Constant changes in time series model I am estimating a time series model using OLS.  The LHS variable has a downtrend across the period. There is certainly autocorrelation on both the LHS and RHS. 
the regression is: 
$us10yr = \alpha + \beta_1 {X} + e_t $
I have split the periods up and have found that the constant has significant variation across the sample. This is probably best seen from the below chart (which shows a rolling 4yr window). 

As you can see from the 4yr rolling estimate: the constant moves a LOT, and typically in the direction of change in the LHS variable (lagging the changes a little). 
I was worried about autocorrelation & some of the diagnostics — so I used HAC (Newey-West) standard errors.
data is here
I have three questions:


*

*What does the trending constant in the recursive estimation suggest?

*Does the use of HAC standard errors help? 

*What estimators ought I to consider? 

 A: A model with no lags or differences and no arima structure is often a NO-NO when you have time series data due to untreated auto-correlation . Time series analysis model identification enables more powerful model structure to be used. See https://autobox.com/pdfs/regvsbox-old.pdf for a good introduction.  Perhaps you could explicitly present not only your model BUT the actual data in a csv format to facilitate further discussion .
EDITED AFTER RECEIPT OF YOUR DATA 246 MONTHLY VALUES (US10yr) starting 2005/1 WITH 3 POSSIBLE PREDICTORS:
What does the trending constant in the recursive estimation suggest?
My ans to question 1
1  in a univariate setting a shifting of the mean 
2) in a causal setting ..not much can be said
I took your 246 values and found that a useful model required Weighted Estimation as the error variance was markedly lower in the most recent data. The model is here  with a supporting scatter plot of Y and your 3rd candidate predictor.
The Actual/Fit and Forecast is here 
If you can predict X3 that will aid your prediction of Y
