Download Diagnostic Tests

Autocorrelation and
Heteroskedasticity
Introduction
• Assess the main ways of remedying
autocorrelation
• Describe the problem of non-constant
error terms
• Assess the main test for heteroskedasticity
• Examine the main remedies for
heteroskedasticity
• Introduce the multivariate approach to
regression analysis
Cochrane-Orcutt methodology
• This approach to remedying
autocorrelation relies on the generalised
difference equation approach
• It is an iterative procedure, the estimates
of the parameters converge towards their
actual value.
• It assumes the autocorrelation follows a
first order autoregressive process
(simplest form)
Cochrane-Orcutt Procedure
yt    xt  ut
ut  ut 1   t
Procedure (Based on previous
slide)
• Run regression and collect the error term
(u).
• Run a second regression of u against u(t1) to obtain an estimate of ρ.
• Form the generalised difference equation,
to obtain new estimates of the parameters
• Re-run this equation, obtain u and repeat
process until there is no significant
difference from one iteration to the next.
Restricted Version of the GDE
• The generalised difference equation
can be written in the following form,
which includes a restriction, the
product of the coefficients on y(-1) and
x is equal to the negative of the
coefficient on x(-1) :
yt   (1   )  yt 1  xt  xt 1   t
Unrestricted version of the GDE
• In the unrestricted version of the equation in
the previous slide, there are no restrictions
on the values of the coefficients:
yt   0  1 yt 1   2 xt   3 xt 1   t
The Common Factor Test
• To determine whether the restricted or unrestricted
versions are best, we can use the common factor test,
under the null hypothesis that the restriction holds, the
test statistic is:
RSS res
T log e
RSS unres
T  number of observations
RSS  Re sidual sum of squares
Heteroskedasticity
• This occurs when a Gauss-Markov
assumption is broken, it indicates nonconstant variance of the error term.
• It is often caused by variables which have
substantial differences in the values of the
observations, i.e. GDP differing between
the USA and Cuba
• With heteroskedasticity the estimator is no
longer BLUE, as it is not the best.
Tests for Heteroskedasticity
• Goldsfeldt-Quandt test (not used much
now)
• Whites Test (used in E-views computer
software).
• LM test (Similar to the LM test for
heteroskedasticity)
White’s Test
• This test follows a similar pattern to the LM test
for autocorrelation discussed earlier.
• Based on the same first equation as in slide 4,
we estimate the model and collect the residual u.
• Square the residual to form the variance of the
residual
• Run a secondary regression of the squared
residual on the explanatory variable and this
variable squared
White’s Test
• This secondary regression appears like the
following:
2
ut
2
  0  1xt   2 xt
 t
White’s Test
• Collect the R-squared statistic and multiply by
the number of observations to obtain the test
statistic.
• This follows a chi-squared distribution, the d of f
equal the number of parameters in the
secondary regression (i.e. 2 in above example)
• The null hypothesis is no heteroskedasticity.
• If there were 2 explanatory variables, you could
include the cross product of these variables, in
the secondary regression.
Remedies for heteroskedasticity
• If the standard deviation of the residual is
known, the heteroskedasticity can be removed
by dividing the regression equation through by
the standard deviation of the residual (weighted
Least Squares)
• If this is not known, as is likely, we need to
stipulate what the standard deviation is equal to.
We can then divide the regression equation
through by this variable to obtain a constant
variance residual.
Remedying Heteroskedasticity
• If we assume the error variance is
proportional to x, the explanatory variable
and we have the usual model:
E (ut )
2
  xt
2
yt    xt  ut
Remedying Heteroskedasticity
• If we divide our model through by the
square root of x:
yt
xt
ut




xt
xt
xt
xt
Remedying Heteroskedasticity
• Taking the new error term, we can show it is
no longer suffering from heteroskedasticity
by proving it is now constant:
ut 2 E (ut ) E (ut )  xt
2
E( ) 




2
xt
xt
xt
( xt )
2
2
2
Multivariate Regression
• Regressions with more than one explanatory
variable are similar to the bi-variate case, we
can interpret the coefficients and t-statistics in
much the same way as before.
• We now have a potential problem with
multicollinearity, when the explanatory variables
are correlated
• The formula for the β contains an expression for
the covariance between the explanatory
variables.
Multivariate Regression Analysis
• The regression coefficients are more likely to be
accurate, the less closely related are the explanatory
variables
• This allows testing for the significance of groups of
variables
• The R-squared increases with the more explanatory
variables
• We need to consider whether all relevant effects are
included in the model
• When estimating the t-tests, we need to account for the
extra degrees of freedom
• There is a trade off between the number of explanatory
variables and the ease of interpreting the model
Conclusion
• The common factor test can be used to
determine the best solution to the problem
of autocorrelation
• In the presence of heteroskedasticity, the
estimator is no longer BLUE.
• White’s test is the most common test for
heteroskedasticity
• Weighted Least Squares can be used to
remedy the heteroskedasticity problem.