1
$\begingroup$

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

Improve this question
$\endgroup$
3
  • $\begingroup$ What are your assumptions about $e_t | X$, where $e_t$ is error and $X$ is vector of IVs? $\endgroup$
    – Dayne
    Oct 21, 2020 at 18:15
  • $\begingroup$ 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 $\endgroup$
    – Aishwarya Deore
    Oct 21, 2020 at 20:46
  • $\begingroup$ But if you have already included AR(1) term in the model why would errors have AR(1) correlation? $\endgroup$
    – Dayne
    Oct 22, 2020 at 1:29

1 Answer 1

Reset to default
0
$\begingroup$

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)

Improve this answer
$\endgroup$
12
  • 1
    $\begingroup$ 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 $\endgroup$
    – Aishwarya Deore
    Oct 22, 2020 at 13:19
  • $\begingroup$ 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. $\endgroup$
    – Dayne
    Oct 22, 2020 at 13:35
  • 1
    $\begingroup$ See this link: stats.stackexchange.com/questions/52458/… $\endgroup$
    – Aishwarya Deore
    Oct 22, 2020 at 13:37
  • $\begingroup$ Thanks for the reference. So the coefficients do get biased if estimated using OLS. $\endgroup$
    – Dayne
    Oct 22, 2020 at 13:42
  • 1
    $\begingroup$ 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 $\endgroup$
    – Aishwarya Deore
    Oct 22, 2020 at 14:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.