F-Tests in Econometrics
There are three types of F-tests commonly used in economics.
(1) zero slopes
(2) linear restrictions
(3) Chow test for regression stability
All three of these tests can be easily computed using the GRETL software. I will show you how to do
this later in class. Let’s begin with zero slopes.
(1) zero slopes – F-test
Suppose that we have the following regression:
1  
Yt  o   X1t    
XK
t
t
1
K1
for t = 1,...,N.
Clearly, we have K coefficients and N observations on data. The zero slopes F-test is
concerned with the following hypothesis:
Ho: β1 = 0, ..., βK-1 = 0
Ha: βj ≠ 0 for some j
How can we test this hypothesis? To test this, we consider two separate regressions.
(Restricted)
(Unrestricted)
Yt  o   R
t
Yt  o   X1t    
X K1   UR
t
1
K1 t
You should study these two equations carefully. Note how that the restricted equation
simply assumes that all the β’s are all zero (except for the constant βo). The error term in
the first equation is  R while the error term in the unrestricted equation is  UR .
t
t
R
UR
When we run these regressions, we will get estimated residuals ̂ and ̂
t
t . In
general, these estimated residuals will be different. However, if Ho is true, then it makes
sense to believe that ˆ R  ̂ UR . That is, they will be nearly the same. Therefore, if
t
t
Ho is true then the following logic holds:
UR 
ˆ R
t  ̂ t
2 ˆ UR 2
(ˆ R
t )  ( t )


N
R )2  N (ˆ UR )2

t
t
t 1
t 1
 (ˆ
SSRR – SSRUR  0
where SSRR 
N
 (ˆ
t 1
R )2 .
t
We next “normalize” this statistic (since any change in the units of measurement of Yt
will also change the units of measurement of εt). We do this by dividing by SSRUR.
{SSR R  SSR UR }
 0. We can now see the F statistic
SSR UR
can be defined in the following way:
It follows that if Ho is true, then
F̂ 
{SSR R  SSR UR } ( N  K )
SSR UR
( K  1)
We multiply the statistic by (N-K)/(K-1) because it allows us to determine the
distribution of the statistic for any linear model and any amount of data. If Ho is true
then the F̂ statistic is distributed as an FK-1 ,N-K distribution. Textbooks on econometrics
will have F distribution tables. You can also find such tables on the Internet.
Our decision rule for the test is as follows:
If F̂ is “large”, then we should reject Ho. If, on the other hand, F̂ is “small”, then we
should not reject Ho. What do we mean by “large” and “small” here? This is given by
the distribution of F̂ . On the next page is a drawing of the F distribution (thanks to my
daughter Sonya -- age 13 -- for making this drawing) with degrees of freedom df1 = K-1,
df2 = N-K. Our notion of “small” would be a value of F-hat that lands us in the
aquamarine colored area. Our notion of “large” would be any value of F-hat that lands us
in the red region.
Our rule becomes reject Ho is F-hat is larger than the critical value and do not reject
Ho if F-hat is smaller than the critical value. Another way of saying this is to reject Ho
if the p-value of F-hat is less than 0.05 and do not reject Ho if the p-value of F-hat is
larger than 0.05.
Reject Ho if F̂o  critical value . Do not reject Ho if F̂o  critical value .
Note that the critical value will change whenever N and K change. Note also that if we
do not reject Ho then we are saying that our regression is not any better at explaining Yt
than the mean of Y. If we reject Ho, then we are saying that the regression model is
better at explaining the variation in Yt than Y.
(2) linear restrictions – F-test
To see how we encounter linear restrictions in econometrics, suppose that we have the
following Cobb-Douglas production function.
e u t
Yt  AL
K
t t
We usually just assume β = 1- α. However, this may not be justified by the data. We
should at least test to see if α + β = 1. Such a test is a test of linear restrictions. In fact it
is a test of exactly ONE restriction – namely α + β = 1.
First we convert the above model into a linear model by taking the natural logarithm of
both sides to get
log( Yt )  log( A)   log( L t )   log( K t )  u t
We can write this again as
log( Yt )  o   log( L t )   log( K t )  u t
1
2
Our hypothesis is that Ho: β2 = 1- β1 with an alternative hypothesis H1: β2 ≠ 1 – β1. How
can we test this hypothesis? To test Ho we need to run two regressions.
(Restricted)
{log( Yt )  log( K t )}  o   {log( L t )  log( K t )}  u R
t
1
(Unrestricted)
log( Yt )  o   log( L t )   log( K t )  u UR
t
1
2
We run these two regressions and get estimated û R and û UR . Once again we know
t
t
R
UR
that if Ho is true then the estimated residuals û  û
. We can therefore form the
t
t
following F-test statistic:
F̂ 
{SSR R  SSR UR } ( N  K )
~ F1,N-K
SSR UR
1
In this case, the unrestricted regression has N-K degrees of freedom. In addition, there is
only 1 restriction used in the restricted regression. Therefore, we multiply the statistic by
the ratio (N-K)/1 and get the F statistic.
Our decision rule is to reject Ho if F̂ > the critical value. We will not reject Ho if the
calculated F̂ < the critical value. Note that the F distribution is now F1 ,N-K.
The critical value shown in the figure can be determined by looking up the value in an Ftable or the Internet. The critical value changes whenever K or N changes.
This leads us to the final type of F-test.
(3) Chow Test for Regression Stability
The last F-test concerns the issue of whether the regression is stable or not. We begin by
assuming a regression model
Yt  o   X1t    
X K1   t
1
K1 t
for t = 1, ..., N.
The question we ask is whether our estimated model will be roughly the same, whether
we estimate it over the first half of the data or the second half of the data.
Consider splitting our data into two equal parts.
t = 1,2,...,N/2
and
t = (N/2)+1, (N/2) + 2, ..., N.
Both intervals have N/2 observations of data.
We next consider two regressions—one on the first data set, the other on the second data
set.
1
X K1   UR
t
K 1 t
Unrestricted Regression on data1: Y  1  1X1    1
t
o
for
1 t
t = 1,2,...,N/2
2
X K1   UR
t
K 1 t
Unrestricted Regression on data2: Yt  2  2 X1t    2
o
for
1
t = (N/2)+,(N/2)+2, ..., N
Our hypothesis is that: Ho: 1o   2o , 11  12 , ..., 1K 1   2K 1
Ha: 1j  2j for some j
This leads to K constraints in Ho. We run a third regression over the entire data set and
impose the K constraints in Ho. Therefore we can write this regression as
Restricted Regression on all data:
Yt  o   X1t    
X K1   Rt
1
K 1 t
Once again we form an F statistic using the SSR from these three regressions above.
F̂ 
{SSR R  SSR UR1  SSR UR 2 } ( N  2 K )
{SSR UR1  SSR UR 2 }
K
We divide by K because there are K constraints in Ho. We multiply by (N-2K) because
the two unrestricted regressions each have (N/2) – K degrees of freedom. Adding these
two together gives (N – 2K).
The F-statistic above is distributed with an F distribution having K and N-2K degrees of
freedom. The graph of the F distribution is shown below.
Like before we will reject Ho if F̂ > the critical value. We will not reject Ho if the
calculated F̂ < the critical value. Note that the F distribution is now FK, N-2K.
This means we should reject Ho if the p-value for F̂ is less than 0.05 and we should not
reject Ho if the p-value for F̂ > 0.05.