Misspecified autoregressive models I want to know the impact of misspecifying lag levels in AR models. Say the true model is AR(P), while in estimation AR(P-1) is estimated and used for forecast.

What will be the impact of estimating AR(P-1) on the estimated coefficients of $\phi_1 - \phi_{p-1}$, can they be consistently estimated. What about the forecast error.
Just think your true model is: 
$$y_t = \phi_0 + \phi_1 y_{t-1} + \phi_2 y_{t-2} + u_t \tag{1}$$

So, under your true model, $u_t$ is uncorrelated with all explanatory variables. 

So, instead of using AR(2), use selected AR(1) model as: 
$$y_t = \phi_0 + \phi_1 y_{t-1} + \epsilon_t \tag{2}$$
So, now, $\epsilon_t = \phi_2 y_{t-2} + u_t$; therefore, $y_{t-1}$ will no longer will be uncorrelated with $\epsilon_t$ (due to correlation between $y_{t-1}$ &amp; $y_{t-2}$). In simple words, this lead to the problem of endogeneity. And we know, OLS estimators are not consistent when residuals are correlated with any of the explanatory variables.