I am trying to create a model for inflation for trading purposes. In his book The Market: Practice and Policy S. Nickell presents a model that relates unemployment, inflation and trade deficit. His final formula is
$$ [ \alpha_1 + \delta_1 \alpha_{12}]u + \alpha_2 \Delta^2 p + \alpha_{12} \delta_2 td = [\alpha_1 + \delta_1 \alpha_{12} ] \hat{u} $$
After he fits this model to data, he finds the coefficients as,
$$ 0.091 \log u + 0.05 u + 1.07 \Delta^2 p + 1.25 td = 0.091 \log \hat{u} + 0.054 \hat{u} - 1.27 \Delta u $$
where $\Delta^2 p$ the rate of change of the price level (ie inflation), $u$ unemployment rate, $td$ trade deficit as proportion of potential output, $\hat{u}$ is natural rate of unemployment. The full derivation can be found at the link below
I am trying to fit his formula to the data for UK, but cannot figure out how to get $\hat{u}$. Nickell seems to indicate this comes from a seperate calculation, I guess a first-pass on data would calculate $\hat{u}$, then with this new column in hand, I could fit all of the variables shown above. How to compute that first pass? Nickell says $\hat{u}$ can be defined "as that unemployment rate which is consistent with constant inflation and balanced trade" i.e. $\Delta^2p = 0$ and $td=0$. I am not sure what to do with this information: if I set $\Delta^2p = 0$, $td=0$ in the first formula above, I have
$$ [ \alpha_1 + \delta_1 \alpha_{12}]u = [\alpha_1 + \delta_1 \alpha_{12} ] \hat{u} $$
which make no sense. What should my approach be for this computation? Any help would be greatly appreciated. Data for UK is below.