Lecture 10 Preview: Multiple Regression Analysis – Introduction Linear Demand Model and the No Money Illusion Theory A Two-Tailed Test: No Money Illusion

Lecture 10 Preview: Multiple Regression Analysis – Introduction

Linear Demand Model and the No Money Illusion TheoryA Two-Tailed Test: No Money Illusion Theory

A One-Tailed Test: Downward Sloping Demand Curve Theory

Constant Elasticity Demand Model and the No Money Illusion Theory

Calculating Prob[Results IF H0 True]: Clever Algebraic Manipulation

Simple and Multiple Regression Analysis

Goal of Multiple Regression Analysis

Cleverly Define a New Coefficient That Equals 0 When H0 Is True

Reformulate the Model to Incorporate the New Coefficient

Estimate the Parameters of the New Model

Use the Tails Probability to Calculate Prob[Results IF H0 True]

Linear Demand Model

Distinction between Simple and Multiple Regression Analysis

Simple and Multiple Regression AnalysisSimple Regression Analysis: A single explanatory variable.

Multiple Regression Analysis: Multiple explanatory variables.

Question: Why study multiple regression analysis?Answer: Typically, a dependent variable is affected by many, not just one, explanatory variables.

Goal of Multiple Regression Analysis

Multiple regression analysis attempts to separate out the individual effect of each explanatory variable.

Downward Sloping Demand Curve Theory RevisitedTheory: Microeconomic theory teaches that while the quantity of a good demanded by a household depends on the good’s own price, other factors also affect demand: household income, the prices of other goods, etc.

An explanatory variable’s coefficient estimate allows us to estimate the change in the dependent variable resulting from a change in that particular explanatory variable while all other explanatory variables remain constant.

Step 0: Construct a model reflecting the theory to be tested Qt = Const + PPt + IIt + CPChickPt + et

Qt = Quantity of beef demanded

Pt = Price of beef (the good’s own price)

It = Household income

ChickPt = Price of chickenTheory: P < 0. An increase in the price of beef (the good’s own price) decreases the

quantity demanded when all other factors that influence demand (income and the price of chicken) remain constant.

When we ran our simple regression assessing the downward sloping theory of demand we included the quantity demanded of beef as the dependent variable and its price as the only explanatory variable. We ignored these other factors.

We need a way to include not only the effect of the price of the good but also the other factors that influence demand.

But economic theory teaches us that these other factors are important too. Consequently, we should not ignore them.

Multiple regression analysis can consider all the factors that our theory suggests are important.

Demand Curve: The demand curve for a good illustrates how the quantity demanded changes when the good’s price changes while all the other factors relevant to demand remain constant.

P

Q

D

All other factors relevant to demand

remain constant

Example: Demand for Beef. Multiple regression analysis allows us to separate out the individual effect of each factor.

“Slope” = P

Dependent Variable: QExplanatory Variable(s): Estimate SE t-Statistic Prob

P 549.4847 130.2611 -4.218333 0.0004I 24.24854 11.27214 2.151192 0.0439ChickP 287.3737 193.3540 1.486257 0.1528Const 159032.4 61472.68 2.587041 0.0176

Number of Observations 24

Beef Consumption Data: Monthly time series data of beef consumption, beef prices, income, and chicken prices from 1985 and 1986. Qt Quantity of beef demanded in month t (millions of pounds)

Pt Price of beef in month t (cents per pound)It Disposable income in month t (billions of chained 1985 dollars)ChickPt Price of chicken in month t (cents per pound)

Year Month Q P I ChickP Year Month Q P I ChickP 1985 1 211,865 168.2 5,118 75.0 1986 1 222,379 159.7 5,219 75.0 1985 2 216,183 168.2 5,073 75.9 1986 2 219,337 152.9 5,247 73.7 1985 3 216,481 161.8 5,026 74.8 1986 3 224,257 149.9 5,301 74.2 1985 4 219,891 157.2 5,131 73.7 1986 4 235,454 144.6 5,313 75.1 1985 5 221,934 155.9 5,250 73.6 1986 5 230,326 151.9 5,319 74.6 1985 6 217,428 157.2 5,137 74.6 1986 6 228,821 150.1 5,315 77.1 1985 7 219,486 152.9 5,138 71.4 1986 7 229,108 156.5 5,339 85.6 1985 8 218,972 151.9 5,133 69.3 1986 8 225,543 164.3 5,343 93.3 1985 9 218,742 147.4 5,152 70.9 1986 9 220,516 160.6 5,348 81.9 1985 10 212,243 160.4 5,180 72.3 1986 10 221,239 163.2 5,344 92.5 1985 11 209,344 168.4 5,189 76.2 1986 11 223,737 162.9 5,351 82.7 1985 12 215,232 172.1 5,213 75.7 1986 12 226,660 160.4 5,345 81.8

Step 1: Collect data, run the regression, and interpret the estimates

Theory: P < 0Model: Qt = Const + PPt + IIt + CPChickPt + et

EViews

Estimated Equation: EstQ = 159,030 549.5P + 24.25I + 287.4ChickP

Dependent Variable: Q

Explanatory Variables: P, I, and ChickP

Question: How can we interpret the coefficient estimates?

http://www3.amherst.edu/~fwesthoff/MITLinks/MIT-BeefDemand-1985-1986.wf1

EstQ = bConst + bPP + bII + bCPChickP

From ToPrice: P P + P while all other explanatory variables remain constant

EstQ + Q = bConst + bP(P + P) + bII + bCPChickP

EstQ + Q = bConst + bPP + bPP + bII + bCPChickP

EstQ = bConst + bPP + bII + bCPChickP Q = bP P

Multiply through by bP

Original equation

Subtract

EstQ = 159,032 549.5P + 24.25I + 287.4ChickP

bP estimates by how much the quantity changes when the price of beef changes while all other explanatory variables remain constant.

bI estimates by how much the quantity changes when income changes while all other explanatory variables remain constant.

bCP estimates by how much the quantity changes when the price of chicken changes while all other explanatory variables remain constant.

NB: The coefficients separate out the individual effect of each explanatory variable.

Quantity: EstQ EstQ + QAfter P changes:

Q = bPP

while all other explanatory variables remain constant

Q = bII while all other explanatory variables remain constant

Q = bCPChickP while all other explanatory variables remain constant

Putting everything together: Q = bPP + bII + bCPChickP

For the moment replace the numerical value of each estimate with its symbol.


P 549.4847 130.2611 -4.218333 0.0004I 24.24854 11.27214 2.151192 0.0439ChickP 287.3737 193.3540 1.486257 0.1528Const 159032.4 61472.68 2.587041 0.0176



Theory: P < 0Model: Q = Const + PP + II + CPChickP + et

Interpretation: If the price of chicken increases by 1 cent, while the price of beef and income remain unchanged, the quantity demanded increases by 287.4 million pounds

Interpretation: If a household’s income rises by $1 billion, while the price of beef and the price of chicken remain unchanged, the quantity demand increases 24.25 million pounds.

Interpretation: If the price of beef increases by 1 cent while income and the price of chicken remain unchanged, the quantity demanded decreases by 549.5 million pounds

Critical Result: The price coefficient estimate equals 549.5.

Q = bPP

Q = 549.5Pwhile all other explanatory variables remain constant

Q = bII

Q = 24.25Iwhile all other explanatory variables remain constant

Q = bCPChickP

Q = 287.4ChickP

while all other explanatory variables remain constant

This evidence supports the downward sloping demand curve theory.

Multiple regression analysis separates out the individual effect of each explanatory variable.

Q = 549.5P + 24.25I + 287.4ChickP

The estimate is negative.

Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses:

H0: P = 0 Cynic is correct: The price of beef (the good’s own price) does not affect the quantity of beef demandedH1: P < 0 Cynic is incorrect: An increase in the price of beef (the good’s own price) decreases the quantity of beef demanded

Step 3: Formulate the question to assess the cynic’s view.

Prob[Results IF H0 True] small

Unlikely that H0 is true

Prob[Results IF H0 True] large

Likely that H0 is true

Do not reject H0

Cynic’s view: Despite the results, the price has no impact on the quantity demanded

Reject H0

Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H0 True]

Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the cynic were correct?

Specific Question: The regression’s own price coefficient estimate was 549.5. What is the probability that the coefficient estimate, bP, in one regression would be 549.5 or less, if H0 were true (if the actual coefficient, P, equaled 0)?

H0 reflects the cynic’s view; H0 challenges the evidence.

H1 reflects the evidence.


P 549.4847 130.2611 -4.218333 0.0004I 24.24854 11.27214 2.151192 0.0439ChickP 287.3737 193.3540 1.486257 0.1528Const 159032.4 61472.68 2.587041 0.0176


Step 4: Use the estimation procedure’s general properties to calculate Prob[Results IF H0 True].

H0: P = 0 Cynic is correct: Price has no impact on the quantity demandedH1: P < 0 Cynic is incorrect: As own price increases, the quantity demanded decreases

bP

t-distribution

OLS estimation procedure unbiased

Mean[bP] = P SE[bP]

If H0 were true

Number of observations

Number of parameters

StandardError

DF = 24 4 = 20= 0 = 130.3

Prob[Results IF H0 True] = .0002

Tails Probability: Probability that the coefficient estimate, bP, resulting from one regression would will lie at least 549.5 from 0, if the actual coefficient, P, equaled 0.

Estimate was 549.5: What is the probability that the coefficient estimate in one regression would be 549.5 or less, if H0 were true (if the actual coefficient, P, equaled 0)?

Mean = 0SE = 130.3DF = 20

-549.5 0

Tails Probability = .0004

.0004/2

Step 5: Decide on the standard of proof, a significance levelThe significance level is the dividing line between the probability being small and the probability being large.





Do not reject H0

Reject H0

Prob[Results IF H0 True]Less Than Significance Level

Prob[Results IF H0 True]Greater Than Significance Level

H0: P = 0 Cynic is correct: Price has no impact on the quantity demandedH1: P < 0 Cynic is incorrect: As own price increases, the quantity demanded decreases

Question: At the “traditional” significance levels of 1, 5, or 10 percent (.01, .05, or .10), do we reject the null hypothesis?

Prob[Results IF H0 True] = .0002

Answer: Yes.

Question: Do these results lend support to the downward sloping demand curve theory?

Answer: Yes.

Another Microeconomic Theory: No Money Illusion Theory

This theory is well grounded. It is based on the theory of utility maximization:

max Utility = U(X, Y)s.t. PXX + PYY = I

Budget constraint:PXX + PYY = I

To maximize utility, we find the highest indifference curve that still touches the budget constraint.

Microeconomic theory teaches that there is no money illusion:

No Money Illusion Theory: If all prices and income change by the same proportion, the quantity of a good demanded will not change.

X-intercept: Y = 0

Y-intercept: X= 0

If all prices and income increase by 1 percent, the quantity of a good demanded will not be affected.

When all prices and income are doubled, the X-intercept, the Y-intercept, and the slope are unaffected.

Now, suppose that all prices and income double:

PXX + PYY = I 2PXX + 2PYY = 2I

If all prices and income double, the quantity of a good demanded will not be affected.

PX 2PX

PY 2PY

I 2I

There is no money illusion.

The budget line is unaffected.

The picture does not change.

The no money illusion theory is based on sound logic. But remember, we must test our theories. Many theories that appear to be sound turn out to be incorrect.

If all prices and income triple, the quantity of a good demanded will not be affected.

Linear Demand Model and the No Money Illusion TheoryThe linear demand model: Q = Const + PP + II + CPChickP “Slope” of demand curve = P

P0

Q0

P1

Q1

2P0

2P1

P2

Q2

2P2

P

Q

Case 1: If initially the price of beef were P0, the quantity demanded would be Q0.

Double Income and Price of Chicken

Double the price of beef from P0 to 2P0 If there were no money illusion, the quantity demanded would remain at Q0.





Now, we can draw the new demand curve when income and the price of chicken doubles.

The slope of the demand curve must change to be consistent with the no money illusion theory.

The linear demand model assumes that the value of P is a constant.

The linear demand model is intrinsically inconsistent with the no money illusion theory.

The value of P is a constant.

“Slope” = P

Constant Elasticity Demand Model:

Claim: When the exponents, the elasticities, sum to 0, there is no money illusion:

P + I + CP = 0 or CP = P I

This model of demand is consistent with the theory when the exponents sum to 0.

Theory: There is no money illusion: When all prices and income increase by the same proportion, the quantity of goods demanded is unaffected.

Testing the No Money Illusion TheoryStep 0: Construct a model reflecting the theory to be tested.

What happens when prices and income are double?

The values of the fractions are unchanged; consequently, the quantity demanded is unchanged – there is no money illusion.

P = Own Price Elasticity of Demand

I = Income Elasticity of Demand

CP = Cross Price Elasticity of Demand

The exponents equal the elasticities:

We can use this model to test the no money illusion theory.

Dependent Variable: LogQExplanatory Variable(s): Estimate SE t-Statistic Prob

LogP 0.411812 0.093532 -4.402905 0.0003LogI 0.508061 0.266583 1.905829 0.0711LogChickP 0.124724 0.071415 1.746465 0.0961Const 9.499258 2.348619 4.044615 0.0006


Step 1: Collect data, run the regression, and interpret the estimatesModel: Theory – No Money Illusion: P + I + CP = 0

Taking logs: log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et

Interpretation of the Estimates:

bI = Estimate for the Income Elasticity of Demand = .51

bP = Estimate for the (Own) Price Elasticity of Demand = .41 A one percent increase in the price of beef (the good’s own price) decreases the quantity of beef demanded by .41 percent while ...

A one percent increase in income increases the quantity of beef demand by .51 percent while ...

bCP = Estimate for the Cross Price Elasticity of Demand = .12 A one percent increase in the price of chicken increases the quantity of beef demanded by .12 percent while ...

bP + bI + bCP = .41 + .51 + .12 Critical Result: The sum of the elasticity estimates equals .22 , not 0. The sum is .22 from 0.

Estimate: A one percent increase in all prices and income results in a .22

percent increase in quantity demanded.

= .22

EViews

This evidence suggests that money illusion is present and that the no money illusion theory is incorrect.

http://www3.amherst.edu/~fwesthoff/MITLinks/PP-BeefDemand-1985-1986.wf1

Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses:

H0: P + I + CP = 0 Cynic’s view is correct: No money illusion

H1: P + I + CP 0 Cynic’s view is incorrect: Money illusion present

Cynic’s view: Despite the results, no money illusion is present.

Lab 10.1

Can we dismiss the cynic’s view as nonsense?

As a consequence of random influences, could we ever expect

the estimate for an individual coefficient to equal its actual value?

the sum of coefficient estimates to equal the sum of their actual values?

In this case, even if the actual elasticities summed to 0, could we ever expect the sum of their estimates to equal 0?

Could the cynic possibly be correct?

No

No

No

Yes

The cynic always challenges the evidence.

The evidence suggests that money illusion exists..

Is this a one or two tail hypothesis test?Theory postulates that the elasticity sum equals a specific value.

Why is a two tail hypothesis appropriate?A two tail hypothesis test.

H0 reflects the cynic’s view, challenging the results. H1 reflects the results.

http://www3.amherst.edu/~fwesthoff/MITLinks/MIT-Lab-10-01.html

Question: How can we calculate Prob[Results IF H0 True]?

Answer: There are three ways: Clever algebraic manipulation

Wald (F-distribution) test

Let statistical software do the work

Step 4: Use the properties of the estimation procedure to calculate Prob[Results IF H0 True].






Do not reject H0

Reject H0



Specific Question: In the regression, the sum of coefficient estimates was .22 from 0. What is the probability that the sum in one regression would be at least .22 from 0, if H0 were true (if the sum of the actual coefficients equaled 0)?

H0: P + I + CP = 0 Cynic’s view is correct: No money illusion

H1: P + I + CP 0 Cynic’s view is incorrect: Money illusion present

Dependent Variable: LogQExplanatory Variable(s): Estimate SE t-Statistic Prob

LogPLessLogChickP 0.411812 0.093532 -4.402905 0.0003LogILessLogChickP 0.508061 0.266583 1.905829 0.0711LogChickP 0.220974 0.275863 0.801027 0.4325Const 9.499258 2.348619 4.044615 0.0006


Testing the Hypothesis – Method 1: Clever Algebraic Manipulation

The Prob column of the regression printout reports the tails probability based on the premise that the actual value of the coefficient equals 0.

Exploit this by cleverly defining a new coefficient so that the null hypothesis can be expressed as the new coefficient equaling 0:

Clever = P + I + CP

Step 0: Reconstruct the model to exploit the “tails probability:”

log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) +et

CP = Clever P I

= log(Const) + Plog(Pt) + Ilog(It) + (Clever P I)log(ChickPt) +et

= log(Const)+ Plog(Pt) + Ilog(It) + Cleverlog(ChickPt) Plog(ChickPt) Ilog(ChickPt)+et

= log(Const) + Plog(Pt) Plog(ChickPt) + Ilog(It) Ilog(ChickPt) + Cleverlog(ChickPt)+et

= log(Const) + P[log(Pt) log(ChickPt)] + I[log(It) log(ChickPt)] + Cleverlog(ChickPt) et

Generate new variables:

NB: Clever = 0 if and only if P + I + CP = 0

LogPLessLogChickP = log(P) log(ChickP) LogILessLogChickP = log(I) log(ChickP)


Critical Result: The estimate is not 0 (more specifically, it is .22 from 0).

No Money Illusion Theory:

P + I + CP = 0

No Money Illusion Theory: Clever = 0

EViews

Is this estimate consistent with the previous regression?

Yes.Previous Regression: bP + bI + bCP = .22 This Regression: bClever = .22

The evidence, the estimate of the elasticity sum (bClever), suggests that the no money illusion theory is incorrect.

http://www3.amherst.edu/~fwesthoff/MITLinks/PP-BeefDemand-1985-1986.wf1

Step 2: Play the cynic, challenge the results, and construct the null and alternative hypotheses:

H0: P + I + CP = 0 or Clever = 0 Cynic is correct: No money illusionH1: P + I + CP 0 or Clever 0 Cynic is incorrect: Money illusion present






Cynic’s view: Sure, bClever, the estimate for the sum of the actual elasticities, does not equal 0 suggesting that money illusion exists, but this is just “the luck of the draw.” In fact, money illusion is not present; the sum of the actual elasticities equals 0.

The null hypothesis, H0, reflects the cynic’s view.



Specific Question: The regression’s coefficient estimate was .22 from 0. What is the probability that the coefficient estimate, bClever, in one regression would be at least .22 from 0, if H0 were true (if the actual coefficient, Clever, equaled 0)?

Do not reject H0

Reject H0

The cynic always challenges the evidence.

The alternative hypothesis, H1, reflects the evidence.

Is this a one or two tail hypothesis test? A two tail hypothesis test.

Dependent Variable: LogQExplanatory Variable(s):

Estimate SE t-Statistic Prob

LogPLessLogChickP 0.411812 0.093532 -4.402905 0.0003LogILessLogChickP 0.508061 0.266583 1.905829 0.0711LogChickP 0.220974 0.275863 0.801027 0.4325Const 9.499258 2.348619 4.044615 0.0006


bClever

t-distribution

OLS estimation procedure unbiased

Mean[bClever] = Clever SE[bClever]

If H0 were true

Number of observations

Number of parameters

StandardError

DF = 24 4 = 20= 0 = .2759

Prob[Results IF H0 True]

Prob Column (Tails Probability): Probability that the coefficient estimate, bClever, resulting from one regression would will be at least .22 from 0, if the actual coefficient, Clever, equaled 0.

Estimate was .22: What is the probability that the coefficient estimate in one regression would be at least .22 from 0, if H0 were true (if the actual coefficient, Clever, equaled 0)?

.4325/2

Mean = 0SE = .2759DF = 20

.220

Tails Probability = .4325

.4325/2

=.4325.22.22

Step 4: Use the properties of the estimation procedure to calculate Prob[Results IF H0 True].

H0: P + I + CP = 0 or Clever = 0 Cynic is correct: No money illusionH1: P + I + CP = 0 or Clever 0 Cynic is incorrect: Money illusion present

Step 5: Decide on the standard of proof, a significance levelThe significance level is the dividing line between the probability being small and the probability being large.





Do not reject H0

Reject H0

Prob[Results IF H0 True]Less Than Significance Level

Prob[Results IF H0 True]Greater Than Significance Level

At the “traditional” significance levels of 1 , 5, o1 10 percent (.01, .05, or .10), do we reject the null hypothesis?

Prob[Results IF H0 True] =.4325

H0: P + I + CP = 0 or Clever = 0 Cynic is correct: No money illusion

H1: P + I + CP = 0 or Clever 0 Cynic is incorrect: Money illusion present

No. We do not reject the null hypothesis at the traditional significance levels.

Do these results lend support to the no money illusion theory? Yes.

Documents

Lecture 10 Preview: Multiple Regression Analysis – Introduction Linear Demand Model and the No Money Illusion Theory A Two-Tailed Test: No Money Illusion