Forecasting model with OLS

Question

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.

Answer 1

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)?