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# Introduction to Econometrics - PowerPoint PPT Presentation

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.

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### Introduction to Econometrics

Lecture 7

Heteroskedasticity and

some further diagnostic testing

• Heteroskedasticity

• Some further diagnostic testing

• Normality of the disturbances

• Multicollinearity

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”

The Homoskedastic Case worthiness as a missing variable?

The Heteroskedastic Case worthiness as a missing variable?

The consequences of heteroskedasticity worthiness as a missing variable?

• 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 worthiness as a missing variable?

• 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 worthiness as a missing variable?

Plot residuals against one variable at a time

Goldfeld-Quandt test ( worthiness as a missing variable?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 worthiness as a missing variable?

• 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 worthiness as a missing variable?

• 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 worthiness as a missing variable?

• 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

Normality of the disturbances worthiness as a missing variable?

• 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 worthiness as a missing variable?

• 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 worthiness as a missing variable?

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) worthiness as a missing variable?

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