Unconventional Monetary Policies in the UK, estimating their impact using shrinkage and persistent volatility

宏观计量经济学和金融计量经济学分析文献的两个主要方面是：1)结构变化和波动性建模；2)高维回归。对于这两个方面，我们提出了一个有可能产生许多贡献的框架。下面，我们简要地描述了研究思路的设置和目标，并说明了我们的方法可以应用的一些潜在的实证应用。首先，实证工作得出结论，波动性的变化是非常持久的，这一特征在接近平稳边界的估计参数中得到说明。这意味着波动性可以以持续的、可能是非平稳的过程为特征。在本内容中，我们建议使用内核波动估计器（KVE），它有可能充分拟合观察到的波动行为。我们的估计器需要一个小的假设集，并且可以区分持久类型的波动性和较低类型的波动性，如ARCH和SV。为了说明这种能力，需要进行蒙特卡罗（MC）模拟。该框架的潜在收益是，首先，我们不会强加线性类型的波动，其次，如果我们的估计器能够正确估计可能是数据中的主要驱动因素的持续波动，则参数估计可以更加精确。其次，在过去十年中，越来越多的大型数据集的可用性可以潜在地帮助计量经济学家，假设它们携带丰富和相关的信息。它们的大小造成了一个“不适定”的问题，即使是普通回归这样的基本方法也无法工作。文献已经提出了许多方法来解决这个问题，这些方法可以分别在稀疏回归和降维框架中分离出来。其主要目的是首先，通过产生具有最大信息内容的因子来降低回归量的维度，在相关性方面，并进一步排除具有最小信息的回归量。这样做的主要途径是扩展经典的最小二乘估计最小化问题，并使用通常在第一和第二规范上引起收缩和维数减少的惩罚。最著名的估计方法是Lasso、Ridge回归、稀疏偏最小二乘法和弹性网。在这个内容中，我们提出了一个不同的估计，推广了上面的内容。具体来说，我们没有强加特定数量的规范惩罚，而是设想了一个包含大量规范惩罚的估计器，其性能将通过交叉验证和样本外预测来评估。这个过程的好处是，虽然不知道包含先验的标准惩罚的数量，但我们产生了一个惩罚方案，可以以不同的方式为每个数据集工作，并可以产生更好的结果。评估该估计器的性能需要使用合成数据集的MC。在经验应用方面，我们的估计器是自然的候选者，首先检查股票指数中存在的波动性，以及从KVE过程中获得的参数估计是否更好地预测。此外，KVE和收缩估计值都可以帮助我们潜在地检验英国央行（Bank of England）采用的非常规货币政策对实体经济的影响，以及这种影响在最初实施后是否会恶化。

Two main strands in econometric analysis literature for macro-econometrics and finance are, i) structural change and volatility modelling and ii) high-dimensional regressions. For these two strands we propose a framework that has the potential to create a number of contributions. Below, we briefly describe the setup and goals of the research idea and illustrates some potential empirical applications where our methodology can be applied. First, empirical work has concluded that variation in volatility is very persistent, a feature illustrated in estimated parameters that lie close to the boundary for stationarity. This implies that volatility can be characterised by persistent and possibly non-stationary processes. In this content we propose the use of a kernel volatility estimator (KVE) that has the potential to adequately fit the observed behaviour of volatility. Our estimator requires a small set of assumptions and can disentangle persistent types of volatilities and lower type such as ARCH and SV. To illustrate this ability, Monte Carlo (MC) simulations are required. The potential gains of this framework are first, that we do not impose linear types of volatility and second, that parameter estimates can be more precise if our estimator is able to correctly estimate the persistent volatility that can be the major driver in the data. Second, the growing availability of large datasets in the last ten years can potentially assist econometricians under the assumption that they carry rich and relevant information. Their size creates an "ill-posed" problem where even basic methods as ordinary regression cannot work. The literature has produced many ways to tackle this problem that can be separated in the sparse regression and dimensionality reduction framework respectively. Its main aim is first, reduce the dimensions of the regressors by producing factors that carry the maximal informative content, in terms of correlations and further to exclude regressors that have minimal information for inference. The main avenue of doing this is to expand the classical least squares estimator minimisation problem with penalties that induce shrinkage and dimensionality reduction, commonly on the first and second norms. The most well known estimators are the Lasso, Ridge regression, Sparse Partial Least Squares and the elastic net. In this content we proposed a different estimator that generalises the above. Specifically, we do not impose a specific number of norm-penalties but instead we envision an estimator that includes a vast number of norm-penalties, whose performance will be assessed by Cross-Validation and out of sample forecasting. The benefit of this procedures is that while agnostic to the number of norm-penalties to include a-priori, we produce a penalisation scheme that can potentially work in a different way for each dataset and can yield better results. Assessing the performance of this estimator requires MC with synthetic datasets. In terms of the empirical applications our estimators are natural candidates to examine first the volatility that exists in stock indexes and whether parameter estimates obtained from the KVE procedure are better in forecasting. Further both the KVE as well as the shrinkage estimator can help us to potentially examine the effects unconventional monetary policies, as employed by the Bank of England, had in the real economy and whether the effect deteriorated after their initial employment.