Quantitative methods

Question 1

Define conditional heteroskedasticity.

Answer 1

Conditional heteroskedasticity occurs when error terms are related to the independent variables.

Question 2

How to test for heteroskedasticity?

Answer 2

Breusch-Pagan chi-square test

Question 3

How to correct Heteroskedasticity?

Answer 3

Heteroskedasticity refers to situations where the variance of the residuals is unequal over a range of measured values.

To correct heteroskedasticity

-Calculate robust standard errors - Correct the standard errors of the model’s estimated coefficient to account for heteroskedasticity.

-Generalized least square - Regession is motified to elimate heteroskedasticity.

Question 4

What is homoskedasticity?

Answer 4

The variance of error terms is constant across all observations.

Question 5

Define what is Autoregressive Conditional Heteroskedasticity ARCH(1) model?

Answer 5

ARCH(1) is a AR(1) model with conditionalheteroskedasticity.

When tested for ARCH(1) by regressing the squared residual against the lagged value of the squared residual. ARCH(1) the lagged squared residual would explain the current squared residual, hence the coefficient would be significantly different to zero.

If ARCH(1), it can predict the variance of the error terms next period using this period’s squared residuals.

Question 6

Given ARCH(1) model how it can predict the variance of the error terms?

Answer 6

It can predict the varinace of the error terms next period using this period’s squared residuals.

Question 7

Define serial correlation (autocorrelation).

Answer 7

Serial correlation occurs when regression errors are correlated across observations, where errors in one period is correlated with errors in other periods.

Question 8

What is the statistical test to determine whether there is a serial correlation for the errors in a regression?

Answer 8

Durbin Watson test.

Step 1 - Find DW statistics
Step 2 - Find the lower bound and upper bound DW critical value in the DW table
Step 3 - Compare

If erros are not serially correlated, then DW will be close to 2.

If statistics < lower bound (2), this indicates a positive correlation.
If lower bound (2) < statistics < upper bound (2), this supports the null hypothesis of no serial correlation.
If statistics > upper bound (2), this indicates a negative correlation.

Durbin watson cannot be used for autoregressive models.

Reference:
https://analystnotes.com/cfa-study-notes-the-durbin-watson-statistic.html

Question 9

What is the meaning of multicollinearity?

Answer 9

When the independent variables are related in a regression.

Question 10

How to identify multicollinearity in a regression?

Answer 10

• High R^2 meaing the equation as a whole is significant.
• invidual t-statistics is low or not be significant.

Question 11

How to resolve multicollinearity in a regression?

Answer 11

This can be elimated by excluding one of the realted variable after which both t-statistics and the regression as a whole is significant.

Question 12

How to calculate the coefficient of determination R^2?

Answer 12

R^2 = SSR / SST
= (Regression sum of square)/ (total sum of square)

It describe the percentage variation in the dependent variable explained by movements in the independent variable.

Question 13

How to calcualte SEE standard error of the estimate?

Answer 13

SEE = sqrt(SSE / (n - k - 1))

SSE = Sum of square error
n = number of observiation
k = number of independent variable

Question 14

How to calculate correaltion coefficient?

Relationship between coefficient of determination R^2 and correlation

Answer 14

correaltion coefficient = sqrt(SSR / SST)

sqrt(R^2) = correlation coefficient

Question 15

How to calculate the F test?

Answer 15

F test = MSR / MSE
= (SSR / K) / (SSE / n - k - 1)

n = number of observiation
k = number of independent variable

Question 16

Test statistc

Answer 16

Test statistic = (Sample mean- Hypothesized mean ) / [Standard deviation of the sample / sqrt (number of sample)]

where Hypothesized mean = 0

Question 17

How to calculate t statistic using correlation coefficient?

Answer 17

t = [correlation coefficient * sqrt(n - k)] / sqrt(1 - correlation coefficient ^2)

n= number of sample
k =independent variable

Question 18

Calculate F-statistic.

Answer 18

F = Mean regression sum of square / Mean square error

Mean regression sum of square = RSS / k

Mean square error = SSE / (n - k - 1)

Question 19

What is the relationship of regression coefficient and p value?

Answer 19

The coefficient with the highest t stats will have the lowest p value.

The t stat = (regression coefficient - Hyp) / Standard error

*Hyp this can usually assume 0

Question 20

How to test for unit root of non-stationary?

Answer 20

Dickey fuller test if it have unit root, which indicate random walk.

Question 21

Define three condition for time series to be covariance stationary.

Answer 21

1 The expected value of the time series is constant and finite over time.

2 The volatility of the time series is constant and finite in all periods.

3 The covariance of the time series with leading or lagged values is constant.

Question 22

Does Durbin watson works for autoregression model?

Answer 22

No

Question 23

Can time series model that is not covariance stationary can be used?

Answer 23

A time series model that is not covariance stationary can still potentially be used by transforming the model into a model that is covariance stationary.