1. Econometric methods of analysis and forecasting of financial markets
Lecture 2. Linear regression model assumptions and its violations

2. This lecture helps to understand:
What are the main assumptions of linear regressions
What if they violate and how to detect it.
How to solve these problems.

3. Contents Linear regression model assumptions Multicollinearity
Endogeneity
Heteroskedasticity
autocorrelation

4. Linear regression model assumptions
Once again from the previous lecture:
E( 𝑢 𝑡 )=0 (zero mean of the errors)
Var 𝑢 𝑡 = 𝜎 2 <∞ (constant and finite variance of the errors)
Cov 𝑢 𝑖 , 𝑢 𝑗 =0 (linearly independent errors)
Cov 𝑢 𝑖 , 𝑥 𝑖 =0 (no relationship between the error and explanatory variable)
𝑢 𝑖 ~𝑁(0, 𝜎 2 ) (normally distributed)

5. Multicollinearity
Multicollinearity is the case when explanatory variables are highly correlated with each other.
Leads to:
R2 will be high but the individual coefficients will have high standard errors;
regression becomes very sensitive to small changes in the specification;
confidence intervals for the parameters become very wide, and significance tests show incorrect results.

6. Multicollinearity Solutions to the problem:
Ignore it, if the model is otherwise adequate
Drop one of the collinear variables
Transform the highly correlated variables into a ratio and include only the ratio
Look at your data and collect more

7. Endogeneity
If one of the explanatory variables is correlated with the error term, then this variable is endogenous. Formally:
𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 +…+ 𝛽 𝑘 𝑥 𝑘 +𝜀
𝑥 𝑗 is endogenous, if 𝑐𝑜𝑟𝑟( 𝑥 𝑗 , 𝜀)≠0 for at least one j
In this case estimators are biased and inconsistent!

8. Endogeneity Potential reasons for endogeneity: omitted variable
functional form misspecification
measurement error (in dependent variable or in explanatory variable)
simultaneity

9. Endogeneity How to solve the problem of endogeneity? proxy variables
IV and 2sls
The main steps:
use tests for form misspecification and try to find the most appropriate model (including quadratic terms and logs)
if you have data for the omitted variable - include it, if not - use proxy variable (in the case you don’t know what proxy to use, take as a proxy lagged dependent variable)
use instrumental variables and 2sls estimation

10. Heteroskedasticity
Assumption 2 (Var 𝑢 𝑡 = 𝜎 2 <∞,constant and finite variance of the errors) is the assumption of homoscedasticity. If it violates, there is the case of heteroskedasticity.

11. Heteroskedasticity
Consequence? Wrong standard errors! In this case it’s impossible to run the tests
How to detect heteroskedasticity?
Breush-Pagan test
𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 +…+ 𝛽 𝑘 𝑥 𝑘 +𝑢 (1)
𝑢 2 = 𝛿 0 + 𝛿 1 𝑥 1 + 𝛿 2 𝑥 2 +…+ 𝛿 𝑘 𝑥 𝑘 +𝑣 (2) 𝐻 0 : 𝛿 1 = 𝛿 2 =…= 𝛿 𝑘 =0 (homoskedasticity)
estimate (1) by OLS, obtain 𝑢 2
Estimate (2) using 𝑢 2 , get the 𝑅 2
Test statistic 𝑛𝑅 2 ~ 𝜒 𝑘 2
If 𝑛𝑅 2 > 𝜒 𝑘 2 then reject 𝐻 0

12. Heteroskedasticity How to detect heteroskedasticity? White test
Estimate (1) by OLS, get 𝑦 and 𝑢 , compute 𝑦 2 and 𝑢 2
Run the regression 𝑢 2 = 𝛿 0 + 𝛿 1 𝑦 + 𝛿 2 𝑦 2 +𝜀, get the 𝑅 2
Test statistic 𝑛𝑅 2 ~ 𝜒 𝑘 2
If 𝑛𝑅 2 > 𝜒 𝑘 2 then reject 𝐻 0

13. Heteroskedasticity Solutions for heteroskedasticity:
Transforming the variables into logs or reducing by some other measure of ‘size’.
Using heteroscedasticity-consistent standard error estimates (‘robust’).

14. Autocorrelation
The violation of the assumption about correlation of error terms results in autocorrelation problem:
cov(εi, εj)≠0 if i≠j
Reasons for autocorrelation:
Omitting relevant variables. As a result they could correlate with the error term (lags of included variables, trend, seasonal effects).
More likely when frequency of data is higher.
Wrong specification of a model.
Why this problem is important?
Estimates by OLS method are biased and inefficient: estimates βi have variance which is not the lowest. Consequently, this estimates are not the most accurate.
Incorrect standard errors of estimates. Consequently, we cannot use it while doing t and F tests and constructing confidence intervals.
How to detect:
t test for AR(1) serial correlation with strictly exogenous regressors
Durbin-Watson test
t test for AR(1) serial correlation without strictly exogenous regressors
testing for higher order serial correlation

15. Autocorrelation Correction of autocorrelation:
If we found out autocorrelation in the model it is obligatory to make a correction. For this purpose we can use the next ways:
Include omitted variables (lag, trend, control for seasonality).
Use more appropriate methods for data with high frequency.
Change model specification.

16. Conclusions We covered
The main violations of linear regressions assumptions
What they cause and how to solve these problems

17. References
Brooks C. Introductory Econometrics for Finance. Cambridge University Press
Cuthbertson K., Nitzsche D. Quantitative Financial Economics. Wiley
Tsay R.S. Analysis of Financial Time Series, Wiley, 2005.
Y. Ait-Sahalia, L. P. Hansen. Handbook of Financial Econometrics: Tools and Techniques. Vol. 1, 1st Edition
Alexander C. Market Models: A Guide to Financial Data Analysis. Wiley
Cameron A. and Trivedi P.. Microeconometrics. Methods and Applications
Lai T. L., Xing H. Statistical Models and Methods for Financial Markets. Springer
Poon S-H. A practical guide for forecasting financial market volatility. Wiley, 2005.
Rachev S.T. et al. Financial Econometrics: From Basics to Advanced Modeling Techniques, Wiley, 2007.