I am trying to estimate a model by using OLS (ordinary least squares) regression, I find that the default rate (my dependent variable) $y_t$ is non-stationary, therefore I take the difference to make it stationary and set up my model as;

$y_t - y_{t-1} \approx \beta_0 + \beta_1 GDP_t + \beta_2 UP_t$,

where $GDP_t$ is some stationary transformation of the gross domestic product at time $t$, and $UP_t$ is some stationary transformation of the unemployment rate at time $t$.

However, I need predicted values of $y_t$. How do I obtain that?

E.g. consider that I estimate my model on the in-sample data below, and I want to get forecasted values of $y_t$ out of sample;

Sample time $y_t$ $y_t - y_{t-1}$ $\hat y_t$ $\hat{y_t - y_{t-1}}$ GDP UP
In-Sample 2020Q1 0.1 - - - - -
In-Sample 2020Q2 0.2 0.1 0.09 + 0.1 0.09 -0.4 0.3
In-Sample 2020Q3 0.5 0.3 0.31 + 0.2 0.31 0.3 0.1
In-Sample 2020Q4 0.3 -0.2 -0.14 + 0.5 -0.14 -0.2 -0.4
Out-of-Sample 2021Q1 - - -0.05 + 0.3 -0.05 0.3 0.2
Out-of-Sample 2021Q2 - - -0.01 + (-0.05 + 0.3) -0.01 0.2 0.1
Out-of-Sample 2021Q3 - - 0.2 + (-0.01 + (-0.05 + 0.3)) 0.2 0.5 0.3

Where $\hat{y_t - y_{t-1}}$ is my predicted values of the LHS, and $\hat y_t$ is computed afterwards as $\hat{y_t - y_{t-1}}$ + $y_{t-1}$ in-sample and out-of-sample as $\hat{y_t - y_{t-1}}$ + $\hat y_{t-1}$. As you see it becomes sort of recursive. When I tried this with real data I noticed I only got reasonable results when using e.g. the last historic value $y_t$ = 0.3, when computing $\hat y_t$ at future time points. For example in 2021Q1 I would take -0.05 + 0.3, and in 2021Q2 I would take -0.01 + 0.3 etc. Why does that give reasonable results while doing as illustrated in the table gives some kind of constant increase/decrease.

  • $\begingroup$ It is not necessary for either the dependent or independent variables to be stationary, only the error term needs to be stationary. Aside from that, you have two broad options: either forecast the regressors as well (possibly jointly with the target), or use what people in econometrics sometimes call "direct forecasting", i.e. build separate models $Y_t = a_h + b_h X_{t-h} + \varepsilon^{(h)}_t$ for each forecast horizon $h$. If you use the actual observed regressors to forecast at the end instead, that's not "out-of-sample". $\endgroup$
    – Chris Haug
    Commented Dec 1, 2021 at 23:39

1 Answer 1


i checked it just fast, seems good to me.


default rate (my dependent variable) yt

delta y; just copied from table:

2021Q1 -0.05 
2021Q2 -0.01
2021Q3  0.2

some kind of constant increase/decrease

hm 3 different numbers, rate was decreasing alot, then decreased a little, then increased..

maybe.. did u computed coefs b0, b1,b2 after every new row?

  • or u used same coeficients as calculated from data included rows from 1 to 4 (first 4 rows)?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.