Will unemployment be useful in forecasting the inflation rate in the same year given the inflation rate in the previous year?

Phillips curves are a class of models which propose connections between the inflation rate and the unemployment rate in an economy. If there is some such relationship, we might expect that knowing both today's unemployment rate and last year's inflation rate allows us to forecast this year's inflation rate better than just relying on last year's inflation rate alone.

To this end, let ( \pi_t, u_t ) denote the inflation rate and the unemployment rate for the United States in the year ( t ) respectively. We can then run the following ordinary least squared regressions from the year ( 2022 ) to the year ( 2031 ) inclusive:

[ (1), , \pi_t = \alpha + \beta \pi_{t-1} + \varepsilon_t ] [ (2), , \pi_t = \alpha + \beta_1 \pi_{t-1} + \beta_2 u_t + \varepsilon_t ]

The adjusted coefficient of determination is a measure of how much of the variance in the dependent variable is explained by the regression that corrects for the number of independent variables so as to control for the effect of overfitting. Ideally, the adjusted coefficient of determination only goes up when we add an additional independent variable to a regression if the additional variable is actually useful in predicting the dependent variable.