RDP 9105: Inflation in Australia: Causes, Inertia and Policy Appendix 3: Tests for Serial Correlation and Heteroskedasticity
July 1991
- Download the Paper 350KB
In Table 1, the abbreviations SC(1), SC(2) and SC(3) refer to the test statistics for first, second and third order serial correlation respectively.
Stewart (1986) outlines a Lagrange Multiplier test for serial correlation in a model which is linear in variables but non-linear in parameters.
Consider the model
where yt is the dependent variable, g is the model specification with variables
Xt and parameters b and ut is the error term. The estimated model yields a
series of parameter estimates
and estimated errors
. The individual error terms
are
standardised by subtracting their sample mean to produce the series
.
The partial derivatives of g (denoted ) with respect to each of the
parameters b are calculated and evaluated at the parameter estimates
. The auxiliary regression (a2.2)
of
on
and
(the
residual series lagged j times) is estimated to test for the presence of serial correlation
specifically of the order j.
The test statistic is equal to T.R2 where R2 and T (the number of observations) refer to the estimation of equation (a3.2). This test statistic is distributed as χ2(1).
The abbreviations HS(1) and HS(2) in table 1 denote tests for different forms of heteroskedasticity of the residuals.
Stewart outlines how the Breusch-Pagan test can be used to test for heteroskedasticity related to a time trend (HS(1)) or the value of predicted dependent variable (HS(2)).
Using the error term series outlined above, equation (a3.3) is
estimated:
where is
the sample variance of
, Zt is the variable potentially
related to the heteroskedasticity of
, κ0 and κ1
are parameters and ut is an error term. For HS(1), Zt is a time trend. For
HS(2), Zt is the predicted value of dependent variable in the primary estimation
(from equation a3.1).
In both cases, the test statistic is equal to T.R2 where R2 and T (the number of observations) refer to the estimation of equation (a3.3). This test statistic is distributed as χ2(1).