernel, lm1; prewhite = false)
optimalbw(Andrews, QuadraticSpectralKernel, lm1; prewhite = false)
Sometime interest lies in estimating the long-run variance of the average of the process. At the moment this can be done by carrying out a regression on a constant (the sample mean of the realization of the process) and using vcov
or stderror
to obtain a consistent variance estimate (or its diagonal elements).
lm2 = glm(u~1, df, Normal(), IdentityLink())
vcov(lm1, ParzenKernel(NeweyWest), prewhite = false)
stderr(lm1, ParzenKernel(NeweyWest), prewhite = false)
As in the HAC case, vcov
and stderr
are the main functions. They know get as argument the type of robust variance being sought
vcov(::DataFrameRegressionModel, ::HC)
Where HC is an abstract type with the following concrete types:
HC0
HC1
(this isHC0
with the degree of freedom adjustment)HC2
HC3
HC4
HC4m
HC5
using CovarianceMatrices
using DataFrames
using GLM
# A Gamma example, from McCullagh & Nelder (1989, pp. 300-2)
# The weights are added just to test the interface and are not part
# of the original data
clotting = DataFrame(
u = log([5,10,15,20,30,40,60,80,100]),
lot1 = [118,58,42,35,27,25,21,19,18],
lot2 = [69,35,26,21,18,16,13,12,12],
w = 9*[1/8, 1/9, 1/25, 1/6, 1/14, 1/25, 1/15, 1/13, 0.3022039]
)
wOLS = fit(GeneralizedLinearModel, lot1~u, clotting, Normal(), wts = array(clotting[:w]))
vcov(wOLS, HC0
vcov(wOLS, HC1)
vcov(wOLS, HC2)
vcov(wOLS, HC3)
vcov(wOLS, HC4)
vcov(wOLS, HC4m)
vcov(wOLS, HC5)
The API of this class of variance estimators is subject to change, so please use with care. The difficulty is that CRHC
type needs to have access to the variable along which dimension the clustering mast take place. For the moment, the following approach works --- as long as no missing values are present in the original dataframe.
using RDatasets
df = dataset("plm", "Grunfeld")
lm = glm(Inv~Value+Capital, df, Normal(), IdentityLink())
vcov(lm, CRHC1(convert(Array, df[:Firm])))
stderr(lm, CRHC1(convert(Array, df[:Firm])))