## Nonspherical Models

### What Are Nonspherical Models?

Consider the linear time series model ${y}_{t}={X}_{t}\beta +{\epsilon }_{t}$, where yt is the response, xt is a vector of values for the r predictors, β is the vector of regression coefficients, and εt is the random innovation at time t.

Ordinary least squares (OLS) estimation and inference techniques for this framework depend on certain assumptions, e.g., homoscedastic and uncorrelated innovations. For more details on the classical linear model, see Time Series Regression I: Linear Models. If your data exhibits signs of assumption violations, then OLS estimates or inferences based on them might not be valid.

In particular, if the data is generated with an innovations process that exhibits autocorrelation or heteroscedasticity, then the model (or the residuals) are nonspherical. These characteristics are often detected through testing of model residuals (for details, see Time Series Regression VI: Residual Diagnostics).

Nonspherical residuals are often considered a sign of model misspecification, and models are revised to whiten the residuals and improve the reliability of standard estimation techniques. In some cases, however, nonspherical models must be accepted as they are, and estimated as accurately as possible using revised techniques. Cases include:

• Models presented by theory

• Models with predictors that are dictated by policy

• Models without available data sources, for which predictor proxies must be found

A variety of alternative estimation techniques have been developed to deal with these situations.