Outline the key differences between machine learning algorithms and conventional statistical models in the context of quantitative trading, and provide an example of when to use one over the other.

Machine learning algorithms and conventional statistical models are both used in quantitative trading, but they have different strengths, weaknesses, and are suitable for different types of tasks. The key differences lie in their approach to learning from data, their flexibility, and their interpretability.

Conventional statistical models, like linear regression, ARIMA models, and logistic regression, are built on established statistical theories and assumptions. These models typically involve a well-defined mathematical structure based on the relationships between variables, and their parameters are estimated by optimizing certain criteria based on statistical properties. For example, linear regression, as discussed earlier, models a linear relationship between independent and dependent variables. ARIMA models capture the serial autocorrelation in time series data, using past values to forecast future ones. These models make strong assumptions about the underlying data distribution and the relationships between variables. For example, linear regression assumes a linear relationship, ARIMA models assume stationarity of time series data, and logistic regression assumes a sigmoid relationship between the outcome and the features. Conventional statistical models are typically easier to interpret. The parameters of these models have clear statistical meanings that can help in understanding how a model works. However, their limited flexibility can be a drawback when dealing with complex, non-linear relationships in financial data, and also the models can become fragile if the underlying statistical assumptions are not met.

In contrast, machine learning algorithms are more data-driven and less constrained by theoretical assumptions. These algorithms are designed to learn patterns from the data with minimal human intervention, and the algorithm structure is determined by the learning process. Machine learning models can be broadly divided into supervised learning (predicting a target based on the features, like regression and classification), unsupervised learning (finding patterns within the data, like clustering and dimensionality reduction), and reinforcement learning (learning through interactions with an environment). For example, Support Vector Machines (SVMs), which can be used for classification and regression, use a kernel function to transform data into a high-dimensional space, which can capture non-linear relationships that would not be possible with linear regression. Neural networks are another class of machine learning models that consist of interconnected layers of nodes that can learn complex relationships from the data through the use of backpropagation algorithm. Machine learning models can typically learn complex nonlinear relationships and deal with many features, and have very high flexibility, but they require a large volume of data and the models can be hard to interpret and also are prone to overfitting. In the context of financial data, a common approach is to use machine learning algorithms for classification tasks (e.g. predicting if a price will go up or down), and to use conventional models for other tasks, such as modeling time series or estimating volatility.

One of the main differences is how the models are trained and evaluated. Conventional models are often trained using statistical tests such as p-values that evaluate the statistical significance of the relationship; while machine learning models use optimization techniques such as gradient descent that minimizes an error function. The machine learning model performance is usually evaluated with error metrics like mean absolute error, root mean squared error, accuracy, precision, or recall. The models may also require different ways of data preprocessing, and feature engineering.

When to choose one over the other depends heavily on the specific problem and the data at hand.

For example, if the trading objective is to model the volatility of a financial asset, a conventional statistical model like an ARCH or GARCH model would be a suitable approach. These models are specifically designed to capture the time-varying nature of volatility, and have a well-established theoretical basis. This model is generally easy to understand and interpret, and can provide insights into how volatility changes over time, and is often the preferred way to estimate volatility. The model is also readily available in most statistical and programming software packages.

On the other hand, if the task is to detect and trade on complex, non-linear patterns in a market, machine learning algorithms might be preferred. For example, if you want to predict whether a stock will go up or down based on various features, including price, volume, news sentiment, social media data, and historical volatility, a machine learning algorithm like a neural network could be a better fit. Neural networks can handle the many inputs and the complex, non-linear interactions between the features. However, it would require a large data set, and a careful choice of the architecture and hyperparameters of the network. Also, this model will be a black box, that is much less interpretable than a simple statistical model.

Another example is the prediction of high-frequency market data. In these scenarios, machine learning algorithms are often preferred, due to their ability to model complex non-linear relationships in the data, but also because they can be used to discover patterns automatically, without the need to define every relationship explicitly. For instance, a strategy trying to perform statistical arbitrage may use a classification model such as SVM to determine which trades to take by processing many parameters that are difficult to model using conventional statistical models.

In summary, conventional statistical models are useful for tasks that require a clear and interpretable model, and when there is a clear theoretical basis for the model structure and the underlying assumptions are met. They are often simpler, easier to implement, and are suitable for datasets that are not very large, and for situations where interpretability is crucial. Machine learning algorithms are better suited for problems involving complex relationships, high-dimensional data, and large data volumes where the interpretability is less of a concern, and where the structure of the model is not easy to define ahead of time. The decision of which to use depends on the particular problem, the data quality, and the trade-offs between model complexity, interpretability, and generalization performance. A prudent quantitative trader should be familiar with both approaches and select the most appropriate tool for each specific task.