# 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.

• 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". Commented Dec 1, 2021 at 23:39

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)?