OSU Economics 444: Elementary Econometrics
Ch.5 Hypothesis Testing
• General Purpose of Hypothesis Testing
Hypothesis testing asks whether this sample provides support for the theory or provides evidence against
the theory. It helps a practitioner to make a decision.
• Approaches to hypothesis testing — Basic features
1) Any approach of hypothesis testing shall take into account that the (estimated) result could have been
obtained by chance.
2) It shall take into account that any particular estimate comes from a distribution of estimates.
3) An estimate shall be interpreted with respect to the standard error of that sampling distribution of
estimates.
• Statistical Inference
1) Hypothesis testing is a component of statistical inference.
2) A model is estimated with a representative sample of the population. Statistical inference attempts to
infer the properties of the population from the sample.
3) A classical statistical inference assumes that the sample data are only information about the population.
4) An alternative, Bayesian statistics approach, adds prior information to the sample to draw statistical
inference. Prior information is presented as distributions on possible values of parameters.
• What hypothesis testing can or can not do:
1) It is impossible to prove that a hypothesis is correct based on hypothesis testing.
2) Hypothesis testing can indicate that a particular sample may conÞrm to a particular hypothesis.
3) One may reject a given hypothesis with a certain degree of conÞdence.
In such a case, the researcher concludes that the sample result would be very unlikely to have observed
if the hypothesized theory were correct.
4) If there is conßicting evidence on the validity of a theory, one needs to collect more sample (information).
•• Steps of the classical hypothesis testing approach:
• Step 1: Develope hypotheses
i) The developed hypotheses usually come from the application of economic theory or are based on common
beliefs.
ii) Specify the hypothesis before the equation is estimated.
iii) Formulate the null hypothesis and the alternative hypothesis.
a) The null hypothesis is a statement of the range of possible values of parameters of a model representing theoretical restrictions.
b) The alternative hypothesis would be expected to occur if the theoretical restrictions are not valid.
c) Notations: H0 : stands for the null hypothesis. H1 : stands for the alternative hypothesis. (or HA :)
d) For example, want to test the coefficient β1 of x1 in a regression equation is zero. Then
H0 : β1 = 0,
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H1 : β1 6= 0.
• Step 2: Calculating estimates in a model and their sampling distributions, in particular, their standard
errors.
• Step 3: Testing possible implications of the hypothesis based on the estimates and their sampling
distribution.
i) Formulate some relevant statistics based on implications of the hypotheses.
ii) Evaluate the test statistics based on estimates and their sampling distribution (using the sample).
iii) Reject or accept the hypotheses.
a) Because estimates are not exactly the population parameters, test statistics involve uncertainty.
Decision based on test statistics could contain errors.
b) There are two kinds of errors, namely, Type I and Type II.
Type I error : We reject a true null hypothesis.
Type II error : We do not reject a false null hypothesis.
•• The t-Statistic for y = β0 + β1 x1 + · · · + βk xk + ².
• Testing βl = b where b is a speciÞed constant, e.g., b = 0:
tl =
(β̂l − b)
SE(β̂l )
where β̂l =OLSE of βl and SE(β̂l ) is the estimated standard error of β̂l .
i) The t-statistic is a random variable with a Student t-distribution with (n − k − 1) degrees of freedom
(under normal ²’s)
ii) To allow only a probability error of c percent (called the level of significance) of rejecting the true
hypothesis (i.e., reject H0 ), take the value tc (critical value) from a t-statistic table.
iii) Reject the H0 : βl = b when |tl | > tc . Otherwise, not.
iv) The above test for H0 : βl = b is a two-side test.
• For testing that H0 : βl < b, use a one-sided test:
Reject that H0 : βl < b if tl > tc ; otherwise, not.
• Testing a linear constraint, e.g., H0 : a1 β1 + a2 β2 = b where b is a speciÞed value, use the following
t-statistic:
t=
δ̂
SE(δ̂)
,
where δ̂ = a1 β̂1 + a2 β̂2 − b and SE(δ̂) is the estimated standard error of δ̂.
i) This statistic is also a t random variable with n − k − 1 degrees of freedom under H0 (and normal ²s).
• An example:
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ŷt = 1.30+4.91x1t + 0.00123x2t − 7.14x3t
(2.38)
(0.00022)
(71.38)
where the Þgures in parentheses are the estimated standard errors;
y = new car sales (in hundreds of thousands of units) in year t
x1 = real U.S. disposable income (in hundreds of billions of dollars)
x2 = the average retail price of a new car in year t (in dollars)
x3 = the number of sports utility vehicales sold in year t (in millions).
The data are annual from 1990 to 1999. So n = 10 and n − k − 1 = 10 − 3 − 1 = 6 degrees of freedom.
For a one sided test, the critical value of a 5% level of signiÞcance = 1.943 (TABLE B-1). (1.943 is also
the critical value of a two-side test with a 10% level of signiÞcance)
For a two sided test, the critical values of a 5% level of signiÞcance are, -2.447 and 2.447.
1) Testing H0 : β1 = 0. Calculate the value of the t-statistic
t1 =
β̂1
SE(β̂1 )
= 4.91/2.38 = 2.1.
Use a two-sided test with a 5% level of signiÞcance. As 2.1 < 2.447, it can not reject H0 : β1 = 0. We
conclude that β1 is not significantly different from zero at 5% level of signiÞcance.
Use a two-sided test with a 10% level of signiÞcance. As 2.1 > 1.943, reject H0 : β1 = 0 and concludes
that β1 is signiÞcantly different from zero at 10% level of signiÞcance.
2) Testing H0 : β1 ≤ 0 (the alternative is H1 : β1 > 0). Calculate the value of the t-statistic
t1 =
β̂1
SE(β̂1 )
= 4.91/2.38 = 2.1.
Use a one-sided test with a 5% level of signiÞcance. As 2.1 > 1.943, reject H0 : β1 ≤ 0 and accept the
alternative that β1 is positive.
•• The F-Test
1) It can be used for testing several (linear equality) constraints on the coefficients βs of a linear regression
model.
2) Construct the F statistic:
a) Suppose that there are q constraints on βs, estimate the regression model by imposing all the q
constraints and also estimate the model without imposing the constraints.
b) Compute the RSS for each of the estimated equation. Let RSS be the residual sum of squares of the
estimated regression equation without imposing the constraints. Let RSSc denote the residual sum
of squares of the estimated regression equation of the restricted model (with constraints imposed).
c) The F -statistic for testing the q constraints is
F =
(RSSc − RSS)/q
.
RSS/(n − k − 1)
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3) The F statistic is a random variable which has a F -distribution with q and (n−k−1) degrees of freedom.
4) With a selected level of signiÞcance, its critical value Fc can be obtained (TABLES B-2, B-3).
Reject H0 if F ≥ Fc .
3) Test the overall significance of a regression equation, i.e., test H0 : β1 = 0, β2 = 0, · · · , βk = 0:
F =
ESS/k
RSS/(n − k − 1)
where ESS is the explained sum of squares of the residuals (of the estimated regressin equation of y on
1, x1 , · · · , xk ).
The overall signiÞcance test has a F -distribution with k and (n − k − 1) degrees of freedom.
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