Lecture#12. Instrumental variables regression Causal parameters III

Transcription

1 Lecture#12 Instrumental variables regression Causal parameters III 1

2 Demand experiment, market data analysis & simultaneous causality 2

3 Simultaneous causality Your task is to estimate the demand function for a homogenous good sold at unit price. How would you do this in an experiment? By changing the price yourself ( at random ) and observing how many units are sold at each price. 3

4 Experiment What does choosing prices at random mean? 1. We offer different randomized prices to individual consumers. 2. We offer different randomized prices each to a group of consumers. 4

5 Experiment We record quantity sold at different prices. We study the outcomes. 5

6 Experiment Let s run experiments. We choose the price at random as follows:. set obs number of observations (_N) was 0, now 10,000. set seed gen p_exp = invnorm(uniform()) * 5 6

7 p_exp q_exp p_exp q_exp

8 p_exp q_exp 8

9 90 p_exp q_exp 9

10 80 90 p_exp q_exp 10

11 80 90 p_exp q_exp 11

12 q_exp p-q linear regr. 12

13 Regression analysis. regr q_exp p_exp Source SS df MS Number of obs = 10,000 F(1, 9998) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE = q_exp Coef. Std. Err. t P> t [95% Conf. Interval] p_exp _cons

14 Robustness analysis. regr q_exp p_exp* Source SS df MS Number of obs = 10,000 F(2, 9997) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE = q_exp Coef. Std. Err. t P> t [95% Conf. Interval] p_exp p_exp _cons

15 Parameters for inverse demand function. sum alpha_exp beta_exp Variable Obs Mean Std. Dev. Min Max alpha_exp 10, beta_exp 10,

16 Market outcomes Assume you are an outside observer of a market say a prospective buyer of a firm or the competition authority. you cannot run experiments. You would still want to know demand (to calculate e.g. price-cost margins, consumer surplus). 16

17 Market outcomes We collect data from the market. We observe pairs (P i, Q i ), i = market. Let s think how such pairs are determined, using a simple monopoly model. 17

18 Market outcome data 18

19 p p_exp q_exp p q q 19

20 p q 20

21 p q 21

22 p q 22

23 p q 23

24 q p-q linear regr. 24

25 80 90 Fitted values market data experimental data 25

26 Regression using market data. regr p q Source SS df MS Number of obs = 10,000 F(1, 9998) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE = p Coef. Std. Err. t P> t [95% Conf. Interval] q _cons

27 Parameter comparison for inverse demand fcn. sum alpha_exp beta_exp alpha_ols beta_ols Variable Obs Mean Std. Dev. Min Max alpha_exp 10, beta_exp 10, alpha_ols 10, e beta_ols 10,

28 Challenge with market data How is price determined? By the firm(s). Firm(s) take market conditions into account: 1. demand 2. (variable) costs of production 3. competition. 28

29 Challenge with market data Price quantity pairs are a leading example of simultaneous causality. This generalizes to more complicated markets with 1. differentiated goods 2. multiproduct firms. 3. endogenous entry and exit 4. dynamic considerations 5. advertising 6. 29

30 Market data - solution Need to address simultaneous causality. need to understand and exploit determinants of price and quantity. How did the experiment solve the problem? By having the researcher shift (=change) prices instead of the firm. 30

31 Linear monopoly Demand function Q i = a bp i + ε i a = average intercept b = slope ε i = market-specific deviation from the average intercept. NOTE: we assume the firm observes all these parameters. 31

32 Linear monopoly Inverse demand function P i = a b 1 b Q i + 1 b ε i = α βq i + ε i 32

33 Linear monopoly Supply function how to get? Need to specify costs of production: constant marginal cost c i = c 0 + c 1 z i + η i 33

34 Linear monopoly c i = c 0 + c 1 z i + η i c 0 = average of marginal cost when w i =0 z i = a component of marginal cost that varies across markets (cost of raw materials / unit of output, cost of labor / unit of output,..) η i = shock to average marginal cost / deviation from the avg. 34

35 Linear monopoly Profit function max Pi π i = (P i -c i )Q i Equilibrium price P i = a 2b c i + 1 2b ε i 35

36 Linear monopoly Equilibrium price is a function of P i = a 2b (c 0+c 1 z i + η i ) + 1 2b ε i 1. (fixed) demand parameters a and b 2. (fixed) supply side parameters c 0, c 1 3. variable cost determinant z i 4. cost shock η i 5. demand shock ε i 36

37 Linear monopoly Equilibrium quantity is a function of Q i = a 2 b 2 c i ε i 1. (fixed) demand parameters a and b 2. (fixed) supply side parameters c 0, c 1 3. variable cost determinant z i 4. cost shock η i 5. demand shock ε i 37

38 Market data - challenge Both eq. price and eq. quantity are functions of 1. demand shock ε i and 2. supply shock η i simultaneous causality. 38

39 Market data - solution We want to learn the demand curve. Could we mimic the experimental approach with market data? Needed: something that shifts firm s (supply) decision at random. Random = without being affected by demand shock ε i. 39

40 Experiment #2 Imagine the firm still sets the price, But we choose (at random) z i = z i exp. Recall that c i = c 0 + c 1 z i + η i we shift firm s marginal cost. 40

41 Experiment #2 Now the firm sets each time the price P i = a 2b (c 0+c 1 z i exp + η i ) + 1 2b ε i equivalent to running an experiment. Change in price due to (known) change in z i exp. 41

42 Experiment #2 Imagine we raise z i exp by 1 unit. By how much does 1. marginal cost c i = c 0 + c 1 z i + η i change? c 1 2. price change? 1 2 c 1 (by eqn on slide #41) 3. demand change? b 1 2 c 1 (by eqn on slide #37) 42

43 Experiment #2 How could we get the slope of the demand function from these changes? Yes, by dividing the change in quantity by the change in demand: b 1 2 c c 1 = b 43

44 Experiment #2 How could we get those numbers? 1. Regress P i on z i exp to get 1 2 c 1. P i = γ 0 + γ 1 z i + e i γ 1 = 1 2 c 1 (by eqn on slide #41) 44

45 Experiment #2 2. Regress Q i on z i exp to get b 1 2 c 1. Q i = μ 0 + μ 1 z i + w i μ 1 = b 1 2 c 1 (by eqn on slide #37) These are called reduced form equations. 45

46 Experiment #2 When would this work? 1. z i exp has to impact firm decision, i.e., have an effect on c i. c 1 cannot be (insignificantly different from) zero. 2. z i exp may not have an effect on Q i directly, but only via c i. 46

47 Let s regress P on z. regr p z Source SS df MS Number of obs = 10,000 F(1, 9998) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE = p Coef. Std. Err. t P> t [95% Conf. Interval] z _cons scalar red_1 = _b[z] 47

48 Let s regress Q on z. regr q z Source SS df MS Number of obs = 10,000 F(1, 9998) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE = q Coef. Std. Err. t P> t [95% Conf. Interval] z _cons scalar red_2 = _b[z] 48

49 Let s calculate b. scalar b_red = red_2 / red_1. scalar list b_red b_red =

50 Instrumental variable Instrumental variable = a variable that causes variation in price (explanatory variable X) but does not affect demand (dependent variable Y) directly. If the variable cost component z i affecting demand directly, varies at random, i.e., without market data allows us to use the experimental approach indirectly. 50

51 Approach #2 Could we proceed differently? 1. Regress P i on z i. Calculate predicted price P i = γ 0 + γ 1 z i. 2. Regress Q i on P i to get b (and a). Equation Q i = a bp i + ε i is a structural equation. 51

52 Regress P on z, created predicted P. regr p z Source SS df MS Number of obs = 10,000 F(1, 9998) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE = p Coef. Std. Err. t P> t [95% Conf. Interval] z _cons predict p_hat (option xb assumed; fitted values) 52

53 Regress Q on predicted P. regr q p_hat Source SS df MS Number of obs = 10,000 F(1, 9998) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE = q Coef. Std. Err. t P> t [95% Conf. Interval] p_hat _cons

54 Why / how do these approaches work? 1. Regress P i on z i to get 1 2 c 1 = cov(p i,z i ) var(z i ). 2. Regress Q i on z i to get b 1 2 c 1 = cov(q i,z i ) var(z i ). b = cov(q i,z i ) cov(p i,z i ) 54

55 Why / how do these approaches work? Regress Q i on P i : b = cov(q i, P i ) var( P i ) = cov(q i, γ 0 + γ 1 z i ) var( γ 0 + γ 1 z i ) b = γ 1cov(Q i, z i ) γ 2 i var(z i ) ; recall γ 1 = cov(p i, z i ) var(z i ) b = var(z i ) cov(p i, z i ) cov(q i, z i ) var(z i ) 55

56 Why / how do these approaches work? b = var(z i ) cov(p i, z i ) cov(q i, z i ) var(z i ) b = cov(q i, z i ) cov(p i, z i ) 56

57 2sls / instrumental variables regression In practice, want to use the so-called Two Stage Least Squares (2sls) or instrumental variables regression command. In Stata, ivregress. Manual and ivregress commands produce same point estimates, but the latter corrects the standard errors. This is important, as the manual approach yields too small se: It ignores the uncertainty in the parameters γ 0 and γ 1 used to calculate P i. 57

58 2sls estimation of demand. ivregress 2sls q (p = z) Instrumental variables (2SLS) regression Number of obs = 10,000 Wald chi2(1) = Prob > chi2 = R-squared =. Root MSE = q Coef. Std. Err. z P> z [95% Conf. Interval] p _cons Instrumented: p Instruments: z 58

59 Requirements for an instrument Think of our normal regression Y = β 0 + β 1 X + u. 1. Instrument relevance: The instrument Z has to affect the (endogenous) explanatory variable X of the equation of interest ( 2nd stage equation ) in the equation X = α 0 + α 1 Z + v 2. Instrument exogeneity: The instrument Z may not be correlated with the error term of the equation of interest, i.e., cov Z, u = 0 59

60 Instrument relevance / Weak instrument Relevance = instrument z needs to be correlated enough with the endogenous explanatory variable X. What happens when cov z, X 0? β 1 becomes undefined! 60

61 Instrument relevance / Weak instrument you want to check that your instrument is relevant = you don t have a weak instrument. Rule of thumb: F-statistic of Z when you regress X on Z (and possible further controls) > 10. Note: with 1 instrument, F-test is the square of the t-test. 61

62 . estat firststage First-stage regression summary statistics Adjusted Partial Variable R-sq. R-sq. R-sq. F(1,9998) Prob > F p

63 Instrument relevance / Weak instrument There are more sophisticated tests. There are ways of allowing for weak instruments. We leave all that for later courses. 63

64 Instrument correlated with error = exogeneity assumption fails. biased estimate of β 1. Similar to omitted variable bias. 64

65 Instrument correlated with error What can be done? 1. Strong story for why no correlation between instrument and error. 2. With multiple instruments, may do tests. 3. There are ways of allowing for (some) correlation to check robustness of your results (for later). 65