Financial prediction and feature construction

Definition Fix a decision time $t$, a horizon $h > 0$ and the physical measure $\mathbb{P}$, since the question concerns the distribution of future observations. A prediction target is a real random variable $Y$ whose value is determined by the information available at time $t + h$, so $Y$ is $\mathcal{F}_{t+h}$-measurable, while in general it is not $\mathcal{F}_t$-measurable. Defining it means stating the quantity, such as a simple or log return, an excess return, a sign or an event indicator, together with its units, the start and end of its interval and the treatment of cash flows. Squared-error prediction also assumes $\mathbb{E}_{\mathbb{P}}[Y^2] < \infty$. Different targets built from the same return have different best predictors.

Definition The label horizon $h$ is the length of the interval $(t, t + h]$ over which the target of the row dated $t$ is measured. For a self-financing position with value $W_t > 0$, the horizon-$h$ total return label is

It becomes known only at $t + h$, so a model fitted at time $s$ can train only on rows with $t + h \le s$. When rows are dated at a spacing $\Delta < h$, the labels of two rows $k\Delta$ apart with $k\Delta < h$ share the interval $(t + k\Delta, t + h]$ and are in general dependent. If daily log returns are independent and identically distributed with finite positive variance, the $h$-day log return labels of rows $k$ days apart, with $0 \le k \le h$, have correlation

because the two sums share exactly $h - k$ daily terms, and labels at least $h$ days apart are uncorrelated.

Definition A feature for a forecast made at time $t$ is a random vector $X_t$, and it is admissible when $\sigma(X_t) \subseteq \mathcal{F}_t$, that is, when it is $\mathcal{F}_t$-measurable. The standard construction is

where $O_t$ collects the observations available by $t$, the as-of view of Point in time financial data and sampling, and $\phi$ is a Borel function chosen without data from after $t$. If $\mathcal{F}_t = \sigma(O_t)$, the Doob-Dynkin lemma shows that every admissible real feature has this form. In the same way, the random variables measurable with respect to $\sigma(X_t)$ are exactly the functions $g(X_t)$ with $g$ Borel, so these are the forecasts that use only the features. A feature vector can discard information. With $\mathcal{G} = \sigma(X_t) \subseteq \mathcal{F}_t$, the best squared-error forecast based on the features is $\mathbb{E}[Y \mid X_t]$, and its error is at least that of $\mathbb{E}[Y \mid \mathcal{F}_t]$, as the squared-error card shows. A rule allowed to use $Y$ itself would have zero error, which is why a feature that reaches past $t$ can make a backtest look better than any admissible rule.

Definition The availability time of a feature value is the latest availability time among everything it uses. That includes the raw observations, their revisions and the data used to fit any parameter of $\phi$, such as a mean, a scale, a rank, a regression coefficient or the membership of a universe. The feature is available at $t$ when that time is at most $t$, and a feature available at $t$ is admissible there. A fitted step is fitted on the data available for fitting and then applied, with those parameters, to later rows. Standardizing with the mean $\bar{x}$ and the sample standard deviation $s_x$ of a full sample $x_1, \dots, x_T$,

gives every row the availability time of the last observation $x_T$.

Definition Suppose $\mathbb{E}_{\mathbb{P}}[\lvert Y \rvert] < \infty$. The conditional mean predictor is

the integrable $\mathcal{F}_t$-measurable random variable with $\mathbb{E}[m_t \mathbf{1}_A] = \mathbb{E}[Y \mathbf{1}_A]$ for every $A \in \mathcal{F}_t$. It exists and is unique up to events of probability zero. When $\mathcal{F}_t$ is generated by a finite partition into cells $A_1, \dots, A_k$ with $\mathbb{P}(A_i) > 0$, it is the cell average

The expectation is taken under the physical measure. A conditional expectation under a pricing measure $\mathbb{Q}$ uses pricing weights and is not a return forecast under $\mathbb{P}$, even though the notation looks almost the same. The properties of conditional expectation used here are those of Expectation conditional expectation and variance.

Definition Squared-error prediction scores a forecast $a$ of $Y$ by the loss $(Y - a)^2$, and a forecasting rule by its population risk $\mathbb{E}_{\mathbb{P}}[(Y - a_t)^2]$. Suppose $\mathbb{E}[Y^2] < \infty$, and let $a_t$ be any $\mathcal{F}_t$-measurable predictor with $\mathbb{E}[a_t^2] < \infty$. Then

To prove it, write $Y - a_t = (Y - m_t) + (m_t - a_t)$ and expand the square. The predictor $m_t$ has a finite second moment by the conditional Jensen inequality, so the cross term is integrable by the Cauchy-Schwarz inequality. Because $m_t - a_t$ is known given $\mathcal{F}_t$,

and the tower property makes the unconditional cross term zero as well. This says that the residual $Y - m_t$ is orthogonal to every square-integrable $\mathcal{F}_t$-measurable variable, which does not require the two to be independent. The last term of the identity is nonnegative and vanishes exactly when $a_t = m_t$ almost surely, so the conditional mean is the unique optimal predictor up to events of probability zero. For a smaller information set $\mathcal{G} \subseteq \mathcal{F}_t$, such as the one generated by the features, the identity with $a_t = \mathbb{E}[Y \mid \mathcal{G}]$ shows that the risk rises by $\mathbb{E}[(m_t - \mathbb{E}[Y \mid \mathcal{G}])^2]$. With $\mathcal{G}$ trivial this is the law of total variance

The theorem identifies a population target for estimation. It does not say that a model fitted to finite data recovers it, and it says nothing about other losses or about trading decisions.