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.