金融代写|金融计量经济学Financial Econometrics代考|A Theory-Based Lasso for Time-Series Dat

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金融代写|金融计量经济学Financial Econometrics代考|Basic Setup and Notation

Wé bégin with thé simplé linéar régréssión mơdél in thé crooss-séction sêtting with $p$ independent variables.

A Theory-Based Lasso for Time-Series Data
5
$$y_{i}=\beta_{0}+\beta_{1} x_{1 i}+\beta_{2} x_{2 i}+\cdots+\beta_{p} x_{p i}+\varepsilon_{i}$$
In traditional least squares regression, estimated parameters are chosen to minimise the residual sum of squares (RSS):
$$R S S=\sum_{i=1}^{n}\left(y_{i}-\beta_{0}-\sum_{j=1}^{p} \beta_{j} x_{i j}\right)^{2}$$
The problem arises when $p$ is relatively large. If the model is too complex or flexible, the parameters estimated using the training dataset do not perform well with future datasets. This is where regularisation is key. By adding a shrinkage quantity to RSS, regularisation shrinks parameter estimates towards zero. A very popular regularised regression method is the lasso, introduced by Frank and Friedman (1993) and Tibshirani (1996). Instead of minimising the RSS, the lasso minimises
$$R S S+\lambda \sum_{j=1}^{p}\left|\beta_{j}\right|$$
where $\lambda$ is the tuning parameter that determines how much model complexity is penalised. At one extreme, if $\lambda=0$, the penalty term disappears, and lasso estimates are the same as OLS. At the other extreme, as $\lambda \rightarrow \infty$, the penalty term grows and estimated coefficients approach zero.

Choosing the tuning parameter, $\lambda$, is critical. We discuss below both the most popular method of tuning parameter choice, cross-validation, and the theory-derived ‘rigorous lasso’ approach.

We note here that our paper is concerned primarily with prediction and model selection with dependent data rather than causal inference. Estimates from regularised regression cannot be readily interpreted as causal, and statistical inference on these coefficients is complicated and an active area of research. ${ }^{1}$

We are interested in applying the lasso in a single-equation time series framework, where the number of predictors may be large, either because the set of contemporaneous regressors is inherently large (as in a nowcasting application), and/or because the model has many lags.

金融代写|金融计量经济学Financial Econometrics代考|Literature Review

The literature on lag selection in VAR and ARMA models is very rich. Lütkepohl (2005) notes that fitting a $\operatorname{VAR}(R)$ model to a VAR $(R)$ process yields a better outcome in terms of mean square error than fitting a $\operatorname{VAR}(R+i)$ model, because the latter results in inferior forecasts than the former, especially when the sample size is small. In practice, the order of data generating process (DGP) is. of course, unknown and we face a trade-off between out-of-sample prediction performance and model consistency. This suggests that it is advisable to avoid fitting VAR models with unnecessarily large orders. Hence, if an upper bound on the true order is known or suspected, the usual next step is to set up significance tests. In a causality context, Wald tests are useful. The likelihood ratio (LR) test can also be used to compare maximum log-likelihoods over the unrestricted and restricted parameter space.

If the focus is on forecasting, information criteria are typically favoured. In this vein, Akaike $(1969,1971)$ proposed using 1-step ahead forecast mean squared error (MSE) to select the VAR order, which led to the final prediction error (FPE) criterion. Akaike’s information criterion (AIC), proposed by Akaike (1974), led to (almost) the same outcome through different reasoning. AIC, defined as $-2 \times \log$-likelihood $+$ $2 \times$ no. of regressors, is approximately equal to FPE in moderate and large sample sizes $(T)$.

Two further information criteria are popular in applied work: Hannan-Quinn criterion (Hannan and Quinn 1979) and Bayesian information criterion (Schwarz 1978). These criteria perform better than AIC and FPE in terms of order selection consistency under certain conditions. However, AIC and FPE have better small sample properties, and models based on these criteria produce superior forecasts despite not estimating the orders correctly (Lütkepohl 2005). Further, the popular information criteria (AIC, BIC, Hannan-Quinn) tend to underfit the model in terms of lag order selection in a small- $t$ context (Lütkepohl 2005).

Although applications of lasso in a time series context are an active area of research, most analyses have focused solely on the use of lasso in lag selection. For example, Hsu et al. (2008) adopt the lasso for VAR subset selection. The authors compare predictive performance of two-dimensional VAR(5) models and US macroeconomic data based on information criteria (AIC, BIC), lasso, and combinations of the two. The findings indicate that the lasso performs better than conventional selection methods in terms of prediction mean squared errors in small samples. In a related application, Nardi and Rinaldo (2011) show that the lasso estimator is model selection consistent when fitting an autoregressive model, where the maximal lag is allowed to increase with sample size. The authors note that the advantage of the lasso with growing $R$ in an $\mathrm{AR}(R)$ model is that the ‘fitted model will be chosen among all possible AR models whose maximal lag is between 1 and $[. . .] \log (\mathrm{n})^{\prime}$ (Nardi and Rinaldo 2011).

金融代写|金融计量经济学Financial Econometrics代考|High-Dimensional Data and Sparsity

The high-dimensional linear model is:
$$y_{i}=\boldsymbol{x}{i}^{\prime} \boldsymbol{\beta}+\varepsilon{i}$$

Our initial exposition assumes independence, and to emphasise independence we index observations by $i$. Predictors are indexed by $j$. We have up to $p=\operatorname{dim}(\boldsymbol{\beta})$ potential predictors. $p$ can be very large, potentially even larger than the number of observations $n$. For simplicity we assume that all variables have already been meancentered and rescaled to have unit variance, i.e., $\sum_{i} y_{i}=0$ and $\frac{1}{n} \sum_{i} y_{i}^{2}=1$, and similarly for the predictors $x_{i j}$.

If we simply use OLS to estimate the model and $p$ is large, the result is very poor performance: we overfit badly and classical hypothesis testing leads to many false positives. If $p>n$, OLS is not even identified.

How to proceed depend on what we believe the ‘true model’ is. Does the true model (DGP) include a very large number of regressors? In other words, is the set of predictors that enter the model ‘dense’? Or does the true model consist of a small number of regressors $s$, and all the other $p-s$ regressors do not enter (or equivalently, have zero coefficients)? In other words, is the set of predictors that enter the model ‘sparse’?

In this paper, we focus primarily on the ‘sparse’ case and in particular an estimator that is particularly well-suited to the sparse setting, namely the lasso introduced by Tibshirani (1996).

In the exact sparsity case of the $p$ potential regressors, only $s$ regressors belong in the model, wherre
$$s:=\sum_{j=1}^{p} \mathbb{1}\left{\beta_{j} \neq 0\right} \ll n$$
In other words, most of the true coefficients $\beta_{j}$ are actually zero. The problem facing the researcher is that which are zeros and which are not is unknown.

We can also use the weaker assumption of approximate sparsity: some of the $\beta_{j}$ coefficients are well-approximated by zero, and the approximation error is sufficiently ‘small’. The discussion and methods we present in this paper typically carry over to the approximately sparse case, and for the most part we will use the term ‘sparse’ to refer to either setting.

The sparse high-dimensional model accommodates situations that are very familiar to researchers and that typically presented them with difficult problems where traditional statistical methods would perform badly. These include hoth settings where the number $p$ of observed potential predictors is very large and the researcher does not know which ones to use, and settings where the number of observed variables is small but the number of potential predictors in the model is large because of interactions and other non-linearities, model uncertainty, temporal and spatial effects, etc.

金融代写|金融计量经济学Financial Econometrics代考|Basic Setup and Notation

5

R小号小号=∑一世=1n(是一世−b0−∑j=1pbjX一世j)2

R小号小号+λ∑j=1p|bj|

金融代写|金融计量经济学Financial Econometrics代考|High-Dimensional Data and Sparsity

s:=\sum_{j=1}^{p} \mathbb{1}\left{\beta_{j} \neq 0\right} \ll ns:=\sum_{j=1}^{p} \mathbb{1}\left{\beta_{j} \neq 0\right} \ll n

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