1. ECONOMETRICS EC331 Prof. Burak Saltoglu
Statistical Inference

2. Topic to be covered Simple versus Multiple regression
Interpretation of coefficients estimates
Hypothesis testing: testing the significance of the model as a whole

3. Testing the Overall Significance of a Regression
With k-Variable regression
if we want to test the hypothesis

4. Testing the Overall Significance of a Regression
We have the test statistic,
k: number of
parameters
α: level of significance
n: sample size

5. Testing the Overall Significance of a Regression
On Eviews estimating three variable case

6. Testing the Overall Significance of a Regression
Results as
Checking
individual
significance
The F statistic
p value

7. Testing the Overall Significance of a Regression
An alternative way:

8. Testing the Overall Significance of a Regression
Then, with
to test the hypothesis
You can calculate the same F as

9. Testing the Overall Significance of a Regression
On Eviews Result Screen

10. Testing the Marginal Contribution of a Variable
How can we measure marginal effects of explanatory variables on
explanatory power of the model?
or alternatively
If you are using the R2 version, make sure dependent variables in
two models identical, otherwise use the former.

11. Testing the Marginal Contribution of a Variable
X2 variable statistically a significant variable to be added or not

12. Testing the Marginal Contribution of a Variable
NEW Regression result is

13. Testing the Marginal Contribution of a Variable,
So adding X2 to our model, significantly increases model’s total
explanatory power,

14. Testing Linear Restrictions on Parameters
Suppose we given the Cobb-Douglas production function
Taking logarithms
So we now have a linear model.How do we test the null of constant
returns to scale

15. Testing Linear Restrictions on Parameters
1) t-Test Approach: Test Statistic will be
If t exceeds critical t value, reject the null of
This method is easy to implement, but via this method, you investigate
a hypothesis after the estimation. Somehow we need
to embed the restriction in to estimation process.

16. Testing Linear Restrictions on Parameters
2) Restricted Least Squares: We can build a new regression upon the
restriction we have
Plugging in
is our restricted regression

17. Testing Linear Restrictions on Parameters
Now, we have unrestricted(UR) and restricted(R) regressions. We have
the test statistic or
n: sample size k: number of parameters in UR regression
q: the number of restrictions

18. Testing Linear Restrictions on Parameters
Let’s do an example with EViews.(another version of example 8.3) The
UR regression is
We are using RSS
version, because
dependent variables
are different in two
regressions

19. Testing Linear Restrictions on Parameters
Restricted regression estimation result

20. Testing Linear Restrictions on Parameters
We have
So since F statistic<F critical we fail to reject null of constant returns to scale.
That is, there is evidence in favour of constant returns to scale in this economy

21. Chow Test
Chow tests can be conducted to see whether there is a structural change
during the sample period.
Suppose you are working with data between and there is the
suspect of a structural change in 1982.
If there is no structural change, you estimate
for whole sample range,

22. Chow Test
But if there is a structural change 1982 and parameters of model has
changed, the correct specification must be
for
for

23. Chow Test The procedure as follows:
1) Estimate whole sample with n1+n2 observations.Note that you are
applying restrictions
Take the RSS and name it restricted RSS, RSSR
2) Estimate two sub-samples, having n1 and n2 observations,
respectively, individually and get RSS1 and RSS2
then

24. Chow Test 3) And test statistic is But assumptions of Chow test are
u1 and u2 are independently distributed—no serial correlation
Both u1 and u2 are distributed by N(0,σ2) – homoscedasticity

25. Chow Test
Let’s perform an example with Eviews.Suppose we are given linearized
money demand function(real values in logarithms):
and have quarterly data between
Suppose, we suspect of a breakdown in the first quarter of 1971 when
Nixon declared that USA abandoned Gold Standard. So we have two sub
samples:
to end of 1970
to end of 2003

26. First fit the whole sample with 176(=n1+n2) observations
Chow Test
First fit the whole sample with 176(=n1+n2) observations

27. Chow Test
Then fit with n1=44 observations

28. Chow Test
and fit 1971 to 2003 with n2=132 observations

29. Chow Test Then RSS for unrestricted regression will be F statistic is
F statistic exceeds the critical value, so we reject the null hypothesis,
so there is evidence of parameter breakdown.

30. Dummy Variables

31. Outline Why Dummty Use of Intercept Dummy Variables Intercept Dummies
Slope Dummy Variables
Comparing Two Regressions with Dummy Variables (Test for Structural Change)

32. Introduction
Dummy Variables are explanatory variables that take one of two values, usually 0 or 1.
Dummy Variables are useful for capturing the qualitative characteristics like gender, race or geographic region.
Interaction: Age and gender,

33. Use of Intercept Dummy Variables
intercept may change for some of the observations in the sample.
Example: US consumption and income during

34. Use of Intercept Dummy Variables
How did consumption change before and after WWII.
Therefore the relation during war times and reconstruction period.
consider the parameter changes during war.
So we should develop a way to incorporate the qualitative factors to our model.
A general a dummy variable is;

35. Intercept Dummy
In our example;
Then our model is;
Therefore the estimated consumption is;

36. Use of Intercept Dummy Variables

37. Intercept Dummy In Matrix Notation

38. Slope Dummy Variables
The new variable is the product of income and the dummy variable
is called interaction variable.
Alternatively it can be called as a slope dummy variable.

39. Slope Dummy Variables C Y

40. Slope Dummy In Matrix Notation

41. Comparing Two Regressions with Dummy Variables
As we have already discussed the Chow test is the way to test structural changes in the relationship.
Same can be tested by using dummy variables. Let us consider our example of consumption and income relationship, if there is a structural change in our data, the dummy variables in the model must be statistically significant.
If they’re significant this shows that there has been structural change in consumption and income relationship during WWII.

42. Comparing Two Regressions with Dummy Variables
First regress the following model;
Obtain t values for and
If the t values exceed the critical value, this means there is structural change otherwise there is not.

43. Some Related Topics Regime Switching Models and Business Cycles
Seasonal Effects (daily, montly hourly dummies)
Test for structural changes:
New regime in variance
New regime in skewness
New regime in kurtosis and even higher moments
Dummies and Autocorrelation

44. END
End of lecture

45. End of the Lecture