# Stambaugh bias definition

How would you go about explaining "Stambaugh Bias" in simple relatively non-technical language?

• This paper also explains the Stambaugh bias: Stambaugh (JFE1999): Oct 9 '18 at 20:00

I'm not sure you can explain this term without using some technical terms, unfortunately. I'll give it my best shot.

Some definitions first:

1. Bias: the difference between the expectation of an estimator and the true value of the parameter you're estimating.
2. OLS: Ordinary Least Squares; a method for solving a regression problem.
3. Autoregressive process (AR): (via Wikipedia)

Stambaugh bias occurs when you perform regression on a lagged stochastic input. Essentially, when you do this, you have to use an estimate for the input (regressor), which requires estimating autocorrelation coefficients. The bias in the autocorrelation coefficients is then proportional to the bias in the slope coefficient's estimate from the OLS. You can correct for this if you know that your method for computing autocorrelation coefficients is biased.

The original paper really isn't too complicated, so long as you know both what an AR process is and how OLS regression works: Paper.

• I thought that I have fair understanding of those concepts until now. Could you please give an example. "How example, if you try to predict such and such using such and such, then such and such coefficients will be biased because of ..." Thanks in advance Apr 19 '11 at 15:56
• also... when you say auto-correlations, do you mean auto-correlations of the variable that I want to predict or predictors which I use to do it? Many thanks Apr 19 '11 at 15:58
• The assumption in Stambaugh bis is that your predictor is an autoregressive process. The inputs into your regressive routine would then be lagged versions of this process. Apr 19 '11 at 16:05
• OK. So if I want to use variable which is AR process (current value can be explained with linear combination of last couple of values)...why does that imply that inputs in my regression model are correlated? I have only one input in my regression model? Apr 19 '11 at 16:18
• How having AR process as a predictor relates to the variable that I want to predict? That's the bit I'm missing. Apr 19 '11 at 16:20

This is the original paper that explains the Stambaugh bias: Stambaugh (JFE1999). It examines the bias in the slope coefficient in the predictive regression of returns on past dividend yields, when the dividend yield process is highly persistent.

Let $$y_t$$ denote the date $$t$$ return and $$x_t=D_t/P_t$$ the date $$t$$ dividend yield. Consider the following regression equations: \begin{align*} y_t &= \alpha+\beta\,x_{t-1}+u_t\\ x_t &= \theta+\rho\,x_{t-1}+v_t \end{align*} for $$t=1,\ldots,T$$ with $$\begin{bmatrix}u_t\\v_t\end{bmatrix}=\mathcal{N}\left(0,\Sigma= \begin{bmatrix} \sigma_u^2&\sigma_{uv}\\ \sigma_{uv}&\sigma_v^2 \end{bmatrix}\right)$$.

The paper shows that the bias in $$\beta$$ is $$\mathbb{E}\left[\hat{\beta}-\beta\right]=\frac{\sigma_{uv}}{\sigma^2_v}\mathbb{E}\left[\hat{\rho}-\rho\right],$$ where $$\hat{\rho}$$ is the OLS estimator of $$\rho$$.

The paper shows that $$\hat{\rho}$$ has a downward bias and $$\sigma_{uv}$$ is negative leading to an upward bias in $$\hat{\beta}$$.

• the bias in $\beta$: did you mean the bias in $\hat\beta^{OLS}$? Oct 25 '19 at 10:59
• here is the ScienceDirect link for the paper: sciencedirect.com/science/article/pii/S0304405X99000410 Oct 26 '19 at 11:57
• Yes, The $\beta$ is the slope coefficient in the OLS regression of $y_t$ on $x_{t-1}$ in the first equation. Oct 26 '19 at 12:15
• The slope coefficient cannot be biased. An estimator for it can be. Hence my suggestion. Regarding the new link, my original comment applies: could you please include a full reference to the paper (in case the link goes dead in the future)? Thank you! Oct 26 '19 at 12:29
• In case you missed it, there was a clause (twice): in case the link goes dead in the future. Naming a link permanent does not guarantee it will exist in the future. Oct 28 '19 at 6:42