# Bias-variance trade-off in case of biased estimators: is the bias zero?

Consider a data generating process (DGP) that is AR(1): $$y_t=\varphi_1 y_{t-1}+\varepsilon_t$$ with $$\varepsilon_t\sim i.i.D(0,\sigma^2)$$ for some distribution $$D$$ with mean zero and variance $$\sigma^2$$. Consider a model that coincides with the DGP up to parameter values that are unknown and need to be estimated from a sample time series. The parameters will be estimated by conditional least squares (CLS), a fairly standard estimator of autoregressive models. Consider the decomposition of the expected squared prediction error at a point $$y_{t-1}=c$$ for some $$c\in R$$ into squared bias, variance and irreducible error? Is the bias zero?

No, the bias is not zero unless $$c=0$$. Bias is the difference between the expected value of the prediction (here $$\hat\varphi_1 y_{t-1}$$) and the deterministic part of the target (here $$\varphi_1 y_{t-1}$$). Given our estimator $$\hat\varphi=\hat\varphi^{CLS}$$ and the point of interest $$y_{t-1}=c$$, \begin{aligned} \text{Bias} &= \mathbb{E}[\hat\varphi_1^{CLS}\cdot c]-\varphi_1\cdot c \\ &= c\cdot(\mathbb{E}[\hat\varphi_1^{CLS}]-\varphi_1). \end{aligned} The last expression in parentheses is nonzero, as CLS estimators of the slope of AR models are known to be biased (see e.g. this or this). Thus even if we have the correct model (a model that coincides with the DGP up to unknown parameter values), the bias may be nonzero if the parameter estimator is biased and yields a biased prediction of the deterministic part of $$y_t$$ at the point of interest.