### 金融代写|金融计量经济学Financial Econometrics代考|Sparsity and the Rigorous or Plug-In Lasso

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## 金融代写|金融计量经济学Financial Econometrics代考|The Penalisation Approach and the Lasso

The basic idea behind the lasso and its high-dimensional-friendly relatives is penalisation: put a penalty or ‘price’ on the use of predictors in the objective function that the estimator minimizes.

The lasso estimator minimizes the mean squared error subject to a penalty on the absolute size of coefficient estimates (i.e., using the $\ell_{1}$ norm):
$$\hat{\boldsymbol{\beta}}{\text {lasso }}(\lambda)=\arg \min \frac{1}{n} \sum{i=1}^{n}\left(y_{i}-\boldsymbol{x}{i}^{\prime} \boldsymbol{\beta}\right)^{2}+\frac{\lambda}{n} \sum{j=1}^{p} \psi_{j}\left|\beta_{j}\right| .$$
The tuning parameter $\lambda$ controls the overall penalty level and $\psi_{j}$ are predictor-specific penalty loadings.

The intuition behind the lasso is straightforward: there is a cost to including predictors, the unit ‘price’ per regressor is $\lambda$, and we can reduce the value of the objective function by removing the ones that contribute little to the fit. The bigger the $\lambda$, the higher the ‘price’, and the more predictors are removed. The penalty loadings $\psi_{j}$ introduce the additional flexibility of putting different prices on the different predictors, $x_{i j}$. The natural base case for standardised predictors is to price them all equally, i.e., the individual penalty loadings $\psi_{j}=1$ and they drop out of the problem. We will see shortly that separate pricing for individual predictors turns out to be important for our proposed estimators.

We can say ‘remove’ because in fact the effect of the penalisation with the $\ell_{1}$ norm is that the lasso sets the $\hat{\beta}_{j} \mathrm{~s}$ for some variables to zero. This is what makes the lasso so suitable to sparse problems: the estimator itself has a sparse solution. The lasso is also computationally feasible: the path-wise coordinate descent (‘shooting’) algorithm allows for fast estimation.

The lasso, like other penalized regression methods, is subject to an attenuation bias. This bias can be addressed by post-estimation using OLS, i.e., re-estimate the model using the variables selected by the first-stage lasso (Belloni and Chernozhukov 2013):
$$\hat{\boldsymbol{\beta}}{\text {post }}=\arg \min \frac{1}{n} \sum{i=1}^{n}\left(y_{i}-\boldsymbol{x}{i}^{\prime} \boldsymbol{\beta}\right)^{2} \quad \text { subject to } \quad \beta{j}=0 \text { if } \tilde{\beta}{j}=0$$ where $\tilde{\beta}{j}$ is the first-step lasso estimator. In other words, the first-step lasso is used exclusively as a model selection technique, and OLS is used to estimate the selected model. This estimator is sometimes referred to as the ‘Post-lasso’ (Belloni and Chernozhukov 2013).

## 金融代写|金融计量经济学Financial Econometrics代考|Cross-Validation

The objective in cross-validation is to choose the lasso penalty parameter based on predictive performance. Typically, the dataset is repeatedly divided into a portion which is used to fit the model (the ‘training’ sample) and the remaining portion which is used to assess predictive performance (the ‘validation’ or ‘holdout’ sample), usually with mean squared prediction error (MSPE) as the criterion. Arlot and Celisse (2010) survey the theory and practice of cross-validation.

In the case of independent data, common approaches are ‘leave-one-out’ (LOO) cross-validation and the more general ‘ $\mathrm{K}$-fold’ cross-validation.

In ‘ $K$-fold’ cross-validation, the dataset is split into $K$ portions or ‘folds’; each fold is used once as the validation sample and the remainder are used to fit the model for some value of $\lambda$. For example, in 10-fold cross-validation (a common choice of $K$ ) the MSPE for the chosen $\lambda$ is the MSPE across the 10 different folds when used for validation. LOO cross-validation is a special case where $K=1$, i.e., every observation is used once as the validation sample while the remaining $n-1$ observations are used to fit the model (Fig. 1).

Cross-validation is computationally intensive because of the need to repeatedly estimate the model and check its performance across different folds and across a grid of values for $\lambda$. Standardisation of data adds to the computational cost because it needs to be done afresh for each training sample; standardising the entire dataset once up-front would violate a key principle of cross-validation, which is that a training dataset cannot contain any information from the corresponding validation dataset. LOO is a partial exception because the MSPE has a closed-form solution for a chosen $\lambda$, but a grid search across $\lambda$ and repeated standardisation are still needed.

## 金融代写|金融计量经济学Financial Econometrics代考|Cross-Validation for Time Series

Cross-validation with dependent data adds further complications because we need to be careful that the validation data are independent of the training data. It is possible that some settings, standard $K$-fold cross-validation is appropriate. Bergmeir et al. (2018) show that standard cross-validation that ignores the time dimension is valid in the pure auto-regressive model if one is willing to assume that the errors are uncorrelated. This implies, for example, that $K$-fold cross-validation can be used with auto-regressive models that include a sufficient number of lags, since the errors will be uncorrelated (if the model is not otherwise misspecified).

In general, however, researchers typically use a version of ‘non-dependent cross validation’ (Bergmeir et al. 2018), whereby prior information about the nature of the dependence is incorporated into the structure of the cross-validation and possibly dependent observations are omitted from the validation data. For example, one approach used with time-series data is 1 -step-ahead cross-validation (Hyndman and Athanasopoulos 2018 ), where the predictive performance is based on a training sample with observations through time $t$ and the forecast for time $t+1$.

Rolling $h$-step ahead $\mathrm{CV}$ is an intuitively appealing approach that directly incorporates the ordered nature of time series-data (Hyndman and Athanasopoulos 2018). ${ }^{4}$ The procedure builds on repeated $h$-step ahead forecasts. The procedure is illustrated in Figs. 2 and 3 .

Figure 2 a displays the case of 1 -step ahead cross-validation. ‘ $T$ ‘ and ‘ $V$ ‘ refer to the training and validation samples, respectively. In the first step, observations 1 to 3 constitute the training data set and observation 4 used for validation; the remaining observations are unused as indicated by a dot (‘:). Figure 2 b illustrates 2 -step ahead

cross-validation. Figures $2 \mathrm{a}$ and $3 \mathrm{~b}$ both illustrate cross-validation where the training window expands incrementally. Figure 3 displays a variation of rolling $\mathrm{CV}$ where the training window is fixed in length.

We use 1-step-ahead rolling CV with a fixed window for the comparisons in this paper.

## 金融代写|金融计量经济学Financial Econometrics代考|The Penalisation Approach and the Lasso

b^套索 (λ)=参数⁡分钟1n∑一世=1n(是一世−X一世′b)2+λn∑j=1pψj|bj|.

lasso 与其他惩罚回归方法一样，存在衰减偏差。这种偏差可以通过使用 OLS 的后估计来解决，即使用第一阶段套索选择的变量重新估计模型（Belloni 和 Chernozhukov 2013）：

b^邮政 =参数⁡分钟1n∑一世=1n(是一世−X一世′b)2 受制于 bj=0 如果 b~j=0在哪里b~j是第一步套索估计器。换句话说，第一步套索专门用作模型选择技术，而OLS用于估计所选模型。该估计器有时被称为“后套索”（Belloni 和 Chernozhukov 2013）。

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