# Assumptions and terminology for dynamic regression with endogenous offset ($y_t=y_{t-1}+\beta X_{t-1}+\epsilon_t$)

I'm dealing with a fairly simple time series regression model with the following basic form:

$y_t=y_{t-1}+\beta X_{t-1}+\epsilon_t$

I'm assuming that observations of $y$ are known without error. $X$ contains exogenous predictor variables, and $\beta$ contains the corresponding parameters to be estimated. $\epsilon$ are the errors.

I'm asking 1) if my terminology for this type of model is correct; 2) what assumptions and consequences apply/ are violated with I fit this model using OLS; 3) same as 2, but fit using MLE.

I expand on these slightly below.

1) Is the model a dynamical linear model, a transfer function, or something else? How do I state that it contains only process errors, not observation error on $y$? Because it only contains process error, it's not a "state space" representation, correct? Related to 2 & 3, I'm wondering if the changes I make when fitting this model with OLS vs. MLE should change how I refer to the model.

2) When fitting using OLS, I take the first difference of the dependent variables in R (i.e., delta.y <- diff(y)) so that the response in my regression basically becomes $\Delta y_t = y_t - y_{t-1}$. In other words, $y_{t-1}$ is treated as a predictor variable that has a coefficient fixed at a value of 1, and thus is treated as an "offset" and subtracted from both sides of the equation before fitting. I want to confirm that the assumption of "strict exogeneity", as described by Mathworks Here, also applies when a coefficient is not associated with "autoregressive" term.

3) The biggest difference, to me, when fitting with MLE vs. OLS is that in the MLE format I do not take the first difference of $y$ ahead of time, and $y_{t-1}$ is actually the predicted value from previous time step, not the observed value. Is this difference meaningful in distinguishing the OLS from MLE? It seems like it affects the interpretation of the error term, as a large innovation at time t will propagate to the next time step (I know this is poorly worded, but hopefully clear enough to understand my meaning). Also, I'm looking for a reference similar to the above OLS Mathworks link for describing the assumptions and consequences of violating those assumption when fitting with MLE.

Any assistance in answering these questions or links that point me in the direction of digestible reading would be greatly appreciated; thank you.

A quick remark: OLS and RE estimators in dynamic models are inconsistent. Since $y_{it}$ is a function $\alpha_i$ and $\alpha_i$ is constant over time $y_{i,t-1}$ is likewise a function of $\alpha_i$. One of the regressors ($y_{i,t-1}$) is correlated with the error term which leads to biased and inconsistent estimators. This bias in turn depends on the assumption about the initial condition. The way forward might be to look for an alternative approach (transformation of the equation) which eliminates the individual effects but which leads to consistent estimates at the same time (for example the IV estimator of Anderson and Hsiao).
• Under some circumstances, OLS can still be consistent estimator even if the expanatory variables are lags of the dependent variable. In particular, if the explanatory variables (considered random variables,not fixed) are contemporaneously uncorrelated with the disturbance $\epsilon_t$, $E(X_t\epsilon_t)=0$, then OLS is consistent and inference is valid in large samples. For some details see for example this post. – javlacalle Apr 29 '15 at 13:35
• In this case, we don't have information about $X_{t-1}$ to see if it is correlated with $\epsilon_t$. Note that $y_{t-1}$ is not an issue, there is no coefficient attached to $y_{t-1}$, the dependent variable is actually $\Delta y_t=y_t - y_{t-1}$. – javlacalle Apr 29 '15 at 13:35