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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.

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Introduction to Econometrics

Lecture 7

Heteroskedasticity and

some further diagnostic testing

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Topics to be covered

  • Heteroskedasticity

  • Some further diagnostic testing

    • Normality of the disturbances

    • Multicollinearity

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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”

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The Homoskedastic Case worthiness as a missing variable?

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The Heteroskedastic Case worthiness as a missing variable?

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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

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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

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Residual plots worthiness as a missing variable?

Plot residuals against one variable at a time

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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

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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

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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)

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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

  • I will say more about this when we look in more detail at dynamic models

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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

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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

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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

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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