Reading Suggestions for June

Well, here we are - it's June already.

Here are my reading suggestions:

• Abadie, A., S. Athey, G. Imbens, & J. Wooldridge, 2017. When should you adjust standard errors for clustering? Mimeo.
• Berk, R., A. Buja, L. Brown, E. George, A. K. Kuchibhotla, W. Su, & L, Shazo, 2019. Assumption lean regression. American Statistician, in press.
• Ghosh, T., M. Ghosh, & T. Kubokawa, 2019. On the loss robustness of least-square estimators, American Statistician, in press.
• Gustafsson, O. & P. Stockhammar, 2019. Variance stabilizing filters. Communications in Statistics - Theory and Methods, in press.
• Kocherlakota, N. R., 2019. A near-exact finite sample theory for an instrumental variable estimator. Mimeo. (Hat-tip to Frank Diebold.)
• Panagiotelis, A., G. Anathasopoulos, R. J. Hyndman, B. Jiang, & F. Vahid, 2019. Macroeconomic forecasting for Australia using a large number of predictors. International Journal of Forecasting, 35, 613-633.

© 2019, David E. Giles

Here's Your Reading List!

As we count the year down, there's always time for more reading!

• Birg, L. and A. Goeddeke, 2014. Christmas economics - A sleigh ride. Discussion Paper No. 220, CEGE, University of Gottingen.
• Geraci, A., D. Fabbri, and C. Monfardini, 2014. Testing exogeneity of multinomial regressors in count data models: Does two stage residual inclusion work? Working Paper 14/03, Health, Econometrics and Data Group, University of York.
• Li, Y. and D. E. Giles, 2014. Modelling volatility spillover effects between developed stock markets and Asian emerging stock markets. International Journal of Finance and Economics, in press.
• Ma, J. and M. Wohar, 2014. Expected returns and expected dividend growth: Time to rethink an established literature. Applied Economics, 46, 2462-2476. 
• Qin, D., 2014. Resurgence of instrument variable estimation and fallacy of endogeneity. Economics Discussion Papers No. 2014-42, Kiel Institute for the World Economy. 
• Romano, J. P. and M. Wolf, 2014. Resurrecting weighted least squares. Working Paper No. 172, Department of Economics, University of Zurich.
• Tchatoka, F.D., 2014. Specification tests with weak and invalid instruments. Working Paper No. 2014-05, School of Economics, University of Adelaide.

© 2014, David E. Giles

All About Spherically Distributed Regression Errors

This post is based on a handout that I use for one of my courses, and it relates to the usual linear regression model,

                                  y = Xβ + ε

In our list of standard assumptions about the error term in this linear multiple regression model, we include one that incorporates both homoskedasticity and the absence of autocorrelation. That is, the individual values of the errors are assumed to be generated by a random process whose variance (σ²) is constant, and all possible distinct pairs of these values are uncorrelated. This implies that the full error vector, ε, has a scalar covariance matrix, σ²I_n. 

We refer to this overall situation as one in which the values of the error term follow a “Spherical Distribution”. Let's take a look at the origin of this terminology.

Spherically Distributed Errors in Regression Models

Let's think about the standard linear regression model that we encounter in our introductory econometrics courses:

 

                                       y = Xβ + ε .                      (1)

 

By writing the model in this form, we've already made two assumptions about the stochastic relationship between the dependent variable, y, and the regressors (the columns of the X matrix). First, the relationship is a parametric one - hence the presence of the coefficient vector, β; and second, the relationship is a linear one. That's to say, the model is linear in these parameters. If it wasn't, we wouldn't be able to write the model in the form given in equation (1).

 

However, the model isn't fully specified until we lay out any assumptions that are being made about the regressors and the random error term, ε. Now, let's consider the full set of (rather stringent) assumptions that we usually begin with:

F-tests Based on the HC or HAC Covariance Matrix Estimators

We all do it - we compute "robust" standard errors when estimating a regression model in any context where we suspect that the model's errors may be heteroskedastic and/or autocorrelated.

More correctly, we select the option in our favourite econometrics package so that the (asymptotic) covariance matrix for our estimated coefficients is estimated, using either White's heteroskedasticity-consistent (HC) estimator, or the Newey-West heteroskedasticity & autocorrelation-consistent (HAC) estimator.

The square roots of the diagonal elements of the estimated covariance matrix then provide us with the robust standard errors that we want. These standard errors are consistent estimates of the true standard deviations of the estimated coefficients, even if the errors are heteroskedastic (in White's case) or heteroskedastic and/or autocorrelated (in the Newey-West case).

That's fine, as long as we keep in mind that this is just an asymptotic result.

Then, we use the robust standard error to construct a "t-test"; or the estimated covariance matrix to construct an "F-test", or a Wald test.

And that's when the trouble starts!