Question

Consider data from the straight-line regression model with \(n\) observations and $$ x_{j}= \begin{cases}0, & j=1, \ldots, m \\ 1, & \text { otherwise }\end{cases} $$ where \(m \leq n .\) Give a careful interpretation of the parameters \(\beta_{0}\) and \(\beta_{1}\), and find their least squares estimates. For what value(s) of \(m\) is \(\operatorname{var}\left(\widehat{\beta}_{1}\right)\) minimized, and for which maximized? Do your results make qualitative sense?

Short answer

\(\beta_0\) is the mean for \(x=0\), \(\beta_1\) is the change to \(x=1\). \(\text{Var}(\hat{\beta}_1)\) minimized when \(m=n/2\), maximized at extremes.

Step-by-step solution

Understand the data and model

We are given a regression model with observations where the independent variable, \(x_j\), takes values 0 for the first \(m\) observations and 1 for the rest. In these cases, the dependent variable can be expressed as \(y_j = \beta_0 + \beta_1 x_j + \epsilon_j\), where \(\epsilon_j\) is an error term with mean zero.

Interpret the parameters \(\beta_0\) and \(\beta_1\)

Since \(x_j = 0\) for \(j = 1, \ldots, m\), the corresponding \(y_j\) are modeled as \(y_j = \beta_0 + \epsilon_j\). For the others where \(j > m\), \(y_j = \beta_0 + \beta_1 + \epsilon_j\). Therefore, \(\beta_0\) represents the expected value of \(y\) when \(x = 0\), and \(\beta_1\) is the change in expected value of \(y\) when \(x\) changes from 0 to 1.

Set up the normal equations for least squares estimates

We use the normal equation \(X'X\beta = X'y\) for our design matrix having a column of ones and a column with the values of \(x_j\). Solving, we find that:\[\hat{\beta}_0 = \frac{\sum_{j=1}^{m} y_j}{m} \quad \text{and} \quad \hat{\beta}_1 = \frac{\sum_{j=m+1}^{n} y_j}{n-m} - \hat{\beta}_0\]

Analyze the variance of \(\hat{\beta}_1\)

The variance of \(\hat{\beta}_1\) is derived from the variance of sample means, factoring the number of observations from groups where \(x_j = 0\) and \(x_j = 1\). This variance is minimized when \(m = n/2\), achieving equal split and maximum variance when \(m\) approaches either end of the range (0 or \(n\)).

Qualitative interpretation

The result makes qualitative sense as equal partitioning of data (balanced design) generally reduces the variance of the parameter estimates, while biased partitioning increases variance due to imbalance in sample sizes in terms of information about behavior at \(x = 0\) and \(x = 1\).

Key concepts

Least Squares Estimates

The least squares estimates are crucial in analyzing the straight-line regression model, as they provide the best-fitting line through the given data points by minimizing the sum of the squared differences between observed and predicted values.
To find the least squares estimates, we use the formula for the regression line: \[ y_j = \beta_0 + \beta_1 x_j + \epsilon_j \]where \(\epsilon_j\) represents the error term.

• For \(x_j = 0\), our formula simplifies to \(y_j = \beta_0 + \epsilon_j\).
• For \(x_j = 1\), it becomes \(y_j = \beta_0 + \beta_1 + \epsilon_j\).

By solving the normal equations derived from the above expressions, the estimates are calculated as:

• \(\hat{\beta}_0 = \frac{\sum_{j=1}^{m} y_j}{m}\)
• \(\hat{\beta}_1 = \frac{\sum_{j=m+1}^{n} y_j}{n-m} - \hat{\beta}_0\)

These estimates serve to effectively interpret the pending parameters, providing a clear guideline on what values \(y\) takes when \(x = 0\) and the change that occurs when \(x\) shifts to 1.

Parameter Interpretation

Interpreting the parameters \(\beta_0\) and \(\beta_1\) of our regression model offers insights into the relationships they share with the data.

• \(\beta_0\) represents the expected value of the dependent variable \(y\) when the independent variable \(x\) equals 0.
  This means if you were to measure \(y\) at the point where \(x\) takes a value of 0, \(\beta_0\) is the mean of those measured values.
• \(\beta_1\) quantifies the change in \(y\) from when \(x\) changes from 0 to 1.
  If \(\beta_1\) is positive, \(y\) increases as \(x\) moves from 0 to 1, but if it's negative, \(y\) decreases.

By this interpretation, these parameters help in understanding the expected outcomes at different settings of \(x\), enabling predictions and careful analysis of data behavior.

Variance Analysis

Variance analysis of \(\hat{\beta}_1\) involves understanding how the estimate's variance changes with different values of \(m\).
Such analysis is important, as variance reflects the reliability and stability of an estimate.

• When \(m = n/2\), variance is minimized. This is because the data is balanced, with equal observations for both segments of \(x\) values, providing a stable basis for estimation.
• On the other hand, variance becomes maximized at extreme values of \(m\) (either \(0\) or \(n\)).
  This implies an imbalance, with all data points skewed towards either \(x = 0\) or \(x = 1\), reducing interpretive clarity and introducing higher instability in estimates.

This qualitative understanding of variance emphasizes the need for balanced data to achieve the most reliable estimations, reflecting the consistency principle in statistical design.