Sampling and Asymptotic Inference for an Implicit Return Statistic
Summary
The document asks about the sampling and large-sample distributions of a statistic defined as the root of an equation involving observed portfolio returns and risk-free rates. By setting the displayed function to zero, the author reduces the problem to finding a parameter value for which the sample average of transformed relative returns equals one. This makes the statistic an implicitly defined estimator based on the observed series.
The author invites assumptions such as independent, identically distributed observations or stationary processes, but does not give a solution, a chosen set of assumptions, or data-based evidence. Under suitable regularity conditions, one would typically study an estimator defined by a sample moment using a law of large numbers and a central limit theorem, then apply an implicit-function or delta-method argument. Existence and uniqueness of the root, dependence over time, and behavior near the singular parameter value require attention; the document leaves these issues open.
Key ideas
- The parameter is defined implicitly by a sample equation in relative returns.
- The root depends on the observed returns and risk-free rates, so it is random across samples.
- The question distinguishes a fixed-sample distribution from behavior as the sample grows.
- Any asymptotic analysis requires assumptions on dependence, moments, and the existence and uniqueness of a root.
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Full text
# Sampling and/or asymptotic distribution of a function
# Sampling and/or asymptotic distribution of a function
Assume we have the following function:
$$f(p) = \frac{1}{(1-p)d}\ln\left(\frac{1}{T}\sum_{t=1}^{T}\left[\frac{1+X_t}{1+Y_t} \right]^{1-p} \right)$$
where
$d$ is a constant
$T$ is a constant
$X_t$ for $t = 1, 2, \cdots, T$ are random variables (it is actually the portfolio's annualized rate of return at time $t$)
$Y_t$ for $t = 1, 2, \cdots, T$ are random variables (it is actually the risk-free rate at time $t$)
$p$ is defined such that it is value that satisfies $f(p) = 0$
What is the sampling and/or asymptotic distribution for the statistic $p$?
By sampling distribution I mean the following:
The solution to $f(p) = 0$ doesn't have a closed-form solution, but it is obvious that the resulting value of $p$ depends on $X_t$ and $Y_t$, so $p$ can be treated as a random variable that depends on the random variables $X_t$ and $Y_t$. Then for every (fixed) $T$ observations of $X_t$ and $Y_t$, we have a corresponding value $p$ that satisfies $f(p) = 0$, what is the sampling distribution of $p$?
By asymptotic distribution I mean the following:
Similar to above, now assume $T$ isn't fixed, then clearly the solution $p$ to $f(p)=0$ implicitly depends on $T$, then what is the asymptotic distribution of $p$ as $T \rightarrow \infty$?
Assume you are allowed the following assumptions to achieve the above:
1) You can make any distributional assumptions regarding $X_t$ and $Y_t$, e.g., $X_t$ and $Y_t$ are independent from each other, also $X_t$, $Y_t$ for $t = 1, 2, \cdots, T$ are independently and identically distributed.
2) Rather than making distributional assumptions about $X_t$ and $Y_t$, assume you can make some assumptions about the processes $\{X_t\}$ and $\{Y_t\}$, e.g., both processes are stationary (or weakly stationary) etc.
3) Any assumption you see fit to yield a solution.
This is what I've tried so far.
If we let $Z_t = \frac {1 + X_t} {1 + Y_t}$, then we have $\displaystyle f(p) = \frac {1} {(1 - p)d} \ln\left(\frac {1} {T} \sum_{t=1}^T Z_t^{1-p} \right)$. Setting this equation to $0$ and rearranging, we have: $$\sum_{t=1}^T Z_t^{1-p} = T$$
The my question becomes:
1) Sampling distribution: For a fixed $T$, what is the distribution of $p$?
2) Asymptotic distribution: For $T \rightarrow \infty$, what is the distribution of $p$?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.