问一道题：NO.PZ2020011101000019 [ FRM I ]

问题如下：

When modeling $ln Y_t$ using a time trend model, what is the relationship between $exp E_T[ln Y_{T+ h}]$ and $E_T[Y_{T+ h}]$ 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 $ln Y_t$ can be stated as:

$ln Y_t = g(t) + \epsilon_t, \epsilon ∼ N(0, \sigma^2)$,

where g(t) is a function of t.

So,

$E_T[ln Y_{T+ h}] = g(T + h)$,

which gives

$exp E_T[ln Y_{T+ h}] = exp [g(T + h)]$,

On the other hand:

$E_T[Y_{T+ h}] = E_T[exp(g(T + h) + \epsilon_{T+ h})] = exp(g(T + h) + E_T[exp \ epsilon_{T+ h})]$,

which equals

$E_T[Y_{T+ h}] = exp[g(T + h)] +\sigma^2/2$

And so:

$E_T [Y_{T+ h}] = exp E_T[ln Y_{T+ h}] +\sigma^2/2$

These will be equal if the variance is zero (in other words, if the process is completely deterministic.