Changing frequency and the effect on autocorrelation

Question

Say I have a model following an AR(1)-structure $x_t = \rho x_{t-1}+\epsilon_t$. This model is valid for hourly observations of $x$. With this assumption, what can be said about the distribution of $x_t$ observed each minute? Is there a relation between the hourly $\rho$ and a minute $\rho$? Does it require some further model assumptions?

Answer 1

Generally, without additional assumptions, there is nothing we can say about the distribution of $x$ sampled at high frequency between the time points in which the actual observations have been sampled at low frequency. Imagine drawing the low-frequency points against time. There is an infinite amount of ways to connect the consecutive points, unless we restrict (by additional assumptions) the shape of the lines connecting them.

When a time series has a subject-matter interpretation, additional assumptions may be natural. E.g. if due to subject-matter reasons the time series is known to be (approximately) a random walk ($\rho=1$) and we have sampled it at low frequency, we might be able to model the high-freqency observations e.g. using a Gaussian process.