how to make a distribution model tolerable of trend?

Question

I'm building an model on different loans' NPL rate. The problem is NPL rates are always affected by the market. When NPL rates move in trend, my model will fail the back-testing.

Assuming $x(t)$ is a random variable that distributed among $[-1, 1]$, with the mean $\mu = 0$ and a standard deviation $\sigma$.

When the sample size $n$ is big, the distribution of observed mean $\bar x$ will be ~ $N(0, \sigma^2/n)$. The back-testing confidence interval for $\bar x$ is $[- z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} , z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} ]$. The model always passes the backtest.

Now, troubles come when $x(t)$ has some trend. Let's say $x(t)$ becomes $x(t) = \sin (t + random(t) )$. Here $x(t)$ still follows the distribution, but when $x(t)$ moves near $+1$, the samples' mean will be around $+1$, increasing the sample size will not bring the $\bar x$ near to $0$, the model fails the backtesting.

Now my problem is, NPL's trend is hard to predict, the cycle sometimes are 6 months, sometimes are 2 years. My NPL model always fails the back-testing because of trend.

Any suggestion please?

Answer 1

In my experience with forecasting, you could try a model of the form $$ X_ t = cycle_t + seasonality_t + residuum_t. $$ Sometimes it is hard to find the cycle but the seasonality could be doable if it has some natural structure (something happening in a certain month each year e.g.). Rob Hyndman explains all these things (and provides an R package) in his free online book. You could have a look at chapter 6. After having calculated the first 2 components you could model the distribution of the residuum.

Answer 2

you could calculate and subtract the trend.

dx=h(m-x)*dt+s*dz
x_(t) - x_(t - 1) = m (1 - exp(- h Dt)) + (exp(- h Dt) - 1) x_(t - 1) + e_t
error e_t normally distributed with

(s_e)^2 = [1 - exp(- 2 h)] (s^2)/2h

Because when Dt->0

x_t - x_(t - 1)  ~h(m-x_(t - 1))dt +  et 

you calculate the parameters h and m from the mean regression after that....?