Question

How we test the hypothesis about population variance σu2 ? Q.2 Consider the following Models which are estimated through OLS Model 1 Y=α+βX+u Model 2 Y∗=ψ+δX∗+v where Y∗=2Y and X∗=X/2 Find the relationship between intercept and slope coefficients of the two estimated models. Q.3 Consider the modelY =β1+β2X2+β3X3+u how we can test the hypothesis that β1=0 and β2+β3=1 simultaneously?

Answer 1

To test the hypothesis about population variance σu2, we can use the F-test. The F-test compares the ratio of two variances: the variance explained by the regression model and the variance unexplained by the regression model.

Step-by-step explanation:

To test the hypothesis about population variance σu2, we can use the F-test. The F-test compares the ratio of two variances: the variance explained by the regression model and the variance unexplained by the regression model. The steps to test the hypothesis using the F-test are as follows:

1. First, estimate the population variance using the sum of squared residuals from the regression model: SSE = Σ(yᵢ - ŷᵢ)².
2. Next, estimate the population variance under the null hypothesis (assuming no relationship between the predictors and the response variable) using the total sum of squares: SST = Σ(yᵢ - ȳ)², where ȳ is the mean of the response variable.
3. Calculate the F-statistic: F = (SST - SSE) / p, where p is the number of predictors in the model.
4. Compare the calculated F-statistic to the critical F-value at the desired significance level. If the calculated F-statistic is greater than the critical F-value, we reject the null hypothesis and conclude that there is sufficient evidence to support a relationship between the predictors and the response variable.