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Lecture 10 Preview: Multiple Regression Analysis – Introduction

Learn the distinction between simple and multiple regression analysis, the goal of multiple regression analysis, and how to interpret coefficient estimates. Explore the application of multiple regression analysis in the context of demand curves.

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Lecture 10 Preview: Multiple Regression Analysis – Introduction

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  1. Lecture 10 Preview: Multiple Regression Analysis – Introduction Simple and Multiple Regression Analysis Distinction between Simple and Multiple Regression Analysis Goal of Multiple Regression Analysis A One-Tailed Test: Downward Sloping Demand Curve Theory Linear Demand Model A Two-Tailed Test: No Money Illusion Theory Linear Demand Model and the No Money Illusion Theory Constant Elasticity Demand Model and the No Money Illusion Theory Calculating Prob[Results IF H0 True]: Clever Algebraic Manipulation 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]

  2. Simple and Multiple Regression Analysis Simple 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. 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. Downward Sloping Demand Curve Theory  Revisited Theory: 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.

  3. 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 All other factors relevant to demand 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. But economic theory teaches us that these other factors are important too. Consequently, we should not ignore them. “Slope” = P D We need a way to include not only the effect of the price of the good but also the other factors that influence demand. Q Multiple regression analysis can consider all the factors that our theory suggests are important. Multiple regression analysis allows us to separate out the individual effect of each factor. Example: Demand for Beef. Step 0: Construct a model reflecting the theory to be tested Qt = Const + PPt + IIt + CPChickPt+ et Qt = Quantity of beef demanded It = Household income Pt = Price of beef (the good’s own price) ChickPt = Price of chicken 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. Theory: P < 0.

  4. Model: Qt = Const + PPt + IIt + CPChickPt+ et Theory: P < 0 EViews Step 1: Collect data, run the regression, and interpret the estimates Beef Consumption Data: Monthly time series data of beef consumption, beef prices, income, and chicken prices from 1985 and 1986. QtQuantity of beef demanded in month t (millions of pounds) PtPrice of beef in month t (cents per pound) ItDisposable 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 Dependent Variable: Q Explanatory Variables: P, I, and ChickP Estimated Equation:EstQ = 159,030  549.5P + 24.25I + 287.4ChickP Question: How can we interpret the coefficient estimates?

  5. EstQ = bConst + bPP + bII + bCPChickP EstQ = 159,032  549.5P + 24.25I + 287.4ChickP For the moment replace the numerical value of each estimate with its symbol. From ToPrice: P P + P while all other explanatory variables remain constant Quantity: EstQ EstQ + Q After P changes: EstQ + Q = bConst + bP(P + P) + bII + bCPChickP Multiply through by bP EstQ + Q = bConst + bPP + bPP + bII + bCPChickP Original equation EstQ= bConst + bPP + bII + bCPChickP Subtract Q= bP P Q = bPP bP estimates by how much the quantity changes when the price of beef changes while all other explanatory variables remain constant. while all other explanatory variables remain constant Q = bII bI estimates by how much the quantity changes when income changes while all other explanatory variables remain constant. while all other explanatory variables remain constant Q = bCPChickP 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. Putting everything together: Q = bPP + bII + bCPChickP

  6. Model: Q = Const + PP + II + CPChickP + et Theory: P < 0 Step 1: Collect data, run the regression, and interpret the estimates 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 Q = bPP while all other explanatory variables remain constant Q = 549.5P Q = bII while all other explanatory variables remain constant 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. Q = 24.25I while all other explanatory variables remain constant Q = bCPChickP Q = 287.4ChickP 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 Q = 549.5P + 24.25I + 287.4ChickP Multiple regression analysis separates out the individual effect of each explanatory variable. Critical Result: The price coefficient estimate equals 549.5. The estimate is negative. This evidence supports the downward sloping demand curve theory.

  7. Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses: Cynic’s view: Despite the results, the price has no impact on the quantity demanded H0: P = 0 Cynic is correct: The price of beef (the good’s own price) does not affect the quantity of beef demanded H1: P < 0 Cynic is incorrect: An increase in the price of beef (the good’s own price) decreases the quantity of beef demanded H0 reflects the cynic’s view; H0 challenges the evidence. H1 reflects the evidence. Step 3: Formulate the question to assess the cynic’s view. 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)? Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H0 True] Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true Reject H0 Do not reject H0

  8. H0: P = 0 Cynic is correct: Price has no impact on the quantity demanded H1: P < 0 Cynic is incorrect: As own price increases, the quantity demanded decreases Step 4: Use the estimation procedure’s general properties to calculate Prob[Results IF H0 True]. 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)? OLS estimation procedure unbiased If H0 were true StandardError Number of observations Number of parameters Mean[bP] = P = 0 SE[bP] = 130.3 DF = 24  4 = 20 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. t-distribution Mean = 0 SE = 130.3 DF = 20 .0004/2 Tails Probability = .0004 Prob[Results IF H0 True] = .0002 bP -549.5 0

  9. H0: P = 0 Cynic is correct: Price has no impact on the quantity demanded H1: P < 0 Cynic is incorrect: As own price increases, the quantity demanded decreases Prob[Results IF H0 True] = .0002 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]Less Than Significance Level Prob[Results IF H0 True]Greater Than Significance Level Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true  Reject H0  Do not reject H0 Question: At the “traditional” significance levels of 1, 5, or 10 percent (.01, .05, or .10), do we reject the null hypothesis? Answer: Yes. Question: Do these results lend support to the downward sloping demand curve theory? Answer: Yes.

  10. Another Microeconomic Theory: No Money Illusion Theory 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. 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 X-intercept: Y = 0 Y-intercept: X= 0 To maximize utility, we find the highest indifference curve that still touches the budget constraint.

  11. When all prices and income are doubled, the X-intercept, the Y-intercept, and the slope are unaffected. The budget line is unaffected. The picture does not change. Now, suppose that all prices and income double: PX 2PXPY 2PYI 2I PXX + PYY = I 2PXX + 2PYY = 2I If all prices and income double, the quantity of a good demanded will not be affected. If all prices and income triple, the quantity of a good demanded will not be affected. If all prices and income increase by 1 percent, the quantity of a good demanded will not be affected. There is no money illusion. 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.

  12. Linear Demand Model and the No Money Illusion Theory “Slope” of demand curve = P The linear demand model: Q = Const + PP+ II + CPChickP The value of P is a constant. P Double Income and Price of Chicken Case 1: If initially the price of beef were P0, the quantity demanded would be Q0. Double the price of beef from P0 to 2P0 2P0 Now, we can draw the new demand curve when income and the price of chicken doubles. If there were no money illusion, the quantity demanded would remain at Q0. Case 2: If initially the price of beef were P1, the quantity demanded would be Q1. 2P1 Double the price of beef from P1 to 2P1 If there were no money illusion, the quantity demanded would remain at Q1. “Slope” = P P0 Case 3: If initially the price of beef were P2, the quantity demanded would be Q2. P1 Double the price of beef from P2 to 2P2 2P2 If there were no money illusion, the quantity demanded would remain at Q2. P2 The slope of the demand curve must change to be consistent with the no money illusion theory. Q Q0 Q1 Q2 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.

  13. Testing the No Money Illusion Theory Step 0: Construct a model reflecting the theory to be tested. Constant Elasticity Demand Model: P = Own Price Elasticity of Demand The exponents equal the elasticities: I = Income Elasticity of Demand CP = Cross Price Elasticity of Demand Theory: There is no money illusion: When all prices and income increase by the same proportion, the quantity of goods demanded is unaffected. Claim: When the exponents, the elasticities, sum to 0, there is no money illusion: What happens when prices and income are double? P + I + CP = 0 or CP = P I The values of the fractions are unchanged; consequently, the quantity demanded is unchanged – there is no money illusion. This model of demand is consistent with the theory when the exponents sum to 0. We can use this model to test the no money illusion theory.

  14. Model: Theory – No Money Illusion: P + I + CP = 0 Step 1: Collect data, run the regression, and interpret the estimates Taking logs: log(Qt) = log(Const) + Plog(Pt) + Ilog(It)+ CPlog(ChickPt) + et EViews Interpretation of the Estimates: 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 demandedby .41 percent while ... bI = Estimate for the Income Elasticity of Demand = .51 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 ... Critical Result: The sum of the elasticity estimates equals .22 , not 0. The sum of the elasticity estimates is .22 from 0. bP + bI + bCP = .41 + .51 + .12 = .22 Estimate: A one percent increase in all prices and income results in a .22 percent increase in quantity demanded. This evidence suggests that money illusion is present and that the no money illusion theory is incorrect.

  15. Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses: The cynic always challenges the evidence. The evidence suggests that money illusion exists.. Cynic’s view: Despite the results, no money illusion is present. 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 H0 reflects the cynic’s view, challenging the results. H1 reflects the results. Is this a one or two tail hypothesis test? A two tail hypothesis test. Why is a two tail hypothesis appropriate? Theory postulates that the elasticity sum equals a specific value. 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? No the sum of coefficient estimates to equal the sum of their actual values? No Lab 10.1 In this case, even if the actual elasticities summed to 0, could we ever expect the sum of their estimates to equal 0? No Could the cynic possibly be correct? Yes

  16. Step 3: Formulate the question to assess the cynic’s view. 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 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: 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)? Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H0 True] Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true Reject H0 Do not reject H0 Step 4: Use the properties of the estimation procedure to calculate Prob[Results IF H0 True]. 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

  17. Testing the Hypothesis – Method 1: Clever Algebraic Manipulation No Money Illusion Theory:P + I + CP = 0 Step 0: Reconstruct the model to exploit the “tails probability:” 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: NB: Clever = 0 if and only if P + I + CP = 0 Clever = P + I + CP CP = Clever P I No Money Illusion Theory: Clever = 0 log(Qt) = log(Const) + Plog(Pt) + Ilog(It)+ CPlog(ChickPt) +et = 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 Step 1: Collect data, run the regression, and interpret the estimates Generate new variables: LogPLessLogChickP = log(P)  log(ChickP) Critical Result: The estimate is not 0 (more specifically, it is .22 from 0). LogILessLogChickP = log(I)  log(ChickP) EViews The evidence, the estimate of the elasticity sum (bClever), suggests that the no money illusion theory is incorrect. Is this estimate consistent with the previous regression? Previous Regression: bP + bI + bCP = .22 This Regression: bClever = .22 Yes.

  18. Step 2: Play the cynic, challenge the results, and construct the null and alternative hypotheses: The cynic always challenges the evidence. 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. 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 The null hypothesis, H0, reflects the cynic’s view. The alternative hypothesis, H1, reflects the evidence. Is this a one or two tail hypothesis test? A two tail hypothesis test. Step 3: Formulate the question to assess the cynic’s view. 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 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)? Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H0 True] Prob[Results IF H0 True] small Prob[Results IF H0 True] large Likely that H0 is true Unlikely that H0 is true Reject H0  Do not reject H0

  19. 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 Step 4: Use the properties of the estimation procedure to calculate Prob[Results IF H0 True]. 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)? OLS estimation procedure unbiased If H0 were true StandardError Number of observations Number of parameters Mean[bClever] = Clever = 0 SE[bClever] = .2759 DF = 24  4 = 20 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. t-distribution Mean = 0 SE = .2759 DF = 20 .4325/2 .4325/2 Tails Probability = .4325 Prob[Results IF H0 True] =.4325 bClever .22 .22 0 .22

  20. 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 Prob[Results IF H0 True] =.4325 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]Less Than Significance Level Prob[Results IF H0 True]Greater Than Significance Level Prob[Results IF H0 True] small Prob[Results IF H0 True] large Unlikely that H0 is true Likely that H0 is true  Reject H0  Do not reject H0 At the “traditional” significance levels of 1 , 5, o1 10 percent (.01, .05, or .10), do we reject the null hypothesis? 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.

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