问题如下:
When modeling lnYt using a time trend model, what is the relationship between expET[lnYT+h] and ET[YT+h] for any forecasting period h? Are these ever the same? Assume the error terms is normally distributed around a mean of zero.
解释:
A time trend model for lnYt can be stated as:
lnYt=g(t)+ϵt,ϵ∼N(0,σ2),
where g(t) is a function of t.
So,
ET[lnYT+h]=g(T+h),
which gives
expET[lnYT+h]=exp[g(T+h)],
On the other hand:
ET[YT+h]=ET[exp(g(T+h)+ϵT+h)]=exp(g(T+h)+ET[exp epsilonT+h)],
which equals
ET[YT+h]=exp[g(T+h)]+σ2/2
And so:
ET[YT+h]=expET[lnYT+h]+σ2/2
These will be equal if the variance is zero (in other words, if the process is completely deterministic.
误差服从idd,那误差的指数怎么求方差啊?能否讲一下sigma平方/2怎么来的呀?