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4 exercise(s) shown, 0 hidden
BBy Bot
May 21'24

Using the results of Chapter, show that the following holds for the multivariate regression model.

1. There exists an estimator $\hat \Theta \in \R^{d \times T}$ such that
[$] \frac{1}{n}\|\X\hat \Theta -\X \Theta^*\|_F^2 \lesssim \sigma^2\frac{rT}{n} [$]
with probability .99, where $r$ denotes the rank of $\X$\,.
2. There exists an estimator $\hat \Theta \in \R^{d \times T}$ such that
[$] \frac{1}{n}\|\X\hat \Theta -\X \Theta^*\|_F^2 \lesssim \sigma^2\frac{|\Theta^*|_0\log(ed)}{n}\,. [$]
with probability .99.
BBy Bot
May 21'24

Consider the multivariate regression model where $\Y$ has SVD:

[$] \Y=\sum_j\hat \lambda_j \hat u_j \hat v_j^\top\,. [$]

Let $M$ be defined by

[$] \hat M=\sum_j\hat \lambda_j \1(|\hat \lambda_j| \gt 2\tau)\hat u_j \hat v_j^\top\,, \tau \gt 0\,. [$]

1. Show that there exists a choice of $\tau$ such that
[$] \frac{1}{n}\|\hat M -\X \Theta^*\|_F^2 \lesssim \frac{\sigma^2\rank(\Theta^*)}{n}(d\vee T) [$]
with probability .99.
2. Show that there exists a matrix $n \times n$ matrix $P$ such that $P\hat M=\X\hat \Theta$ for some estimator $\hat \Theta$ and
[$] \frac{1}{n}\|\X\hat \Theta -\X \Theta^*\|_F^2 \lesssim \frac{\sigma^2\rank(\Theta^*)}{n}(d\vee T) [$]
with probability .99.
3. Comment on the above results in light of the results obtain in Section.
BBy Bot
May 21'24

Consider the multivariate regression model and define $\hat \Theta$ be the any solution to the minimization problem

[$] \min_{\Theta \in \R^{d \times T}} \Big\{ \frac{1}{n}\|\Y-\X\Theta\|_F^2 + \tau \|\X\Theta\|_1\Big\} [$]

1. Show that there exists a choice of $\tau$ such that
[$] \frac{1}{n}\|\X \hat \Theta -\X \Theta^*\|_F^2 \lesssim \frac{\sigma^2\rank(\Theta^*)}{n}(d\vee T) [$]
with probability .99.

Hint: Consider the matrix
[$] \sum_j \frac{\hat\lambda_j + \lambda_j^*}{2}\hat u_j \hat v_j^\top [$]
where $\lambda^*_1\ge \lambda^*_2 \ge \dots$ and $\hat\lambda_1\ge \hat\lambda_2\ge \dots$ are the singular values of $\X\Theta^*$ and $\Y$ respectively and the SVD of $\Y$ is given by
[$] \Y=\sum_j\hat \lambda_j \hat u_j \hat v_j^\top [$]
2. Find a closed form for $\X\hat\Theta$.
BBy Bot
May 21'24

In the Markowitz theory of portfolio selection [1], a portfolio may be identified to a vector $u \in \R^d$ such that $u_j \ge 0$ and $\sum_{j=1}^d u_j=1$. In this case, $u_j$ represents the proportion of the portfolio invested in asset $j$. The vector of (random) returns of $d$ assets is denoted by $X \in \R^d$ and we assume that $X\sim \sg_d(1)$ and $\E[XX^\top]=\Sigma$ unknown.t In this theory, the two key characteristics of a portfolio $u$ are it's reward $\mu(u)=\E[u^\top X]$ and its risk $R(u)=\var{X^\top u}$. According to this theory one should fix a minimum reward $\lambda \gt 0$ and choose the optimal portfolio

[$] u^*=\argmin_{u\,: \mu(u) \ge \lambda} R(u) [$]

when a solution exists for a given. It is the portfolio that has minimum risk among all portfolios with reward at least $\lambda$, provided such portfolios exist. In practice, the distribution of $X$ is unknown. Assume that we observe $n$ independent copies $X_1, \ldots, X_n$ of $X$ and use them to compute the following estimators of $\mu(u)$ and $R(u)$ respectively:

[] \begin{align*} \hat \mu(u)&=\bar X^\top u =\frac{1}{n}\sum_{i=1}^n X_i^\top u\,, \\ \hat R(u)&= u^\top \hat \Sigma u, \quad \hat \Sigma = \frac{1}{n-1}\sum_{i=1}^n (X_i-\bar X)(X_i -\bar X)^\top\,. \end{align*} []

We use the following estimated portfolio:

[$] \hat u=\argmin_{u\,: \hat \mu(u) \ge \lambda} \hat R(u) [$]

We assume throughout that $\log d \ll n \ll d$\,.

1. Show that for any portfolio $u$, it holds
[$] \big|\hat \mu(u) - \mu(u)\big| \lesssim \frac{1}{\sqrt{n}}\,, [$]
and
[$] \big|\hat R(u) - R(u)\big| \lesssim \frac{1}{\sqrt{n}}\,. [$]
2. Show that
[$] \hat R(\hat u) - R(\hat u) \lesssim \sqrt{\frac{\log d}{n}}\,, [$]
with probability .99.
3. Define the estimator $\tilde u$ by:
[$] \tilde u=\argmin_{u\,: \hat \mu(u) \ge \lambda- \eps} \hat R(u) [$]
find the smallest $\eps \gt 0$ (up to multiplicative constant) such that we have $R(\tilde u) \le R(u^*)$ with probability .99.

References

1. Markowitz, Harry. "Portfolio selection". The journal of finance 7. Wiley Online Library.