2. Continuous Probability Distribution, Samples, Estimators, Hypothesis Testing

Question 1

What is the normal distribution characterised by?

Answer 1

The mean and the variance

Question 2

What % of values are within 1 standard deviation from the mean?

Answer 2

68%

Question 3

Properties of chi squared distribution

Answer 3

1. Asymmetric, right skewed. The skewness decreases with degrees of freedom
2. The mean is given by n and the variance is given by 2n
3. A sum of independent chi square random variables is also chi square distributed

Question 4

What is the t distribution?

Answer 4

The ratio of independent standard normal and square root scales chi squared random variables

Question 5

Properties of t distribution

Answer 5

1. Has thicker tails when n is small
2. When n is large it is basically the standard normal
3. Values of t distribution are used as critical values in hypotheses tests
4. Mean is 0 and variance is n/(n-2)

Question 6

Properties of F distribution

Answer 6

Asymmetric and right skewed. As n1 and n2 increase, the F distribution approaches the normal

Question 7

Unbiased

Answer 7

An estimator is unbiased if it’s sampling distribution equals the parameter of interest

Question 8

Efficiency

Answer 8

An estimator is efficient if it has the smallest possible variance

Question 9

Type 1 error

Answer 9

We reject Ho even when it is true. The probability of this happening is equal to the significance level

Question 10

Type 2 error

Answer 10

Failing to reject Ho when H1 is true

Question 11

What happens to the power of the test as n approaches infinity?

Answer 11

The power of the test approaches 1, we say the test is consistent

Question 12

What is a residual?

Answer 12

The deviation of Yt from the estimated relationship relationship for a given observation

Question 13

How do we derive OLS estimators of alpha hat and beta hat?

Answer 13

Take partial derivatives of the sum of squared deviations wrt alpha hat and beta hat. Set each derivative to 0. Solve for alpha hat and beta hat

Question 14

OLS assumptions

Answer 14

1. Explanatory variables x are uncorrelated with errors
2. Errors have zero conditional mean
3. Errors have constant variance (no heteroscedasticity)
4. Errors are uncorrelated with each other (no serial correlation)

Question 15

Which one of the OLS assumptions is the one that is likely to cause us issues?

Answer 15

1. Explanatory variables x are uncorrelated with errors. If this assumptions doesn’t hold then the estimators become biased and inconsistent

Question 16

What is the extra assumption of the multivariate linear regression model

Answer 16

No perfect collinearity