# 金融代写|金融实证代写Financial Empirical 代考|FINE703

## 金融代写|金融实证代写Financial Empirical 代考|Statistical Background

Let us label the original domain of the data as the “parent space,” and all variables will be written in bold. The “transformation space” arises from application of a one-to-one mapping $\varphi$, which is chosen so as to reduce heteroscedasticity, skewness, and kurtosis in an effort to produce data that is closer to having a Gaussian structure. For Gaussian time series variables, the additive group structure is extremely natural: optimal mean square error estimates of quantities of interest (such as future values, missing values, unknown signals, etc.) are linear in the data, and hence are intimately linked to the addition operator. Errors in estimation are assessed by comparing estimator and target via subtraction-this applies to signal extraction, forecasting, and any other Gaussian prediction problem. Therefore the additive operator is quite natural for relating quantities in the transformation space.
It is for the above reasons (the linearity of estimators when the data is Gaussian) that the sum of signal and noise estimates equals the data process; no other algebraic operation is natural for relating Gaussian signal and noise. Given an observed time series $\left{\mathbf{X}_t\right}$ in the parent space, say for $1 \leq t \leq n$, the analyst would select $\varphi$ via exploratory analysis such that $X_t=\varphi\left(\mathbf{X}_t\right)$ is representable as a sample from a Gaussian process. Most of the classical results on signal extraction (Bell, 1984; McElroy, 2008) and projection (Brockwell and Davis, 1291) are interpretable in terms of a Gaussian distribution. Mơré précisély, thé éstimates coommonly uséd in time series applications minimize the mean squared prediction error among all linear estimators, and are also conditional expectations when the process is Gaussian. If $\varphi$ does not produce a Gaussian distribution, at a minimum it should reduce skewness and kurtosis in the marginal distributions.

Also, it is necessary that $\varphi$ be invertible, and it will be convenient for it to be a continuously differentiable function. Denoting the joint probability distribution function (pdf) of the transformed data by $p_{X_1, \cdots, X_n}\left(x_1, \cdots, x_n\right)$, the joint pdf of the original data is then
$$p_{\mathbf{X}1, \cdots, \mathbf{X}_n}\left(\mathbf{x}_1, \cdots, \mathbf{x}_n\right)=p{X_1, \cdots, X_n}\left(x_1, \cdots, x_n\right) \cdot \Pi_{t=1}^n \frac{\partial \varphi\left(\mathbf{x}t\right)}{\partial x} .$$ Of course, here $x_t=\varphi\left(\mathbf{x}_t\right)$. If we select a parametric family to model $p{X_1, \cdots, X_n}$, e.g., a multivariate Gaussian pdf, then (1) can be viewed as a function of model parameters rather than of observed data, and we obtain the likelihood. It is apparent that the Jacobian factor does not depend on the parameters, and hence is irrelevant for model fitting purposes. That is, the model parameter estimates are unchanged by working with the likelihood in the parent space.

## 金融代写|金融实证代写Financial Empirical 代考|Algebraic Structure of the Parent Space

Given an additive operation in the transformed space, e.g., $x_t+x_{t-1}$, it is crucial to define a corresponding composition rule $\oplus$ in the parent domain such that $\varphi$ is a group homomorphism. A group is a set together with an associative composition law, such that an identity element exists and every element has an inverse (Artin, 1991). A homomorphism is a transformation of groups such that the laws of composition are respected. The groups under consideration are $\mathscr{R}=(\mathbb{R},+)$ for the transformed space, and $\mathscr{G}=\left(\varphi^{-1}(\mathbb{R}), \oplus\right)$ for the parent space. Consider the situation of latent components in the transformed space, where $X_t=S_t+N_t$ is a generic signal-noise decomposition. Then the components in the parent space are $\varphi^{-1}\left(S_t\right)=\mathbf{S}_t$ and $\varphi^{-1}\left(N_t\right)=\mathbf{N}_t$, which can be quantities of interest in their own right. How do we define an algebraic structure that allows us to combine $\mathbf{S}_t$ and $\mathbf{N}_t$, such that the result is always $\mathbf{X}_t$ ? What is needed is a group operator $\oplus$ such that
$$\mathbf{S}_t \oplus \mathbf{N}_t=\mathbf{X}_t=\varphi^{-1}\left(S_t+N_t\right)=\varphi^{-1}\left(\varphi\left(\mathbf{S}_t\right)+\varphi\left(\mathbf{N}_t\right)\right)$$
This equation actually suggests the definition of $\oplus$ : any two elements $\mathbf{a}, \mathbf{b}$ in the parent group $B G$ are summed via the rule
$$\mathbf{a} \oplus \mathbf{b}=\varphi^{-1}(\varphi(\mathbf{a})+\varphi(\mathbf{b})) .$$
This definition “lifts” the additive group structure of $\mathscr{R}$ to $\mathscr{G}$ such that: (1) $\varphi^{-1}(0)=$ $1_G$ is the unique identity element of $\mathscr{G} ;(2) \mathscr{G}$ has the associative property; (3) the unique inverse of any $\mathbf{a} \in \mathscr{G}$ is given by $\mathbf{a}^{-1}=\varphi^{-1}(-\varphi(\mathbf{a}))$. These properties are verified below, and establish that $\mathscr{G}$ is indeed a group. Moreover, the group is Abelian and $\varphi$ is a group isomorphism.

First, a $\oplus \varphi^{-1}(0)=\varphi^{-1}(\varphi(\mathbf{a})+0)=\mathbf{a}$, which together with the reverse calculation shows that $\varphi^{-1}(0)$ is an identity; uniqueness similarly follows. Associativity is a book-keeping exercise. For the inverse, note that $\mathbf{a} \oplus \varphi^{-1}(-\varphi(\mathbf{a}))=$ $\varphi^{-1}(\varphi(\mathbf{a})-\varphi(\mathbf{a}))=\varphi^{-1}(0)=1 \mathscr{G}$. This shows that $\mathscr{G}$ is a group, and commutativity follows from (3) and the commutativity of addition; hence, $\mathscr{G}$ is an Abelian group. Finally, $\varphi$ is a bijection as well as a homomorphism, i.e., it is an isomorphism.

## 金融代写|金融实证代写Financial Empirical 代考|Statistical Background

$$p_{\mathbf{X} 1, \cdots, \mathbf{X}n}\left(\mathbf{x}_1, \cdots, \mathbf{x}_n\right)=p X_1, \cdots, X_n\left(x_1, \cdots, x_n\right) \cdot \Pi{t=1}^n \frac{\partial \varphi(\mathbf{x} t)}{\partial x} .$$

## 金融代写|金融实证代写Financial Empirical 代考|Algebraic Structure of the Parent Space

$$\mathbf{S}_t \oplus \mathbf{N}_t=\mathbf{X}_t=\varphi^{-1}\left(S_t+N_t\right)=\varphi^{-1}\left(\varphi\left(\mathbf{S}_t\right)+\varphi\left(\mathbf{N}_t\right)\right)$$

$$\mathbf{a} \oplus \mathbf{b}=\varphi^{-1}(\varphi(\mathbf{a})+\varphi(\mathbf{b})) .$$

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