Chow Test
Kevin Ardon
Colt Van Eaton
Dayana Zhappassova
Savings and Personal Disposable Income
• Estimating a savings function that relates savings (Y) to disposable
personal income (X)
• We can run a simple regression and obtain OLS estimates.
• However, by doing so we are assuming the relationship between
savings and the disposable personal income did not change over 26
years.
• Might not be a valid assumption, considering the recession in 1982 in
the USA
• Might disturb the relationship between savings and disposable
personal income
Savings and Personal Disposable Income
Savings and Personal Disposable Income
Savings and Personal Disposable Income
• To check for a structural break we can divide our sample into two
time periods
• 1970-1981
• 1982-1995
• Now we can run three possible regressions:
• 𝑌𝑡 = 𝜆1 + 𝜆2 𝑋𝑡 + 𝑢1𝑡
• 𝑌𝑡 = 𝛾1 + 𝛾2 𝑋𝑡 + 𝑢2𝑡
• 𝑌𝑡 = 𝛼1 + 𝛼2 𝑋𝑡 + 𝑢𝑡
𝑛1 = 12
𝑛2 = 14
𝑛 = (𝑛1 + 𝑛2 ) = 26
(1)
(2)
(3)
Savings and Personal Disposable Income
• The third regression assumes that there is no structural break
• Assumes that the intercept and slope coefficients remain the same
over the entire time period
• In that case 𝛼1 = 𝜆1 =𝛾1 and 𝛼2 = 𝜆2 =𝛾2
• The first two regressions assume that the intercepts and the slope
coefficients can be different.
Structural Break Test
• To test for the structural break we can use the Chow Test
• This allows us to test the hypothesis that some or all of the regression
coefficients are different in different subsets of the data.
• There are two ways of using the Chow test to test for the structural
change
• Unrestricted approach
• Restricted approach
Unrestricted Approach
• Begin by partitioning the data into two parts (assuming there is only
one structural break)
• The regression model is
𝜖1
𝑌1
𝑋1 0 𝛽1
•
=
+ 𝜖
𝑌2
0 𝑋2 𝛽2
2
• 𝑌 = 𝑋𝛽 + 𝜖
• Estimate two regression models using OLS
𝑋1
• b=
0
0
𝑋2
−1
𝑋1′ 𝑌1
.
′
𝑋2 𝑌2
Unrestricted Approach
• Testing for structural change using the standard F test
• 𝐻0 ∶ 𝑅𝛽 = 𝑞 or 𝛽1 = 𝛽2
• 𝐻𝐴 ∶ 𝑅𝛽 ≠ 𝑞 or 𝛽1 ≠ 𝛽2
• Test the hypothesis using the standard F statistic
𝐹=
𝑅𝑏 − 𝑞
′
𝑅 𝑋′𝑋
−1 ′ −1
𝑅
𝑠2
• where 𝑅 = (𝐼𝑘 | − 𝐼𝑘 ) and 𝑞 = 0.
(𝑅𝑏 − 1)/𝑘
~𝐹 𝑘, 𝑛 − 2𝑘 .
Restricted Approach
• This approach allows us to impose a restriction on a model and
perform the test using the unrestricted and restricted residuals
• Unrestricted regression allows the coefficients to be different in the
two periods:
𝒚1
𝑿1
• 𝒚 =
0
2
𝜺1
0 𝜷1
+ 𝜺
𝑿2 𝜷2
2
• Once again, using OLS find the unrestricted least squares estimator
for the two equations separately
′
𝑿
𝑿1
1
′
−1 ′
• 𝒃= 𝑿𝑿 𝑿𝒚=
0
0
𝑿′2 𝑿2
−1
𝑿1′ 𝒚1
𝒃1
=
′
𝒃2
𝑿2 𝒚2
• Compute the total SSE from the two regressions:
• 𝒆′ 𝒆 = 𝒆1′ 𝒆1 + 𝒆′2 𝒆2
Restricted Approach
• The restricted coefficient vector can be obtained by building in the
restriction into the model
• If the two coefficients are the same then the model can be written as
𝒚1
𝜺1
𝑿1
• 𝒚 =
𝜷+ 𝜺
𝑿
2
2
2
• Restricted estimator can be obtained simply by stacking the data and
estimating a single regression
• The residual sum of squares from the restricted regression 𝒆′∗ 𝒆∗
Restricted Approach
Then we can use the F statistic to test the hypothesis
(𝒆′∗ 𝒆∗ − 𝒆1′ 𝒆1 )/𝑘
𝐹= ′
𝒆1 𝒆1 /(𝑛1 + 𝑛2 − 2𝑘)
Savings and Personal Disposable Income
Savings and Personal Disposable Income
Savings and Personal Disposable Income
Example
For the data presented in Table 8.9 the regression results are presented
in the table below:
𝑌𝑡 = 1.0161 + 0.0803 𝑋𝑡
𝑡 = 0.0873 9.6015
𝑅2 = 0.9021
𝑅𝑆𝑆1 = 1785.032
𝑌𝑡 = 153.4947 + 0.0148 𝑋𝑡
𝑡 = 4.6922 1.7707
𝑅2 = 0. 2971
𝑅𝑆𝑆2 = 10,005.22
𝑌𝑡 = 62.4226 + 0.0376 𝑋𝑡 + ⋯
𝑡 = 4.8917 8.8937 + ⋯
𝑅2 = 0.7672
𝑅𝑆𝑆3 = 23,248.30
(1.a)
𝑑𝑓 = 10
(2.a)
𝑑𝑓 = 12
(3.a)
𝑑𝑓 = 24
Savings and Personal Disposable Income
Example
• Applying the formula from the restricted approach we get:
(23,248.30 − 11,790.252)/2
𝐹=
(11,790.252)/22
= 10.69
• The critical value for 2 and 22 degrees of freedom for 99% confidence
level is 5.72
• Therefore reject 𝐻0 (which means there was a structural change in
savings-income relation over period 1970-1995)
MATLAB
Chow Predictive Test
• In some circumstances, the data series may not be long enough to
test for structural changes
• Running a test with fewer observations then coefficients can not fit
the two separate models.
• Valid way for testing with insufficient observations:
1. Estimate the regression, using the full data set, and compute the restricted
sum of squared residuals, 𝒆′∗ 𝒆∗
2. Use the longer sub-period to estimate the regression and compute the
unrestricted sum of squares, 𝒆1′ 𝒆1
• Assumming 𝑛2 < 𝐾
Chow Predictive Test
The F statistic is then computed using,
(𝒆′∗ 𝒆∗ − 𝒆1′ 𝒆1 )/𝑛2
𝐹=
𝒆1′ 𝒆1 /(𝑛1 − 𝑘)
Limitation of Chow Test
There are some limitations when using the Chow test
1. The assumptions underlying the test must be fulfilled
a. Error terms are normally distributed and variances are homoscedastic
• If heteroscedastic → use Wald test
2. Chow test will not show if the two regressions are different on
account of the intercepts, slopes or both
Dummy Variable Alternative
To find whether the difference between the two regressions lies from
the difference in slopes, intercept or both:
• Pool in all the observations
• Run one multiple regression
𝑌𝑡 = 𝛼1 + 𝛼2 𝐷𝑡 + 𝛽1 𝑋𝑡 + 𝛽2 𝐷𝑡 𝑋𝑡 + 𝑢𝑡
Where
𝑌=
𝑋=
𝑡=
𝐷=
savings
income
time
1 for observations in 1982-1995
0, otherwise(i.e., for observations in 1970-1981)
Limitation of Chow Test (cont’d)
The final limitation of the Chow test
3. Chow test assumes we know the point of structural break
a. If point of structural break is unknown, alternative method is needed
• Recursive Least Squares
THANK YOU