Dynamic models
Non-linearities and structural change
The linear regression model: functional form and
structural breaks
Ragnar Nymoen
Department of Economics, UiO
16 January 2009
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
Overview
A little bit more about dynamics
Extending inference to parameters of interest that are
non-linear functions of regression coe¢ cients
Modelling and testing of structural breaks.
Main reference is Greene Ch 6.1-6.4. See course page syllabus
for overlapping reference to Biørn and Kennedy
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
Models with short and long-run derivative coe¢ cients
We ended the lecture 1 slide set with a note on the dynamic model:
= β2 yt 1 + εt ,
t = 1, . . . T .
yt
εt
N (0, 1),
which we found can be estimated consistently by OLS despite the
correlation between yt and past disturbances.
As, noted this suggest that OLS can be used to estimate
dynamic models that contain both exogenous regressors and a
lagged regressor.
The autoregressive distributed lag model, ADL, is
yt = β1 + β2 yt
1
+ β3 xt + β4 xt
1
+ εt ,
εt
N (0, σ2 ), (1)
ADLs have obvious relevance in economics (eq ECON
34310/4410):
Impact, dynamic and long-run multipliers.
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
Greene has numerous references to regression models that are
ADLs, cf Example 4.7 on p 69 .
In ADLs, one parameter of interest is the long-run multiplier:
B2 =
β3 + β4
1 β2
B2 is a non-linear function of the parameters of the regression
modeI, so can we test an hypothesis about this parameter of
interest, i.e. H0 : B2 = B2o ?
The answer is: based on an asymptotically valid computation
of the variance of B̂2 = b13 +bb24 we can. We use “ ^” to denote
the estimator of the long-run multiplier here.
Greene makes reference to the “delta method” on page 68,
but we state directly a results due to Bårdsen (1989):
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
First, re-write the ADL as
∆yt
= β1 + ( β2
= β1 + αyt
+ β3 ∆xt + ( β3 + β4 )xt
1 + β3 ∆xt + γt 1 xt 1 + εt
1)yt
1
1
+ εt
(2)
where ∆ is the di¤erence operator, so ∆yt = yt yt 1 and
α = ( β2 1) and γ = ( β3 + β4 ).
(1) and (2) gives identical SSE s, so statistically they are the
same models. (Although R 2 very di¤erent)
(2) easier to use since
b3 + b4
γ̂
,
B̂1 =
1 b2
α̂
and Var [B̂1 ] can be obtained as
\
Var
[B̂1 ]
1
α̂
+2
2
\
Var
(γ̂) +
1
α̂
γ̂
( α̂)2
γ̂
( α̂)2
2
\
Var
(α̂) (3)
\
Cov
(γ̂, α̂).
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
This method applies to models with K 1 exogenous
regressors, and with higher order lags in both the dependent
variable and in the xs.
Estimates of long-run derivative coe¢ cients and their
variances are part of the output of PC Give.
But Bårdsen’s formula is convenient if you use other software,
since only need the covariance matrix of the estimates.
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
Example: Table 4.7, p 69 in Greene
Consider long-run elasticity of gasoline demand with respect to
income. Income is variable number 3 in the model, so
B̂3 =
Var\
[B̂3 ] =
0.164097
= 0.97047
0.169090291
1
0.169090291
0.164097
(0.169090291)2
1
+2
0.169090291
= 0.026349
+
2
(0.0030279)
2
0.0020943
0.164097
(0.169090291)2
( 0.0021881)
which comes close to the estimate of the from the delta method
reported by Greene on page 70.
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
Linearity in parameters and “intrinsic linearity”
Have already made the point that linearity in parameters, not
linearity in parameters, is the de…ning trait of the linear
regression model.
Can extend the relevance of the model to the case where our
parameters of interest are one-one functions of the coe¢ cients
of the regression model.
Greene p 119 call this “intrinsic linearity”.
The long-run multiplier is an example!
Many other in econometrics
CES production function Green p 119, and Ch 16.64
The natural rate of unemployment,ie
π t = β1 + β2 ut + εt
where ut is the rate of unemployment and π t is in‡ation. The
β
Phillips curve natural rate is u phil = β1
2
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
If we have a hypothesis about when a structural break occurs
we can test that hypothesis
Let T1 denote the last period with the “old” regime and let
T1 + 1 denote the …rst period of the “new”;
yt = β1 + β2 X2t + εt , t = 1, 2, . . . , T1 and
yt = γ1 + γ2 X2t + ei , i = T1 + 1, 2, . . . , T .
then
H0 : β1 = γ1 , β2 = γ2 vs H1 : β1 6= γ1 , β2 6= γ2 .
In the multivariate case:
H0 : β 1 = γ 1 , β 2 = γ 2 , β 3 = γ 3 , . . . , β K = γ K
There are two well known statistics for these cases, both due
to Chow (1960) and referred to as Chow tests.
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
2-sample Chow-test
SSE1 is for the …rst sample (t = 1, 2, .., T1 ) SSE2 is for the
second.
SSEU = SSE1 + SSE2 . SSER is the SSE when the whole sample is
used, i.e under H0
FChow 2 =
(T
SSER SSEU
SSEU
4)
2
F (2, T
4).
In general
FChow 2 =
SSER SSEU
SSEU
T
2K )
K
F (K , T
ECON 4610: Lecture 2
2K )).
Dynamic models
Non-linearities and structural change
Predictive Chow-test
Consider T T1 < K . Same SSER (full sample) but SSEU is
only on the basis of the …rst T1 observations. This predictive
Chow-test is given as
FChowP =
SSER SSEU
SSEU
T1 K
T T1
F (T
T1 , T1
K)
If we have no clear idea about the dating of a regime shift,
graphs with the whole sequence of predictive Chow tests are
useful.
Chow tests rely on constant and equal variances of the
disturbances. Hence, good practice to plot the sequence of s 2
as a function of t.
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
Testing by dummies
Dummy variables are ‡exible tools for modelling and testing
parameter changes in both cross-section and time series data,
Greene ch. 6.2.
Consider temporary change in period T1 in β1 :
H0 : β1 = γ1 , vs H1 : β1 6= γ1 , for t = T1
Test H0 with t-statistic of λ1 = 0 in
yi = β1 + λ1 Dt + β2 xt + εt ,
t = 1, 2, . . . , T
where Dt = 1 when t = T1 and 0 elsewhere.
ECON 4610: Lecture 2
Dynamic models
Non-linearities and structural change
If the break also a¤ects the slope, use
Yi = β1 + λ1 Dt + β2 Xt + λ2 Xt Dt + εt , t = 1, 2, . . . , T1
to test
H0 : λ1 = λ2 = 0 vs H1 : λ1 6= 0, or λ2 6= 0.
The F statistic is distributed F (2, T
3 regressors and an intercept.
4) , since SSEU is based on
ECON 4610: Lecture 2