Introduction to Econometrics

1
Introduction to Econometrics

• Lecture 7
• Heteroskedasticity and
• some further diagnostic testing

2
Topics to be covered

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

3
Econometric problems
4
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

5
Consumption function example (cross-section
data) credit worthiness as a missing variable?
6
The Homoskedastic Case
7
The Heteroskedastic Case
8
The consequences of heteroskedasticity

• OLS estimators are still unbiased (unless there
  are also omitted variables)
• However OLS estimators are no longer efficient or
  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

9
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

10
Residual plots
Plot residuals against one variable at a time
11
Goldfeld-Quandt test (JASA, 1965)

• Suppose it looks as if sui suXi
• i.e. the error variance is proportional to the
  square of one of the Xs
• 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 gt Ftables

12
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

13
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 wont solve the problem
  use robust Heteroskedastic Consistent Standard
  Errors (due to Hal White, Econometrica 1980)

14
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

15
Normality of the disturbances

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

16
Reset test

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

17
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

18
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 VIFJgt5