Suppose that the time series data $(y_1, y_2,…, y_n)$ can be explained through a regression model with $k$ explanatory variables:
(1) $y_t = b_0+b_1x_{1t}+b_2x_{2t}…+b_kx_{kt} + \epsilon_t,\ t=1,2,…,n$
where $(\epsilon_1, \epsilon_2, .., \epsilon_n) \sim N(0,\ \Sigma)$. When serial correlation exists in the residual time series $\epsilon$, we can solve the model (1) through generalized least squares. For simplification, let us assume the correlation structure of the residuals $\epsilon$ is AR(1).
Occasionally I see in literature that, for the same data, some people model the serial correlation of AR(1) with a different model
(2) $y_t = \phi y_{t-1}+b_1x_{1t}+b_2x_{2t}…+b_kx_{kt} + \epsilon_t,\ t=1,2,…,n$
with the assumption of white noise for the residuals: $(\epsilon_1, \epsilon_2, .., \epsilon_n) \sim N(0,\ \sigma^2I)$.
I suppose that the underlying assumption would be different for the model (2). Here are my questions:
1) What exactly is the difference in terms of assumptions between the two models?
2) How to justify the adoption of one model over the other?
3) What is the impact of different choices on statistical inferences about those explanatory variables $x_i$?
4) Any literature that discuss the choice between the two models?
Best Answer
Both models can be used for processes with serially correlated errors, and will produce the same coefficients under the conditions that you outlined. However, in the presence of autocorrelation in errors, the model (2) will not have correct variances of coefficients. You may end up over estimating the statistical significance of coefficients. What do you do about this?
First of all, this is not always an issue. In forecasting applications it may not matter at all. Who cares what's the variance of my coefficient if the model produces good forecasts?
When this does matter, you may take many different paths. For instance, in the lecture notes take a look at Newey-West HAC estimator. It corrects for autocorrelation of residuals and estimates a better covariance matrix of coefficients. GLS is another approach, as well as ARIMA and exponential smoothing and the list goes on.
There's no universal algorithm to choose which approach is best for your problem.