# Correcting for auto-correlation when using a lagged DV in the regression

I am conducting a regression where in I have data at the quarterly level for 19 companies (I have data ranging from 2007-2019 so about 30-50 quarters for each company). My regression model in STATA is as follows:

DV (in quarter t+1) = constant + IV (in quarter t) + Controls (in quarter t) + Lag DV (i.e DV in quarter t) + error

IV stands for independent variable and DV for dependent variable. The lagged DV is just a control variable and not my main variable of interest. The main variable of interest is the IV (in quarter t). I run the above using quarter and firm fixed effects and robust standard errors.

Question - does the inclusion of lagged DV bias all coefficients or just the coefficient on lagged DV? I know I should control for some sort of autocorrelation but how can I do that (eg. using prais command?). Is there anything else I can do to test the robustness of my results.

Any help is appreciated

• What are your assumptions about $e_t | X$, where $e_t$ is error and $X$ is vector of IVs? Commented Oct 21, 2020 at 18:15
• The assumption is that they are uncorrelated - else I would have an omitted variable bias. Plus I have no reason to believe that the error term would be correlated with any of the independent variables - but since I have a lag of DV now, there would be an AR(1) correlation between the error terms Commented Oct 21, 2020 at 20:46
• But if you have already included AR(1) term in the model why would errors have AR(1) correlation? Commented Oct 22, 2020 at 1:29

Not a complete answer but maybe we can analyze this by partitioned regression:

$$\mathbf{y_t}=\mathbf{X}\beta+\phi \mathbf{y_{t-1}}+\mathbf{e_t}$$

OLS estimate, $$\hat{\beta}$$ would be:

$$\hat{\beta}=(\mathbf{X'X})^{-1}\mathbf{X'}(\mathbf{y_t-\hat{\phi}y_{t-1}})=(\mathbf{X'X})^{-1}\mathbf{X'}(\mathbf{X}\beta+(\phi-\hat{\phi})\mathbf{y_{t-1}+e_t})$$

Therefore,

\begin{align} E(\hat{\beta})&=\beta+\phi (\mathbf{X'X})^{-1}\mathbf{X'}\big(E(\mathbf{y_{t-1}})-E(\hat{\phi}\mathbf{y_{t-1}})\big) \\&=\beta + \beta\frac{\phi}{1-\phi}+(\mathbf{X'X})^{-1}\mathbf{X'}E(\hat{\phi}\mathbf{y_{t-1}}) \\&=\frac{\beta}{1-\phi}+ (\mathbf{X'X})^{-1}\mathbf{X'}E(\hat{\phi}\mathbf{y_{t-1}}) \end{align}

I am unable to show that the term $$E(\hat{\phi}\mathbf{y_{t-1}})$$ would not be $$0$$, but I think because $$\hat{\phi}$$ is a function of $$y_t$$, there will be a lingering variance term in this and so it would not be $$0$$. If correct, then it seems the other parameter estimates are also biased.

(I tried searching for OLS estimation and inference of ARIMAX models but couldn't find anything; also the above results are based on simple OLS and not for robust standard errors)

• Thank you so much for this. I do read elsewhere that inclusion of lagged DV can bias the coefficients of other DVs. My expectation is for the coefficient of the main X to be negative. My coeffs are -.03 and -0.05 when I do and don't control for lagged DV respectively. Both are significant. If the bias is downward (i.e. smaller magnitude like -0.03) it is still okay for me since I am interested in the sign and significance and not the magnitude Commented Oct 22, 2020 at 13:19
• Actually, I hope someone else gives a thorough solution. From what I have done, I am unable to conclude whether it will be biased or not. Commented Oct 22, 2020 at 13:35
• See this link: stats.stackexchange.com/questions/52458/… Commented Oct 22, 2020 at 13:37
• Thanks for the reference. So the coefficients do get biased if estimated using OLS. Commented Oct 22, 2020 at 13:42
• More importantly - the coeffs get downward biased and can become insignificant. However, in my case, they remain significant and since I am only interested in prediction direction and not commenting much on the magnitude, I don't think using OLS should be a problem Commented Oct 22, 2020 at 14:51