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I have two sets of data points:

  • Each data point (y_t) in the first set represents the annualized return for the next ten years of monthly return data, i.e., this is forward-looking data, so my last data point is from October 2014
  • Each data point (x_t) in the second set represents the average value from the trailing ten years on a monthly basis, i.e., this is backwards-looking data, so I have data up until this past month in 2024

The second set of data is an economic indicator that many use to describe the stock market as expensive or cheap by historical standards that I'd like to try and use to forecast the ten-year annualized return. Empirically, there is a clear linear relationship, but the nature of both time series is making me question how to properly specify my model.

Given that both the response and explanatory variables rely on so much adjacent data and that residuals of a basic OLS show autocorrelation exists (empirically and Durbin-Watson), I'm struggling on where to go next. First order differencing adds in stationarity evidenced by a significant Dickey-Fuller test statistic, but I'm unsure whether I need to also add in lag variables for either/both my response and explanatory variables and how far to go out with those lags given the extent of the forward and backward reach to create the data sets. ACF and PACF charts heavily imply I need more than one lag variable, and I've started reading up on how to implement the Cochrane-Orcutt procedure but am having trouble translating and applying academic notes on the internet to python code for my scenario.

After coming across a somewhat relevant post (Correcting for autocorrelation in simple linear regressions in R) that delves into the pros and cons of error adjusting vs model changes, I'm wondering if the transformations I've been going after are completely necessary. Does using heteroskedasticity and autocorrelation robust standard errors satisfy the requirements for forecasting or is it more likely that invalid coefficients from model misspecification would void any confidence in predictions?

If anyone has come across this and has any reading suggestions or python packages that I may have overlooked, it would be greatly appreciated.

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  • $\begingroup$ I'm not sure if it can help you but check out the auto-regressive distributed lag a specific case of which is referred to as the koyck distributed lag. In that model, the effect of a predictor is allowed to sort of slow down over time and not just disappear. That may help you to some extent because it handles the case where adjacent responses are highly correlated. But I have no idea how to handle the explanatory variables also being correlated with each other ? I would see how correlated they are and maybe knowing the strength can allow you to drop a variable ? $\endgroup$
    – mlofton
    Commented Nov 13 at 5:14
  • $\begingroup$ Perhaps you will find something relevant in the threads tagged with overlapping-data. $\endgroup$
    – Richard Hardy
    Commented Nov 13 at 14:25
  • $\begingroup$ quant.stackexchange.com those guys might have an opinion $\endgroup$
    – Ggjj11
    Commented Nov 13 at 15:23

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