Quantitative Methods for
Economics, Finance and
Management
(A86050 – F86050)
Matteo Manera – matteo.manera@unimib.it
Marzio Galeotti – marzio.galeotti@unimi.it
1
This material is taken and adapted from
Guy Judge’s Introduction to Econometrics page
at the University of Portsmouth Business School
http://judgeg.myweb.port.ac.uk/ECONMET/
2
Econometric “problems”
Heteroskedasticity
Normality of the disturbances
Multicollinearity
Autocorrelation
Bias
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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?
Scatter diagram and fitted line
Consumption
400
300
200
100
0
0
100
200
Income
300
400
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 efficient or
minimum variance
The formulae used to estimate the coefficient
standard errors are no longer correct (show)
• 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
Suppose it looks as if σui = σuXi
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 χ2 and F forms are available
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
Transform the variables so that the transformed
model has a homoskedastic error (show this
case)
Where respecification won’t solve the problem
use robust Heteroskedastic Consistent Standard
Errors (due to White, 1980)
Normality of the disturbances
Test null hypothesis of normality
Use χ2 test with 2 degrees of freedom
At 5% level reject H0 if χ2 > 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
Regress Y on X’s and powers of fitted Y’s
If these additional regressors are significant
(judged by an 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
Specification and mis-specification tests
Suppose we have in mind a model that we might call our
“maintained hypothesis”
Yi = b0 + b1 X1i + b2 X2i +..........+ bkXki+ ui
Tests of all types of restrictions on the b values are
sometimes called specification tests
Our maintained hypothesis also makes a number of
assumptions about the
disturbance term u.
We must check the validity of these assumptions. These tests
are called mis-specification tests (or diagnostic tests)