Introduction

CovarianceMatrices.jl](https://github.com/gragusa/CovarianceMatrices.jl/) implements several covariance estimator for stochastic process. Using these covariances as building blocks, the package extends GLM.jl to alllow obtaining robust covariance estimates of GLM's coefficients.

Three classes of estimators are considered:

1. HAC
   • heteroskedasticity and autocorrelation consistent (Andrews, 1996; Newey and West, 1994)
2. VARHAC
3. HC
   • heteroskedasticity consistent (White, 1982)
4. CRVE
   • cluster robust (Arellano, 1986)

Long run covariance

For what follows, let $\{X_t\}$ be a stochastic process, i.e. a sequence of random vectors. We will assume throughout that the r.v. are $p$-dimensional. The sample average of the process is defined as $math \bar{X}_n = \frac{1}{n} \sum_{t=1}^n X_t$ It is often the case that the sampling distribution of $\sqrt{n}\bar{X}_n$ can be approximated (as $n\to\infty$) by the distribution of a standard multivariate normal distribution centered at $\mu$ and long-run variance-covariance $V$. In other words, the following holds $math \sqrt{n}V^{-1/2}(\bar{X}_n - \mu_n) \xrightarrow{d} N(0, I_{p}),$ where $math \mu = E(X_t), \quad \text{and} \quad V \equiv \lim_{n\to\infty} \mathrm{Var}\left(\sqrt{n}\bar{X}_n\right).$

Estimation of $V$ is central in many applications of statistics. For instance, we might be interested in constructing asymptotically valid conficence intervals for a linear combination of the unknown expected value of the process $\mu$, that is, we are interested in making inference about $c'\mu$ for some $p$-dimensional vector $c$. For any random random matrix $\hat{V}$ tending in probability (as $n\to\infty$) to $V$, a confidence interval for $c'\bar{X}$ with asymptotic coverage $(\alpha\times 100)\%$ is given by $math \left[c'\bar{X}_n - q_{1-\alpha/2} c'\hat{V}c/sqrt{n}, c'\bar{X}_n + q_{\alpha/2} c'\hat{V}c/sqrt{n}\right]$ where $q_{\alpha}$ is the $\alpha$-quantile of the standard normal distribution.

CovarianceMatrices.jl provides methods to estimate $V$ under a variety of assumption on the correlation stracture of the random process. We know explore them one by one starting from the simplest case.

Serially uncorrelated process

If the process is uncorrelated, the variance covariance reduces to $math V_n = E\left[\sqrt{n}\left(\frac{1}{n}\sum_{t=1}^n \left(X_t - \mu\right)\right].$ An consistent estimator of $V$ is thus given by $math \hat{V}_n = \frac{1}{n} \sum_{t=1}^n \left(X_t - \bar{X}_n\right).$ Given X::AbstractMatrix with size(X)=>(n,p) containing $n$ observations on the $p$ dimensional random vectors, an estimate of $V$ can be obtained by lrvar:

Vhat = lrvar(Uncorrelated(), X)

Uncorrelated is the type signalling that the random sequence is assumed to be uncorrelated.

Api

Serially correlated process

Correlated process (time-series)

Correlated process (spatial)

In this case,