Interpretation of short and long run reactions in ADL(1,1) This is a task from my statistical minor. 
The question proposes that this equation is a model, where $x$ are health expenditures and $y$ is life expectancy in some country. If so, why is $\phi(0)$ negative? What does the long-run reaction mean here and is the negative sign of $\psi$ means that $y$ decreases in log run as $x$ increases, which would be strange? (It may be that I have done an arithmetical mistake)
What does the returning-to-stability period T mean here?
Let us have Autoregressive Distributed Lag Model
$$(1-0.1L)y_t= 10 - \frac{1}{2}x_{t}+0.500 x_{t-1} - 0.072 x_{t-1} $$
$$(1-0.1L)y_t= 10 - \frac{1}{2}x_{t}+0.428 x_{t-1}  $$
$$(1-0.1L)y_t= 10 - \frac{1}{2}(x_{t}-0.856 x_{t-1}) $$
$$y_t = \frac{100}{9}- \frac{1-0.856L}{2(1-0.1L)}x_t$$
Equivalently 
 $$ \Delta y_t =-\frac{1}{2}\Delta x_t - (1-0.1)(y_{t-1}- \frac{10}{0.9} + \frac{0.72}{0.9}x_{t-1})$$
The stability returning period is $T = 1/0.9 = 1.1$
I am asked to find short-run and long-run reactions of the model.
$$y_t= \frac{100}{9} - \frac{1}{2}x_t + \frac{1}{2} \sum_{i=1}^{\infty}{0.756}^i x_{t-i} $$
That means that short-run reaction $\phi(0) = -\frac{1}{2}$ and $\phi(i) = 0.756^i/2$
The long-run reaction $\psi = \frac{-0.5+0.428}{1-0.1} = -\frac{72}{90}$
 A: I anticipate that I have not reviewed all the calculations punctually, however, I want to answer the initial question with my interpretation of the results that you get (which are not necessarily wrong or counterintuitive in my opinion, and I will explain why as follows)

What does the long-run reaction mean here and is the negative sign of ψ means that y decreases in log run as x increases, which would be strange? (It may be that I have done an arithmetical mistake) What does the returning-to-stability period T mean here?

Personally I have been historically more inclined to watch this 
$$ \Delta y_t =-\frac{1}{2}\Delta x_t - (1-0.1)(y_{t-1}- \frac{10}{0.9} + \frac{0.72}{0.9}x_{t-1})$$
as an Error Correction Model, provided that the variables x and y are I(1) and cointegrated (i.e. the term $ -(1-0.1)(y_{t-1}- \frac{10}{0.9} + \frac{0.72}{0.9}x_{t-1})$ is significant). If you look at the ECM above, the interpretation is that:
1) The long-run relationship between health expenditures and life expectancy is $ (y_{t-1} = \frac{10}{0.9} - \frac{0.72}{0.9}x_{t-1})$ which means that possible eventual errors/divergences from the long-run equilibrium relationship mentioned above (divergences are $(y_{t-1} - \frac{10}{0.9} + \frac{0.72}{0.9}x_{t-1})$) will be corrected over time (as the coefficient $-(1-0.1)$ is negative) to resume the convergence to such relationship in the long-run. Indeed, when the error is positive because today life expectancy is far above what historically justified based on the current expenditures (i.e. $y_{t-1}>- \frac{10}{0.9} + \frac{0.72}{0.9}x_{t-1})$), then the product $-(1-0.1)(y_{t-1}- \frac{10}{0.9} + \frac{0.72}{0.9}x_{t-1})$ will be negative. The interpretation of the latter may be that, over time, the lack of today adequate expenditures compared to life expectancy will weight on the future expectancy.
2) Periods denoted by positive increase in health expenditures ($\Delta x_{t}<0$) are usually periods where life expectancy is shrinking ($\Delta y_{t}<0$) (assuming that we are at the long term equilibrium so that the error is 0 and the term described above goes to 0, otherwise you have to include the effect of the error term while forecasting $\Delta y_{t}$). So there is a simultaneous association between the two. But, over the long term, the relationship is positive and illustrated at point above. 
To illustrate the effect of the past-period error term, we could say that, if we are in a period where the life expectancy is far below the historical relationship with the health expenditure pursued by the state (i.e. the state is spending a lot but the health system is not yet gaining from that and the life expectancy is low, so the error is negative), then, over time, the things will rebalance (the life expectancy will get better) and this will have an effect on the present variation of the current life expectancy ($\Delta x_{t}$) given the current variation in the health expenditure ($\Delta y_{t}$).
