Multiple Regression Model: Hypotheses Testing

1
Multiple Regression ModelHypotheses Testing

• y b0 b1 x1 b2 x2 . . . bk xk u

2
Lecture 4 THE MULTIPLE REGRESSION MODEL
HYPOTHESES TESTINGProfessor Victor Aguirregabiria

• OUTLINE
• Introduction and Examples
• Sampling Distribution of OLS
• Hypotheses on one parameter t-test
• Hypotheses on a linear combination of parameters
  t-test
• Hypotheses on Multiple Linear Restrictions F-test

3
1. Introduction and Examples

• Typically, an estimated econometric model is used
  to make predictions and to make decisions.
• In both cases, it is important to know how much
  uncertainty (or how much confidence) we have in
  the predictions and decisions we made based on
  the estimated model.
• There are different sources of uncertainty (right
  model?). Here we concentrate on the uncertainty
  generated by the fact that we have estimates and
  not the true parameters.

4
Example 1 Demand Elasticity and Pricing

• Consider the example (in Lecture 2) of the
  manager interested in the estimation of the
  elasticity of demand.
• The demand model is

• Where ? is the elasticity of demand.

5
Example 1 Demand Elasticity and Pricing

• Given the estimated model, the manager will use
  it to decide its profit maximizing price.

• The manager may decide to take this estimate to
  choose her optimal price

6
Example 1 Demand Elasticity and Pricing

• However, this decision is ignoring that there is
  uncertainty associated with the estimate, and
  therefore there is uncertainty in the profit.
• The profit function is

• But the true ? is unknown.

7
Example 1 Demand Elasticity and Pricing

• Note that from the point of view of the manager,
  the estimated elasticity is a known value (say
  3.45) and the true value of ? is a random
  variable (with mean 3.45).
• Is this uncertainty important for the choice of
  the optimal price?
• Yes, the optimal price that accounts for
  uncertainty is different than the price that
  ignores it.

8
Example 1 Demand Elasticity and Pricing

• How can the manager take account this uncertainty
  when deciding prices?
• There are different possible approaches.
• To treat profits as a random variable
  (conditional on the estimate of ?), and maximize
  expected profits.
• Construct confidence intervals for the true ? and
  use them to construct confidence intervals on
  optimal prices.

9
Example 2 Credit Scores and Mortgages

• Consider a bank deciding whether to give a
  mortgage loan to a person.
• To make this decision, the bank uses a MRM to
  predict the probability of repayment.
• The dependent variable Y is an indicator, 0 o 1,
  of the repayment of a loan. The explanatory
  variables X are socio-economic characteristics of
  borrower, the aggregate economy, and the local
  economy.

10
Example 2 Credit Scores and Mortgages

• Using data of previous mortgages, the bank
  estimates this MRM.
• The fitted values of this model can be
  interpreted as predictions for the probability of
  repayment of a mortgage.
• Given the estimated model, the bank has to decide
  whether to approve a mortgage to a new applicant.

11
Example 2 Credit Scores and Mortgages

• Given the values of the X regressors for this
  applicant, the fitted value (credit score) of
  this applicant is 0.76.
• What does the bank do with this credit score?
• The bank has to choose either Y1 (approval) or
  Y0 (not approval).
• There is a possible error associated with each
  decision.

12
Example 2 Credit Scores and Mortgages

• Suppose that the bank considers the scenario
  (hypothesis) that the person will pay the
  mortgage (Y1).
• If the bank accepts the hypothesis, it can make
  an ERROR TYPE I reject when it is true.
• If the bank rejects the hypothesis, it can make
  an ERROR TYPE II accept when it is false.
• Each error has a probability.

13
2. SAMPLING DISTRIBUTION OLS ESTIMATOR

• In order to do hypothesis testing, we need to
  know the sampling distribution of the OLS
  estimator (not just the mean and variance).
• We add another assumption (beyond the
  Gauss-Markov assumptions).
• Assumption 6 Normality.
• u is independent of x1, x2,, xk and u is
  normally distributed with zero mean and variance
  s2
• u Normal(0,s2)

14
COMMENTS ON NORMALITY ASSUMPTION

• Assumptions 1 to 6 are called the Classical
  Assumptions.
• Under the Classical Assumptions, OLS is not only
  BLUE, but it is the minimum variance unbiased
  estimator.
• Normality of u implies that, conditional on x
  yx
  Normal(b0 b1x1 bkxk, s2)
• Under Classical Assumptions the OLS estimator is
  Normally distributed

15
The homoskedastic normal distribution with a
single explanatory variable
y
f(yx)
.
E(yx) b0 b1x
.
Normal distributions
x1
x2
16
COMMENTS ON NORMALITY ASSUMPTION

• Do we really need to assume that u is normally
  distributed to have that the OLS is normal?
• Not really.
• When we have large samples (and under Assumptions
  1 to 5) the Central Limit Theorem implies that
  the OLS estimator is closed to be normally
  distributed.
• We say that the OLS is asymptotically normal.
  This means that the normal distribution is a good
  approximation when the sample is large even when
  u is not normal.

17
3. Hypotheses about one parameter t-test
18
3. Hypotheses about one parameter t-test

• Knowing the sampling distribution for the
  standardized estimator allows us to carry out
  hypothesis tests.
• Start with a null hypothesis
• For example, H0 bj 0
• If accept null, then accept that xj has no effect
  on y, controlling for other xs

19
3. Hypotheses about one parameter t-test
20
3. Hypotheses about one parameter t-test

• Besides our null, H0, we need an alternative
  hypothesis, H1, and a significance level.
• H1 may be one-sided, or two-sided
• H1 bj gt 0 is one-sided
• H1 bj lt 0 is one-sided
• H1 bj ? 0 is a two-sided alternative
• If we want to have only a 5 probability of
  rejecting H0 if it is really true, then we say
  our significance level is 5

21
3. Hypotheses about one parameter t-test

• Having picked a significance level, a, we look
  up the (1 a)th percentile in a t distribution
  with n k 1 df and call this c, the critical
  value
• We can reject the null hypothesis if the t
  statistic is greater than the critical value
• If the t statistic is less than the critical
  value then we fail to reject the null

22
One-Sided Alternatives (cont)
yi b0 b1xi1 bkxik ui H0 bj
0 H1 bj gt 0
Fail to reject
reject
(1 - a)
a
c
0
23
T-test One-sided alternatives (cont)

• Because the t distribution is symmetric, testing
  H1 bj lt 0 is straightforward. The critical
  value is just the negative of before
• We can reject the null if the t statistic lt c,
  and if the t statistic gt than c then we fail to
  reject the null
• For a two-sided test, we set the critical value
  based on a/2 and reject H1 bj ? 0 if the
  absolute value of the t statistic gt c

24
T-test Two-sided alternatives

• Because the t distribution is symmetric, for a
  two-sided test, we set the critical value based
  on a/2 and reject H1 bj ? 0 if the absolute
  value of the t statistic gt c

25
Two-Sided Alternatives
yi b0 b1Xi1 bkXik ui H0 bj
0 H1 bj ? 0
fail to reject
reject
reject
(1 - a)
a/2
a/2
-c
c
0
26
Summary for H0 bj 0

• Unless otherwise stated, the alternative is
  assumed to be two-sided.
• If we reject the null, we typically say xj is
  statistically significant at the a level.
• If we fail to reject the null, we typically say
  xj is statistically insignificant at the a
  level

27
Testing other hypotheses

• A more general form of the t statistic recognizes
  that we may want to test something like
• H0 bj aj
• In this case, the appropriate t statistic is

28
Confidence Intervals

• We can construct a confidence interval using the
  same critical value as was used for a two-sided
  test.
• A (1 - a) confidence interval is defined as

29
Computing p-values for t tests

• An alternative to the classical approach is to
  ask, what is the smallest significance level at
  which the null would be rejected?
• So, compute the t statistic, and then look up
  what percentile it is in the appropriate t
  distribution this is the p-value.
• The smaller the p-value, the stronger is the
  evidence in favor of the null hypothesis.

30

• 3. TESTING A LINEAR RELATIONSHIP OF PARAMETERS
• Suppose instead of testing whether b1 is equal to
  a constant, you want to test if it is equal to
  another parameter, that is H0 b1 b2
• Use same basic procedure for forming a t
  statistic

31
Testing Linear Combo (cont)
32
Testing a Linear Combo (cont)

• So, to use formula, need s12, which standard
  output does not have
• Many packages will have an option to get it, or
  will just perform the test for you
• In Stata, after reg y x1 x2 xk you would type
  test x1 x2 to get a p-value for the test
• More generally, you can always restate the
  problem to get the test you want

33
Multiple Linear Restrictions

• Everything weve done so far has involved
  testing a single linear restriction, (e.g. b1 0
  or b1 b2 )
• However, we may want to jointly test multiple
  hypotheses about our parameters
• A typical example is testing exclusion
  restrictions we want to know if a group of
  parameters are all equal to zero

34
Testing Exclusion Restrictions

• Now the null hypothesis might be something like
  H0 bk-q1 0, ... , bk 0
• The alternative is just H1 H0 is not true
• Cant just check each t statistic separately,
  because we want to know if the q parameters are
  jointly significant at a given level it is
  possible for none to be individually significant
  at that level

35
Exclusion Restrictions (cont)

• To do the test we need to estimate the
  restricted model without xk-q1,, , xk
  included, as well as the unrestricted model
  with all xs included
• Intuitively, we want to know if the change in
  SSR is big enough to warrant inclusion of
  xk-q1,, , xk

36
The F statistic

• The F statistic is always positive, since the
  SSR from the restricted model cant be less than
  the SSR from the unrestricted
• Essentially the F statistic is measuring the
  relative increase in SSR when moving from the
  unrestricted to restricted model
• q number of restrictions, or dfr dfur
• n k 1 dfur

37
The F statistic (cont)

• To decide if the increase in SSR when we move to
  a restricted model is big enough to reject the
  exclusions, we need to know about the sampling
  distribution of our F stat
• Not surprisingly, F Fq,n-k-1, where q is
  referred to as the numerator degrees of freedom
  and n k 1 as the denominator degrees of
  freedom

38
The F statistic (cont)
f(F)

• Reject H0 at a
• significance level
• if F gt c

fail to reject
reject
a
(1 - a)
0
c
F
39
The R2 form of the F statistic

• Because the SSRs may be large and unwieldy, an
  alternative form of the formula is useful
• We use the fact that SSR SST(1 R2) for any
  regression, so can substitute in for SSRu and
  SSRur

40
Overall Significance

• A special case of exclusion restrictions is to
  test H0 b1 b2 bk 0
• Since the R2 from a model with only an intercept
  will be zero, the F statistic is simply

41
General Linear Restrictions

• The basic form of the F statistic will work for
  any set of linear restrictions
• First estimate the unrestricted model and then
  estimate the restricted model
• In each case, make note of the SSR
• Imposing the restrictions can be tricky will
  likely have to redefine variables again

42
F Statistic Summary

• Just as with t statistics, p-values can be
  calculated by looking up the percentile in the
  appropriate F distribution
• Stata will do this by entering display fprob(q,
  n k 1, F), where the appropriate values of F,
  q,and n k 1 are used
• If only one exclusion is being tested, then F
  t2, and the p-values will be the same