Lagged Liquidity, Autocorrelation, and Fixed-Effects Panel Models
Summary
The document examines whether a stock-liquidity regression should include yesterday’s liquidity when its residuals appear serially correlated. The setting is German stocks modeled with price, free float, and book-to-market variables, alongside time fixed effects. One response argues that autocorrelation points to a missing lagged dependent variable and that current liquidity may depend strongly on its previous value. Another response distinguishes residual autocorrelation from omitted-variable endogeneity: in a static model, autocorrelated errors can impair efficiency and conventional inference without necessarily biasing the coefficient estimates.
For the static panel specification, the latter response recommends clustering standard errors by firm over time to address inference. It cautions that adding lagged liquidity creates a dynamic panel model, where fixed effects can bias estimates, and a wrongly specified dynamic structure can add further bias. The discussion is not a definitive resolution: it presents competing answers, and the original Durbin-Watson diagnostic’s suitability for panel data is itself uncertain. Model choice should reflect the research question and the assumptions behind the panel estimator and standard errors.
Key ideas
- Residual autocorrelation alone does not establish that a lagged dependent variable is required.
- A static panel model can have autocorrelated errors without the regressors becoming endogenous.
- Firm-level clustering over time can address inference affected by within-firm serial dependence.
- Adding lagged liquidity creates a dynamic panel model and may introduce fixed-effects bias.
- A dynamic specification can add bias if its assumed time-series structure is incorrect.
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# Lagged dependent variable, yes or no? # Lagged dependent variable, yes or no? I read conflicting opinions about the inclusion of lagged dependent variables in modeling, and I guess it is partly up to the researcher and depending on the scope and goal of the research. I'm currently modeling the liquidity of German stocks, with panel data regression (fixed time effects), and my independent variables are price (logged), freefloat number of shares (logged) and book-to-market-value. Using E-views, my results are OK, except for a Durbin-Watson value around 1.5. Assuming Durbin Watson is valid for paneldata (but for the separate stocks, DW is also too low), we have autocorrelation in the errors. This is a problem because: - Estimates of the regression coefficients are inefficient. - Forecasts based on the regression equations are sub-optimal. - The usual significance tests on the coefficients are invalid. [source: Granger] Including a lagged dependent variable, i.e. liquidity from the day before, solves this issue and as expected increases the R^2 a bit more. But I am not really sure if this is the way to go. This is modeling liquidity where liquidity of the previous day is the most important factor... Another option would be that I'm missing a independent variable? Specifically the papers of Achen (To Lag or Not to Lag? Re-evaluating the Use of Lagged Dependent Variables in Regression Analysis) and Wilkins (Why Lagged Dependent Variables Can Suppress the Explanatory Power of Other Independent Variables) talk about these issues. ## Answer by bronc (score 1, accepted) https://quant.stackexchange.com/a/9866 If there is autocorrelation than you need to add the lagged dependent variable. By not including it, your regression is suffering from the omitted variable bias. You say that by doing this you will be "modelling liquidity where liquidity of the previous day is the most important factor" but since your regression "demands" adding the LDV (due to the AC) then most likely this period's liquidity is strongly dependent on last period's liquidity. ## Answer by Wille (score 2) https://quant.stackexchange.com/a/32568 Excluding an autoregressive term from a regression (a regular univariate time series regression or as in your case a static panel data regression) is not an omitted variables problem. For the univariate case most standard textbooks in statistics cover that autocorrelation in the residuals still leaves the OLS estimate unbiased and consistent (but inference will be incorrect) see e.g. p. 265 in Greene (2002). This also holds true for static panel data models, see e.g. p. 92 in Baltagi (2008). More technically leaving out the AR(1) term leads to cov(e_it,e_it-1)≠0 but not to cov(X_it,e_it)≠0 which is the endogeneity problem caused by omitted variables. What you should do is to allow for clustering of standard errors wihtin firms (over time) that will take care of your inference problem. Clustering across firms should not be a problem since you include time fixed effects, see Pedersen (2009) for more on calculating standard errors in panel data. If you include lagged liquidity you will have a dynamic panel data model which will have biased estimated with firm or time fixed effects are included. This bias is called the Nickell bias after Nickell (1981) and when you have time fixed effects the bias is of magnitude O(1/N). One important thing to consider is that if you go for a dynamic model and you specify the dynamics incorrectly (say the true time series structure is not an AR(1) model) then you will have introduced an additional source of bias from dynamic misspecification, see (Lee, 2012). To sum up: If you go for the static model at least your estimates will be unbiased but inefficent and inference will be incorrect (but easily fixed by allowing for clustering). Refs Baltagi, Badi H. (2008). "Econometric Analysis of Panel Data." Fourth edition, Wiley. Greene, William (2002). “Econometric Analysis”. Sixth edition, Prentice Hall. Lee, Yoonseok. (2012). "Bias in Dynamic Panel Models under Time Series Misspecification." Journal of Econometrics 169, 54-60. Nickell, Stephen. "Biases in dynamic models with fixed effects." Econometrica: Journal of the Econometric Society (1981): 1417-1426. Petersen, Mitchell A. "Estimating standard errors in finance panel data sets: Comparing approaches." Review of financial studies 22.1 (2009): 435-480.
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