L11 - Testing Linear Restrictions 2

Question 1

How do you write Linear Restrictions in Matrix Form for a bivariate model?

Answer 1

• Where R is a matrix of coefficient, θ is a vector or parameters and r is a vector os restrictions

Question 2

How do you write Linear Restrictions in Matrix Form for a multivariate model?

Answer 2

Question 3

What are the three classical approaches to testing restrictions?

Answer 3

1. Likelihood ratio
2. Wald
3. Langrange Multiplier

The classical approaches to testing can be explained best in the context of maximum likelihood estimation.

Question 4

What is the Notation needed for a maximum Likelihood Problem with a single parameter?

Answer 4

Question 5

What is the Likelihood Ratio approach when testing restrictions?

Answer 5

The Likelihood ratio test is based on the difference between the log- likelihoods of the restricted and unrestricted cases

Question 6

What is the Wald approach when testing restrictions?

Answer 6

The Wald test is based on the difference between the restricted and unrestricted parameter estimate –> horizontal difference

Question 7

What is the Lagrange Multiplier approach when testing restrictions?

Answer 7

The Lagrange multiplier test is based on the slope of the log-likelihood function at the restricted parameter value

Question 8

How are all three test statistics distributed?

Answer 8

All three test statistics follow a chi-squared distribution under the null hypothesis with degrees of freedom equal to the number of restrictions.

Question 9

What is the Wald Test an example of?

Answer 9

where k is the number of parameters to be estimated including slope and intercept

Question 10

What is the Likelihood Ratio Test similar to?

Answer 10

Question 11

How can you convert the t-test into a chi-squared distribution?

Answer 11

• you square it, this will give you a chi squared distribution with 1 degree of freedom ( considering that you are only testing for a value of β –> thus only one restriction)

Question 12

Is the F-statistic positive or Negative?

Answer 12

Restrictions always increase the residual sum of squares. ( As the unrestricted value is already at the minimum)

Therefore the F statistic is always positive. if (RRSS-URSS > 0 then the F statistic is positive)

Question 13

How are Regression Model Misspecified?

Answer 13

A regression model may be misspecified for a number of reasons:

• An incorrect choice of functional form –> assume its linear, but the relationship mat not be
• Omitted variable bias –> when we set up the model we may not have the correct set of X variables ;–> left out revelant variables
• Inclusion of irrelevant variables
• Measurement error in the regressors –> X variables not measured correctly (have negative variables but take logs of them –> which give zero because it cant be measured)
• Correlation of the independent variables with the errors

Question 14

How can you test if there is an Omitted Variable Bias?

Answer 14

The error term disappears in the 3 line because the expected value of the error term is equal to 0

• There is only two condition where we can have an unbiased estimator:
• If β₂= 0
• if E(X_i1X_i2)= 0 (they are correlated)

if this happen we still have an unbiased OLS estimator

Question 15

How can you test for Inclusion of Irrelevant Variables?

Answer 15

Variance of β₁(hat) > Variance of the OLS estimator –> the denominator of the Variance of β₁(hat) is multiplied by a correlation coefficient (which is alwys between -1 and 1) thus the denominator should always be smaller allowing for the variance to be bigger for β₁(hat)

Normally we want a variance as small as possible but when we do include irrelevant variables it is greater than the variance of the OLS estimator so it is unbiased but inefficient

• the degrees of inefficiency we depend on the correlation of the right hand size variables –> more correlated the higher the inefficiency will be

Question 16

Why is it best to start with a general model and test restrictions?

Answer 16

The reasons for this are as follows:

1. If the unrestricted model is general enough then it should provide a valid basis for tests of restrictions.
2. If the restricted model is incorrect then tests based on it will not be valid.
3. There may be several alternative ways to modify an incorrect model.