# Does using lagged independent variables makes sense?

While it seems quite common to calculate a lagged version of the dependent variable and to use it on the right hand side of a model (e.g., autoregressive models), I have rarely seen that lagged versions of independent variables are included in a model. Is there a reason for that?

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The models with lagged independent variables are called distributed lag models. Usually introductory econometrics texts have a section or chapter dedicated to them. They were more popular in the 1980s, and actually Christopher Sims got his Nobel prize in economics for the work on such models (see the article Money, Income and Causality). Nowadays they are used less frequently, but still they can be useful in time series regressions when it is clear that the lags of independent variable have an effect on the dependent variable.

Also note that autoregressive models can be written as infinite distributed lag models, i.e. model

$$y_t = \rho y_{t-1}+\beta x_t + u_t$$

is equivalent to

$$y_t = \beta \sum_{j=0}^\infty \rho^j x_{t-j}+ \sum_{j=0}^\infty \rho^j u_{t-j},$$

which means that autoregressive models actually use lagged independent variables, albeit not in a direct manner.

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Thanks. Your answer is really good. It would be great, if you could give a comment on the reasons of why distributed lag models became less popular in comparison to autoregressive models. Thank you. – Julian Jan 6 at 5:57