Monday, October 15 Handout: Multiple Regression Analysis Introduction

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1 Amherst College Department of Economics Economics 360 Fall 2012 Monday, October 15 Handout: Multiple Regression Analysis Introduction Review Simple and Multiple Regression Analysis o Distinction between Simple and Multiple Regression Analysis o Goal of Multiple Regression Analysis A One-Tailed Test: Downward Sloping Demand Theory o Linear Demand Model A Two-Tailed Test: No Money Illusion Theory o Linear Demand Model and the No Money Illusion Theory o Constant Elasticity Demand Model and the No Money Illusion Theory o Calculating True]: Clever Algebraic Manipulation Cleverly Define a New Coefficient That Equals 0 When H 0 Is True Reformulate the Model to Incorporate the New Coefficient Estimate the Parameters of the New Model Use the Tails Probability to Calculate True] Simple and Multiple Regression Analysis Simple Regression:. Multiple Regression:. 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. P 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 price, other factors, also affect demand: household income, the price of other goods, etc. Demand Curve: The demand curve for a good reveals how the D quantity demanded changes when the good s price changes while all the other factors relevant to demand remain constant. Multiple regression analysis allows us to consider all the factors that theory suggests are important and separate out the individual effect of each factor. Illustration: Demand for Beef Step 0: Construct a model reflecting the theory to be tested "Slope" = β P All other factors relevant to demand remain constant Q t = β Const + β P P t I t ChickP t + e t where Q t = Quantity of beef demanded P t = Price of beef (the good s own price) I t = Household income ChickP t = Price of chicken Downward Sloping Demand Theory: β P < 0. Q

2 2 Step 1: Collect data, run the regression, and interpret the results: Beef Consumption Data: Monthly time series data of beef consumption, beef prices, income, and chicken prices from 1985 and Q t P t I t ChickP t Quantity of beef demanded in month t (millions of pounds) Price of beef in month t (cents per pound) Disposable income in month t (billions of chained 1985 dollars) Price of chicken in month t (cents per pound) Year Month Q P I ChickP Year Month Q P I ChickP , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Dependent Variable: Q Explanatory Variables: P, I, and ChickP Dependent Variable: Q Method: Least Squares Sample: 1985M M12 Included observations: 24 Coefficient Std. Error t-statistic Prob. P I CHICKP C Estimated Equation: EstQ = 159, P I ChickP To interpret these estimates, let us for the moment replace the numerical value of each estimate with the italicized lower case Roman letter b, b, that we use to denote the estimate. That is, replace 159,032 with b Const, with b P, with b I, and with b CP : EstQ = b Const + b P P + b I I + b CP ChickP Claim: When all other explanatory variables remain constant ΔQ = b P ΔP or b P = ΔQ where ΔP = Change in price ΔP ΔQ = Estimated change in quantity of beef demanded A little algebra explains why. We begin with the equation estimating our model: EstQ = b Const + b P P + b I I + b CP ChickP

3 3 Now, increase the price by ΔP; the change in the price causes the estimated quantity to change by ΔQ: From To Price: P P + ΔP All other explanatory variables remain constant; Quantity: EstQ EstQ + ΔQ that is, I and ChickP remain constant EstQ = b Const + b P P + b I I + b CP ChickP Substituting EstQ + ΔQ = b Const + b P (P + ΔP) + b I I + b CP ChickP EstQ + ΔQ = b Const + b P P + b P ΔP + b I I + b CP ChickP Multiplying through by b P EstQ = b Const + b P P + b I I + b CP ChickP Original equation ΔQ = b P ΔP ΔQ = b P ΔP Subtracting the equations Simplifying Dividing by ΔP ΔQ ΔP = b P while all other explanatory variables constant ΔQ = b P ΔP or b P = ΔQ ΔP while all other explanatory variables remain constant b P estimates the change in the quantity of beef demanded when the price of beef (the good s own price) changes while all other explanatory variables (income and the price of chicken) remain constant. Applying the same logic to the income and chicken price coefficients: ΔQ = b I ΔI or b I = ΔQ while all other explanatory variables remain constant ΔI b I estimates the change in the quantity of beef demanded when income changes while all other explanatory variables (the price of beef and the price of chicken) remain constant. ΔQ ΔQ = b CP ΔChickP or b CP = while all other explanatory variables remain constant ΔChickP b CP estimates the change in the quantity of beef demanded when the price of chicken changes while all other explanatory variables (the price of beef and income) remain constant. What happens when all explanatory variables change simultaneously? Total estimated change in quantity of beef demanded resulting from a change in Price of Beef (the good s own price) Income Price of Chicken ΔQ = + + Each term estimates the change in the dependent variable, quantity of beef demanded, resulting from a change in each individual explanatory variable.

4 4 Interpreting the Coefficients in Our Regression Estimated effect of a change in the price of beef (the good s own price): ΔQ = b p ΔP = ΔP while all other explanatory variables remain constant Interpretation: The estimate of the coefficient for the price of beef (the good s own price) equals ; that is, we estimate that if the price of beef increases by 1 cent while income and the price of chicken remain unchanged, the quantity of beef demanded by about million pounds. Estimated effect of a change in income: ΔQ = b I ΔI = ΔI while all other explanatory variables remain constant Interpretation: The estimate of the income coefficient equals ; that is, we estimate that if income increases by 1 billion dollars while the price of beef and the price of chicken remain unchanged, the quantity of beef demanded by about million pounds. Estimated effect of a change in the price of chicken: ΔQ = b CP ΔChickP = ΔChickP while all other explanatory variables remain constant Interpretation: The estimate of the chicken price coefficient equals ; that is, we estimate that if the price of chicken increases by 1 cent while the price of beef and income remain unchanged, the quantity of beef demanded by about million pounds. Putting the three estimates together: ΔQ = b p ΔP + b I ΔI + b CP ΔChickP or ΔQ = ΔP + ΔI + ΔChickP Critical Result: The estimate of the coefficient for the price of beef equals. The sign of the coefficient estimate suggests that an increase in the price of beef the quantity of beef demanded. This evidence the downward sloping demand curve theory. Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses Cynic s view: Sure, the estimate of the coefficient for the price of beef (the good s own price) suggests that the demand curve is downward sloping, but this is just the luck of the draw. In fact, the actual coefficient equals 0; the price of beef actually has no effect on the quantity of beef demanded. H 0 : β P = 0 H 1 : β P < 0 Cynic s view is correct: The price of beef (the good s own price) has no impact on quantity demanded. Cynic s view is incorrect: An increase in the price decreases quantity demanded. The null hypothesis, H 0, challenges the evidence, supporting the cynic s view. The alternative hypothesis, H 1, reflects the evidence.

5 5 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 obtained (or even stronger), if the cynic is correct and price actually has no impact? Specific Question: What is the probability that the coefficient estimate, b P, in one regression would be or less, if H 0 were true (if the actual price coefficient, β P, equals 0)? Answer: Prob[Results IF Cynic Correct] or equivalently True] Probability Distribution Student t-distribution Mean = SE = DF = b P Step 4: Use the general properties of the estimation procedure, the properties of the probability distribution of the estimates, to calculate True]. OLS estimation Assume H 0 Standard Number of Number of procedure unbiased is true error Observations Parameters Mean[b P ] = = SE[b P ] = DF = = True] = =. Using EViews Prob. Column Dependent Variable: Q Method: Least Squares Sample: 1985M M12 Included observations: 24 Coefficient Std. Error t-statistic Prob. P I CHICKP C 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. True] True] Less Than Significance Level Greater Than Significance Level True] True] that H 0 is true that H 0 is true H 0 H 0 At the traditional significance levels of 1, 5, or 10 percent (.01,.05, or.10), would you reject the null hypothesis? Do these results lend support to the downward sloping demand curve theory?

6 6 No Money Illusion Theory Microeconomic theory teaches that as a consequence of utility maximization, 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. For example, suppose that all prices and income double: max Utility = U(X, Y) P X 2P X max Utility = U(X, Y) s.t. P X X + P Y Y = I P Y 2P Y s.t. 2P X X + 2P Y Y = 2I How is the budget constraint affected? I 2I P X X + P Y Y = I 2P X X + 2P Y Y = 2I X-intercept: Y = 0 X-intercept: Y = 0 = Y-intercept: X = 0 Y-intercept: X = 0 = Since the intercepts have not changed, the budget constraint line has not changed; consequently, the solution to the household s constrained utility maximizing problem will not change. Y I/P Y Indifference curve Solution A proportional increase in prices and income does not affect the quantity demanded; no money illusion exists. I/P X Budget constraint Slope = P X /P Y X That is, If all prices and income double, the quantity of a good demanded will. If all prices and income triple, the quantity of a good demanded will. If all prices and income increase by 1 percent, the quantity of a good demanded will. 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.

7 7 Linear Demand Model and Money Illusion Theory The linear demand model: Q t = β Const + β P P t I t ChickP t + e t Note that the slope of the demand curve is β P. 1 β P is a constant value. No Money Illusion Theory: When all prices and income increase by the same proportion, the quantity demanded remains constant. Now suppose that income and the price of chicken double. Consider three cases: 2P 0 P Case 1: If initially the price Case 2: If initially the price Case 3: If initially the price of beef equals P 0 of beef equals P 1 of beef equals P 2 Double Income and the Price of Chicken P Double Income and the Price of Chicken P Double Income and the Price of Chicken P 0 "Slope" = β P 2P 1 "Slope" = β P "Slope" = β P P 1 Q 0 Q Q 1 2P 2 P 2 Q Q 2 Q If there is no money illusion, when the price of beef (the good s own price) doubles, the quantity demanded must remain the same in each case. P Double Income and the Now, draw a demand curve through the three points. Price of Chicken Question: Has the slope of the new demand curve changed? Reconsider the equation for the linear demand curve: Q = β Const + β P P I ChickP β P reflects the slope of the demand curve. β P is a constant value. Question: Is the linear demand model is consistent with the no money illusion theory? "Slope" = β P Question: Can we use a model that is intrinsically inconsistent with the theory to test it? Q 1 Actually, this is not quite correct. Since the dependent variable price is plotted on the y-axis rather than the x-axis, the slope is 1/β P. This does not affect the validity of our argument, however. The important point is that the linear model implicitly assumes that the slope of the demand curve is constant; that is, the slope of the demand curve is not affected by changes in other factors relevant to demand.

8 8 Constant Elasticity Demand Model and the No Money Illusion Theory Q = β Const P β P I β I ChickP β CP β P = (Own) Price Elasticity of Demand = dq P dp Q = Percent change in the quantity of beef demanded resulting from a one percent change in the price of beef (the good s own price). β I = Income Elasticity of Demand = dq I di Q = Percent change in the quantity of beef demanded resulting from a one percent change in income. dq ChickP β CP = Cross Price Elasticity of Demand = dchickp Q = Percent change in the quantity of beef demanded resulting from a one percent change in the price of chicken. Question: Can the constant elasticity demand model be consistent with our money illusion theory? Claim: When the exponents sum to 0, that is, when the sum of the actual elasticities equals 0, there is no money illusion: β P = 0 β CP = β P β I A little algebra: Q = β Const P β P I β I ChickP β CP Substituting for β CP = β Const P β P I β I ChickP β P β I Splitting the exponent = β Const P β P I β I ChickP β P ChickP β I Moving negative exponent factors to the denominator P = β Const ( β P )( I βi ) ChickP β P ChickP β I Simplifying = β Const ( P ChickP )β P I ( ChickP )β I When all prices and income double, what happens to the quantity demanded? This model of demand is consistent with the money illusion theory whenever the elasticities, the exponents, sum to. Apply Hypothesis Testing Step 0: Construct a model reflecting the theory to be tested: Q = β Const P β P I β I ChickP β CP No Money Illusion Theory: The sum of the actual elasticities equals : β P =.

9 9 Step 1: Collect data, run the regression, and interpret the results: Constant elasticity demand model: Q = β Const P β P I β I ChickP β CP Use logarithms to convert the original equation into a linear equation: log(q t ) = log(β Const log(p t ) log(i t ) Generate new variables: LogQ, LogP, LogI, and LogChickP. Dependent variable: LogQ Explanatory variables: LogP, LogI, and LogChickP Dependent Variable: LOGQ Method: Least Squares Sample: 1985M M12 Included observations: 24 Coefficient Std. Error t-statistic Prob. LOGP LOGI LOGCHICKP C Estimated Equation: LogQ = Interpreting the Estimates b P = Estimate for the (Own) Price Elasticity of Demand = We estimate that a one percent increase in the price of beef (the good s own price) the quantity of beef demanded by percent when income and the price of chicken remain constant. b I = Estimate for the Income Elasticity of Demand = We estimate that a one percent increase in income the quantity of beef demanded by percent when the price of beef and the price of chicken remain constant. b CP = Estimate for the Cross Price Elasticity of Demand = We estimate that a one percent increase in the price of chicken the quantity of beef demanded by percent when the price of beef and income remain constant. Question: What does the sum of the elasticity estimates equal? b P + b I + b CP = + + = The estimates suggest that if all prices and income increase by 1 percent, the quantity of beef demanded will by percent. Critical Result: The sum of the elasticity estimates equals. The sum equal 0; the sum is from 0. This evidence suggests that money illusion present; the evidence suggests that the no money illusion theory is.

10 10 Step 2: Play the cynic and challenge the evidence; construct the null and alternative hypotheses. Cynic s view: Sure, the sum of the elasticity estimates 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. Let us now construct the null and alternative hypotheses: H 0 : β P = 0 Cynic s view is correct: Money illusion not present H 1 : β P 0 Cynic s view is incorrect: Money illusion present 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? 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? 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: What is the probability that the sum of the coefficient estimates in one regression would be at least.22 from 0, if H 0 were true (if the sum of the actual elasticities equaled 0)? Answer: Prob[Results IF Cynic Correct] or equivalently True] True] small that H 0 is true True] large that H 0 is true H 0 H 0 Step 4: Use the general properties of the estimation procedure, the properties of probability distribution of the estimates, to calculate True]. How can we calculate this probability? We shall discuss three approaches: Clever algebraic manipulation Wald (F-distribution) test Letting statistical software do the work

11 11 Calculating True] Method 1: Clever Algebraic Manipulation Step 0: Reconstruct the model to exploit the tails probability. The Prob column of the EViews printout reports the tails probability based on the premise that the actual value of the coefficient 0. We can exploit this by cleverly defining a new coefficient so that the null hypothesis is expressed as the new coefficient equaling 0: β Clever = β P β Clever = 0 if and only if Solving, β CP = log(q t ) = log(β Const log(p t ) log(i t ) = log(β Const log(p t ) log(i t ) + (β Clever β P β I ) = log(β Const log(p t ) log(i t ) + β Clever ) β P ) β I = log(β Const log(p t ) β P ) log(i t ) β I )+ β Clever = log(β Const [log(p t ) )] [log(i t ) )] + β Clever Step 1: Run the regression, and interpret the results: Generate two new variables: LogPLessLogChickP = log(p) log(chickp) LogILessLogChickP = log(i) log(chickp) log(q t ) = log(β Const [log(p t ) )] [log(i t ) )] + β Clever LogQ t = c + β P LogPLessLogChickP t LogILessLogChickP t + β Clever LogChickP t + e t Dependent variable: LogQ Explanatory variables: LogPLessLogChickP, LogILessLogChickP, and LogChickP Dependent Variable: LOGQ Method: Least Squares Sample: 1985M M12 Included observations: 24 Coefficient Std. Error t-statistic Prob. LOGPLESSLOGCHICKP LOGILESSLOGCHICKP LOGCHICKP C Question: What was the sum of the elasticity estimates in the previous regression, when the explanatory variables were the logarithms of the price of beef (the good s own price), income, and the price of chicken?. Are the two regressions consistent?. Critical Result: b Clever, the estimate for the sum of the actual elasticities, equals. The estimate equal 0; the estimate is from 0. This evidence suggests that money illusion is ; the evidence suggests that the no money illusion theory is. Step 2: Play the cynic and challenge the results; reconstruct the null and alternative hypotheses. Cynic s View: Sure, b Clever, the sum of the elasticity estimates 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. H 0 : β P = 0 β Clever = 0 Cynic is correct: Money illusion not present H 1 : β P 0 β Clever 0 Cynic is incorrect: Money illusion present

12 12 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 obtained (or even stronger), if the cynic is correct and no money illusion was present? Specific Question: What is the probability that the coefficient estimate in one regression, b Clever, would be at least.22 from 0, if H 0 were true (if the actual coefficient, β Clever, equals 0)? Student t-distribution Mean = SE = DF = b Clever Answer: Prob[Results IF Cynic Correct] or equivalently True] True] small True] large that H 0 is true that H 0 is true H 0 H 0 Step 4: Use the general properties of the estimation procedure, the properties of probability distribution of the estimates, to calculate True]. OLS estimation Assume H 0 Standard Number of Number of procedure unbiased is true error Observations Parameters Mean[b Clever ] = = SE[b Clever ] = DF = = True] = Using EViews Prob. Column Dependent Variable: LOGQ Method: Least Squares Sample: 1985M M12 Included observations: 24 Coefficient Std. Error t-statistic Prob. LOGPLESSLOGCHICKP LOGILESSLOGCHICKP LOGCHICKP C Step 5: Decide on the standard of proof, a significance level At the traditional significance levels of 1, 5, or 10 percent (.01,.05, or.10), would you reject the null hypothesis? Do these results lend support to the no money illusion theory?