# Inclusion of lagged dependent variable in regression

I'm very confused about if it's legitimate to include a lagged dependent variable into a regression model. Basically I think if this model focuses on the relationship between the change in Y and other independent variables, then adding a lagged dependent variable in the right hand side can guarantee that the coefficient before other IVs are independent of the previous value of Y.

Some say that the inclusion of LDV will biase downward the coefficient of other IVs. Some others say that one can include LDV which can reduce the serial correlation.

I know this question is pretty general in terms of which kind of regression. But my statistical knowledge is limited and I really have a hard time figuring out if I should include a lagged dependent variable into a regression model when the focus is the change of Y over time.

Are there other approaches to deal with the influence of Xs on the change of Y over time? I tried different change scores as DV as well, but the R squared in that situation is very low.

• what do you hope to achieve with your model? Maximizing R-squared is rarely a good model-selection criteria. Mar 17, 2013 at 21:08
• This model will be used for prediction. It is true that although the R-squared differ a lot, the predicted values are actually the same using Y or the change of Y. However, given the low R-squared value using the change of Y as DV, does it mean that the current set of IVs is not able to explain the change very well and there must be some omitted variables? Mar 17, 2013 at 23:12

The decision to include a lagged dependent variable in your model is really a theoretical question. It makes sense to include a lagged DV if you expect that the current level of the DV is heavily determined by its past level. In that case, not including the lagged DV will lead to omitted variable bias and your results might be unreliable. In such a scenario, including the lagged DV, will take out a lot of your variance and is likely to make your other DV's effects less significant (which means both make the $\beta$s smaller and the standard errors bigger). However, what it will allow you to do is say that those IVs that still influence your outcome have an effect controlling for past value of the DV. An alternative approach to this is to use the difference between your outcome variable at period $t$ and $t-1$ as your DV for period $t$.

However, doing any of these imply answering an important question: what is the right lag structure for your DV? You can get some information about this by observing the correlation between your outcome variable with itself for different lag values (e.g. correlation between Y and Y$t-1$, Y and Y$t-2$, etc.).

• This is not correct. Whether the lagged DV explains the variance in the current DV is not relevant to omitted variable bias. Yes, it must explain the DV - but its covariance with the explanatory variable(s) of interest is more important. OVB is not a result of failing to include a DV predictor, it's a result of your predictor being correlated with the error term. May 5, 2022 at 23:27
• @socialscientist is correct and my answer above is wrong. As they suggest, OVB is relevant in the context of estimating a specific coefficient, it is not a model level characteristic. Jun 30 at 13:26

I recommend two articles:

1. Achen C. H. (2001) Why lagged dependent variables can suppress the explanatory power of other independent variables (link)
2. Keele, L. and Kelly N. J. (2005) Dynamic models for dynamic theories: the ins and outs of lagged dependent variables (link).

The upshot is that including a lagged dependent variable can have a large influence on the coefficients of the remaining variables. Sometimes this is appropriate (for the Dynamic models of Keele and Kelly) and sometimes not. As others have said, it's important to think about the process being modeled.

Including lagged dependent variables can reduce the occurrence of autocorrelation arising from model misspecification. Thus accounting for lagged dependent variables helps you to defend the existence of autocorrelation in the model. The past value affects the present in the model, requires theoretical foundation, and best fit up the model as per required.

• Not sure I did justice to the last sentence; please edit if the meaning can be clarified. Welcome to Cross Validated BTW! Jul 18, 2014 at 8:37
• Autocorrelation can be an artifact of the way data are collected. I obtained data by digitising a plot which meant the data were sorted. This sorting and the non-linear relationship caused autocorrelation in the residuals. Jun 13, 2016 at 21:15

Some say that the inclusion of LDV will biase downward the coefficient of other IVs.

To be more specific, using OLS with the inclusion of a LDV can bias your coefficient downwards. Consider the model $$Y_t=\alpha + \lambda Y_{t-1} + U_t$$ for $$t=1,...,T$$ with $$-1<\lambda<1$$ and the $$U_t$$s are identically distributed normal random variables with mean zero. Under this model, the bias of the OLS estimator of $$\lambda$$ is given by $$-(1+3\lambda)/T$$ (Kendall, 1954). This is clearly a problem and one should not apply OLS directly when incorporating LDVs as without considering this.

The crux of the issue is that including an LDV breaks a core assumption of OLS: observations are independent. I highly recommend reading Maeshiro (1996) as he asks (and answers to some extent by providing options for what to do instead) the same question as you've presented here.

References:

1. Kendall, Maurice G. "Note on bias in the estimation of autocorrelation." Biometrika 41.3-4 (1954): 403-404. (link)
2. Maeshiro, Asatoshi. "Teaching regressions with a lagged dependent variable and autocorrelated disturbances." The Journal of Economic Education 27.1 (1996): 72-84. (link)

Yes, you should be wary of Nickell bias in a small T large N situation (Nickell, S. (1981). Biases in dynamic models with fixed effects. Econometrica: Journal of the Econometric Society, 1417-1426.)

You might wish to look at Dynamic Panel Data models like Arellano-Bond or Blundell-Bond estimators.

What makes me intrigued about this question is not knowing more about the specification of the model or the estimation technique for it. I mention that because although using a lagged DV among the IVs may be theoretically important and methodologically necessary, it may also introduce a risky amount of endongeneity in the model, depending on the substantial relation between variables and time units and, also, on the AR order that may exist in the model. Unless you (and us) have more details on the variables and on the estimation, I would not feel confortable to recomend lagging the DV unless you are thinking of some instrumental variable technique or something like Arellano-Bond estimation.

Please, give us more details so we may know better on what kind of model we are talking about.