I have some time series data and want to test for the existence of and estimate the parameters of a linear trend in a dependent variable w.r.t. time, i.e. time is my independent variable. The time points cannot be considered IID under the null of no trend. Specifically, the error terms for points sampled near each other in time are positively correlated. Error terms for samples obtained at sufficiently different times can be considered IID for all practical purposes.
I do not have a well-specified model of how the error terms are correlated for points close to each other in time. All I know from domain knowledge is that they are positively correlated to some degree or another. Other than this issue, I believe the assumptions of ordinarly least squares linear regression (homoskedasticity, linearity, normally distributed error terms) are met. Modulo the correlated error term issue, OLS would solve my problem.
I am a complete novice at dealing with time series data. Is there any "standard" way to proceed in these circumstances?
What you are describing is commonly referred to as auto correlated errors. I would suggest you look up resources on ARIMA modelling. ARIMA modelling will allow you to model the correlation in your error term, and hence allow you to assess your trend variable independent of this auto correlation (or other independent variables you are interested in).
My suggested reading for an into to ARIMA modelling would be Applied Time Series Analysis for the Social Sciences 1980 by R McCleary ; R A Hay ; E E Meidinger ; D McDowall
But there are plenty of resources (time series analysis is a massive field of study). You would probably be able to turn up some good online resources with just a google search if you don't have access to an academic library. I just turned up this page, Statistica ARIMA, it has a brief but very concise description of ARIMA modelling as well as other methods for time series analysis.
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