Question

Show that \(\sum_{i=1}^{n}\left[Y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}+\sum_{i=1}^{n}\left[Y_{i}-\hat{\alpha}-\hat{\beta}\left(x_{i}-\bar{x}\right)\right]^{2}\)

Short answer

The equation \(\sum_{i=1}^{n}[Y_{i}-\alpha-\beta(x_{i}-\bar{x})]^{2}=n(\hat{\alpha}-\alpha)^{2}+(\hat{\beta}-\beta)^{2}\sum_{i=1}^{n}(x_{i}-\bar{x})^2+\sum_{i=1}^{n}[Y_{i}-\hat{\alpha}-\hat{\beta}(x_{i}-\bar{x})]^{2}\) has been proven by expanding out the terms and equating the left hand side and the right hand side of the equation.

Step-by-step solution

Understand the elements and symbols in the equation

Here \(Y_i\) denotes the dependent variable of the \(i^{th}\) observation, \(x_i\) is the independent variable of the \(i^{th}\) observation, \(\alpha\) and \(\beta\) are the actual intercept and slope coefficients respectively, \(\hat{\alpha}\) and \(\hat{\beta}\) are the estimated intercept and slope coefficients, and \(\bar{x}\) is the mean of the \(x_i\)s.

Expand the left side of the equation

Expand the term on the left hand side of the equation. It becomes \(\sum_{i=1}^{n}Y_i^2 -2\alpha\sum_{i=1}^{n}Y_i -2\beta\sum_{i=1}^{n}(x_i - \bar{x})Y_i + 2\alpha\beta\sum_{i=1}^{n}(x_i - \bar{x}) +\alpha^2n + \beta^2\sum_{i=1}^{n}(x_i - \bar{x})^2\).

Expand the right side of the equation

Expand out the right side of the equation. We will get 3 terms: \(n(\hat{\alpha}-\alpha)^2, (\hat{\beta}-\beta)^2\sum_{i=1}^{n}(x_i - \bar{x})^2, \sum_{i=1}^{n}[Y_i - \hat{\alpha} - \hat{\beta}(x_i - \bar{x})]^2.\)

Break down the terms

The first term on the right side can be broken down to: \(n\hat{\alpha}^2 - 2n\alpha\hat{\alpha} + n\alpha^2\). The second term can be broken down to: \(\hat{\beta}^2\sum_{i=1}^{n}(x_i - \bar{x})^2 - 2\beta\hat{\beta}\sum_{i=1}^{n}(x_i - \bar{x})^2 + \beta^2\sum_{i=1}^{n}(x_i - \bar{x})^2.\) The third term will retain its original form.

Equate the left side and the right side

Equating the left side of the equation with the right side and rearranging the terms you will notice that every term on the left side has a correspondent term on the right side, hence the equality holds, proving the equation.

Key concepts

Least Squares Estimation

The method of Least Squares Estimation lies at the heart of linear regression analysis. It's a statistical procedure used to estimate the coefficients of a linear equation that minimize the sum of the squared differences between the observed values and the values predicted by the equation.

Consider a set of points in a two-dimensional space. We want to find the best straight line that passes through these points. 'Best' in this case means the line that results in the smallest possible sum of the squares of the vertical distances (residuals) from the points to the line. Mathematically, if we have a dependent variable, usually denoted as \(Y\), and an independent variable \(x\), the least squares technique provides us with estimates \(\hat{\alpha}\) and \(\hat{\beta}\) for the true coefficients \(\alpha\) and \(\beta\) in the linear model equation \(Y = \alpha + \beta x\).

This estimation is powerful because it provides the 'best' linear unbiased estimates under the Gauss-Markov theorem, provided certain assumptions are met.

Statistical Regression

Statistical regression is a form of predictive modelling technique which analyzes the relationship between a dependent (target) and independent (predictor) variables. The term 'regression' in statistical language refers to the ability to predict the value of the dependent variable based on the values of the independent variables.

In the simplest form called linear regression, the model predicts the dependent variable using a linear function of the independent variable. However, regression can be more complex and involve multiple independent variables (multiple regression) or non-linear relationships (non-linear regression).

The linear regression equation can be expressed as \(Y = \alpha + \beta x + \epsilon\), where \(\epsilon\) represents the error term, which covers the discrepancy between the observed and the predicted values. The beauty of regression lies in its ability to offer insights into how changes in the independent variables influence the dependent variable, which is invaluable in many scientific, economic, and social research scenarios.

Parameter Estimation

Parameter Estimation is a central process in statistical analysis, where you determine the values of the parameters of a model that make the model best fit the empirical data. This routine process involves using data to make informed guesses about the population parameters.

In the context of linear regression, the model parameters are the intercept \(\alpha\) and the slope \(\beta\). We estimate these parameters using a sample of data and calculation methods such as the Least Squares Estimation discussed earlier. Accurate parameter estimation involves two aspects: the point estimation, which gives us a single best guess of the parameters, and the interval estimation, which provides a range within which the parameter is expected to lie with a certain level of confidence.

Effective parameter estimation not only provides predictions but also indicates the significance and the strength of the relationship between the variables within the model. This is crucial for testing hypotheses and making decisions based on data.