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# 统计代写|回归分析代写Regression Analysis代考|The Assumptions of the Classical Regression Model

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## 统计代写|回归分析代写Regression Analysis代考|The Assumptions of the Classical Regression Model

A definitive source for the definition of a statistical model is the article “What is a Statistical Model?” by Peter McCullagh (2002), who defines a statistical model as “… a set of probability distributions on the sample space …” Translating, a statistical model is simply an assumption that your sample data are produced randomly by a particular probabilistic process that lies in a prescribed set of possible probabilistic processes. This “prescribed set of possible probabilistic processes” is what is meant by a “statistical model.”

For a simple example, one often refers to the “normal model” in statistics. This model does not prescribe a particular normal distribution as the model for the DGP; instead, it states that the data are randomly generated from a particular normal distribution within the general class of $\mathrm{N}\left(\mu, \sigma^2\right)$ distributions.

There are several assumptions that you make when you analyze the data using regression models. The first and most important assumption is that the data are produced probabilistically, which is specifically stated as $Y \mid X=x \sim p(y \mid x)$. Different types of regression models then make further assumptions regarding the prescribed sets of distributions, and regarding the prescribed way that these distributions are related to $x$. The assumptions are important because they determine the adequacy of the model.

Adequacy of the regression model refers to the closeness of the approximation of the model, as a producer of data, to the real data-generating process.

## 统计代写|回归分析代写Regression Analysis代考|Randomness

Statistical models, including regression models, are statements about how the potentially observable data are produced, in general. They are quantifications of your subject matter theory. If you are writing a research paper in a scientific discipline, you will typically explain all this theory in words that state how and why such generalities occur. Your statistical model is simply a concise, mathematical and probabilistic summary of all that general theory. Your research hypotheses, which are also statements about how your data will appear (or might have appeared), are also defined in terms of your statistical model for your data-generating process.

Usually, you do not see any “randomness” assumption explicitly stated in research articles or other texts. Instead, the assumption is implicit, which you will often see stated in a model form such as
$$Y=\beta_0+\beta_1 X+\varepsilon$$
Implicit in that model formulation is that $\varepsilon$ is random. This assumption is necessary because the data $Y$ are not a deterministic function of $X$. If your relationships are in fact deterministic, then stop reading this book immediately! You should read a book on differential equations instead.

Anticipating multiple regression, where there is one or more $X$ variables, we introduce the boldface term $X$ to denote a set of possible $X$ variables: $X=\left(X_1, X_2, \ldots, X_k\right)$.

## 统计代写|回归分析代写Regression Analysis代考|Randomness

$$Y=\beta_0+\beta_1 X+\varepsilon$$

## MATLAB代写

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