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An Upper Bound for Functions of Estimators in High Dimensions
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The delta method is one of the most widely used theorems in econometrics and statistics. It is a very simple and a useful idea. It can provide limits for complicated functions of estimators as long as the function is differentiable. The limit of the function of estimators can be obtained from the limit of the estimators, with the same rate of convergence. In the case of finite-dimensional parameter estimation, since the derivative at the parameter value is finite, rates of convergence of both estimators and function of estimators are the same.
In the case of high dimensional parameter estimation, we show that this is not the case, and the rates of convergence may change. We show that the structure of the derivative of the function is the key. An upper bound random variable is provided for the functions of estimators in high dimensions. We show this upper bound on functions of estimators may converge faster, slower, or at the same rate as estimators. Even though a new delta theorem is not provided, this upper bound can get the rate of convergence of functions of estimators. For example, the variance of a portfolio is a quadratic function of the portfolio weight. Our theorem implies that the convergence rate of the portfolio variance is slower than that of the estimated weight when the number of assets is diverging (see Example 3 in Section 3). From now on we denote the number of assets as $p_n$ since they grow with sample size.
Our result is useful when the number of parameters $p_{n}$ is larger than the sample size $n$, where $p_{n}$ grows with $n$. The reason is that proofs in high dimensional problems usually depend on knowing the rate of convergence of the bound, which may be a lasso-type estimation, oracle inequality, or problems in high dimensional portfolio analysis.
After the main theorem, we illustrate our point in three examples: first by examining a linear function of estimators that is heavily used in econometrics, second a new debiased lasso type of estimator, and third by analyzing the out-of-sample variance of a large portfolio of assets in finance.
Section 2 provides our theorem. Section 3 has three examples. Section 4 provides a discussion of how the results may be tied to nonparametric analysis and many weak instrument asymptotics. Appendix shows the proof and provides more examples.
Let $\beta_{0}=(\beta_{1,0},\cdots,\beta_{p_n,0})'$ be a $p_n\times1$ parameter vector with an estimator $\hat{\beta}=(\hat{\beta}_{1},\cdots,\hat{\beta}_{p_n})'$. Define a function $f(.)$, $f:K\subset R^{p_n}\to R^{m}$. To be specific $m$ is a constant unless noted otherwise.
We provide two conditions for our theorem. First, denote a column vector of $p_{n}$ zeros by $0_{p_{n}}$. Let $f_{d}(.)$ represent the $m\times p_{n}$ matrix of derivatives.
Condition C1. For all $h\neq0_{p_{n}}$, and $h$ is a $p_{n}\times1$ vector \[ \lim_{h\to0}\frac{\|f(\beta_{0}+h)-f(\beta_{0})-f_{d}(\beta_{0})h\|_{2}}{\|h\|_{2}}=0,\] where $\|.\|_{2}$ is the Euclidean norm for a generic vector, and $f_{d}(\beta_{0})$ is an $m\times p_{n}$ matrix, whose $(i,j)$ th cell consists of $\partial f_{i}/\partial\beta_{j}$ evaluated at $\beta_{0}$, for $i=1,\cdots,m$, $j=1,\cdots,p_{n}$.
Second, define the rate of convergence of an estimator as $r_{n}$, a positive sequence in $n$, and when $n\to\infty$, $r_{n}\to\infty$.
Condition C2. \[ r_{n}\|\hat{\beta}-\beta_{0}\|_{2}=O_{p}(1).\]
Condition C1 is a high-level assumption that shapes our function of interest. It restricts the function $f$ to be differentiable. Hence, C1 rules out continuous functions that are non-differentiable at $\beta_{0}$. Condition C2 gives a convergence rate for the estimator of interest. Several examples about $\hat{\beta}$ and $r_{n}$ that satisfy C2 are provided in Section 3.
The following theorem is the main theoretical result. Given the convergence rate of the high-dimensional estimator, it provides an upper bound (random variable) for the estimation error of functions of the high-dimensional estimator. This is useful in econometric theory since it can give us an idea about what the rate of convergence of functions of estimators might be. Let $k_{n}$ and $d_{n}$ be positive sequences in $n$ so that both $k_{n}\to\infty$ and $d_{n}\to\infty$ as $n\to\infty$. Take a generic matrix, $D$, of dimensions $ m \times p_n$: $\|| D \||_{2}$ denotes the Frobenius norm of the matrix $D$: \[ \|| D \||_2 = \sqrt{\sum_{i=1}^m \| d_i \|_2^2},\] where $d_i$ is a $p_n \times 1$ vector, and its transpose $d_i'$ is the $i$th row of $D$.
Remarks. 1. The theorem does not provide a limit for functions of estimators, so this is not the delta theorem.
2. Part (a) shows under what conditions we can get the same rate of convergence for the functions of estimators compared with the rate of convergence of $\hat{\beta}-\beta_{0}$. Example 2 illustrates this point.
3. Part (b) shows that the $l_{2}$ norm of the partial derivative function may change with the dimension of the parameter vector. When $r_n > k_n$, and $k_{n}\to\infty$, then the upper bound $L_{n}$ converges at a slower rate than estimators of parameters. With $r_{n}/k_{n}\to0$, then the upper bound is diverging which tells us that the function of estimators may diverge, too.
4. Part (c) shows that the function of the estimators converges to zero in probability faster than the rate of convergence of estimators $r_{n}$.
5. Theorem (ref) also holds with $l_{1}$ and $l_{\infty}$ norms. These new norm results can be shown when Conditions C1 and C2 hold in $l_{1}$ and $l_{\infty}$ norms. We discuss this in Part A of the Appendix.
We now provide three examples that highlight the contribution. The first one is related to the linear functions of estimators, the second one considers the Debiased Conservative Lasso (DCL) of Caner and Kock (2018), and the third one is related to the out-of-sample variance of large portfolios.
Example 1.
This example considers lasso, which is one of the benchmark methods in machine learning. It is a penalized least squares estimator with $l_{1}$ penalty. The penalty induces sparsity in the model, which can prevent overfitting (Hastie, Tibshirani, Friedman, 2009, Tibshirani, 1996).
Let us denote $\beta_{0}$ as the true value of vector ($p_{n}\times1$) of coefficients. The number of the true nonzero coefficients is denoted by $s_{0}$, and $s_{0}>0$. A simple linear model is:
for $t=1,\cdots,n$. For simplicity, we assume that $u_{t}$ are iid errors with zero mean and finite variance and $x_{t}$ is a set of $p_{n}$ deterministic regressors. The iid assumption on $u_{t}$ is to keep our illustration simple and the following result still holds under more general assumptions on $u_{t}$ and $x_{t}$ as shown in Assumption 1 of Caner and Kock (2018).
The lasso estimator is defined as
where $\beta_{j}$ is the $j$ th element of $\beta$, $\lambda_{n}$ is a positive tuning parameter, and it is established that $\lambda_{n}=O(\sqrt{\frac{logp_{n}}{n}})$. Corollary 6.14 or Lemma 6.10 of Buhlmann and van de Geer (2011) shows that, for lasso estimators $\hat{\beta}$, with $p_{n}>n$
where
Given ((ref)), we may be interested in the large sample behavior of $D(\hat{\beta}-\beta_{0})$, where $D$ is an $m\times p_{n}$ matrix. The $D$ matrix can be thought of putting restrictions on $\beta_{0}$. We want to see whether $D(\hat{\beta}-\beta_{0})$ has a different rate of convergence from $\hat{\beta}-\beta_{0}$. From our Theorem (ref)(a), it is clear that $f_{d}(\beta_{0})=D$. Basically in the case of inference, this matrix and the vectors show how many of $\beta_{0}$ will be involved with the restrictions. If we want to use $s_{0}$ elements in each row of $D$ to test $m$ restrictions, then $\||D\||_{2}=O(\sqrt{s_{0}})$. Note that this corresponds to using $s_{0}$ elements in $\beta_{0}$ for testing $m$ restrictions. In other words, if $k_{n}=s_{0}$ and $s_{0}\to\infty$ as $n\to\infty$, then Theorem (ref)(b) implies that \[ L_{n}=O_{p}(\frac{\sqrt{s_{0}}}{r_{n}}).\] This means that even with a fixed number of $m$ restrictions, the upper bound random variable has a slower rate of convergence than the estimators, so it is possible that functions of estimators also converge slower to a limit.
Remark. In high dimensions, a common assumption is to impose $\|d_{i}\|_{2}=1$. See, for example, Caner, Han and Lee (2018) and Caner and Kock (2018). In that case, \[ \||D\||_{2}=\sqrt{m},\] where $m$ is fixed, but $p_{n}$ is growing with $n$. The rate of convergence of $D(\hat{\beta}-\beta_{0})$ will be still $r_{n}$, which is in ((ref)). Thus, there will be no slowdown of the rate of convergence and this is a sharp rate since $\||f_{d}(\beta_{0})\||_{2}=\||D\||_{2}=\sqrt{m}$. This is not an upper bound on $f_{d}(\beta_{0})$.
Example 2. Another estimator that is recently analyzed in the context of $p_{n}>n$ is the DCL of Caner and Kock (2018). Consider the model in example 1 above in the matrix form: \[ Y=X\beta_{0}+u,\] where $Y$ is an $n\times1$ vector, $X$ is an $n\times p_{n}$ matrix, $\beta_{0}$ is a $p_{n}\times1$ vector that consists of $s_{0}$ nonzero parameters, $u$ is an $n\times1$ error vector. $\hat{\beta}_{CL}$ is the conservative lasso estimator defined as \[ \hat{\beta}_{CL}=argmin_{\beta\in R^{p_{n}}}\{\|Y-X\beta\|_{2}^{2}/n+2\lambda_{n}\sum_{j=1}^{p_{n}}\hat{w}_{j}|\beta_{j}|\},\] where $\hat{w}_{j}=\lambda_{n}/max(|\hat{\beta}_{j}|,\lambda_{n})$, $max(a,b)$ chooses maximum of two elements $a$ or $b$, and $\hat{\beta}_{j}$ is the lasso estimator defined in example 1. The DCL is uniformly consistent, and it has a standard normal asymptotic limit and an asymptotically valid uniform confidence band, unlike the lasso and the conservative lasso. These are established in Theorem 3 of Caner and Kock (2018). The formula for the DCL estimator $\hat{b}$ is: \[ \hat{b}=\hat{\beta}_{CL}+\hat{\Theta}X'(Y-X\hat{\beta}_{CL})/n,\] where $\hat{\Theta}$ is an approximate estimate for precision matrix, which will be abstracted away in this paper. Information and detailed properties are described in Section 3.2 of Caner and Kock (2018). Specifically, Caner and Kock (2018) derive the rate of convergence for Wald and $\chi^{2}$ type of tests. They also show that the confidence bands on DCL are contracting at the optimal rate of $n^{-1/2}$. By Theorem 2 of Caner and Kock (2018), we have \[ n^{1/2}(\hat{b}_{j}-\beta_{0j})=O_{p}(1),\] for $j=1,\cdots,p_{n}$. If we want to test $m$ restrictions on $\beta_{0}$, with $H=[h_{1},...,h_{m}]^{\prime}$ representing the restriction matrix of $m\times p_{n}$ dimension, then \[ \||H\||_{2}=\sqrt{\sum_{i=1}^{m}\|h_{i}\|_{2}^{2}}=\sqrt{m},\] given $\|h_{i}\|_{2}=1$ in Theorem 2 of Caner and Kock (2018). Conditions C1 and C2 are satisfied. If $m$ is a fixed number, then $H(\hat{\beta}-\beta_{0})$ converges to the limit at rate $n^{1/2}$, and a $\chi^{2}$ type test converges to the limit at rate $n$ as in (21) of Caner and Kock (2018).
Example 3. One of the main issues in finance is the analysis of portfolio variance. If we denote the portfolio allocation vector by $w$ ($p_{n}\times1$), and the covariance matrix of asset returns by $\Sigma$, then the portfolio variance is $w'\Sigma w$. The out-of-sample estimate of this portfolio variance is $\hat{w}'\Sigma\hat{w}$. This estimate can be seen in Ledoit and Wolf (2017) and Ao et al. (2019). The number of assets, $p_{n}$, grows with $n$, which is the time span of the portfolio, and $p_{n}>n$. Let $Eigmax(A)$ denote the maximum eigenvalue of a matrix $A$ and $C>0$ be a positive constant.
We analyze the global minimum portfolio weights. Define $1_{p_{n}}$ as the $p_{n}$ vector of ones. The weights are computed as
\[ w=\frac{\Sigma^{-1}1_{p_{n}}/p_{n}}{1_{p_{n}}'\Sigma^{-1}1_{p_{n}}/p_{n}}.\]
From Theorem 3.3 of Callot et al. (2019), the estimated weights are:
\[ \hat{w}=\frac{\hat{\Theta}1_{p_{n}}/p_{n}}{1_{p_{n}}'\hat{\Theta}1_{p_{n}}/p_{n}},\] where $\hat{\Theta}$ is the nodewise regression estimate of $\Sigma^{-1}$. Take $\beta_{0}=w$, and $\hat{\beta}=\hat{w}$. So our parameter is of dimension $p_{n}$, and it is growing with $n$ and larger than $n$. Our interest centers on the out-of sample portfolio variance estimation, given that we can estimate weights consistently, with a known rate of convergence. First, we start with verifying Condition C1. See that $f(\hat{\beta})=\hat{\beta}'\Sigma\hat{\beta}$, and $f(\beta_{0})=\beta_{0}'\Sigma\beta_{0}$. Condition C2 also holds for this example, and we will show that after verifying Condition C1.
For C1, we have
\[ \|f(\beta_{0}+h)-f(\beta_{0})-f_{d}(\beta_{0})h\|_{1}=\|(\beta_{0}+h)'\Sigma(\beta_{0}+h)-\beta_{0}'\Sigma\beta_{0}-2\beta_{0}'\Sigma h\|_{1}=\|h'\Sigma h\|_{1}.\] Next
where the second inequality follows from $\|h\|_{1}\ge\|h\|_{2}$, and for the convergence to zero we use assumption $Eigmax(\Sigma)\le C<\infty$, and the fact that $\|h\|_{1}\to0$ implies $\|h\|_{2}\to0$. Thus, Condition C1 is satisfied. Then Condition C2 is satisfied since, by Theorem 3.3 of Callot et al. (2019) we have \[ r_{n}\|\hat{w}-w\|_{1}=O_{p}(1),\] where $r_{n}=\frac{\sqrt{n}}{\sqrt{logp_{n}}}\frac{1}{\bar{s}^{3/2}}$, with $\bar{s}=\max_{1\le j\le p}s_{j}$, and $s_{j}$ is the number of nonzero cells in the $j$ th row of the precision matrix. Now, we derive the rate of convergence of the out-of-sample variance estimator. Note that \[ \|f_{d}(\beta_{0})\|_{1}=\|2\beta_{0}'\Sigma\|_{1}=2\|\Sigma w\|_{1},\] since $\Sigma$ is a $p_{n}\times p_{n}$ symmetric matrix and $\beta_{0}=w$. Define $\sigma_{i,j}$ as the $(i,j)$ th element of $\Sigma$ matrix. By ((ref))
where the third line uses $\|w\|_{1}=O(\sqrt{\bar{s}})$ by Theorem 3.3 of Callot et al. (2019), and the last line uses the assumption $[\max_{1\le j\le{p_{n}}}\sum_{k=1}^{p_{n}}|\sigma_{i,j}|]\le C<\infty$. Clearly, we can define $k_{n}:=O(2\|\Sigma w\|_{1})$, which implies $k_{n}=O(\sqrt{\bar{s}})$ by the inequality above. Applying Theorem (ref)(b), we obtain a slow rate for the out-of-sample variance estimator compared with the weight estimation\[ \frac{\sqrt{n}}{\sqrt{logp_{n}}}\frac{1}{\bar{s}^{2}}|\hat{w}'\Sigma\hat{w}-w'\Sigma w|=O_{p}(1),\] when $\bar{s}$ is growing with $n$.
Here we provide a brief discussion about our results in some econometrics problems. We can see three areas related to our technique that may be beneficial to the researchers. The first area is the many weak instruments literature. In the case of many weak instruments as in Newey and Windmeijer (2009) and Caner (2014), we can derive the rate of convergence for the estimation of the sample moments from the estimation of the parameters in the structural equation. If we have $\hat{\beta}$ as the generalized empirical likelihood estimator, with $r_{n}\|\hat{\beta}-\beta_{0}\|_{2}=O_{p}(1)$, then we can have $r_{n}d_{n}\|\hat{g}(\hat{\beta})-\hat{g}(\beta_{0})\|_{2}=O_{p}(1)$, with \[ \hat{g}(\hat{\beta})=\frac{1}{n}\sum_{i=1}^{n}Z_{i}(y_{i}-x_{i}'\hat{\beta}),\] where $Z_{i}$ is an $m\times1$ vector of instruments, $m$ is growing with $n$. Let $y_{i}$ be the outcome variable, and $x_{i}$ represent the control and endogenous variables ($p_{n}\times1$ vector). This satisfies our Theorem (ref)(c). The details of this example are provided in the appendix. This extends the results of Example 1 in Section 2 of Caner (2014), and the linear model of Newey and Windmeijer (2009, p. 690-698).
The second one is the portfolio analysis, where we illustrate our results through an example in the main text. Since there are a lot of nonlinear functions in parameters of interest, and it is neither obvious nor trivial to get the rates of these functions in the modern portfolio theory when the number of assets, $p_{n}$, is larger than the time span of the portfolio. Our technique can help. It analyzes the partial derivative of the function at the parameter and automatically finds the rate of convergence through that.
The third area is nonparametric estimation. Our theorem can be applied to obtain the convergence rate of the estimate of the nonparametric function. Consider, for example, the series estimation of the following model \[ y_{i}=g(x_{i})+\varepsilon_{i},\ \mathrm{with}\ E(\varepsilon_{i}|x_{i})=0,\] where the unknown function $g(x)$ can be approximated by a linear combination of basis functions $h^{p_{n}}(x)$ and $p_{n}$ is increasing with $n$. Let $\mathcal{S}$ be the compact support of $x$. Assume that
Define $\beta^{p_{n}}\equiv\arg\min_{\beta}\sup_{x\in\mathcal{S}}|g(x)-h^{p_{n}}(x)^{\prime}\beta|.$ The estimator of $\beta^{p_{n}}$, denoted by $\hat{\beta}$, is obtained by a linear regression of $y_{i}$ on $h^{p_{n}}(x_{i})$. Newey (1997) shows that $\|\hat{\beta}-\beta^{p_{n}}\|_{2}=O_{p}\left(\sqrt{p_{n}}/\sqrt{n}+p_{n}^{-\alpha}\right)$, which gives the rate for Condition C2. For Condition C1, consider the linear function $f(\beta)=h^{p_{n}}(x_{0})^{\prime}\beta$ for a given $x_{0}\in\mathcal{S}$, so $f(\beta^{p_{n}})$ is the approximation of $g(x_{0})$ and $f(\hat{\beta})$ is the series estimator $h^{p_{n}}(x_{0})^{\prime}\hat{\beta}$. We are interested in the convergence rate of $|f(\hat{\beta})-f(\beta^{p_{n}})|$. Since $f_{d}(\beta^{p_{n}})=h^{p_{n}}(x_{0})^{\prime}$, Condition C1 holds automatically. By our Theorem 2.1(b) we have
where $\zeta(p_{n})=\sup_{x\in\mathcal{S}}\|h^{p_{n}}(x)\|_{2}$. \footnote{Under some regularity conditions, it follows that $\zeta(p_{n})=O(p_{n})$ for power series and $\zeta(p_{n})=O(\sqrt{p_{n}})$ for spline series (Corollary 15.1, Li and Racine, 2007). } Together with ((ref)), the rate in ((ref)) implies \[ |h^{p_{n}}(x_{0})^{\prime}\hat{\beta}-g(x_{0})|\le|f(\hat{\beta})-f(\beta^{p_{n}})|+\sup_{x\in\mathcal{S}}|h^{p_{n}}(x)^{\prime}\beta^{p_{n}}-g(x)|=O_{p}\left(\zeta(p_{n})(\sqrt{p_{n}}/\sqrt{n}+p_{n}^{-\alpha})\right),\] which is the well known convergence rate for the series estimator. Note that our theorem actually gives a sharp upper bound on the convergence rate of the series estimator.
In this section, we study the degree of conservativeness of the bound derived by Theorem (ref) via simulation. We consider the lasso estimator discussed in Example 1. The model is generated using (ref), where $x_{i}\sim iid\ N(0,I_{p_{n}})$, $u_{i}\sim iid\ N(0,s_{0})$, and $\beta_{0}=(\mathbf{1}_{1\times s_{0}},0_{1\times(p_{n}-s_{0})})^{\prime}$. We set $s_{0}\in\{5,10\}$, $p_{n}\in\{50,100,200,300\}$, and $n\in\{100,200,300\}$. Since $\lambda_{n}=O(\sqrt{\log(p_{n})/n})$, we select the optimal tuning parameter from the set $\{\lambda_{n}=c\sqrt{\log(p_{n})/n},\ c=0.1,0.25,0.5,1,2,3,4,5,6,7,8,9,10\}$ by minimizing the following information criterion, \[ \lambda^{*}=\arg\min_{\lambda_{n}}[\log\hat{\sigma}^{2}(\lambda_{n})+\frac{\hat{s}(\lambda_{n})}{n}\log(n)\log\log(p_{n})],\] where $\hat{s}(\lambda_{n})$ is the number of nonzero entries in the lasso estimator given by (ref) using tuning parameter $\lambda_{n}$, and $\hat{\sigma}^{2}(\lambda_{n})$ is the corresponding mean squared residuals. The term $\log\log(p_{n})$ follows the design of Caner, Han and Lee (2018) to deal with the high dimensionality.
We focus on the inference of the first $s_{0}$ parameters in $\beta_{0}$, so $f(\beta_{0})=D\beta_{0}$ with $D=(I_{s_{0}},0)$. We compute the ratio $\||f_{d}(\beta_{0})\||_{2}\|\hat{\beta}-\beta_{0}\|_{2}/\|f(\hat{\beta})-f(\beta_{0})\|_{2}=s_{0}\|\hat{\beta}-\beta_{0}\|_{2}/\|D\hat{\beta}-D\beta_{0}\|_{2}$ to see the tightness of the bound. Table 1 reports the average of this ratio over 1000 replications. It is evident that the averaged ratio is very close to $s_{0}$, which implies that most of the zero elements in $\beta_{0}$ are estimated as zero by the lasso. In the (infeasible) oracle case where all zero parameters are exactly estimated as zero, the ratio would be equal to $s_{0}$. We clearly see that our upper bound grows with sparsity, and will not be that tight unless the model is very sparse. However, the main problem is that in high dimensional econometrics, it is complicated to get closed-form solutions; hence we rely on upper bounds. For example, the oracle inequalities, $l_1$ norm results are all upper bounded.
We provide an upper bound for the functions of the estimators in high dimensions. We also show three examples to illustrate the main theorem. It is possible to extend our result to more relevant econometrics issues in the age of big data. In summary, our method can be useful for obtaining the rate of convergence of functionals of estimators, since it uses the partial derivative of the function at $\beta_{0}$. Our bound can be beneficial in high dimensional scenarios where it may be difficult to have direct proof of the rate of convergence.\\
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