Question

b) Explain the purposes of the t test and the F test in multiple regression.

d) When can we experience autocorrelation in our data and how do we determine whether there is a first-order correlation (Explain)?

Answer 1

b ) F test is used to test whether overall model is a good fit or not . Say b1,b2 ,b3 are coefficients of independent variable , b​​​​​​0​​​​​ is intercept and Y is dependent variable

Then model is Y = b​​​​​​0+ b1X​​​​1 + b2X​​​​2+ b3X 3

We want to test the hypothesis that whether the independent variable play significant role in predicting the dependent variable or not . In other words we can say that we want to test if model given above is different from intercept only model.

Null hypothesis : b​​​​​​1= b​​​​​​2= b​​​3 = 0

Alternative Hypothesis : at least one of them differs from zero .

This hypothesis we test using F statistic. Therefore it is known as omnibus test.

Just for explaining I have taken 3 variables but it can be generalized to k variables.

t - test

F test doesn't tell you which of the coefficients are differing from zero . So using t test we test every coefficient of independent Variable individually and also intercept is tested.

Null hypothesis : b​​​​​​1 = 0

Alternative : b​​​​​​1 not equal to zero.

Similarly we test for every coefficient using t test.

d ) Autocorrelation occurs when observations are very much similar in particular time. Say you are taking observations of sales in a particular month. Sale on day 1 might be samilar to sale on day 2 ,day 3 but is different from sale on day 25. So in one month period first week is showing similar sales value because there was some festival compared to other days in week. Observation 1 is closer to observation 2 , observation 3 , than observations that are far. So we can see certain pattern in data . This is known as Autocorrelation and we can experience it in time series data. We have exemplified same in above example.

But There might be cases where you can experience in cross sectional data . Can We say that IQ in class 1 students is similar but different from Class 5 students. there will be Similarity in IQ of class 5 students.

We can detect Autocorrelation using Durbin Watson test and observing patterns in Autocorrelation plots.