Mean signed deviation

In statistics, the mean signed difference (MSD), also known as mean signed deviation and mean signed error, is a sample statistic that summarises how well a set of estimates $$\hat{\theta}_i$$ match the quantities $$\theta_i$$ that they are supposed to estimate. It is one of a number of statistics that can be used to assess an estimation procedure, and it would often be used in conjunction with a sample version of the mean square error.

For example, suppose a linear regression model has been estimated over a sample of data, and is then used to extrapolate predictions of the dependent variable out of sample after the out-of-sample data points have become available. Then $$\theta_i$$ would be the i-th out-of-sample value of the dependent variable, and $$\hat{\theta}_i$$ would be its predicted value. The mean signed deviation is the average value of $$\hat{\theta}_i-\theta_i.$$

Definition
The mean signed difference is derived from a set of n pairs, $$( \hat{\theta}_i,\theta_i)$$, where $$ \hat{\theta}_i$$ is an estimate of the parameter $$\theta$$ in a case where it is known that $$\theta=\theta_i$$. In many applications, all the quantities $$\theta_i$$ will share a common value. When applied to forecasting in a time series analysis context, a forecasting procedure might be evaluated using the mean signed difference, with $$\hat{\theta}_i$$ being the predicted value of a series at a given lead time and $$\theta_i$$ being the value of the series eventually observed for that time-point. The mean signed difference is defined to be
 * $$\operatorname{MSD}(\hat{\theta}) = \frac{1}{n}\sum^{n}_{i=1} \hat{\theta_{i}} - \theta_{i} .$$

Use Cases
The mean signed difference is often useful when the estimations $$\hat{\theta_i}$$ are biased from the true values $$\theta_i$$ in a certain direction. If the estimator that produces the $$\hat{\theta_i}$$ values is unbiased, then $$\operatorname{MSD}(\hat{\theta_i})=0$$. However, if the estimations $$\hat{\theta_i}$$ are produced by a biased estimator, then the mean signed difference is a useful tool to understand the direction of the estimator's bias.