Raising the bar

Optimal forecast properties

In this post we consider the properties described in section 11.3.1 "Testable implications of forecast optimality" of the Guide (FN3142 Quantitative finance study guide by A. Patton).

There are two tools to keep in mind.

Fact 1. The conditional expectation $latex E_{t}Y_{t+h}$ is the best prediction of a future random variable $latex Y_{t+h}$ in the sense that it minimizes $latex E_{t}\left( Y_{t+h}-f\left( I_{t}\right) \right) ^{2}$ over all functions of the current information set $latex I_{t}$.

Fact 2. According to the Wold representation theorem every covariance-stationary time series $latex Y_{t}$ can be written as the sum of two time series, one deterministic and one stochastic. We will use a simplified version of that representation:

(1) $latex Y_{t}=\varepsilon _{t}+b_{1}\varepsilon _{t-1}+...=\sum_{j=0}^{\infty }b_{j}\varepsilon _{t-j},$

where $latex b_{0}=1,$ $latex \sum_{j=0}^{\infty }\left\vert b_{j}\right\vert ^{2}<\infty $ and $latex \left\{ \varepsilon _{t}\right\} $ is white noise in the sense that

$latex E_{t}\varepsilon _{t+h}=0$ for $latex h\geq 1,$ $latex Var\left( \varepsilon _{t}\right) =\sigma ^{2}$ for all $latex t$ and $latex Cov\left( \varepsilon _{t},\varepsilon _{s}\right) =0$ for all $latex t\neq s.$

In general, forecasts can arise from different procedures. We accept $latex \hat{Y} _{t+h|t}=E_{t}Y_{t+h}$ as the forecast. Because of $latex Y_{t+h}=E_{t}Y_{t+h}+(Y_{t+h}-E_{t}Y_{t+h})$ it is natural to think of $latex e_{t+h,t}=Y_{t+h}-E_{t}Y_{t+h}$ as the forecast error, see Conditional-mean-plus-remainder representation.

Properties

See Canonical form for time series for similar arguments.

1). $latex \hat{Y}_{t+h|t}$ is conditionally constant, $latex E_{t}\hat{Y}_{t+h|t}=E_{t}(E_{t}Y_{t+h})=\hat{Y}_{t+h|t},$ and the forecast error is conditionally zero, $latex E_{t}e_{t+h|t}=E_{t}\left( Y_{t+h}-E_{t}Y_{t+h}\right)=0.$

2). Using (1) we have

$latex \hat{Y}_{t+h|t}=E_{t}Y_{t+h}=\sum_{j=0}^{\infty }b_{j}\varepsilon _{t+h-j}$

$latex =\sum_{j=0}^{h-1}b_{j}E_t\varepsilon _{t+h-j}+\sum_{j=h}^{\infty }b_{j}E_t\varepsilon _{t+h-j}=\sum_{j=h}^{\infty }b_{j}\varepsilon _{t+h-j}.$

This implies

(2) $latex e_{t+h,t}=Y_{t+h}-E_{t}Y_{t+h}=\sum_{j=0}^{h-1}b_{j}\varepsilon _{t+h-j}=b_{0}\varepsilon _{t+h}+...+b_{h-1}\varepsilon _{t+1},$

that is, the number of terms in the error grows. Therefore, variance of the optimal forecast error,

$latex Var\left( e_{t+h,t}\right) =\sigma ^{2}\sum_{j=0}^{h-1}\left\vert b_{j}\right\vert ^{2}$

is increasing in $latex h$.

This property asserts that the distant future is harder to forecast than the near future. If it were easier to forecast next month than it is to forecast tomorrow, we should simply throw away what we've learned in the last month and forecast tomorrow using only information from a month ago.

3). Consider lags of orders $latex k\geq h$ of the error (2):

$latex L_{k}\left( b_{0}\varepsilon _{t+h}+...+b_{h-1}\varepsilon _{t+1}\right) =b_{0}\varepsilon _{t+h-k}+...+b_{h-1}\varepsilon _{t+1-k}.$

For example, with $latex k=h$ it is

(3) $latex b_{0}\varepsilon _{t}+...+b_{h-1}\varepsilon _{t+1-h}.$

We see that the sums in (2) and (3) have non-overlapping subscripts and therefore have correlation zero. The same is true for all $latex k\geq h.$ Thus, optimal forecast errors have zero autocorrelation for lags greater than or equal to $latex h$.

This implies that the forecast error should not be forecastable, otherwise the forecast itself could be improved.