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Introduction to Econometrics

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Introduction to Econometrics

Lecture 7

Heteroskedasticity and

some further diagnostic testing

Topics to be covered

- Heteroskedasticity
- Some further diagnostic testing
- Normality of the disturbances
- Multicollinearity

Econometric problems

Heteroskedasticity

What does it mean? The variance of the error term is not constant

What are its consequences?The least squares results

are no longer efficient and t tests and F tests results may be misleading

How can you detect the problem?Plot the residuals against each of the regressors or use one of the more formal tests

How can I remedy the problem? Respecify the model – look for other missing variables; perhaps take logs or choose some other appropriate functional form; or make sure relevant variables are expressed “per capita”

Consumption function example (cross-section data): credit worthiness as a missing variable?

The Homoskedastic Case

The Heteroskedastic Case

The consequences of heteroskedasticity

- OLS estimators are still unbiased (unless there are also omitted variables)
- However OLS estimators are no longer efficientor minimum variance
- The formulae used to estimate the coefficient standard errors are no longer correct
- so the t-tests will be misleading (if the error variance is positively related to an independent variable then the estimated standard errors are biased downwards and hence the t-values will be inflated)
- confidence intervals based on these standard errors will be wrong

Detecting heteroskedasticity

- Visual inspection of scatter diagram or the residuals
- Goldfeld-Quandt test
- suitable for a simple form of heteroskedasticity

- Breusch-Pagan test
- a test of more general forms of heteroskedastcity

Residual plots

Plot residuals against one variable at a time

Goldfeld-Quandt test (JASA, 1965)

- Suppose it looks as ifsui = suXi
i.e. the error variance is proportional to the square of one of the X’s

- Rank the data according to the culprit variable and conduct an F test using RSS2/RSS1
where these RSS are based on regressions using the first and last [n-c]/2 observations [c is a central section of data usually about 25% of n]

- Reject H0 of homoskedasticity if Fcal > Ftables

Breusch-Pagan test

- Regress the squared residuals on a constant, the original regressors, the original regressors squared and, if enough data, the cross-products of the Xs
- The null hypothesis of no heteroskedasticity will be rejected if the value of the test statistic is “too high” (P-value too low)
- Both c2 and F forms are available in PcGive

Remedies

- Respecification of the model
- Include relevant omitted variable(s)
- Express model in log-linear form or some other appropriate functional form
- Express variables in per capita form

- Where respecification won’t solve the problem use robust Heteroskedastic Consistent Standard Errors (due to Hal White, Econometrica 1980)

ARCH

- Note: with time series data, particularly high-frequency data (for example daily or hourly financial data) a special form of heteroskedasticity called Autoregressive Conditional Heteroskedasticty (ARCH) may be present
- We can see it graphically as excessive volatility of the time series in certain short bursts
- I will say more about this when we look in more detail at dynamic models

Normality of the disturbances

- Test null hypothesis of normality
- Use 2 test with 2 degrees of freedom
- At 5% level reject H0 if 2 > 5.99
- non-normality may reflect outliers or a skewed distribution of residuals

Reset test

- originated by Ramsey (1969)
- tests for functional form mis-specification
- run regression and get fitted values
- now regress Y on X’s and powers of fitted Ys
- if these additional regressors are significant (judged by F test) then the original model is mis-specified

Multicollinearity

What does it mean? A high degree of correlation amongst the

explanatory variables

What are its consequences?It may be difficult to separate out

the effects of the individual regressors. Standard errors may

be overestimated and t-values depressed.

Note: a symptom may be high R2 but low t-values

How can you detect the problem?Examine the correlation

matrix of regressors - also carry out auxiliary regressions

amongst the regressors.

Look at the Variance Inflation Factors

- NOTE:
- be careful not to apply t tests mechanically without checking for multicollinearity
- multicollinearity is a data problem, not a misspecification problem

Variance Inflation Factor (VIF)

Multicollinearity inflates the variance of an estimator

VIFJ = 1/(1-RJ2)

where RJ2 measures the R2 from a regression of Xj on the other X variable/s

serious multicollinearity problem if VIFJ>5