Statistical methods

Question 1

What is the chain rule of probability?

Answer 1

P( X = x n Y = y ) = P( Y = y ) x P( X = x | Y = y )
or
P( X = x n Y = y ) = P( X = x ) x P( Y = y | X = x )

Question 2

What is P( X = x | Y = y ) using the chain rule of probability

Answer 2

P( X = x n Y = y ) / P( Y = y)

Question 3

If X and Y are independent, what do we know?

Answer 3

P( X = x | Y = y ) = P( X = x )

Question 4

If X and Y are independent, what is P( X = x n Y = y ) ?

Answer 4

P( X = x ) x P( Y = y )

Question 5

What is covariance?

Answer 5

Measures how two random variables vary together

Question 6

What is the law of total probability

Answer 6

P( X = x ) = Sum of P( X = x n Y = y)

Question 7

What is variance?

Answer 7

Measures the average deviation of a variable away from its expected value

Question 8

If a new random variable: Z = X + Y, then what is var(Z)

Answer 8

var (Z) = var (X) + var (Y) + 2cov (X,Y)

Question 9

What is correlation?

Answer 9

tells you how close the data is to lying on a straight line

Question 10

If we are in a binomial distribution, and we are trying to approximate P( X >= n), what do we need to do?

Answer 10

Convert to the normal distribution and use the continuity correction

Question 11

If we are in a normal distribution and want P( X > n ), what do we need to do?

Answer 11

Standardise using Z = (X - mean) / standard deviation

Question 12

What is the sample mean?

Answer 12

1/n x (sum of Xi)

Question 13

An estimator, mu hat, is unbiased if:

Answer 13

E ( mu hat) = mu

Question 14

What is the size of the bias

Answer 14

E (mu hat) - mu

Question 15

If we assume observations are iid, what assumptions have we made?

Answer 15

E (Xi) = Mu x
var (Xi) = sigma^2 x
cov (Xi,Xj) = 0 for all i,j

Question 16

What is the central limit theorem

Answer 16

Provided n is sufficiently large, the sample mean (x bar) would be approximately distributed centred around the expectation E(X), with variance var(X) / n

Question 17

When do we use the z-distribution

Answer 17

When n is large and we know the population variance

Question 18

What is the sample variance

Answer 18

(1 / n - 1) x sum of (Xi - X bar)^2

Question 19

When is an estimator considered ‘good’

Answer 19

If it is unbiased
If it is efficient (low variance)

Question 20

What is the size of the error when estimating the population parameter

Answer 20

Error = Mu hat - Mu

Question 21

What is the average error

Answer 21

E (Mu hat - Mu) = E (Mu hat) - Mu

Question 22

What is the mean square error

Answer 22

MSE (Mu hat) = var (Mu hat) + bias^2 (Mu hat)

Question 23

When is an estimator consistent in terms of MSE

Answer 23

lim (n to infinity) MSE (Mu hat) = Mu

Question 24

When is an estimator consistent (Weak law of large numbers)

Answer 24

plim (Mu hat) = E (X) = Mu x

Question 25

What is the formula used in t-distributions

Answer 25

( X bar - (E (X) | Ho is true)) / sqrt( sample variance / n )

Question 26

What is the strong law of large numbers

Answer 26

lim (n to infinity) of sample mean = E (X)

Question 27

If we are asked to carefully interpret a given set of data, what are the three steps we must follow?

Answer 27

1. Does the line provide a good fit to the data - what is the meaning of the R^2 2. Are the coefficients statistically significant? 3. What is the economic significance of the estimated coefficients?

Question 28

If I have a linear relationship, what is the interpretation of Beta1

Answer 28

Marginal impact of a change in X on Y

Question 29

If I have a semi-log relationship (log / ln of Y), what is the interpretation of Beta1

Answer 29

Beta1 represents the proportional change in Y due to a marginal change in X. For example, if Beta1 = 0.5, this means that if X increases by 1, then Y would increase by a factor of 0.5 (50% increase in Y)

Question 30

If I have a log-log relationship (logs on both sides), what is the interpretation of Beta1

Answer 30

Beta1 is the elasticity of Y with relation to X. This means that Beta1 represents the percentage change in Y due to a 1% increase in X. For example, if Beta1 = 0.5, a 1% increase in X is associated with a 0.5% increase in Y.

Question 31

If I have a log explanatory variable relationship (log only on the explanatory variable), what is the interpretation of Beta1

Answer 31

The change in the dependent variable, Y, is equal to Beta1 multiplied by the proportional change in X. For example, if Beta1 = 0.5, then a 1% increase in X will result in an increase in Y of 0.005.

Question 32

What is the general rule of thumb when detecting OV bias

Answer 32

If you introduce new variables, the coefficients on existing variables will change. If they change by more than about 2 times the standard error, it is strong evidence of OV bias.

Question 33

If we have omitted a variable, what was the true DGP before

Answer 33

Yi = Beta0 + Beta1Xi + Beta2Zi + ei

Question 34

What is the size of the bias introduced when omitting a variable Z in large samples

Answer 34

Beta2 x (cov(X,Z) / var(X))

Question 35

If R^2 = 0.05, how well does the model fit the data

Answer 35

Approximately 5% of the variation in the dependent variable is being explained by the variation in the explanatory variable

Question 36

What are the conditions for being a well-defined probability density function

Answer 36

- It must always be positive for all possible values - The integral (area underneath) must be equal to 1

Question 37

If E (ui) = E (ui | {X} ) = 0, what does this tell us?

Answer 37

E (u) = 0 cov (u , X) = 0

Question 38

What is the weak law of large numbers?

Answer 38

plim (sample mean (X bar) ) = E (X)

Question 39

How do you calculate the expectation of a uniform distribution using integration

Answer 39

E (X) = integral between a and b of: (1 / b-a)x dx