Modeling quarterly default rate (non stationary, autorregressive time series)

Question

I am a student writing my thesis on default rate modeling. My major is finance, so I'm not really experienced in econometrics. I'm trying to create a model for quarterly corporate default rates (percentage of defaulted corporations in the previous 12 months) with macroeconomical variables (GDP growth, interest rate, debt/GDP, etc.)

My problem is, that according to ADF and KPSS my variables are non-stationary. Differenciating doesn't really help, and second differences hold no explanatory value. The variables are not cointegrated. I'm using a sample of 17 years (60-70 values, depending on the variable lags). They should be stationary on the long term (100 years), but my sample is very narrow.

Because I'm using quaretly data of annual default rates, the dependent variable is also heavily autocorrelated. Using annual data shows no autocorrelation, but 16 variables are few for a model.

My goal is to examine the significance of the independent variables and their lags in different sectors, and also trying out different methods for modeling (linear, linear with lags, log-linear, diffeneciated, and all with Hodrick-Prescott filter).

In real life it is common practice by banks to just use linear regression, which I did, and got awesome results, but it turns out at the end that the data needs to be stationary and non-autocorrelated for this model.

The only appropriate model I found was ARDL which suggests 4 lags and it shows that all four lags of the dependent variable is significant, so it's not really helpful for me.

I'm running out of ideas what kind of models to use or how to explain why I used linear regression.

Please help, I'm getting desperate.

edit: Data used

edit2: I tried to use Newey-West and Hansen-Hodrick Standard Errors to correct the issue of (overlapping) autocorrelation, the coefficients and the Durbin-Watson value did not change, only the p ant t values in a minimal way.

edit3: MY final regression model is ln⁡(hp_DR)= -0,3349+0,0182*hp_debt/GDP(t-2))+ + 0,0639* hp_interest(t-11)- 0,0182* hp_GDP(t-9)+ μ,

DR is the default rate in percentage, debt/GDP is corporate debt/GDP rate in %, and interest rate is government 10 year bond yields. All with Hodrick-Prescott filter (hp) and appropriate lags.

default rate plot:

Answer 1

Have you thought about using a PCA, clustering or random forest for determining independent variable significance?

For forecasting, you could use an ARIMAX model with exogenous inputs. Just fill in the appropriate parameters, (p,d,q). AR models are commonly used for econometric forecasts. You could also use random forest to do this forecast, but with the lack of data it might be an issue.

One last thought, you might want to transform the data to percent change before performing the independent variable tests. Although, leave the data as is for the ARIMAX forecast.

Update

The answer to your most recent comment is a bit lengthy so I am making it an update. Also I think I have a better picture of what you are try to accomplish.

You should not be using a simple linear model with non-stationary series to explain the variance of the dependent variable. You should be using % change / differences. Using a non-stationary series you will get a great r-squared, but it is incorrect because the model is not picking up the variance, it is only seeing the trend. Below is a quick example to illustrate using the SPY ETF. As you can see the price series regression looks great, .99 r-squared , but it is a lie! When you do the return series, you get the real answer that it is a bad model with no predictive or explanatory power.

Linear Regression using price series

Call:
lm(formula = spy.close.px[-1] ~ spy.close.px[-nrow(spy.close.px)])

Residuals:
     Min       1Q   Median       3Q      Max 
-11.5588  -0.7205   0.0493   0.8592  12.8073 

Coefficients:
                                   Estimate Std. Error  t value            Pr(>|t|)    
(Intercept)                       0.0445310  0.1024606    0.435               0.664    
spy.close.px[-nrow(spy.close.px)] 0.9999792  0.0005781 1729.656 <0.0000000000000002 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.704 on 2991 degrees of freedom
Multiple R-squared:  0.999, Adjusted R-squared:  0.999 
F-statistic: 2.992e+06 on 1 and 2991 DF,  p-value: < 0.00000000000000022

Linear Regression using returns

Call:
lm(formula = spy.returns[-1] ~ spy.returns[-nrow(spy.returns)])

Residuals:
      Min        1Q    Median        3Q       Max 
-0.105063 -0.004327  0.000488  0.005280  0.133375 

Coefficients:
                                  Estimate Std. Error t value  Pr(>|t|)    
(Intercept)                      0.0002255  0.0002262   0.997     0.319    
spy.returns[-nrow(spy.returns)] -0.0804997  0.0182356  -4.414 0.0000105 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.01237 on 2990 degrees of freedom
  (1 observation deleted due to missingness)
Multiple R-squared:  0.006475,  Adjusted R-squared:  0.006143 
F-statistic: 19.49 on 1 and 2990 DF,  p-value: 0.00001049