HYPOTHESIS TESTING
INTRODUCTION
Economic theory and other sources of prior information imply restrictions on the parameters of
a statistical model. There are 3 types of restrictions. 1) Fixed value restriction. 2) Linear
restriction. 3) Nonlinear restriction. These restrictions can be used in 3 ways. 1) Imposed. 2)
Tested. 3) Tested and if found to be valid imposed.
A fixed value restriction specifies the value of one or more individual parameters. A
linear restriction specifies a linear relationship between the values of two or more parameters.
A nonlinear restriction specifies a nonlinear relationship between the values of two or more
parameters.
IMPOSING RESTRICTIONS
To impose a restriction is to assume that a restriction is true. You then impose the restriction
on the parameters of the statistical model; that is, you require it to be satisfied. The major
advantage of imposing restrictions is that you can obtain more precise estimates of the
parameters of the model. The major disadvantage of imposing restrictions is that if they are
incorrect then you will get biased estimates of the parameters of the model.
TESTING RESTRICTIONS
An hypothesis is a statement regarding the relationship between two or more variables that is
capable of falsification. A restriction is a formal expression of an hypothesis. To test a
restriction is to test an hypothesis. A maintained hypothesis is an hypothesis that is assumed to
be true, and therefore is not tested but imposed. A testable hypothesis is an hypothesis that is
not assumed to be true, and therefore is to be tested. A nested hypothesis is an hypothesis
that can be can be expressed in the form of two alternative models: 1) unrestricted model, 2)
restricted model. A non-nested hypothesis is an hypothesis that cannot be expressed in the
form of a restricted and unrestricted model.
Steps Involved in Testing an Hypothesis
1. Specify the null and alternative hypothesis.
2. Derive the appropriate test statistic and the sampling distribution of the test statistic under
the null hypothesis.
3. Choose a level of significance and find the critical value(s) for the test statistic.
4. Use the sample data to calculate the actual value (calculated value) of the test statistic.
5. Compare the actual value of the test statistic to the critical value(s) and accept or reject the
null hypothesis.
Statistical Tests
Step #2 in testing an hypothesis is to derive the appropriate test statistic and the sampling
distribution of the test statistic under the null hypothesis. This involves choosing the
appropriate statistical test. A statistical test has two important components. 1) Test statistic. 2)
Sampling distribution of the test statistic. There are two major categories of statistical tests. 1)
Small sample (finite sample) tests. 2) Large sample (asymptotic) tests.
Small Sample Tests
A small sample test has a test statistic that has a known, exact, sampling distribution for any
sample size. There are two important small sample tests.1) Small sample t-test. 2) Small
sample F-test. A small sample t-test has a t-statistic that has a known, exact, t-distribution. A
small sample F-test has an F-statistic that has a known, exact, F-distribution.
Large Sample (Asymptotic) Tests
A large sample test has a test statistic that has an unknown sampling distribution in finite
samples. However, the test statistic has a known asymptotic (limiting) sampling distribution for
samples that are infinitely large. In finite samples, the asymptotic sampling distribution is used
as an approximation of the true, unknown sampling distribution for the test statistic. Therefore,
a large sample test is not an exact test, rather it is an approximate test. The larger the sample
size the better the approximation, and therefore the more valid the large sample test.
There are 5 important large sample test. 1) Asymptotic t-test. 2) Approximate F-test. 3)
Likelihood ratio test. 4) Wald test. 5) Lagrange multiplier test. The asymptotic t-test has an
asymptotic t-statistic that has an approximate t-distribution in finite samples. The approximate
F-test has an F-statistic that has an approximate F-distribution in finite samples. The Likelihood
ratio, Wald, and Lagrange multiplier tests have test statistics that have an approximate chisquare distribution in finite samples.
Small Sample t-Test
A small sample t-test can be used to test a single fixed value restriction or a single linear
restriction in the classical linear regression model. The test statistic is the t-statistic which has a
t-distribution
i^ - i
t =  ~ t(T-K)
s.e.( i^)^
where i^ is the estimate of the ith individual parameter, i is the hypothesized value of the ith
individual parameter, s.e.( i^)^ is the estimated standard error of the estimate i^, and t(T-K) is
the t-distribution for T-K degrees of freedom.
To calculate the t-statistics, use the sample data to estimate the unrestricted model.
The estimates of i^ and s.e.( i^)^ are then used to calculate the actual value of the t-statistic.
Asymptotic t-Test
An asymptotic t-test can be used to test a single fixed value restriction, single linear restriction,
or single nonlinear restriction in all types of statistical models. The test statistic is the
asymptotic t-statistic which has an approximate t-distribution
i^ - i
t =  ~ t(T-K)
s.e.( i^)^
where i^ is the estimate of the ith individual parameter, i is the hypothesized value of the ith
individual parameter, s.e.( i^)^ is the estimated asymptotic standard error of the estimate i^,
and t(T-K) is the approximate t-distribution for T-K degrees of freedom.
To calculate the t-statistics, use the sample data to estimate the unrestricted model.
The estimates of i^ and s.e.( i^)^ are then used to calculate the actual value of the t-statistic.
Small Sample F-Test
The small sample F-test can be used to test a single fixed value restriction, two or more joint
single fixed value restrictions, a single linear restriction, and two or more joint linear restrictions
in the classical linear regression model. The test statistic is the F-statistic, which has an Fdistribution.
Two alternative approaches can be used to calculate the F-statistic. 1) Unrestrictedrestricted model approach. 2) Unrestricted model approach.
To use unrestricted-restricted model approach, it is necessary to estimate two models:
an unrestricted model and a restricted model. The restricted model is the model that imposes
the restriction(s) that define the null hypothesis. Information is obtained from these to models
to calculate the F-statistic. The F-statistic for this approach is written as
(RSSr - RSSu)/J
F =  ~ F(J, T-K)
RSSu/(T-K)
RSSr is the residual sum of squares from the restricted model; RSSu is the residual sum of
squares from the unrestricted model; J is the number of restrictions being tested; (T-K) is the
degrees of freedom for the unrestricted model; and F(J, T-K) is the F-distribution with J degrees
of freedom in the numerator and T-K degrees of freedom in the denominator.
To use the unrestricted model approach, it is necessary to estimate the unrestricted
model only. Information is obtained from this model to calculate the F-statistic. However, this
approach requires that the restriction(s) that define the null hypothesis and the F-statistic be
written in matrix format.
One or more linear restrictions can be written in matrix format as R = r, where R is a
JxK matrix of zero and/or nonzero numbers;  is a Kx1 vector of parameters; r is a Jx1 vector of
zero and/or nonzero numbers; J is the number of linear restrictions being tested; K is the
number of parameters in the regression model. The matrix R selects the appropriate
parameter(s) from the vector . Each element in the Jx1 vector R defines a linear
combinations of parameters. The vector r is a vector of hypothesized values for the J linear
combinations.
The F-statistic can be written in matrix format as
(R^- r)T[RCov^( ^)RT]-1(R^- r)
F = 
J
~ F(J, T-K)
Cov^( ^) is the estimated variance-covariance matrix for the unrestricted model, and all other
symbols have been defined above.
Likelihood Ratio Test
A likelihood ratio test can be used to test any type of restriction or restrictions in any type of
model as long as two conditions are satisfied. 1) An estimator that produces maximum
likelihood estimates is used to estimate the parameters of the model. 2) You are able to
estimate two models: an unrestricted model and a restricted model. The test statistic is the LR
statistic, which has an approximate chi square distribution
LR = 2(ln Lu – ln Lr) ~ 2(J)
ln Lu is the maximized value of the log likelihood function for the unrestricted model; ln L r is the
maximized value of the log likelihood function for the restricted model; and 2(J) is the chi
square distribution for J degrees of freedom, where J is the number of restrictions being tested.
Wald Test
The Wald test can be used to test any type of restriction or restrictions in any type of model. To
do a Wald test you need to estimate the unrestricted model only. However, the Wald test
requires that the restriction(s) that define the null hypothesis and the Wald-statistic be written
in matrix format.
One or more linear and/or nonlinear restrictions can be written in matrix format as R()
= r, where R(), read R function beta, is a Jx1 column vector of linear and/or nonlinear
combinations of parameters;  is a Kx1 vector of parameters; r is a Jx1 vector of zero and/or
nonzero numbers; J is the number of restrictions being tested; K is the number of parameters
in the regression model.
The Wald statistic has an approximate chi square distribution and is given by
W = (R(^) - r)T [GCov^(^)GT]-1 (R(^) - r) ~ 2(J)
All symbols except G have been defined previously. G is a JxK matrix of partial derivatives of
the vector R(^); that is,
 R1/1, R1/2, …, R1/k 
G = | R2/1, R2/2, …, R2/k |
| ……………………………... |
 RJ/1, RJ/2, …, RJ/k  JxK
Lagrange Multiplier Test
The Lagrange multiplier test can be used to test any type of restriction or restrictions in any
type of model. To do the Lagrange multiplier test you must estimate the restricted model and
then perform an auxiliary regression using the OLS estimator.
The test statistic is the LM statistic, which has an approximate chi square distribution. It
is given by
LM = TR2 ~ 2(J)
where T is the sample size and R2 is the unadjusted R2 statistic from an auxiliary regression.