- Are improper linear models used in practice or are they some kind of curiosity described from time to time in scientific journals? If so, in what areas are they used?
- Are there other examples of such models?
- Finally, would standard errors, $p$-values, $R^2$ etc. taken from OLS for such models be correct, or should they be corrected somehow?
Background: Improper linear models are described from time to time in the literature. In general, such models can be described as
$$ y = a + b \sum_i w_i x_i + \varepsilon $$
what makes them different from regression is that $w_j$'s are not coefficients estimated in the model, but are weights that are
- equal for each variable $w_i = 1$ (unit-weighted regression),
- based on correlations $w_i = \rho(y, x_i)$ (Dana and Dawes, 2004),
- chosen randomly (Dawes, 1979),
- $-1$ for variables negatively related to $y$, $1$ for variables positively related to $y$ (Wainer, 1976).
Also it is common to use some kind of feature scaling, like converting variables into $Z$-scores. So, this kind of model can be simplified to univariate linear regression
$$ y = a + b v + \varepsilon $$
where $v = \sum w_i x$, and can be simply estimated using OLS regression.
Dawes, Robyn M. (1979). The robust beauty of improper linear models in decision making. American Psychologist, 34, 571-582.
Graefe, A. (2015). Improving forecasts using equally weighted predictors. Journal of Business Research, 68(8), 1792-1799.
Wainer, Howard (1976). Estimating coefficients in linear models: It don't make no nevermind. Psychological Bulletin 83(2), 213.
Dana, J. and Dawes, R.M. (2004). The Superiority of Simple Alternatives to Regression for Social Science Predictions. Journal of Educational and Behavioral Statistics, 29(3), 317-331.