Liquidity puzzle in the representative household model

In a cash in advance representative household model, the nominal interest rate may be determined by
$$frac{1}{1 + i_t}= βE_tleft(frac{M_t}{M_{t+1}}right)$$
where $beta$ is the discount rate.

1. I’m interpreting this model as if $M_t$ is known at time t. Is this a valid interpretation?
2. How from a temporary increase of the current money supply, which does not affect the expectation of the future money supply, can the impact on the nominal interest rate be given by $frac{∂i_t}{∂M_t} = −frac{1}{M_t E_t(M_t/M_{t+1})}< 0$
3. Assume that the growth rate of the money supply follows a linear first-order, stationary autoregressive stochastic process of the form $$ln(M_t/M_{t−1})= µ(1 − λ) + λ ln(M_{t−1}/M_{t−2}) + ε_t$$, where $0 <λ<1$, and $ε_t$ is a white noise process. In this setting, how can a positive shock in $ε_t$ lead to an increase in $i_t$?

In the last two questions, I’m interested in the mathematical details. I understand the economic importance of this example. If what I ask in 3. is true, then we have an opposite effect from what we would expect (the liquidity effect = increase in the money supply, decrease in nominal interest rate) and hence the puzzle. However, the mathematics is escaping me, somehow… There’s some assumptions I may not be considering that allows us to reach the conclusions above.

All this was taken from the book Dynamic Macroeconomics by Alogoskoufis (great book, has taught me much so far. It’s just that I’m not a good economist, nor mathematician. =D).