Regression Analysis - Statistics Previous Year Questions

Let \(Y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \epsilon_i\) for \(i = 1, ..., 10\), where \(x_{1i}\)'s and \(x_{2i}\)'s are fixed covariates, and \(\epsilon_i\)'s are uncorrelated random variables with mean 0 and unknown variance \(\sigma^2\). Here \(\beta_0, \beta_1\) and \(\beta_2\) are unknown parameters. Further, define \(\hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_{1i} + \hat{\beta}_2 x_{2i}\), where \((\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2)\) is the unbiased least squares estimator of \((\beta_0, \beta_1, \beta_2)\). Then an unbiased estimator of \(\sigma^2\) is

Consider the regression model \(Y_i = \beta_0 + \beta_1 x_i^2 + \epsilon_i, \ i = 1, 2, \ldots, n \ (n \geq 2) ;\) where \(\beta_0\) and \(\beta_1\) are unknown parameters and \(\epsilon_i\)'s are random errors. Let \(y_i\) be the observed value of \(Y_i\), \(i = 1, \ldots, n\). Using the method of ordinary least squares, the estimate of \(\beta_1\) is

Consider the simple linear regression model \( Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \; i = 1, 2, ..., n \; (n \geq 3) \), where \( \beta_0 \) and \( \beta_1 \) are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance \( \sigma^2 > 0 \). Suppose that \( \hat{\beta}_0 \) and \( \hat{\beta}_1 \) are the ordinary least squares estimators of \( \beta_0 \) and \( \beta_1 \), respectively. Define \( \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \), \( S_1 = \sum_{i=1}^n (x_i - \bar{x})^2 \) and \( S_2 = \sum_{i=1}^n y_i (x_i - \bar{x}) \), where \( y_i \) is the observed value of \( Y_i \), \( i = 1, 2, ..., n \). Then for a real constant \( c \), the variance of \( \hat{\beta}_0 + c \hat{\beta}_1 \) is

Consider the multiple regression model
\( Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon, \)
where \( \epsilon \) is normally distributed with mean 0 and variance \( \sigma^2 > 0, \) and \( \beta_0, \beta_1, \beta_2, \beta_3 \) are unknown parameters. Suppose 52 observations of \( (Y, X_1, X_2, X_3) \) yield sum of squares due to regression as 18.6 and total sum of squares as 79.23. Then, for testing the null hypothesis \( H_0: \beta_1 = \beta_2 = \beta_3 = 0 \) against the alternative hypothesis \( H_1: \beta_i \neq 0 \) for some \( i = 1, 2, 3, \) the value of the test statistic (rounded off to three decimal places), based on one way analysis of variance, is ______

Consider the following regression model
\[ y_k = \alpha_0 + \alpha_1 \log_e k + \epsilon_k, \quad k = 1, 2, \ldots, n, \]
where \( \epsilon_k \)'s are independent and identically distributed random variables each having probability density function \( f(x) = \frac{1}{2} e^{-|x|}, \; x \in \mathbb{R} \). Then which one of the following statements is true?

Consider the simple linear regression model \(y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \quad i = 1, 2, \ldots, n,\) where \(\beta_0\) and \(\beta_1\) are unknown parameters, \(\varepsilon_i\)'s are uncorrelated random errors with mean 0 and finite variance \(\sigma^2 > 0\). Let \(\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\) and \(\widehat{\beta_1}\) be the least squares estimator of \(\beta_1\). Then which one of the following statements is true?