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i have some question about serial correlation:

  1. What are the effects of residual serial correlation when the regressors are strictly exogenous?

  2. What are the effects of residual serial correlation when the regressors are lagged dependent variables?

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The big difference is that when the regressors are lags of the dependent variable, the OLS estimator will be inconsistent. In the case of exogenous regressors, the OLS estimator is consistent.

In both cases, the presence of serial correlation will misestimate the standard errors (OLS standard errors will underestimate true standard errors in case of positive serial correlation, and overestimate in case of negative serial correlation) and hence misestimate the t-statistic respectively.

Please note that you test differently for serial correlation depending on whether your regressors are exogenous or not: The standard Durbin-Watson test is perfectly fine when regressors are exogenous but cannot be used with regressors that are lags of the dependent variable.

The question presumes a time series context. Please note that serial correlation occurs in cross-sections as well, therefore your point 1 has a broader context as it may refer to cross sections as well, whereas your point 2 applies only when there are repeated observations over time.

In a time series context, the presence of serial correlation suggests 2 different messages:

  1. Model is potentially mis-specified
  2. Standard errors will need to be adjusted

Using Hansen or Newey-West serial-correlation robust standard errors only helps with the second point/concern.

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  • $\begingroup$ High, do you have a good resource to read more about what you have said? I have just learned that b0 and b1 estimates are still unbiased and that the standard errors are fine, but the t-statistic no longer follows a normal distribution and thus you don't know if your coefficients are significant. You seem to go more in depth making the distinction between exogenous and endogenous. How do we adjust for the 3 situations? 1) Model only has exogenous variables. 2) Model is a mix of exogenous and endogenous variables. 3) Model only has endogenous variables? $\endgroup$
    – confused
    Commented Jun 7, 2020 at 12:15
  • $\begingroup$ Davidson & Mackinnon (pages 275-280) has a good discussion on it. $\endgroup$
    – ColorStatistics
    Commented Oct 6, 2021 at 19:59
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Serial correlation supposedly breaks one of the major assumptions of linear regression-that the residuals are independent. If they are serially correlated, they are not independent. What that implies is that the statistical significance of your regression coefficients will not be entirely reliable.

However, the above problem is really minute. You just have to recalculate your regression coefficients standard errors with Newey-West Robust Standard Errors. The latter can typically be calculated with any quantitative software (R, Python, SAS, MatLab, etc.). Then, you recalculate the statistical significance of your regression coefficients with these Robust Standard Errors that adjust for both heteroskedasticity and serial correlation issues.

If you do not have access to such software, or if you do not know how to code them (Newey West in R is a bit of a workout), there is a simpler solution. Just look at the t stat of your regression coefficients. To be statistically significant their t stat typically hs to be greater than 2.0. If your t stat are greater than 3.0, these regression coefficients will most probably still be statistically significant when using Robust Standard Errors (RSEs). In other words, RSEs are fairly rarely greater than 1.5 times greater than regular standard errors. If you want to be totally sure you could use a criteria of a t stat of 4 which would allow RSEs to be 2x larger than regular standard errors.

When you have an autoregressive model, the meaning and impact of serial correlation is very much the same. However, it is somewhat more problematic. By definition, using lagged Ys should in most cases entirely eliminate serial correlation. I have seen modelers keeping on adding additional lags of Y to eliminate serial correlation. I think such models are typically uninformative and questionably specified as they take much of the explanatory power away from the true exogenous variables that better explain the behavior of your dependent variable.

One thing to keep in mind, serial correlation does not bias the regression coefficients. It simply mildly questions the statistical significance of your variable. However, in certain circumstances you could very well have a very well specified model with a combination of variables that are statistically significant and some that are not. If a variable is not statistically significant but is very well supported by logic, theory, etc. such a variable may be a lot more meaningful than one that is very statistically significant but whose meaning is absurd or who has the wrong sign. That's probably one of the most important thing to remember on the subject.

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  • $\begingroup$ Are you not assuming that the error process is free of deterministic structure such as Pulses, Level/step shifts , Seasonal Pulses,Local Time Trends AND that the error variance is homogeneous over time in order for what you have written to be true? $\endgroup$
    – IrishStat
    Commented Jan 6, 2018 at 19:18
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    $\begingroup$ I focused my answer on the specific question. $\endgroup$
    – Sympa
    Commented Jan 6, 2018 at 23:19
  • $\begingroup$ I know that both : lagged and strictly exogenous are 2 things that can correct the problem of serial correlation, but I'm not clear about the way they do. Please explain to me if you know it. Thank you! $\endgroup$
    – Thu Hương
    Commented Jan 7, 2018 at 3:13
  • $\begingroup$ Well exogenous variables do not typically resolve serial correlations. Lagged Y variables most often do. But, as you experience sometimes you have to add more than one lag. And, as mentioned such models become increasingly meaningless. I think when you add lags of Y, they have a mean reverting impact on the residuals. And, therefore it attenuates the serial correlation. $\endgroup$
    – Sympa
    Commented Jan 7, 2018 at 19:20

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