Introduction
Recursive residuals
Structural change
Recursive residuals
by Tabar and Andrea
CUSUM and CUSUMSQ
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Least-squares estimator
In the context of Least-Squares Estimation (i.e. OLS residuals),
residuals can be find with this equation
e = Y − Xb.
They are mainly used as a fitting criterion to make the sample
regression line as close as possible to the data points by minimizing
the squared-sum of the residuals, that is
min e0 e = (Y − Xb)0 (Y − Xb).
b
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Recursive residuals
Also recursive residuals are defined as the difference between Y and
Xb, but calculated without using all the observations.
We can calculate recursive residuals by using the first t − 1
observations.
The t th recursive residual is defined as the difference between Yt and
Xt bt−1 .
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Recursive residuals
Also recursive residuals are defined as the difference between Y and
Xb, but calculated without using all the observations.
We can calculate recursive residuals by using the first t − 1
observations.
The t th recursive residual is defined as the difference between Yt and
Xt bt−1 .
The t th standardized recursive residual is obtained by dividing the t th
recursive residual by its forecast variance.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Pros
Unlike OLS residuals, standardized recursive residuals have different
interesting proprieties.
• They are homoskedastic
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Pros
Unlike OLS residuals, standardized recursive residuals have different
interesting proprieties.
• They are homoskedastic
• They are indipendent of one another
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Pros
Unlike OLS residuals, standardized recursive residuals have different
interesting proprieties.
• They are homoskedastic
• They are indipendent of one another
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Pros
Unlike OLS residuals, standardized recursive residuals have different
interesting proprieties.
• They are homoskedastic
• They are indipendent of one another
Both properties make standardized recursive residuals attractive to
calculate some regression diagnostics.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Cons
A note of caution is presented by Kennedy: “. . . because the behavior
of recursive residuals in a mis-specified model is very different from
that of the OLS residuals, (. . . ) test procedures based on the
recursive residuals should be viewed as complementary to tests
based on OLS residuals”.
Introduction
Recursive residuals
Structural change
Recursive residuals
Suppose that the sample contains a total of T observations.
CUSUM and CUSUMSQ
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Recursive residuals
Suppose that the sample contains a total of T observations. The t th
recursive residual is the ex-post prediction error for yt when the
regression is estimated using only the first t − 1 observations
et = yt − xt0 bt−1 .
(1)
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Recursive residuals
Suppose that the sample contains a total of T observations. The t th
recursive residual is the ex-post prediction error for yt when the
regression is estimated using only the first t − 1 observations
et = yt − xt0 bt−1 .
(1)
The forecast variance of this residual is
0
σft2 = σ 2 [1 + xt0 (Xt−1
Xt−1 )−1 xt ].
(2)
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Recursive residuals
The t th stanadardized recursive residual is
et
wt = q
0
0 X
2
−1 x
1 + xt (Xt−1
t−1 )
t
(3)
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Structural change
Recursive residuals can be used both to test for non-linearity and to
test for structural change.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
Structural change
Recursive residuals can be used both to test for non-linearity and to
test for structural change.
Kennedy provides a simple explanation of the use of recursive
residuals to test for non-linearity based on the concept of the
U-shaped that suggest that there is a structural change.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
CUSUM and CUSUMSQ
To test a structural stability of the model there are different tests
based on recursive residuals. The two most important are the
CUSUM and the CUSUM-OF-SQUARES, with the data ordered
chronologically, rather than according to the value of an explanatory
variable.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
CUSUM and CUSUMSQ
To test a structural stability of the model there are different tests
based on recursive residuals. The two most important are the
CUSUM and the CUSUM-OF-SQUARES, with the data ordered
chronologically, rather than according to the value of an explanatory
variable.
The CUSUM test id based on a plot of the sum of the recursive
residuals. If this sum goes outide a critical bound, one concludes that
there was a structural break at the point at which the sum began its
movement toward the bound.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
CUSUM and CUSUMSQ
To test a structural stability of the model there are different tests
based on recursive residuals. The two most important are the
CUSUM and the CUSUM-OF-SQUARES, with the data ordered
chronologically, rather than according to the value of an explanatory
variable.
The CUSUM test id based on a plot of the sum of the recursive
residuals. If this sum goes outide a critical bound, one concludes that
there was a structural break at the point at which the sum began its
movement toward the bound.
The CUSUM-OF-SQUARES test is similar to the cusum test, but plots
the cumulative sum of squared recursive residuals, expressed as a
fraction of these squared residuals summed over all observations.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
CUSUM
The CUSUM test is based on the cumulated sum of the residuals:
T
X
wt
σ̂
Wt =
j=k+1
with
PT
2
σ̂ =
j=k +1 (wt
− w̄)2
T −k −1
and
PT
w̄ =
j=k+1
T −k
wt
,
where k is the minimum sample size for which we can fit the model.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
CUSUM
The CUSUM test is performed by plotting Wt against t. Under the null
hypotesis, Wt , the cumulative sum, with constant parameter has a
mean of zero, E(Wt ) = 0, and variance equal to the number of
residuals being summed, in fact each term has variance 1, and they
are indipendent. But with nonconstant parameter, Wt will tend to
diverge from the zero mean value. The significance of the departure
from the zero line may be assessed by reference to a pair of straight
lines that pass through the points
√
(k, ±a t − k)
and
√
(t, ±3a t − k),
where a is a parameter depending on the significance level α chosen.
Introduction
Recursive residuals
Structural change
CUSUM and CUSUMSQ
CUSUMSQ
The second test statistic, the CUSUMSQ, is based on cumulative
sums of squared residuals:
Pt
k +1
wj2
k+1
wj2
St = PT
with
t = k + 1, · · · , T
t −k
, which goes to zero at
T −k
t = k. The significance of departures from the expected value line is
assessed by reference to a pair of lines drown parallel to the E(St )
line at a distance cs above and below. This value depends on both
the sample size T − k and the significance level α.
The expected value of St is E(St ) =