# Regression part I

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## Outline

• The following topics will be covered in this lecture:
• An introduction to regression
• Simple linear regression
• Multiple linear regression
• Least squares solution
• Basic hypothesis testing
• Multiple regression in R
• Goodness of fit

## Introduction to regression

• Regression models are extremely important in describing relationships between variables.

• Linear regression is a simple, but powerful tool in investigating linear dependencies.

• It relies, however, on strict distributional assumptions in terms of how the relationship varies with respect to the regressors.
• Nonparametric regression models are widely used, because fewer assumptions about the data at hand are necessary.

• At the beginning of every empirical analysis, it is better to look at the data without assumptions about the family of distributions.

• Nonparametric techniques allow describing the observations and finding suitable models, when the sample size is sufficiently large and representative to explain the true population.

### Introduction to regression

• Regression models aim to find the most likely values of a dependent variable $$Y$$ for a set of possible values $$\{x_i\}_{i = 1}^n$$ of the explanatory variable $$X$$.

• We write a proposal for how the variables $$Y$$ and $$X$$ vary together as

\begin{align} Y = g(X) + \epsilon & & \epsilon \sim F_\epsilon , \end{align}

• where $$g(X)= \mathbb{E}\left[Y \vert X =x \right]$$ is an arbitrary function.

• The $$g(X)$$ is included in the model with the intention of capturing the mean of the process that corresponds to a particular value of $$X=x$$.

• If we believed that $$Y$$ had no dependency on the value of $$X=x$$, we could simply model this with respect to the average of the measured $$Y$$.
• The $$\epsilon$$ is a random noise term, representing variation around the deterministic part of the relationship.

• The natural aim is to keep the values of the $$\epsilon$$ as small as possible;

• that is to reduce the overall variation around the signal so that $$g(X)$$, the systematic part, explains as much of the relationship as possible.
• Parametric models assume that the dependence of $$Y$$ on $$X$$ can be fully explained by a finite set of parameters and that $$F_\epsilon$$ has a prespecified form with parameters to be estimated.

### Introduction to regression

• Nonparametric methods do not assume any form:

• neither for $$g(X)$$ nor for $$F_\epsilon$$, which makes them more flexible than the parametric methods.
• The fact that nonparametric techniques can be applied where parametric ones are inappropriate prevents the nonparametric user from employing a wrong method.

• These methods are particularly useful in fields like quantitative finance, where the underlying distribution is in fact unknown.

• However, as fewer assumptions can be exploited, this flexibility comes with the need for more data.

• Particularly, nonparametric methods can be of high variance in how they estimate the trend in the data.

• This can leave the methods susceptible to overfitting when the sample size is not large enough to differentiate the noise due to sampling error versus the true population level trend.

## Introducing linear models • In past mathematics courses, we have seen many examples of linear models.
• Suppose that we wish to model a relationship between two variables, $$x$$ and $$y$$ to the left.
• We will call $$y$$ a dependent variable, or the response variable.
• On the other hand, we will call $$x$$ an independent variable, an explanatory variable or a predictor variable for the response.
• Q: can you propose a valid linear model for the relationship between the response and the predictor?

### Introducing linear models – continued • A: actually, any line that passes through the point is a valid linear model.
• Particularly, this relationship is underconstrained and there exists infinitely many choices of linear models;
• given the current data, any choice is as valid as any other.

### Introducing linear models – continued • Q: given the data on the left, can you propose a valid linear model for the relationship between $$x$$ and $$y$$?