Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Statistics & Exploratory Data Analysis Simple and Multiple Linear Regression
There are two sets of observations on a random vector (X, Y). Consider a simple linear regression model with an intercept for regressing Y on X. Let \(\hat{\beta}_i\) be the least squares estimate of the regression coefficient obtained from the i - th (i = 1, 2) set consisting of n observations (n1, n2 > 2). Let \(\hat{\beta}_0\) be the least squares estimate obtained from the pooled sample of size n1 + n2. If it is known that \(\hat{\beta}_1\) > \(\hat{\beta}_2\) > 0, which of the following statements is true?
1
\(\hat{\beta}_2\) < \(\hat{\beta}_0\) < \(\hat{\beta}_1\)
2
\(\hat{\beta}_0\) may lie outside (\(\hat{\beta}_2\), \(\hat{\beta}_1\)), but it cannot exceed \(\hat{\beta}_1\) + \(\hat{\beta}_2\)
3
\(\hat{\beta}_0\) may lie outside (\(\hat{\beta}_2\), \(\hat{\beta}_1\)), but it cannot be negative
4
\(\hat{\beta}_0\) can be negative
5
Question Not Attempted