### 金融代写|金融计量经济学Financial Econometrics代考|The Rigorous Lasso for Time-Series Data

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## 金融代写|金融计量经济学Financial Econometrics代考|The Rigorous Lasso for Time-Series Data

We propose two estimators, the HAC-lasso and AC-lasso, that extend the rigorous lasso to the pure time-series setting. These estimators are, in effect, special cases of the rigorous lasso for dependent data presented in Chernozhukov et al. (2019).

We first present the HAC-lasso and then AC-lasso as a special case. For simplicity we consider the contemporaneous high-dimensional model, using $t$ to denote observations numbered $1, \ldots, n$ but not including lags:
$$y_{t}=\boldsymbol{x}{t}^{\prime} \boldsymbol{\beta}+\varepsilon{t}$$
The HAC-lasso uses the HAC (heteroskedastic- and autocorrelation-consistent) covariance estimator to estimate the variance of the $j$ th element of the score vector. The implementation we propose is a simplified version of the estimator in Chernozhukov et al. (2019). The simplification follows from the additional assumption that the score is autocorrelated up to order $q$ where $q$ is finite, fixed and known a priori. The form of autocorrelation of this $M A(q)$ process can be arbitrary. Denote the HAC sample autocovariance $s$ of the score for predictor $j$ hy $\Gamma_{j s}^{H A C}$ :
$$\Gamma_{j s}^{H A C}:=\frac{1}{n} \sum_{t=s+1}^{n}\left(x_{t j} \varepsilon_{t}\right)\left(x_{t-s, j} \varepsilon_{t-s}\right)$$
The sample variance of the score for predictor $j$ is
$$\Gamma_{j 0}^{H A C}:-\frac{1}{n} \sum_{i=1}^{n}\left(x_{t j} \varepsilon_{t}\right)^{2}$$
The variance of the $j$ th element of the score vector can be consistently estimated using the truncated kernel with bandwidth $q$ (Hayashi 2000 , p. 408), and hence the HAC ideal penalty loading is
$$\psi_{j}^{H A C}=\sqrt{\Gamma_{j 0}^{H A C}+2 \sum_{s=1}^{q} \Gamma_{j s}^{H A C}}$$

## 金融代写|金融计量经济学Financial Econometrics代考|Monte Carlo

In this section, we present results of Monte Carlo simulations to assess the performance of the HAC-lasso estimator. We focus attention on the HD-C model with only contemporaneous predictors and $p=K$; our motivation is that this resembles the nowcasting application we discuss in the next section. The underlying data generation process for the dependent variable with $p$ explanatory variables is:
$$y_{t}=\beta_{0}+\sum_{j=1}^{p} \beta_{j} x_{t j}+\varepsilon_{t} .$$

A total of $p=100$ predictors are generated, but only the first $s$ predictors are nonzero. Therefore, in all specifications, the coefficients on the predictors $\beta_{j}$ are defined as:
$$\beta_{j}=\mathbb{1}{j \leq s} \forall j=1, \ldots, p$$
where we set the number of non-zero predictors to $s=5 . \beta_{0}$ is a constant and set to 1 in all simulations.
The error component $\varepsilon_{t}$ for the dependent variable is an MA(q) process:
\begin{aligned} &\varepsilon_{t}=\sum_{r=0}^{q} \theta_{r} \eta_{t-r} \ &\eta_{t} \sim N\left(0, \sigma_{\eta}^{2}\right) \end{aligned}
We use three DGPs with $q=0, q=4$, and $q=8$. For all DGPs, the MA coefficient $\theta_{r}$ is fixed such that $\theta_{r}=\theta=1, \quad \forall l=1, \ldots, q$. The standard deviation varies across $\sigma_{\eta}=[0.5 ; 1 ; 2 ; 4 ; 5]$.
The predictors $x_{t j}$ follow an $A R(1)$ process:
$$x_{i j}=\pi_{j} x_{t-1, j}+\xi_{t j}, \quad \forall j=1, \ldots, p$$
The AR coefficients across all predictors are the same with $\pi_{j}=\pi=0.8$.
The random component $\xi_{t}=\left(\xi_{t 1}, \ldots, \xi_{t p}\right)^{\prime}$ is multivariate normal, generated as:
$$\xi_{t}=M V N\left(0, \Sigma_{\xi}\right),$$
where $\Sigma_{\xi}$ is a $p \times p$ covariance matrix. In this approach, we specify error components that are independent over time, and that are either also contemporaneously independent or correlated across $p$. In a first step the Monte Carlo specifies uncorrelated error components for the predictors $x$ and $\Sigma_{\xi}$ is diagonal with elements $\sigma_{\xi^{(1)}}^{2}-\cdots-\sigma_{\xi^{(p)}}^{2}-1$.

## 金融代写|金融计量经济学Financial Econometrics代考|Application to Nowcasting

In this section, we illustrate how the properties of the HAC-lasso and AC-lasso estimators are particularly useful for model consistency for fore- and nowcasting and that it produces competitive nowcasts at low computational cost.

The objective of nowcast models is to produce ‘early’ forecasts of the target variable which exploits the real time data publication schedule of the explanatory data set. Such real time data sets are usually in higher frequency and are published with a considerably shorter lag than the target variable of interest. Nowcasting is particularly relevant for central banks and other policy environments where key economic indices such as GDP or inflation are published with a lag of up to 7 weeks

with respect to their reference period. ${ }^{12}$ In order to conduct informed forward-looking policy decisions, policy makers require accurate nowcasts where it is now common to combine, next to traditional macroeconomic data, ever more information from Big Data sources such as internet search terms, satellite data, scanner data, etc. (Buono et al. 2018).

A data source which has garnered much attention in the recent nowcast literature is Google Trends (GT), Google’s search term indices. GT provides on a scale of 1-100, for a given time frame and location, the popularity of certain search terms entered into the Google search engine. Due to their timeliness as compared to conventional macro data and ability to function as an index of sentiment of demand and supply (Scott and Varian 2014), they have celebrated wide spread use in nowcasting applications in many disparate fields of economics (see Choi and Varian (2012), and Li (2016) for surveys). They have proven especially useful in applications where searches are directly related to the variable of interest, such as unemployment data where internet search engines provide the dominant funnel through which job seekers find jobs (Smith 2016). Only recently has Google Trends been applied to nowcasting such aggregate economic variables as GDP (Kohns and Bhattacharjee 2019).

## 金融代写|金融计量经济学Financial Econometrics代考|The Rigorous Lasso for Time-Series Data

HAC-lasso 使用 HAC（heteroskedastic-and autocorrelation-consistent）协方差估计器来估计j分数向量的第 th 个元素。我们提出的实现是 Chernozhukov 等人的估计器的简化版本。（2019）。简化源于额外的假设，即分数是自相关的q在哪里q是有限的、固定的和先验已知的。这个自相关的形式米一个(q)过程可以是任意的。表示 HAC 样本自协方差s预测器的分数j他ΓjsH一个C :

ΓjsH一个C:=1n∑吨=s+1n(X吨je吨)(X吨−s,je吨−s)

Γj0H一个C:−1n∑一世=1n(X吨je吨)2

ψjH一个C=Γj0H一个C+2∑s=1qΓjsH一个C

## 金融代写|金融计量经济学Financial Econometrics代考|Monte Carlo

bj=1j≤s∀j=1,…,p

e吨=∑r=0qθr这吨−r 这吨∼ñ(0,σ这2)

X一世j=圆周率jX吨−1,j+X吨j,∀j=1,…,p

X吨=米在ñ(0,ΣX),

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