e-book Modeling Time-Dependent Economic Data by Boosting Methods

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Special Issue: Density Forecasting in Economics and Finance In fact, the modelling and the prediction of volatility is one of the central topics in asset pricing. density also allowing for time-dependent higher-order moments. Our empirical analysis of daily FTSE data demonstrates the importance of.
Table of contents

• Current teaching
• How To Identify Patterns in Time Series Data: Time Series Analysis
• Submission history
• Academic Commons
• Academic Commons

Random forests can be thought of as being related to kNN methods with adaptive weighting Lin and Jeon, , where the predicted outcome of an out of sample observation is given by its neighbours defined by a weighting of its characteristics. Gradient boosted trees are additive models consisting of the sum of trees trained by repeatedly fitting shallow trees on the residuals Efron and Hastie, : Given their additive structure, boosted trees are closely related to generalised additive models GAMs in traditional econometrics. However, estimation of GAMs is less efficient than gradient boosting when working with a large number of explanatory variables.

These methods are currently among the most effective prediction techniques applied in many different areas Hastie, Tibshirani and Friedman, ; Efron and Hastie, : Additionally, they provide a ranking of the importance of each explanatory variable. Next to tree-based methods, NNs are the most widely used, effective supervised ML approaches currently available. Sarle provides an early comparison between NNs and statistical models, including a overview of ML jargon.

Goodfellow et al. Characteristically, the mapping consists of layers building a chain like structure of functions. Deep neural networks refer to a NN with many layers. Typical choices for the activation function g k are a rectified linear unit relu or a tanh transformation function. The weights of the NN are trained by minimising a loss function, such as mean squared error for regression or cross-entropy for classification.

Note both linear and logit regression are special cases of a NN, when the NN has only one layer, y is of dimension one and we use a linear or logistic activation function, respectively. From this perspective, NNs are already widely used in our profession!

CNNs are well placed to process grid-like data such as 1D time-series data or 2D image data.

Current teaching

CNNs get their name from the use of a convolutional operator in at least one of their layers, which is then called a convolutional layer. In a convolutional layer, by contrast, each unit looks only at a small fraction of units from the previous layer thus, a sparse interconnection and uses the same parameters at different locations parameter sharing , thereby significantly reducing the number of parameters it needs to estimate. Intuitively, a convolutional layer in a time series model can be thought of as a collection of filters that are shifted across the time sequence; for example, one filter that detects cyclical behaviour and another that calculates a moving average.

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A crucial distinction between CNNs and classical time series models is that CNNs learn the parameters of the filter, i. In an image processing application, for example, a filter may learn to detect vertical edges in small locations of the image, while another filter detects horizontal edges, corners and curved lines.

Each filter is then moved across the image to create a feature map one from each filter specifying where the features are present in the image. The next convolutional layer then combines the features edges, corners etc. RNNs are an alternative to CNNs for processing sequential data, handling dynamic relationships and long-term dependencies. During each time step, new incoming explanatory variables are encoded and combined with the past information in the cell state vector. Importantly, the model itself learns i in which way information is encoded and ii which encoded information can be forgotten i.

As with CNNs, this approach differs from a classical AR process as it does not require the analyst to specify the lag structure and can capture more complex relationships.

Both can be applied in a context of either very long time series or in a panel context with many short time series. Along with prediction, another common use of ML is data grouping or clustering based on the characteristics of observations. Unsupervised approaches aim to discover the joint probability of x instead of E y x.

Submission history

Hence, they can be applied in situations where we lack labels, i. These approaches are often used to reduce the dimensionality of data. Principal component analysis PCA is an unsupervised learning approach familiar to econometricians. As the number of potential descriptors increases, reducing dimensionality becomes more important.

Unsupervised learning can also be applied to pre-train neural networks see below.

In these settings, the primary goal is to learn relevant relationships in the unlabelled data that can then be used in a second step for a supervised learning task. Traditional dimensionality reduction approaches such as PCA rely on linear partitions of the variable space. ML approaches such as Autoencoders facilitate non-linear unsupervised learning. In general, an autoencoder is a NN consisting of an encoder and a decoder function.

Autoencoders can be interpreted as a non-linear generalisation of PCA Hinton and Salakhutdinov, Typically, autoencoders are simply fully connected neural networks, with the twist that the outputs are their own inputs making them an unsupervised approach. While copying input data to itself is not helpful on its own, restricting the internal layers of the neural net can provide an useful encoding of the data.

Thus, the autoencoder is forced to learn an internal representation of x in a lower dimensional space. To be successful, the NN needs to be able to compress data with minimal information loss, by capturing only the most important features of x. Regularised autoencoders or denoising autoencoders are alternative specifications, see Goodfellow et al. Even though many ML methods are more complex than their linear regression counterparts, this is not an inherent problem of ML tools but it rather reflects an unavoidable tradeoff between flexibility and interpretability faced by any method.

As soon as we aim to reflect non-linearities, interactions or heterogeneity, model interpretation becomes more difficult. Consider tobit models that add flexibility to a linear regression to model censored observations at the cost that coefficients cannot be interpreted directly, and marginal effects depend on all explanatory variables. Quantile regression or locally weighted regression allow for more flexibility at the cost of complicating interpretability in the sense that the models generate a large number of marginal effects.

This trade-off between flexibility versus interpretability also holds for simulation methods. For example, simple analytical economic models might be superior in terms of interpretability compared to, say, a complex computable general equilibrium model. While interpretability is fundamental for causal analysis, it can also be helpful for pure prediction tasks. Interpretability is helpful for debugging models or assessing whether the estimated relationships are plausible. Interpretability is also crucial to assess whether ML algorithms are discriminatory, for example when used by banks to determine lending or to give guidance for sentencing to courts Molnar, Numerous methods exist for interpretation, both in ML and econometrics.

A good overview is presented by Molnar , from which much of the following discussion is drawn. One of the primary approaches for interpretation is to plot the implicit marginal effects of one or more specific characteristics such as often used for interpreting output from tobit or logit models. Both partial dependence plots Hastie, Tibshirani and Friedman, : and accumulated local effects plots Apley, compare outcomes of one or two variables against their predicted outcomes, whereas individual conditional expectation plots Goldstein et al.

Other methods use the model results to simulate marginal effects, as is frequently done with large simulation models for policy analysis. Shapley value explanations do this systematically, estimating the marginal effects by computing predictions drawn from the distributions of the other characteristics with and without the characteristic of interest.

When used for the whole distribution of the data, this is referred to as a global surrogate model. Local interpretable model-agnostic explanations LIME focus on understanding the predictions for a single data point Ribeiro, Singh and Guestrin, Another general approach is to determine how much influence each explanatory variable has on the resulting prediction. For tree-based models the relative importance of predictor variables can be assessed by ranking the importance of different predictors Hastie, Tibshirani and Friedman, : Fisher, Rudin and Dominici extended this approach to any other model.

Flipping the idea of sensitivity tests on its head, a common approach in ML is to determine the smallest change of an explanatory variable that causes a change to a certain model prediction Molnar, We explore the potential of ML by first highlighting specific limitations of current econometric and simulation methods, and identify areas where ML approaches may help fill those gaps. While some ML methods can be used to address multiple limitations, the limitations or challenges themselves differ. We hope that by highlighting multiple situations where ML methods may be useful will facilitate their broader use.

As noted above, much of ML is focused on prediction and predictive tasks are highlighted in Sections 3. However, in Section 3. Throughout the entire section we highlight current and potential applications of these methods in agricultural and applied economics. The choice of model complexity should depend on the phenomenon under study and the specific research question. As noted above, many phenomena in agricultural and environmental economics are inherently non-linear, resulting from underlying biophysical, social or economic processes. For example, the effect of weather variables on yield Schlenker and Roberts, , groundwater extraction on pumping costs Burness and Brill, or health effects of pollution Graff Zivin and Neidell, are all likely to contain non-linearities.

Other times we are interested in estimating relationships between observations, over time, space or social networks, and our current approaches usually impose some restrictive structure, such as pre-determined neighbours and structure of interaction in spatial econometrics, without strong grounds to justify these assumptions. Often, we are interested in estimating specific aspects of heterogeneity. For example, we may be specifically interested in the distributional effects of an intervention, such as the case of who reduces consumption in response to food warnings Shimshack, Ward and Beatty, , or which children benefit from maternal health interventions Kandpal, Pre-determining flexible ML processes to identify key dimensions or groups avoids this potential bias, and instead allows the data to determine heterogeneous responses across the population.

Economic theory rarely gives clear guidance about the specific functional form of the object one is trying to estimate. In many settings, it only provides information about shape restrictions such as curvature or monotonicity. Choosing a model that cannot capture non-linearities, interactions or heterogeneity and distributional effects might result in misspecification bias. This misspecification bias increases with the degree of non-linearity of the underlying process Signorino and Yilmaz, While we worry a lot about potential endogeneity and think intensively about finding appropriate instruments or natural experiments to minimise potential bias in our estimates of treatment, we are often readily willing to make strong assumptions on functional form that themselves can introduce bias into our estimates.