Chapter 23: Simultaneous Equation Models Identification

Transcription

1 Chapter 23: Simultaneous Equation Models Identification Chapter 23 Outline Review o Demand and Supply Models o Ordinary Least Squares (OLS) Estimation rocedure o Reduced Form (RF) Estimation rocedure One Way to Cope with Simultaneous Equation Models Two Stage Least Squares (TSLS): An Instrumental Variable (IV) Two Step Approach A Second Way to Cope with Simultaneous Equation Models o 1st Stage: Use the exogenous variable(s) to estimate the endogenous variable(s). o 2nd Stage: In the original model, replace the endogenous variable with its estimate. Comparison of Reduced Form (RF) and Two Stage Least Squares (TSLS) Estimates Statistical Software and Two Stage least Squares (TSLS) Identification of Simultaneous Equation Models: Order Condition o Taking Stock o Underidentification o Overidentification o Overidentification and Two Stage Least Squares (TSLS) Summary of Identification Issues Chapter 23 rep uestions Beef Market Data: Monthly time series data relating to the market for beef from 1977 to t uantity of beef in month t (millions of pounds) t Real price of beef in month t ( cents per pound) Inc t Real disposable income in month t (thousands of chained 2005 dollars) Chick t Real rice of whole chickens in month t ( cents per pound) Feed t Real price of cattle feed in month t ( cents per pounds of corn cobs) Consider the model for the beef market that we used in the last chapter: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t

2 2 Exogenous Variables: Feed t and Inc t 1. We shall now introduce another estimation procedure for simultaneous equation models, the two stage least squares (TSLS) estimation procedure: 1st Stage: Use the exogenous variable(s) to estimate the endogenous variable(s). o Dependent variable: The endogenous variable(s), the problem variable(s). o Explanatory variable(s): All exogenous variables. 2nd Stage: In the original model, replace the endogenous variable with its estimate. o Dependent variable: Original dependent variable. o Explanatory variables: 1st stage estimate of the endogenous variable and the relevant exogenous variables. Naturally, begin by focusing on the first stage. 1st Stage: Use the exogenous variable(s) to estimate the endogenous variable(s). o Dependent variable: The endogenous variable(s), the problem variable(s). In this case, the price of beef, t, is the endogenous variable. o Explanatory variable(s): All exogenous variables. In this case, the exogenous variables are Feed t and Inc t. Using the ordinary least squares (OLS) estimation procedure, what equation estimates the problem variable, the price of beef? [Link to MIT-BeefMarket wf1 goes here.] Est = Generate a new variable, Est, that estimates the price of beef based on the 1st stage. 2. Next, consider the demand model: D D D D D Demand Model: t = βconst + β t + βi Inct + et and the second stage of the two stage least squares (TSLS) estimation procedure: 2nd Stage: In the original model, replace the endogenous variable with its estimate. o Dependent variable: Original dependent variable. In this case, the original variable is the quantity of beef, t. o Explanatory variables: 1st stage estimate of the endogenous variable and the relevant exogenous

3 3 variables. In this case, the estimate of the price of beef and income, Est t and Inc t. a. Using the ordinary least squares (OLS) estimation procedure, estimate the Est coefficient of the demand model. b. Compare the two stage least squares (TSLS) coefficient estimate for the demand model with the estimate computed using the reduced form (RF) estimation procedure in Chapter Now, consider the supply model: S S S S S Supply Model: t = βconst + β t + βffeedt + et and the second stage of the two stage least squares (TSLS) estimation procedure: 2nd Stage: In the original model, replace the endogenous variable with its estimate. o Dependent variable: Original dependent variable. In this case, the original variable is the quantity of beef, t. o Explanatory variables: 1st stage estimate of the endogenous variable and the relevant exogenous variables. In this case, the estimate of the price of beef and income, Est t and Feed t. a. Using the ordinary least squares (OLS) estimation procedure, estimate the Est coefficient of the supply model. b. Compare the two stage least squares (TSLS) coefficient estimate for the supply model with the estimate computed using the reduced form (RF) estimation procedure in Chapter Reconsider the following simultaneous equation model of the beef market and the reduced form (RF) estimates: Demand and Supply Models: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t and Inc t Reduced Form (RF) Estimates uantity Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc rice Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc a. Focus on the reduced form (RF) estimates for the income coefficients:

4 4 1) The reduced form (RF) income coefficient estimates, a I and a I, allowed us to estimate the slope of which curve? Demand Supply 2) If the reduced form (RF) income coefficient estimates were not available, would we be able to estimate the slope of this curve? b. Focus on the reduced form (RF) estimates for the feed price coefficients: 1) The reduced form (RF) feed price coefficient estimates of these coefficients, a F and a F, allowed us to estimate the slope of which curve? Demand Supply 2) If the reduced form (RF) feed price coefficient estimates were not available, would we be able to estimate the slope of this curve? Review Demand and Supply Models For the economist, arguably the most important example of a simultaneous equations model is the demand/supply model: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t and Inc t roject: Estimate the beef market demand and supply parameters It is important to emphasize the distinction between endogenous and exogenous variables in a simultaneous equation model. Endogenous variables are variables whose values are determined within the model. In the demand/supply example, both quantity and price are determined simultaneously within the model; the model is explaining both the equilibrium quantity and the equilibrium price as depicted by the intersection of the supply and demand curves. On the other hand, exogenous are determined outside the context of the model; the values of exogenous variables are taken as given. The model does not attempt to explain how the values of exogenous variables are determined. Endogenous variables Variables determined within the model: uantity and rice. Exogenous variables Variables determined outside the model.

5 5 Unlike single regression models, an endogenous variable can be an variable in simultaneous equation models. In the demand and supply models the price is such a variable. Both the quantity demanded and the quantity supplied depend on the price; hence, the price is an variable. Furthermore, the price is determined within the model; the price is an endogenous variable. The price is determined by the intersection of the supply and demand curves. The traditional demand/supply graph clearly illustrates that both the quantity, t, and the price, t, are endogenous, both are determined within the model. rice S = Equlibrium uantity = Equilibrium rice D uantity Figure 23.1: Demand/Supply Model Ordinary Least Squares (OLS) Estimation rocedure In our last lecture, we showed why simultaneous equations cause a problem for the ordinary least squares (OLS) estimation procedure: Simultaneous Equations and Bias: Whenever an variable is also an endogenous variable, the ordinary least squares (OLS) estimation procedure for the value of the variable s coefficient is biased. In the demand/supply model, the price is an endogenous variable. When we used the ordinary least squares (OLS) estimation procedure to estimate the value of the price coefficient in the demand and supply models we observed that a problem emerged. In each model, price and the error term were correlated resulting in bias; the price is the problem variable:

6 6 Figure 23.2: Correlation of rice and Error Terms So, where did we go from here? We explored the possibility that the ordinary least squares (OLS) estimation procedure might be consistent. After all, is not half a loaf better than none? We took advantage of our Econometrics Lab to address this issue. Recall the distinction between an unbiased and a consistent estimation procedure: Unbiased: The estimation procedure does not systematically underestimate or overestimate the actual value; that is, after many, many repetitions the average of the estimates equals the actual value. Consistent but Biased: As consistent estimation procedure can be biased. But, as the sample size, as the number of observations, grows: o The magnitude of the bias decreases. That is, the mean of the coefficient estimate s probability distribution approaches the actual value. o The variance of the estimate s probability distribution diminishes and approaches 0. Unfortunately, the Econometrics Lab illustrates the sad fact that the ordinary least squares (OLS) estimation procedure is neither unbiased nor consistent.

7 7 We then considered an alternative estimation procedure: the reduced form (RF) estimation procedure. Our Econometrics Lab taught us that while the reduced form (RF) estimation procedure is biased, it is consistent. That is, as the sample size grows, the average of the coefficient estimates gets closer and closer to the actual value and the variance grew smaller and smaller. Arguably, when choosing between two biased estimates, it is better to use the one that is consistent. This represents the econometrician s pragmatic, half a loaf is better than none philosophy. We shall now quickly review the reduced form (RF) estimation procedure. Reduced Form (RF) Estimation rocedure One Way to Cope with Simultaneous Equation Models We begin with the simultaneous equation model and then constructed the reduced form (RF) equations: Demand and Supply Models: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t and Inc t Reduced Form (RF) Estimates uantity Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc rice Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc We use the ordinary least squares (OLS) estimation procedure to estimate the reduced form (RF) parameters and then use the ratio of the reduced form (RF) estimates to estimate the slopes of the demand and supply curves: [Link to MIT-BeefMarket wf1 goes here.]

8 8 Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Feed Inc Const Estimated Equation: Est = 138, Feed Inc a F = a I = Table 23.1: OLS Regression Results uantity Reduced Form (RF) Equation Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Feed Inc Const Estimated Equation: Est = Feed Inc a F = a I = Table 23.2: OLS Regression Results rice Reduced Form (RF) Equation Estimated Slope Estimated Slope of the Demand Curve of the Supply Curve Ratio of Reduced Form (RF) Ratio of Reduced Form (RF) Feed rice Income Coefficient Estimates Coefficient Estimates D D af S S ai Estimate of β = b = Estimate of β = b = a a = F = = = I

9 9 Two Stage Least Squares (TSLS): An Instrumental Variable (IV) Two Step Approach A Second Way to Cope with Simultaneous Equation Models Another way to estimate simultaneous equation model is the two stage least squares (TSLS) estimation procedure. As the name suggests the procedure involves two steps. As we shall see, two stage least squares (TSLS) uses a strategy that is similar to the instrumental variable (IV) approach. 1st Stage: Use the exogenous variable(s) to estimate the endogenous variable(s). o Dependent variable: The endogenous variable(s), the problem variable(s). o Explanatory variable(s): All exogenous variables. 2nd Stage: In the original model, replace the endogenous variable with its estimate. o Dependent variable: Original dependent variable. o Explanatory variables: 1st stage estimate of the endogenous variable and the relevant exogenous variables. We shall now illustrate the two stage least squares (TSLS) approach by consider the beef market. Beef Market Data: Monthly time series data relating to the market for beef from 1977 to t uantity of beef in month t (millions of pounds) t Real price of beef in month t ( cents per pound) Inc t Real disposable income in month t (thousands of chained 2005 dollars) Chick t Real rice of whole chickens in month t ( cents per pound) Feed t Real price of cattle feed in month t ( cents per pounds of corn cobs) Consider the model for the beef market that we used in the last chapter: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t and Inc t

10 10 The strategy for the first stage is similar to the strategy used by instrumental variable (IV) approach. The endogenous variable is the source of the bias; consequently, the endogenous variable is the problem variable. The endogenous variable is the dependent variable. The variables are all the exogenous variables. In our example, price is the endogenous variable; consequently, price becomes the dependent variable in the first stage. The exogenous variables, income and feed price, are the variables. 1st Stage: Use the exogenous variable(s) to estimate the endogenous variable(s). o Dependent variable: The endogenous variable(s), the problem variable(s). In this case, the price of beef, t, is the endogenous variable. o Explanatory variable(s): All exogenous variables. In this case, the exogenous variables are Feed t and Inc t. [Link to MIT-BeefMarket wf1 goes here.] Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Feed Inc Const Estimated Equation: Est = Feed Inc Table 23.3: OLS Regression Results TSLS 1st Stage Equation Using these regression results we estimate the price of beef based on the exogenous variables, income and feed price. The strategy for the second stage is also similar to the instrumental variable (IV) approach. We return to the original model and replace the endogenous variable with the estimate from Stage 1. The dependent variable is the original dependent variable, quantity. The variables are Stage 1 s estimate of the price and the relevant exogenous variables. In our example, we have two models, one for demand and one for supply; accordingly, we first apply the second stage to demand and then to supply.

11 11 2nd Stage: In the original model, replace the endogenous variable with its estimate. Demand Model o Dependent variable: Original dependent variable. In this case, the original variable is the quantity of beef, t. o Explanatory variables: 1st stage estimate of the endogenous variable and the relevant exogenous variables. In this case, the estimate of the price of beef and income, Est t and Inc t. Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Est Inc Const Estimated Equation: Est D = 149, Est Inc D I b = b = Table 23.4: OLS Regression Results TSLS 2nd Stage Demand Equation

12 12 We estimate the slope of the demand curve to be Supply Model Dependent variable: Original dependent variable. In this case, the original variable is the quantity of beef, t. Explanatory variables: 1st stage estimate of the problem endogenous variable and any relevant exogenous variable. In this case, the estimated of the price of beef and income, Est t, and Feed t. Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Est Feed Const Estimated Equation: Est S = 108, Est 1,305.3 Feed S S b = b F = 1,305.3 Table 23.5: OLS Regression Results TSLS 2nd Stage Supply Equation We estimate the slope of the demand curve to be Comparison of Reduced Form (RF) and Two Stage Least Squares (TSLS) Estimates Compare the estimates from the reduced form (RF) approach with the estimates from the two stage least squares (TSLS) approach: rice Coefficient Estimates: Estimated Slope of D S Demand Curve ( b ) Supply Curve ( b ) Reduced Form (RF) Two Stage Least Squares (TSLS) Table 23.6: Comparison of Reduced Form (RF) and Two Stage Least Squares (TSLS) rice Coefficient Estimates The estimates are identical. In this case, the reduced form (RF) estimation procedure and the two stage least squares (TSLS) estimation procedure produce identical results.

13 13 Software and Two Stage Least Squares (TSLS) Many statistical packages provide an easy way to apply the two state least squares (TSLS) estimation procedure so that we do not need to generate the estimate of the endogenous variable ourselves. Getting Started in EViews EViews makes it very easy for us to use the two stage least squares (TSLS) approach. EViews does most of the work for us eliminating the need to generate a new variable: In the Workfile window, highlight all relevant variables: q p feedp income Double click on one of the highlighted variables and click Open Equation. In the Equation Estimation window, click Options and then select TSLS Two Stage Least Squares (TSNLS and ARIMA). In the Instrument List box, enter the exogenous variables: feedp income In the Equation Specification box, enter the dependent variable followed by the variables (both exogenous and endogenous) for each model: o To estimate the demand model enter q p income o To estimate the supply model enter q p feedp [Link to MIT-BeefMarket wf1 goes here.] Two Stage Least Squares (TSLS) Dependent Variable: Instrument(s): Feed and Inc Explanatory Variable(s): Estimate SE t-statistic rob Inc Const Estimated Equation: Est D = 149, Est Inc D D b = b I = Table 23.7: TSLS Regression Results Demand Model

14 14 Two Stage Least Squares (TSLS) Dependent Variable: Instrument(s): Feed and Inc Explanatory Variable(s): Estimate SE t-statistic rob Feed Const Estimated Equation: Est S = 108, Est 1,305.3 Feed S S b = b F = 1,305.3 Table 23.8: TSLS Regression Results Supply Model Note that these are the same estimates that we obtained when we generated the estimates of the price on our own. Identification of Simultaneous Equation Models: Order Condition Taking Stock Let us step back for a moment to review our beef market model: Demand and Supply Models: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t and Inc t Reduced Form (RF) Estimates uantity Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc rice Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc We can use the coefficient interpretation approach to estimate the slopes of the demand and supply in terms of the reduced form (RF) estimates: i

15 15 Figure 23.3: Reduced Form (RF) and Coefficient Interpretation Approach Identified Intuition: Critical Role of the Exogenous Variable Absent from the Model In each model there is one exogenous variable absent and one endogenous variable. This one to one correspondence allows us to estimate the coefficient of the endogenous variable, price. D D D D D Demand Model: t = βconst + β t + βi Inct + et Changes in the feed price, the exogenous variable absent from the demand model, allow us to estimate the slope of the demand curve. The supply curve shifts, but the demand curve remains stationary. Consequently, the equilbria trace out the stationary demand curve. S S S S S Supply Model: = β + β + β Feed + e t Const t F t t

16 16 Changes in income, the exogenous variable absent from the demand model, allow us to estimate the slope of the supply curve. The demand curve shifts, but the supply curve remains stationary. Consequently, the equilibria trace out the stationary supply curve. Key oint: In each case, changes in the exogenous variable absent in the model allow us to estimate the value of the price coefficient, the model s endogenous variable. Order Condition The Order Condition formalizes this relationship: Number of exogenous variables absent from the model Number of endogenous variables included in the model Less Than Equal To Greater Than ã é Model Model Model Underidentified Identified Overidentified No RF Estimate Unique RF Estimate Multiple RF Estimates The reduced form estimation procedure for our beef market example is identified. For both the demand model and the supply model, the number of exogenous variable absent from the model equaled the number of endogenous variables in the model: Exogenous Variables: Feed and Inc. A total of 2 exogenous variables. Demand Model Supply Model Exogenous Endogenous Exogenous Endogenous variables variables variables variables variables variables included absent included included absent included = = equal to 1 1 less than 1 Unique RF Estimate Unique RF Estimate

17 17 Underidentification We shall now illustrate the underidentification problem. Suppose that no income information was available. Obviously, if we have no income information, we cannot include Inc as an variable in our models: Demand and Supply Models: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t and Inc t Reduced Form (RF) Estimates uantity Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc rice Reduced Form (RF) Estimates: Est = a + a Feed + a Inc Const F I Let us now apply the order condition by comparing the number of absent exogenous variables and endogenous variables in each model: Exogenous Variable: Feed. A total of 1 exogenous variable. Demand Model Supply Model Exogenous Endogenous Exogenous Endogenous variables variables variables variables variables variables included absent included included absent included = = equal to 1 0 less than 1 Unique RF Estimate No RF Estimate The order condition suggests that we should be able to estimate the coefficient of the endogenous variable,, in the demand model. not be able to estimate the coefficient of the endogenous variable,, in the supply model. The coefficient interpretation approach explains why. We can still estimate the slope of the demand curve, however, by calculating the ratio of the reduced form (RF) feed price coefficient estimates, a and F a F. We shall use the

18 18 coefficient estimate approach to explain this phenomenon to take advantage of the intuition it provides. Figure 23.4: Reduced Form (RF) and Coefficient Interpretation Approach Underidentified There is both good news and bad news when we have feed price information but no income information: Good news: Since we still have feed price information, we still have information about supply curve shifts. The shifts in the supply curve cause the equilibrium quantity and price to move along the demand curve. In other words, shifts in the supply curve trace out the demand curve; hence, we can still estimate the slope of the demand curve. Bad news: On the other hand, since we have no income information, we have no information about demand curve shifts. Without knowing how the demand curve shifts we have no idea how the equilibrium quantity and price move along the supply curve. In other words, we cannot trace out

19 19 the supply curve; hence, we cannot estimate the slope of the supply curve. To use the reduced form (RF) approach to estimate the slope of the demand curve, we first use ordinary least squares (OLS) to estimate the parameters of the reduced form (RF) equations: [Link to MIT-BeefMarket wf1 goes here.] Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Feed Const Estimated Equation: Est = 239, Feed a = F Table 23.9: OLS Regression Results uantity Reduced Form (RF) Equation Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Feed Const Estimated Equation: Est = Feed Inc a F = Table 23.10: OLS Regression Results rice Reduced Form (RF) Equation Then, we can estimate the slope of the demand curve by calculating the ratio of the feed price estimates: D af Estimated "slope" of the demand curve = b = = = 1,566.5 a F

20 20 Now, let us use the two stage least squares (TSLS) estimation procedure to estimate the slope of the demand curve: Two Stage Least Squares (TSLS) Dependent Variable: Instrument(s): Feed Explanatory Variable(s): Estimate SE t-statistic rob Const Estimated Equation: Est = 461,631 1,566.5 D b = 1,566.5 Table 23.11: TSLS Regression Results Demand Model In both cases, the estimated slope of the demand curve is 1,566.5.

21 21 Similarly, an underidentification problem would exist if income information was available, but feed price information was not. Demand and Supply Models: D D D D D Demand Model: t = βconst + β t + βi Inct + et S S S S S Supply Model: t = βconst + β t + βffeedt + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t and Inc t Reduced Form (RF) Estimates uantity Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc rice Reduced Form (RF) Estimates: Est = a + a Feed + a Inc Const F I Again, let us now apply the order condition by comparing the number of absent exogenous variables and endogenous variables in each model: Number of exogenous variables absent from the model Number of endogenous variables included in the model Less Than Equal To Greater Than ã é Model Model Model Underidentified Identified Overidentified No RF Estimate Unique RF Estimate Multiple RF Estimates Exogenous Variable: Inc. A total of 1 exogenous variable. Demand Model Supply Model Exogenous Endogenous Exogenous Endogenous variables variables variables variables variables variables included absent included included absent included = = less than 1 1 equal to 1 No RF Estimate Unique RF Estimate

22 22 The order condition suggests that we should still be able to estimate the coefficient of the endogenous variable,, in the supply model. not be able to estimate the coefficient of the endogenous variable,, in the demand model. The coefficient interpretation approach explains why. Figure 23.5: Reduced Form (RF) and Coefficient Interpretation Approach Underidentified Again, there is both good news and bad news when we have income information, but no feed price information: Good news: Since we have income information, we still have information about demand curve shifts. The shifts in the demand curve cause the equilibrium quantity and price to move along the supply curve. In other words, shifts in the demand curve trace out the supply curve; hence, we can still estimate the slope of the supply curve. Bad news: On the other hand, since we have no feed price information, we have no information about supply curve shifts. Without knowing how the supply curve shifts we have no idea how the equilibrium quantity and

23 23 price move along the demand curve. In other words, we cannot trace out the demand curve; hence, we cannot estimate the slope of the demand curve. To use the reduced form (RF) approach to estimate the slope of the supply curve, we first use ordinary least squares (OLS) to estimate the parameters of the reduced form (RF) equations: [Link to MIT-BeefMarket wf1 goes here.] Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Inc Const Estimated Equation: Est = 111, Inc a I = Table 23.12: OLS Regression Results uantity Reduced Form (RF) Equation Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Inc Const Estimated Equation: Est = Inc a I = Table 23.13: OLS Regression Results rice Reduced Form (RF) Equation Then, we can estimate the slope of the supply curve by calculating the ratio of the income estimates: S ai Estimated "slope" of the supply curve = b = = = 2,091.7 ai Once again, two stage least squares (TSLS) provide the same estimate:

24 24 Two Stage Least Squares (TSLS) Dependent Variable: Instrument(s): Inc Explanatory Variable(s): Estimate SE t-statistic rob Const Estimated Equation: Est = 140, ,091.7 S b = 2,091.7 Table 23.14: TSLS Regression Results Supply Equation Conclusion: When a simultaneous equations model is underidentified, we cannot estimate all its parameters. For those parameters we can estimate, however, the reduced form (RF) estimation procedure and the two stage least squares (TSLS) estimation procedures are equivalent. Overidentification While an underidentification problem arises when too little information is available, an overidentification problem arises when, in some sense, too much information is available. To illustrate this suppose that in addition to the feed price and income information, the price of chicken is also available. Since beef and chicken are substitutes, the price of chicken would appear as an exogenous variable in the demand model. The simultaneous equation model and the reduced form (RF) estimates would become: Demand and Supply Models: Demand Model: t = βconst + β t + βi Inct + βcchickt + et Supply Model: t = βconst + β t + βffeedt + + et D S Equilibrium: t = t = t Endogenous Variables: t and t Exogenous Variables: Feed t, Inc t, and Chick t D D D D D D S S S S S

25 25 Reduced Form (RF) Estimates uantity Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc + acchick rice Reduced Form (RF) Estimates: Est = aconst + affeed + ai Inc + acchick Let us now apply the order condition by comparing the number of absent exogenous variables and endogenous variables in each model: Exogenous Variables: Feed, Inc, and Chick A total of 3 exogenous variables. Demand Model Supply Model Exogenous Endogenous Exogenous Endogenous variables variables variables variables variables variables included absent included included absent included = = equal to 1 2 greater than 1 Unique RF Estimate Multiple RF Estimate The order condition suggests that we should be able to estimate the coefficient of the endogenous variable,, in the demand model. encounter some complications when estimating the coefficient of the endogenous variable,, in the supply model. The reduced form (RF) estimation procedure provides multiple estimates. We shall now explain why the multiple estimates result.

26 26 Figure 23.6: Reduced Form (RF) and Coefficient Interpretation Approach Overidentified Now we have two exogenous factors that shift the demand curve: income and the price of chicken. Consequently, there are two ways to trace out the supply curve. There are now two different ways to use the reduced form (RF) estimates to estimate the slope of the supply curve: Ratio of the reduced form (RF) Ratio of the reduced form (RF) income coefficients chicken feed coefficients Estimated slope Estimated slope of supply curve: of supply curve: S ai S ac b = b = a a I C

27 27 We shall now go through the mechanics of the reduced form (RF) estimation procedures to illustrate the overidentification problem. First, we use the ordinary least squares (OLS) to estimate the reduced form (RF) parameters: [Link to MIT-BeefMarket wf1 goes here.] Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Feed Inc Chick Const Estimated Equation: Est = 138, Feed Inc Chick a F = a I = a C = Table 23.15: OLS Regression Results uantity Reduced Form (RF) Equation Ordinary Least Squares (OLS) Dependent Variable: Explanatory Variable(s): Estimate SE t-statistic rob Feed Inc Chick Const Estimated Equation: Est = Feed Inc Chick a F = a I = a C = Table 23.16: OLS Regression Results rice Reduced Form (RF) Equation

28 28 Then, we use the reduced form (RF) estimates to compute the estimates for the slopes of the demand and supply curves: Estimated slope Estimated slope Estimated slope of demand curve of the supply curve of the supply curve Ratio of reduced Ratio of reduced Ratio of reduced form (RF) feed price form (RF) income form (RF) chicken price coefficient estimates coefficient estimates coefficient estimates D af S ai S ac b = = = b = = = b = = = af ai ac The reduced form (RF) estimation procedure produces two different estimates for the slope for the supply curve. This is what we mean by overidentification. Overidentification and Two Stage Least Squares (TSLS) While reduced form (RF) estimation procedure cannot resolve the overidentification problem, two stage least squares (TSLS) approach can. The two squares least squares estimation procedure provides a single estimate of the slope of the supply curve. The following regression printout reveals this: [Link to MIT-BeefMarket wf1 goes here.] Two Stage Least Squares (TSLS) Dependent Variable: Instrument(s): Feed, Inc, and Chick Explanatory Variable(s): Estimate SE t-statistic rob Inc Chick Const Estimated Equation: Est D = 149, Est Inc D D D b = b I = b C = Table 23.17: TSLS Regression Results Demand Model The estimated slope of the demand curve is This is the same estimate as computed by the reduced form (RF) estimation procedure.

29 29 Two Stage Least Squares (TSLS) Dependent Variable: Instrument(s): Feed, Inc, and Chick Explanatory Variable(s): Estimate SE t-statistic rob Feed Const Estimated Equation: Est S = 112, Est 1,290.6Feed S S b = b F = 1,290.6 Table 23.18: TSLS Regression Results Supply Model Two stage least squares (TSLS) provides a single estimate for the slope of the supply curve: b = S Table compares the estimates that result when using the two different estimation procedures: rice Coefficient Estimates: Estimated Slope of Demand Curve ( b ) Supply Curve ( b ) Reduced Form (RF) Based on Income Coefficients 1,051.2 Based on Chicken rice Coefficients Two Stage Least Squares (TSLS) Table 23.19: Comparison of RF and TSLS Estimates Note that the demand model is not overidentified and both the reduced form (RF) estimation procedure and the two stage least squares (TSLS) estimation procedure provide the same estimate for the slope of the demand curve. On the other hand, the supply model is overidentified. The reduced form (RF) estimation procedure provides two estimates for the slope of the supply curve; the two stage least squares (TSLS) estimation procedure provides only one. D S

30 30 Summary of Identification Issues Number of exogenous variables absent from the model Number of endogenous variables included in the model Less Than Equal To Greater Than ã é Model Model Model Underidentified Identified Overidentified No RF estimate Unique RF estimate Multiple RF estimates RF and TSLS estimates identical RF and TSLS estimates identical Unique TSLS Estimate 1 Again, recall that the coefficients do not equal the slope of the demand curve, but rather the reciprocal of the slope. They are the ratio of run over rise instead of rise over run. This occurs as a consequence of the economist s convention of placing quantity on the horizontal axis and price on the vertical axis. To avoid the awkwardness of using the expression the reciprocal of the slope repeatedly, we shall we shall place the word slope within double quotes to indicate that it is the reciprocal.