Question

Suppose that the relationship between \(E(y)\) and \(x\) is a straight line. What would you know about the value of \(\beta_{2} ?\)

Short answer

Given the linear relation between the expected value of \(y\) and \(x\), and considering that this relationship does not contain any quadratic terms, we can conclude that the value of \(\beta_{2}\) is zero. This is based on the fact that \(\beta_{2}\) would be the coefficient for a quadratic term in a non-linear relationship, which does not exist in this case. Thus, \(\beta_{2} = 0\).

Step-by-step solution

Write down the linear regression model

The given relationship between the expected value of \(y\) and \(x\) is a straight line, which can be represented by a linear model. A linear regression model is given by the following equation: $$ y = \beta_0 + \beta_1 x + \epsilon$$ where \(y\) is the dependent variable, \(x\) is the independent variable, \(\beta_0\) is the intercept, \(\beta_1\) is the slope, and \(\epsilon\) is the error term.

Find the expected value of \(y\)

Since we are given the relationship between the expected value of \(y\) and \(x\), we need to find the expected value of \(y\). This can be calculated by taking the expectation of both sides of the linear regression model: $$ E(y) = E(\beta_0 + \beta_1 x + \epsilon) $$

Simplify the expected value equation

We can now simplify the expected value equation using the rules of expectation, noting that \(\beta_0\) and \(\beta_1\) are constants, and that the expected value of the error term, \(\epsilon\), is equal to zero: $$ E(y) = E(\beta_0) + E(\beta_1 x) + E(\epsilon) = \beta_0 + \beta_1 E(x) = \beta_0 + \beta_1 x $$

Compare the expected value equation to the given relationship

We are given that the relationship between \(E(y)\) and \(x\) is a straight line. Comparing this with our derived expected value equation: $$ E(y) = \beta_0 + \beta_1 x $$, we can see that the relationship is indeed a straight line.

Identify the value of \(\beta_{2}\)

Since we are given a relationship between \(E(y)\) and \(x\) in the form of a straight line, we can see that there is no quadratic term in the equation, which would be represented by a variable with an exponent of 2, such as \(x^2\). Therefore, there is no \(\beta_2 x^2\) term in the equation. In other words, the coefficient for the quadratic term, \(\beta_{2}\), must be equal to zero because it does not exist in the given linear relationship: $$ \beta_{2} = 0$$

Summary

Given that the relationship between \(E(y)\) and \(x\) is a straight line, we determined that the value of \(\beta_{2}\) must be equal to zero, as there is no quadratic term in the linear relationship.

Key concepts

Expected Value in Linear Regression

The concept of expected value plays a fundamental role in the field of statistics, especially when discussing linear regression models.

The expected value of a variable is essentially its long-term average if the same process is repeated many times. In a linear regression context, when we assume that the relationship between the expected value of the dependent variable, denoted as \(E(y)\), and the independent variable, \(x\), is a straight line, we are unfolding a very specific pattern of relationship.

This aspect forms the basis of predicting outcomes: for any given value of \(x\), we can estimate the corresponding average outcome of \(y\). If we line up these averages for different values of \(x\), we end up with a straight line, characterized by a slope and an intercept - the backbone of our linear regression model.

This straight line provides a predictive framework, making \(E(y)\) a representation of the central or 'expected' path that our observations are predicted to follow as \(x\) changes.

Independent and Dependent Variables

Diving deeper, each linear regression model consists of variables that play specific roles. These are termed 'independent' and 'dependent' variables.

An independent variable, labeled as \(x\) in our linear regression equation, is a value you manipulate or observe to see how it affects the dependent variable. Think of it like the input or cause in your study.

The dependent variable, on the other hand, designated as \(y\), is what you measure or predict - the output or effect. This is the value you expect to change as a result of alterations in the independent variable.

By plotting these variables against each other, with the independent variable on the x-axis and the dependent variable on the y-axis, the linear regression model attempts to draw a straight line that best fits the data points. This line is a visualization of the predicted relationship between \(x\) and \(y\).

Error Term

The error term, symbolized by \(\epsilon\), addresses the element of uncertainty or randomness in our linear regression model. It's not a mistake in calculation, but rather the recognition that not all variations can be explained by the linear relationship between the independent and dependent variables.

In simpler terms, think of the error term as the 'wiggle space' that the actual data points have around our estimated straight line. It represents the unexplained factors that affect the dependent variable, \(y\), beyond what the independent variable, \(x\), can account for.

Statistically, the expected value of the error term is zero, suggesting that while individual observations may deviate from the predicted line, they do so in an unpredictable and unbiased manner, balancing out in the long run.

Understanding the error term is crucial, as it leads to better interpretations of the model and its accuracy. The smaller and more random the error term, the stronger the predictive power of our model.