Question

1.

1.1 What is a continuous variable, a discreet variable and an indicator variable (or, a binary or dummy variable)? What is the condition that ensures that two random variables are statistically independent? Is it the same as having a zero covariance? [10%]

1.2 Suppose ?? ~??(??, ?? 2 ) , with mean ?? and variance ?? 2 . What is the expected value and variance of the variable ?? = ????? ??? ? [10%]

1.3 What is collinearity? What are the implications of collinearity? How can you identify and mitigate it? Provide a detailed discussion and use examples for the explanation. [30%]

Answer 1

ans 1.1

A variable is a quantity whose value changes.

A discrete variable is a variable whose value is obtained by counting.

Examples: number of students present, number of red marbles in a jar etc.

A continuous variable is a variable whose value is obtained by measuring.

Examples:height of students in class, weight of students in class, time it takes to get to school etc

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

?         A random variable is denoted with a capital letter

?         The probability distribution of a random variable X tells what the possible values of X are and how probabilities are assigned to those values

?         A random variable can be discrete or continuous

A discrete random variable X has a countable number of possible values.

Example: Let X represent the sum of two dice.

A continuous random variable X takes all values in a given interval of numbers.

?         The probability distribution of a continuous random variable is shown by a density curve.

?         The probability that X is between an interval of numbers is the area under the density curve between the interval endpoints

?         The probability that a continuous random variable X is exactly equal to a number is zero

# two random variables are independent if the realization of one does not affect the probability distribution of the other.

mathematically,

let there be two random variables X and Y with cumulative distribution functions and and probability densities and. then, they are considered to be independent iff the combined random variable (X, Y) has a joint cumulative distribution function, such that

or equivalently, if the joint density exists, such that

If X and Y are independent, then their covariance is zero. This follows because under independence,

hence, the idependence condition is same as the zero covariance condition.

ans 1.2

ans 1.3

implications of collinearity:-

1. OLS estimators are still BLUE but variances and covariances are large .

2. As a result of which the confidence intervals are wider because of which we tend to accept the null hypotheis that the true parameter is having zero effect on the regressand.

3. t values of the regression coefficients becomes insignificant because of large variances.

4.R^2 value becomes very high .

5. OLS cofficients and standard errors become sensitive to small changes in the data .

IDENTIFICATION:-

•High R2 but few significant t ratios
•High pair wise correlation /partial corrleation among Xs.
•Auxiliary regression: Regress Xi on other Xs and compare Rx 2 from this regression with R2 from original equation. If higher than original (Rx2 >R2), the problem is troublesome.

Compute F:

F= [R²_x /(k-2)] / [(1- R²_x)/ (n-k+1)]

where k – number of variables in auxiliary regression (i.e., number of x variables in original)
If Fc > Ft at chosen level of significance, then particular x is collinear with other Xs.

MITIGATION TECHNIQUES :-

•Dropping unimportant variable(s)
•Transformation of variable: log or first difference
•Adding more observations (increase n-sample size)
•Panel data modelling
•Specification using a priori information

If x1 and x2 are highly collinear and we know that ?2 = 0.1 ?1 from previous studies then, instead of estimating

• Y = ?0 + ?1 X1 + ?2 X2 + u , estimate ,
• Y = ?0 + ?1 X1 + 0.1 ?1 X2 + u

= ?0 + ?1 Xi + u
where Xi = X1 + 0.1 X2
After estimating the above regression , calculate ?2.