Linear Regression

The problem: simple returns can be misleading, as it overlooks compounding effects on returns

They are only appropriate to use when analysing short-term investment performance and when changes in asset prices are small.

A better approach: using log returns. They are often used in financial modelling because they can be added up and interpreted as continuously compounded returns.

• Covariance measures whether 2 variables on average move in the same direction, in opposite directions or have zero association

• Correlation measures the strength of this relationship from -1 to 1

Both ONLY measure the linear relationship between two variables and fail to capture any strong non-linear relationship

• the linear regression describes the relationship between a given variable and one or more other variables

1) Number of influences on Y is too large to fit in one model

2) errors in the way Yt is measured (which can’t be modelled)

3) random outside influences

R^2 is the typical measure for the goodness of fit of our model. It measures how much of our variance in Y is decribed by the model.

Problem is: R^2 never falls when more regressors are added to the regression

Solution: R^2 adjusted formula (which takes k into account)

Still, its only a “soft rule”

1) expected value of error terms = 0

2) variance of errors is standard deviation

3) Covariance between errors is 0

4) Covariance between an error and variable (of one observation) is 0

5) errors are normally distributed over zero and o°2

• the coefficient estimates are wrong

• The associated standard errors are wrong

• The distributions that are assumed for the test statistics are inappropriate

Assumption means, the average value of errors is zero (errors balance out)

• If we have a constant alpha in our regression, this assumption will never be violated

But if violated, R^2 can be negative, then the sample average of y explains more of the variation in y than the explanatory variables.

The Assumption is that the variance of the error terms is o^2

This means we assume homoscedasticity

If errors do not have a constant variance, we call it heteroscedasticity

How can we detect heteroscedasticity ?

• Goldfeld - Quandt Test

It tests for heteroscedasticity in a model.

To test that, the sample is split into two groups T1 and T2 and the residual variances are calculated.

The H0 is that the variances of T1 and T2 are equal. H1 is that they are not equal.

Problem: Where to split the sample

It is better than GQ because it makes few assumptions about the form of heteroscedasticity

It runs an auxiliary regression to test for heteroscedasticity.

First, there is unconditional and conditional heteroscedasticity.

• unconditional does not cause serious problems with OLS regression

• Conditional (if ignored): OLS estimators will still give unbiased and consistent coefficient estimates.

  BUT they are no longer BLUE

• Standard errors could be wrong and any inferences made could be misleading

• Standard errors will be too large for the intercept alpha

• Type I error increases

• transform variables in logs

• Use whites estimates

assumption 3: errors are uncorrelated with one another

Otherwise we have autocorrelation

The Durbin Watson test (DW)

1) There must be a constant term

2) regressors must be non-stochastic

3) there must be no lags of dependent variable

DW can ONLY TEST 1st order autocorrelation

Thus - use Breusch - Godfrey Test

• estimators no longer BLUE

• Wrong statistical inferences

• R^2 likely to be inflated

• Type I error increased

1) Use Newey and West standard error estimates

2) if form of autocorrelation is known - use a GLS approach