Forecasting model with OLS 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.

i checked it just fast, seems good to me.

yt−yt−1≈β0+β1GDPt+β2UPt


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)?

