Download Wednesday, October 10 Handout: One-Tailed Tests, Two

Amherst College
Department of Economics
Economics 360
Fall 2012
Wednesday, October 10 Handout: One-Tailed Tests, Two-Tailed Tests, and
Logarithms
Preview
• A One-Tailed Hypothesis Test: The Downward Sloping Demand Curve
• One-Tailed versus Two-Tailed Tests
• A Two-Tailed Hypothesis Test: The Budget Theory of Demand
• Summary: One-Tailed and Two-Tailed Tests
• Logarithms: A Useful Econometric Tool
o Linear Model
o Log Dependent Variable Model
o Log Explanatory Variable Model
o Log-Log (Constant Elasticity) Model
One Tail Hypothesis Test: Downward Sloping Linear Demand Curve
Theory: A higher price decreases the quantity demanded; demand curve is downward sloping.
Step 0: Construct a model reflecting the theory to be tested:
GasConst = βConst + βPPriceDollarst + et
where GasCons
= Quantity of Gasoline Demanded
PriceDollars = Price (Chained 1990 Dollars)
βP reflects the change in the quantity demanded resulting from a ________________ in the price.
The theory suggests that βP should be ______________.
A higher price __________ the quantity demanded; the demand curve is ____________ sloping.
Step 1: Collect data, run the regression, and interpret the estimates
We have collected data for the price of gasoline and gasoline consumption in the U.S.
during the 1990’s:
GasConst
PriceDollarst
Year
1990
1991
1992
1993
1994
U. S. gasoline consumption in year t (millions of gallons per day)
Real price of gasoline in year t (dollars per gallon – chained 2000 dollars)
Real Price
($ per gallon)
1.43
1.35
1.31
1.25
1.23
Gasoline
Consumption
(Millions of gals)
303.9
301.9
305.3
314.0
319.2
Year
1995
1996
1997
1998
1999
Real Price
($ per gallon)
1.25
1.31
1.29
1.10
1.19
Gasoline
Consumption
(Millions of gals)
327.1
331.4
336.7
346.7
354.1
2
Dependent Variable: GASCONS
Included observations: 10
Coefficient
PRICEDOLLARS
-151.6556
C
516.7801
Std. Error
47.57295
60.60223
t-Statistic
-3.187853
8.527410
Prob.
0.0128
0.0000
Estimated Equation: EstGasCons = _______ − _______PriceDollars.
Interpretation: We estimate that a $1 increase in the real price of gasoline ______________
the quantity of gasoline demanded by _______ million gallons.
Critical Result: The coefficient estimate equals _______. The _____________ sign of the
coefficient estimate suggests that a higher price ____________ the quantity demanded.
This evidence ____________ the downward sloping demand theory.
Step 2: Play the cynic, challenge the evidence, and construct the null and alternative hypotheses.
Cynic’s view: The price actually has no effect on the quantity of gasoline demanded; the
negative coefficient estimate obtained from the data was just “the luck of the draw.” In
fact, the actual coefficient, βP, equals 0.
Now, we construct the null and alternative hypotheses:
Cynic’s view is correct: Price has no effect on quantity demanded.
H0: βP = 0
H1: βP < 0
Cynic’s view is incorrect: A higher price decreases quantity demanded.
Step 3: Formulate the question to assess the cynic’s view.
Question for the Cynic:
• Generic Question: What is the probability that the results would be like those we
actually obtained (or even stronger), if the null hypothesis were actually true (if the
cynic is correct and the price actually has no impact on the quantity demanded)?
• Specific Question: The regression’s coefficient estimate was −151.7. What is the
probability that the coefficient estimate in one regression would be −151.7 or less, if
H0 were actually true (if the actual coefficient, βP, equals 0)?
Answer: Prob[Results IF Cynic Correct] or Prob[Results IF H0 True]
The magnitude of this probability determines whether we accept or reject the null hypothesis:
Prob[Results IF H0 True] small
Prob[Results IF H0 True] large
↓
___________ that H0 is true
↓
___________ that H0 is true
↓
___________ H0
↓
___________ H0
3
Step 4: Use the general properties of the estimation procedure, the probability distribution of
the estimates, to calculate Prob[Results IF H0 True].
•
•
•
•
Since ordinary least squares
estimation procedure for the
coefficient value is unbiased, the
mean of the probability
distribution for the estimate equals
_______________________.
If the null hypothesis were true,
the actual price coefficient would
equal _______.
The standard error equals _______.
The degrees of freedom equal
__________________.
OLS estimation
procedure unbiased
Assume H0
is true
Mean[bP] = ___ = ___
Dependent Variable: GASCONS
Included observations: 10
Coefficient
PRICEDOLLARS
-151.6556
C
516.7801
Student t-distribution
Mean = ______
SE = ______
DF = ______
______
______
bP
______
______
Standard
error
______
______
Number of
Observations
Number of
Parameters
DF = ___ − ___ = ___
SE[bP] = ____
Std. Error
47.57295
60.60223
t-Statistic
-3.187853
8.527410
Prob.
0.0128
0.0000
Recall that the Prob. column reports the “tails probability:”
Tails Probability: The probability that the coefficient estimate, bP, resulting from one
regression would lie at least _________ from ____, if the actual coefficient, βP, equals ____.
Prob[Results IF H0 True] =
Step 5: Decide on the standard of proof, a significance level
The significance level is the dividing line between the probability being small and the
probability being large.
Prob[Results IF H0 True]
Prob[Results IF H0 True]
Less Than Significance Level
Greater Than Significance Level
↓
↓
Prob[Results IF H0 True] large
Prob[Results IF H0 True] small
↓
Unlikely that H0 is true
↓
Likely that H0 is true
↓
Reject H0
↓
Do not reject H0
Do we reject the null hypothesis at a 10 percent (.10) significance level? ______
Do we reject the null hypothesis at a 5 percent (.05) significance level? ______
Do we reject the null hypothesis at a 1 percent (.01) significance level? ______
Do the results lend support to the downward sloping demand curve theory? ______
4
Two Tailed Hypothesis Test: Budget Theory of Demand
Budget Theory of Demand: Total expenditures for gasoline are constant. That is, when the
gasoline price changes, demanders adjust the quantity demanded so as to keep their total
gasoline expenditures constant:
P×Q = Constant
Question: What economic concept is relevant here?
Claim: The price elasticity of demand is the relevant concept. To understand why we begin
with the verbal definition of price elasticity:
Verbal Definition: The price elasticity of demand equals the percent change in the
quantity demanded resulting from a one percent change in price.
How can we convert the verbal definition of the price elasticity of demand into a rigorous
mathematical definition?
Price Elasticity = _____________ Change in Quantity resulting from a 1 ___________
Change in the Price
=
Percent Change in the Quantity
Percent Change in the Price
Calculating percent changes. If X increases from 200 to 220, there
is a 10 percent increase: 200 → 220
20
Percent Change in X = 200 × 100 = .1× 100 = 10 percent
We can now generalize this: X → X + ΔX
ΔX
Percent Change in X = X × 100
ΔQ
Q × 100
=
ΔP
P × 100
=
ΔQ P
ΔP Q
dQ P
= dP Q
Substituting for the percent changes
Simplifying
Taking limits as ΔP approaches 0
5
Step 0: Construct a model reflecting the theory to be tested:
Constant Price Elasticity Model: Q = βConstP
dQ P
Price Elasticity of Demand = dP Q
βP
βP−1
= βConst βP P
βP−1
= βConst βPP
= βP
P
βP−1
P
P
Q
Taking the derivative of Q with respect to P
P
βConstPβP
P
βP
Substituting βConstP
βP
Substituting βConstP
βP
= βP
for Q.
for Q.
Simplifying.
The price elasticity of demand just equals the value of βP.
Question: What does the budget theory of demand postulate about βP?
P×Q = Constant
Solving for Q
Q
= _____________
β
Compare this equation to the constant price elasticity demand model: Q = βConstP P.
Clearly,
βConst = ___________
βP = ___
Answer: The budget theory of demand postulates that the price elasticity of demand
equals −1.0.
Budget Theory of Demand: βP = ___
Logarithms allow us to converting a constant price elasticity model into a linear model:
βConst PβP
Q
=
log(Q) = log(βConst) +
βPlog(P)
Step 1: Collect data, run the regression, and interpret the estimates
We have already collected the data, but now we must generate two new variables: the
logarithm of quantity and the logarithm of price:
• LogQ = log(GasCons)
• LogP = log(PriceDollars)
Dependent Variable: LOGQ
Included observations: 10
LOGP
C
Coefficient
-0.585623
5.918487
Std. Error
0.183409
0.045315
t-Statistic
-3.192988
130.6065
Prob.
0.0127
0.0000
Interpretation: We estimate that a 1 percent increase in the price _________________ the
quantity demand by _______ percent. That is, the estimate for the price elasticity of
demand equals ________.
6
Critical Result: The coefficient estimate equals ______. The coefficient estimate does not
equal _____; the estimate lies __________ from _____.
Theory
______
Evidence
______
Price
Elasticity
______
0
The critical result is that the estimate lies ________ from where the theory claims it
should be.
This evidence suggests that the budget theory of demand is _______________.
Step 2: Play the cynic, challenge the evidence, and construct the null and alternative hypotheses.
The cynic always challenges the evidence:
Cynic’s view: Sure the coefficient estimate from regression suggests that the price
elasticity of demand does not equal −1.0, but this is just “the luck of the draw.” In
fact, the actual price elasticity of demand equals −1.0.
We shall now construct the null and alternative hypotheses to address this question:
H0: βP = −1.0
Cynic’s view is correct: Actual price elasticity of demand equals −1.0
H1: βP ≠ −1.0
Cynic’s view is incorrect: Actual price elasticity of demand does not
equal −1.0
Following the cynic’s lead, the null hypothesis always challenges the evidence. On
the other hand, the alternative hypothesis is consistent with the evidence.
Step 3: Formulate the question to assess the cynic’s view.
Question for the Cynic:
• Generic Question: What is the probability that the results would be like those
we actually obtained (or even stronger), if the cynic is correct and the actual price
elasticity of demand equals −1.0?
• Specific Question: The regression’s coefficient estimate was −.586: What is the
probability that the coefficient estimate, bP, in one regression would be at least
.414 from −1.0, if H0 were actually true (if the actual coefficient, βP, equals −1.0)?
Answer: Prob[Results IF Cynic Correct] or Prob[Results IF H0 True]
7
Step 4: Use the general properties of the estimation procedure, the probability distribution of
the estimates, to calculate Prob[Results IF H0 True].
•
•
•
•
Since ordinary least squares
estimation procedure for the
coefficient value is unbiased, the
mean of the probability distribution
for the estimate equals
_______________________.
If the null hypothesis were true, the
actual price coefficient would equal
_____.
The standard error equals _______.
The degrees of freedom equal
__________________.
OLS estimation
procedure unbiased
Student t-distribution
Mean = ______
SE = ______
DF = ______
______
bCbP
______
Assume H0
is true
Mean[bP] = ___ = ___
______
Standard
error
______
______
Number of
Observations
______
Number of
Parameters
DF = ___ − ___ = ___
SE[bP] = ____
Question: Can we use can use the “tails probability” as reported in the regression
printout to compute this probability?
Answer: ________
Tails Probability: The probability that the coefficient estimate, bP, resulting from one
regression would lie at least _________ from ____, if the actual coefficient, βP, equals ____.
NB: The tails probability is calculated on the premise that the actual value of the coefficient
equals _____.
Econometrics Lab:
Prob[Results IF H0 True] ≈
Left
Tail
↓
_____
Right
Tail
↓
+
_____
=
_____
8
Hypothesis Testing: Using Regression Printouts with Clever Algebraic Manipulations
Clever Definition: βClever = βP + 1.0
βP = −1.0 if and only if βClever = 0
Now, recall our equation for the constant price elasticity model:
where
LogQ = log(GasCons)
LogQ
= c + βPLogP
LogP = log(Price)
Next, a little algebra:
βClever = βP + 1.0,
βP = βClever − 1.0.
Substituting βClever − 1.0 for βP
LogQ
= c + (βClever − 1.0) LogP
LogQ
LogQ + LogP
= c + βCleverLogP − LogP
= c + βCleverLogP
LogQPlusLogP = c + βCleverLogP
where LogQPlusLogP = LogQ + LogP
We can now express the hypotheses in terms of βC. Recall that
βP = −1.0
if and only if
βClever = 0:
H0: βP = −1.0 ⇒ H0: βClever = 0 Actual price elasticity of demand equals −1.0
H1: βP ≠ −1.0 ⇒ H1: βClever ≠ 0 Actual price elasticity of demand does not equal −1.0
Dependent Variable: LOGQPLUSLOGP
Included observations: 10
Coefficient
LOGP
0.414377
C
5.918487
Std. Error
0.183409
0.045315
t-Statistic
2.259308
130.6065
Prob.
0.0538
0.0000
Critical Result: The coefficient estimate equals ________. The coefficient estimate
___________________ equal _____; the estimate is _______ from _____.
Question for the Cynic:
• Specific Question: The regression’s coefficient estimate was .414: What is the
probability that the coefficient estimate, bClever, in one regression would be at least
.414 from 0, if H0 were actually true (if the actual coefficient, βClever, equals 0)?
Answer: Prob[Results IF H0 True]
9
Next, calculate Prob[Results IF H0 True] focusing on βClever:
• Since ordinary least squares
estimation procedure for the
coefficient value is unbiased, the
mean of the probability
distribution for the estimate
______
equals _______________________.
• If the null hypothesis were true,
the actual price coefficient would
equal _________.
• The standard error equals ______.
______
• The degrees of freedom equal
__________________.
OLS estimation
procedure unbiased
Assume H0
is true
Mean[bClever] = ___ = ___
Standard
error
Student t-distribution
Mean = ______
SE = ______
DF = ______
______
bCbClever
______
______
Number of
Observations
SE[bClever] = ____
______
Number of
Parameters
DF = ___ − ___ = ___
Tails Probability: The probability that the coefficient estimate, bP, resulting from one
regression would lie at least _________ from ____, if the actual coefficient, βClever, equals
____.
Answer: Prob[Results IF H0 True] = ________
Question: Is this the same answer as we calculated with the Econometrics Lab? _____
Step 5: Decide upon the standard of proof, what constitutes proof beyond a reasonable
doubt. Decide on the significance level, the dividing line between small and large probability:
Prob[Results IF H0 True]
Less Than Significance Level
↓
Prob[Results IF H0 True] small
Prob[Results IF H0 True]
Greater Than Significance Level
↓
Prob[Results IF H0 True] large
↓
Unlikely that H0 is true
↓
Likely that H0 is true
↓
Reject H0
↓
Do not reject H0
Do we reject the null hypothesis at a 10 percent (.10) significance level? ______
Do we reject the null hypothesis at a 5 percent (.05) significance level? ______
Do we reject the null hypothesis at a 1 percent (.01) significance level? ______
What is your assessment of the budget theory of demand?
10
Summary: One Tailed Versus Two Tailed Tests – Which Is Appropriate?
Theory: Coefficient is less than or
greater than a specific value (often 0)
↓
One tailed test appropriate
Theory: Coefficient
equals a specific value
↓
Two tailed test appropriate
Theory: β > c or β < c
Theory: β = c
Probability Distribution
Probability Distribution
H0: β = c
H1: β ≠ c
H0: β = c
H1: β > c
b
c
Probability Distribution
b
c
Prob[Results IF H0 True] =
Probability of obtaining results like those
we actually got (or even stronger), if H0 is true
H0: β = c
H1: β < c
Prob[Results IF H0 True]
b
c
Logarithms: A Useful Econometric Tool
Small
Large
Reject H0
Do not reject H0
1
Logarithms provide a very convenient way to fine tune our theories by expressing them in
terms of percentages rather than “natural” units.
Linear Model: yt = βConst + βxxt + et
Coefficient estimate: Estimates the (natural) unit change in y resulting from a one
(natural) unit change in x
Log Dependent Variable Model: log(yt) = βConst + βxxt + et
Coefficient estimate multiplied by 100: Estimates the percent change in y resulting from
a one (natural) unit change in x
Log Explanatory Variable Model: yt = βConst + βxlog(xt) + et
Coefficient estimate divided by 100: Estimates the (natural) unit change in y resulting
from a one percent change in x
Log-Log (Constant Elasticity) Model: log(yt) = βConst + βxlog(xt) + et
Coefficient estimate: Estimates the percent change in y resulting from a one percent
change in x
1
The log notation refers to the natural logarithm (logarithm base e), not the logarithm base 10.
11
Using Logarithms – An Illustration: Wages and High School Education
Basic Theory: Additional years of high school education increase the wage.
Wage and Education Data: Cross section data of wages and education for 212 workers
included in the March 2007 Current Population Survey residing in the Northeast region of
the United States who have completed the ninth, tenth, eleventh, or twelfth grades, but have
not continued on to college or junior college.
Waget
HSEduct
Wage rate earned by worker t (dollars per hour)
Highest high school grade completed by worker t (9, 10, 11, or 12 years)
Linear model: Waget = βConst + βEHSEduct + et
This model includes no logarithms. Wage is expressed in dollars (natural units) and
education in years (natural units).
Dependent Variable: Wage
Explanatory Variable: HSEduc
Dependent Variable: WAGE
Included observations: 212
HSEDUC
C
Coefficient
1.645899
-3.828617
Std. Error
0.555890
6.511902
t-Statistic
2.960834
-0.587941
Prob.
0.0034
0.5572
Estimated Equation: Wage = −3.83 + 1.65HSEduc.
Coefficient Interpretation:
One ___________________ increase in HSEduc
↓
Increases Wage by ________ ___________________
Log dependent variable model: LogWaget = β Const + βEHSEduct + et
The dependent variable (LogWage) is expressed in terms of the logarithm of dollars; the
explanatory variable (HSEduc) is expressed in years (natural units).
Dependent Variable: LogWage
Explanatory Variable: HSEduc
Dependent Variable: LOGWAGE
Included observations: 212
Coefficient
HSEDUC
0.113824
C
1.329791
Std. Error
0.033231
0.389280
t-Statistic
3.425227
3.416030
Prob.
0.0007
0.0008
Estimated Equation: LogWage = 1.33 + .114HSEduc.
Coefficient Interpretation:
One ___________________ increase in HSEduc
↓
Increases Wage by ________ ___________________
12
Log explanatory variable model: Waget = βConst + βELogHSEduct + et
The dependent variable (Wage) is expressed in terms of dollars (natural units); the
explanatory variable (LogHSEduc) is expressed in terms of the log of years.
Dependent Variable: Wage
Explanatory Variable: LogHSEduc
Dependent Variable: WAGE
Included observations: 212
LOGHSEDUC
C
Coefficient
17.30943
-27.10445
Std. Error
5.923282
14.55474
t-Statistic
2.922270
-1.862242
Prob.
0.0039
0.0640
Estimated Equation: Wage = −27.1 + 17.31LogHSEduc.
Coefficient Interpretation:
One ___________________ increase in HSEduc
↓
Increases Wage by ________ ___________________
Log-log (constant elasticity) model: LogWaget = β Const + β ELogHSEduct + et
Both the dependent and explanatory variables are expressed in terms of logs. This is just
the constant elasticity model that we discussed earlier.
Dependent Variable: LogWage
Explanatory Variable: LogHSEduc
Dependent Variable: LOGWAGE
Included observations: 212
Coefficient
LOGHSEDUC
1.195654
C
-0.276444
Std. Error
0.354177
0.870286
t-Statistic
3.375868
-0.317647
Prob.
0.0009
0.7511
Estimated Equation: LogWage = −.28 + 1.20LogHSEduc.
Coefficient Interpretation:
One ___________________ increase in HSEduc
↓
Increases Wage by ________ ___________________