Stationarity & Macroeconomic Forecasting I'm working with a time series of macroeconomic data (independent variables) and bank loss rates (dependent variable) to show how bank losses vary based on the state of the economy. What I'm doing is two-fold: I need to identify the most predictive economic indicators for my model and then I need to regress those variables against bank losses using OLS. The ultimate application of this is to forecast loss rates in varying economic climates.
One topic that has been coming up over and over again in terms of my model structure is regarding the stationarity of the data. As most on this board know, time series of economic data rarely pass stationarity tests. My question to everyone is how I should proceed in the event that my model does not satisfy stationarity test results. 
Let's assume for the example below that I have a simply univariate model where Loss = a + B*UnemploymentRate + e.  Please note that the univariate model is only for example - when I'm constructing the actual model, I use multiple economic variables in my regression. 
Below is an example of my model construction process, and was hoping for feedback on what's wrong with my process
1: Cleaning the dependent data using decompose() in R
Because I'm using quarterly data, I want to remove seasonality as well as remove noise. I use the decompose() function to do this. My final dependent variable is the trend series shown in the plot below:

I don't clean the economic variable because I'm using the seasonally adjusted unemployment rate that's provided by Bureau of Labor Statistics. 
2: Regress the variables
When I regress unemployment rate against bank losses, I get the following regression output:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)       -1.236e-03  2.077e-04  -5.954 4.96e-08 ***
Unemployment Rate  3.762e-04  3.077e-05  12.225  < 2e-16 ***
Multiple R-squared:  0.6242,    Adjusted R-squared:   0.62 

The fitted results (red) vs. dependent variable (black) is plotted below. For a single factor model, the fitted results mostly capture the trend in my data:
3: Validate the Results
Finally, I perform validation tests by testing my model residuals for a unit root using the PP test. As shown by the test below, the residuals have a unit root. However, if I were to perform a multivariate regression with more than one economic variable, I would greatly reduce the size of the residuals and mostly eliminate the autocorrelation of the residuals. 
Phillips-Perron Unit Root Test

data:  na.omit(varDiff$residuals)
Dickey-Fuller Z(alpha) = -11.018, Lag parameter = 3, p-value = 0.47033

Third Party Feedback
When having internal reviewers evaluate my model, the review teams are latching on to the non-stationarity of my independent and dependent variables. Both variables show a clear presence of a unit root, and fail all test measures for stationarity. My question is two fold: 1) does the non-stationarity of my model variable's render these models void? 2) If so, what measures should I take to correct for the non stationarity?
I've thought about the second point a bit, and the two solutions I keep reading are cointegration and differencing. I've tried cointegration and these economic factors are not passing cointegration tests. 
On the differencing side of the spectrum, below are the OLS results when I difference both the dependent and indendent variable: diff(Loss) = a + B*diff(UnemploymentRate) + e
                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)             3.011e-06  1.530e-05   0.197    0.844    
diff(Unemployment Rate) 3.055e-04  4.439e-05   6.883 7.93e-10 ***
Multiple R-squared:  0.3474,     Adjusted R-squared:   0.34 

As you can see, the variable is still highly significant, but the R-squared drops to unacceptable levels. From a review standpoint, differencing solves the stationarity issue but creates a new finding as the reviewers would jump on the low fit statistics.
Given everything I've layed out, I want to see what modeling techniques I should consider moving forward?
 A: A few suggestions:
1) There might be a cointegrating relationship. You could fit an error correction model. In this case you would still do the regression in differences, but include terms for the lagged level of the cointegrating relationship, as well as lags for the differences of each of the variables. 
2) Fitting models in differences instead of levels almost always leads to a lower R squared. However, R squared it not the main criteria of statistical analysis. It matters more if the model helps you predict bank loss rates better. 
3) I would push back against the idea that unemployment has a unit root. Over long periods of time, unemployment is fairly mean-reverting. If you're only using about 25 years worth of data, you might not see it. You could also try to find other data sources that could proxy your time series of bank loss rates.
4) Unemployment typically follows regimes: if it is going up, it keeps going up, and if it is going down, it keeps going down. This effect is driven by the state of the economy, monetary policy, and the growth of labor supply. Extracting information about the regime might improve the analysis. One benefit of this analysis is that we have more information, courtesy of the BLS, about the unemployment rate than we do about bank losses.
5) Include more independent variables if you can and it is reasonable.  
