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## 15.2 Dynamic Causal Effects

This section of the book describes the general idea of a dynamic causal effect and how the concept of a randomized controlled experiment can be translated to time series applications, using several examples.

In general, for empirical attempts to measure a dynamic causal effect, the assumptions of stationarity (see Key Concept 14.5) and exogeneity must hold. In time series applications up until here we have assumed that the model error term has conditional mean zero given current and past values of the regressors. For estimation of a dynamic causal effect using a distributed lag model, assuming a stronger form termed strict exogeneity may be useful. Strict exogeneity states that the error term has mean zero conditional on past, present and future values of the independent variables.

The two concepts of exogeneity and the distributed lag model are summarized in Key Concept 15.1.

### The Distributed Lag Model and Exogeneity

The general distributed lag model is \begin{align} Y_t = \beta_0 + \beta_1 X_t + \beta_2 X_{t-1} + \beta_3 X_{t-2} + \dots + \beta_{r+1} X_{t-r} + u_t, \tag{15.2} \end{align} where it is assumed that

1. $X$ is an exogenous variable, $E(u_t\vert X_t, X_{t-1}, X_{t-2},\dots) = 0.$

• a $X_t,Y_t$ have a stationary distribution.
• b $(Y_t,X_t)$ and $(Y_{t-j},X_{t-j})$ become independently distributed as $j$ gets large.
2. Large outliers are unlikely. In particular, we need that all the variables have more than eight nonzero and finite moments — a stronger assumption than before (four finite nonzero moments) that is required for computation of the HAC covariance matrix estimator.

3. There is no perfect multicollinearity.

The distributed lag model may be extended to include contemporaneous and past values of additional regressors.

On the assumption of exogeneity

• There is another form of exogeneity termed strict exogeneity which assumes $E(u_t\vert \dots, X_{t+2},X_{t+1},X_t,X_{t-1},X_{t-2},\dots)=0,$ that is the error term has mean zero conditional on past, present and future values of $X$. Strict exogeneity implies exogeneity (as defined in 1. above) but not the other way around. From this point on we will therefore distinguish between exogeneity and strict exogeneity.

• Exogeneity as in 1. suffices for the OLS estimators of the coefficient in distributed lag models to be consistent. However, if the the assumption of strict exogeneity is valid, more efficient estimators can be applied.